Planning an experimental study. Planning a psychological experiment. Experiments with reproducible conditions
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1 Ministry of Education of the Russian Federation EAST SIBERIAN STATE TECHNOLOGICAL UNIVERSITY Department Metrology, standardization and certification BASIS OF EXPERIMENT PLANNING Methodological guide for students of the specialties "Metrology and metrological support" and "Standardization and certification (by industry) Food Industry)» Compiled by: Khamkhanov K.M. UlanUde, 00
2 CONTENTS Introduction... Basic definitions.. Optimization parameters.... Optimization parameter requirements.. Problems with multiple output parameters 3. Generalized optimization parameter. 3.. The simplest ways to construct a generalized response. 3.. Desirability scale 3.3. Generalized desirability function. 4. Factors Characteristics of factors. 4. Requirements for factors Choice of levels of variation of factors and zero point. 5. Choice of models.. 6. Complete factorial experiment 6.. Complete factorial experiment of type k 6.. Properties of a complete factorial experiment of type k Calculation of regression coefficients. 7. Fractional factorial experiment Minimization of the number of experiments Fractional replica Choice of half replicas. Generating ratios and constitutive contrasts 8. Measurement errors of optimization criteria and factors Randomization .. 9. Screening experiments 9. A priori ranking of factors (psychological experiment) 9.. Random balance method Non-full-block designs (taking into account qualitative factors and expert assessments) 0. Planning example experiment Selection of factors 0.. Conducting the experiment 0.3. Full factorial experiment 0.4. Search for the optimum by the steep ascent method 0.5. Description of the optimum region 0.6. Building graphical dependencies Applications.. 88
3 INTRODUCTION Traditional research methods are associated with experiments that require large expenditures, effort and money, because are "passive" based on the alternate variation of individual independent variables in conditions where the rest tend to remain unchanged. Experiments, as a rule, are multifactorial and are associated with optimizing the quality of materials, finding optimal conditions for conducting technological processes, developing the most rational equipment designs, etc. The systems that serve as the object of such research are very often so complex that they cannot be theoretically studied in a reasonable time. Therefore, despite the significant amount of research work performed, due to the lack of a real opportunity to sufficiently fully study a significant number of research objects, as a result, many decisions are made on the basis of random information, and therefore are far from optimal. Based on the foregoing, it becomes necessary to find a way to lead research work at an accelerated pace and ensuring decision-making that is close to optimal. In this way, the statistical methods of planning an experiment were proposed by the English statistician Ronald Fisher (late twenties). He was the first to show the feasibility of simultaneously varying all the factors, as opposed to the widely used one-factor experiment. In the early 1960s, a new direction in experiment planning appeared, connected with the optimization of the processes of planning an extreme experiment. The first work in this field was published in 1995 by Box and Wilson in England. BoxWilson's idea is extremely simple. The experimenter is invited to set up successive small series of experiments, in each of which all factors are simultaneously varied according to certain rules. The series are organized in such a way that, after mathematical processing of the previous one, it is possible to choose the conditions for conducting (i.e., planning) the next series. So consistently, step by step, the optimum area is reached. The use of experiment planning makes the behavior of the experimenter purposeful and organized, significantly contributes to the increase in labor productivity and the reliability of the results obtained. An important advantage is its versatility, suitability in the vast majority of areas of research. In our country, the planning of the experiment has been developing since 960 under the leadership of V.V. Nalimov. However, even a simple planning procedure is very insidious, which is due to a number of reasons, such as incorrect application of planning methods, the choice of a non-optimal research path, lack of practical experience, insufficient mathematical preparedness of the experimenter, etc. The purpose of this work is to familiarize readers with the most commonly used and simple methods of planning an experiment, to develop skills for practical application. The problem of process optimization is considered in more detail.
4 . BASIC DEFINITIONS Experimental planning, like any branch of science, has its own terminology. For ease of understanding, consider the most common terms. An experiment is a purposeful impact on the object of study in order to obtain reliable information. Most scientific research is related to experiment. It is carried out in production, in laboratories, in experimental fields and plots, in clinics, etc. The experiment can be physical, psychological or model. It can be directly carried out on the object or on its model. The model usually differs from the object in scale, and sometimes in nature. The main requirement for the model is a fairly accurate description of the object. Recently, along with physical models, abstract mathematical models are becoming more widespread. By the way, planning an experiment is directly related to the development and study of a mathematical model of the object of study. Experiment planning is a procedure for choosing the number and conditions for conducting experiments that are necessary and sufficient to solve the problem with the required accuracy. The following is essential here: striving to minimize the total number of experiments; simultaneous variation of all variables that determine the process, according to special rules algorithms; the use of a mathematical apparatus that formalizes many of the experimenter's actions; choosing a clear strategy that allows you to make informed decisions after each series of experiments. The tasks for which the design of an experiment can be used are extremely diverse. These include: the search for optimal conditions, the construction of interpolation formulas, the choice of significant factors, the evaluation and refinement of the constants of theoretical models, the choice of the most acceptable hypotheses from a certain set of hypotheses about the mechanism of phenomena, the study of diagrams, the composition of the property, etc. The search for optimal conditions is one of the most common scientific and technical problems. They arise at the moment when the possibility of carrying out the process is established and it is necessary to find the best (optimal) conditions for its implementation. Such problems are called optimization problems. The process of solving them is called the optimization process or simply optimization. The choice of the optimal composition of multicomponent mixtures and alloys, increasing the productivity of existing plants, improving product quality, reducing the cost of obtaining it are examples of optimization problems. Next comes the concept of the object of study. To describe it, it is convenient to use the concept of a cybernetic system, which is schematically shown in Fig. .. Sometimes such a scheme is called a "black box". The arrows on the right depict the numerical characteristics of the research goals. We denote them with the letter Y (y) and call them optimization parameters. There are other names in the literature: optimization criterion, objective function, black box output, etc. To conduct an experiment, it is necessary to be able to influence the guidance of the "black box". We denote all methods of such influence by the letter X (x) and call them factors. They are also called black box inputs. 88
5 x y x y x k Fig... When solving the problem, we will use mathematical models of the study. By a mathematical model, we mean an equation that relates an optimization parameter to factors. This is the equation in general view can be written as follows: y = ϕ(x, x,..., x), k where the symbol ϕ (), as usual in mathematics, replaces the words: “function of”. Such a function is called a response function. Each factor can take on one of several values in the experiment. These values are called levels. To facilitate the construction of the "black box" and the experiment, the factor must have a certain number of discrete levels. A fixed set of factor levels defines one of the possible black box states. At the same time, this is a condition for conducting one of the possible experiments. If you enumerate all possible sets of states, then you get a lot of different states of the "black box". At the same time, this will be the number of possible different experiences. The number of possible experiments is determined by the expression = where the number of experiments; p number of levels; k number of factors. Real objects are usually of great complexity. So, at first glance, simple system with five factors at five levels has 35 states, and for ten factors at four levels there are already over a million. In these cases, the implementation of all experiments is almost impossible. The question arises: how many and what experiments should be included in the experiment in order to solve the problem? This is where experiment design comes into play. Performing research by designing an experiment requires the fulfillment of certain requirements. The main ones are the conditions for the reproducibility of the results of the experiment and the controllability of the experiment. If we repeat some experiments at irregular intervals and compare the results, in our case, the values of the optimization parameter, then the spread of their values characterizes the reproducibility of the results. If it does not exceed some given value, then the object satisfies the requirement of reproducibility of results. Here we will consider only such objects where this condition is satisfied. The design of an experiment involves active intervention in the process and the possibility of choosing in each experiment those levels of factors that are of interest. Therefore, such an experiment is called active. An object on which an active experiment is possible is called a controlled object. In practice, there are no absolutely controlled objects, because they are acted upon as p k, y m
6 controllable and uncontrollable factors. Uncontrolled factors affect the reproducibility of the experiment and are the cause of its violation. In these cases, it is necessary to switch to other research methods. 88. OPTIMIZATION PARAMETERS The choice of optimization parameters (optimization criteria) is one of the main stages of work at the stage of preliminary study of the object of study, because correct setting problem depends on the correct choice of the optimization parameter, which is a function of the goal. The optimization parameter is understood as a characteristic of the goal, given quantitatively. The optimization parameter is a reaction (response) to the influence of factors that determine the behavior of the selected system. Real objects or processes are usually very complex. They often require the simultaneous consideration of several, sometimes very many, parameters. Each object can be characterized by the entire set of parameters, or by any subset of this set, or by one single optimization parameter. In the latter case, other characteristics of the process no longer act as an optimization parameter, but serve as limitations. Another way is to construct a generalized optimization parameter as a function of a set of initial ones... REQUIREMENTS FOR THE OPTIMIZATION PARAMETER An optimization parameter is a feature by which the process is optimized. It should be quantitative, set by a number. The set of values that an optimization parameter can take is called its domain of definition. Domains of definition can be continuous and discrete, limited and unlimited. For example, the reaction yield is an optimization parameter with a continuous bounded domain. It can vary from 0 to 00%. The number of defective products, the number of grains on an alloy section, the number of blood cells in a blood sample are examples of parameters with a discrete domain of definition limited from below. A quantitative estimation of the optimization parameter is not always possible in practice. In such cases, a technique called ranking is used. At the same time, the optimization parameters are assigned ranks on a pre-selected scale: two-point, five-point, etc. The rank parameter has a discrete limited domain of definition. In the simplest case, the area contains two values (yes, no; good, bad). This may correspond, for example, to good products and defects. So, the first requirement: the optimization parameter must be quantitative. The second requirement is that the optimization parameter must be expressed as a single number. Sometimes this happens naturally, like recording an instrument reading. For example, the speed of a car is determined by the number on the speedometer. Often you have to do some calculations. This is the case when calculating the yield of a reaction. In chemistry, it is often required to obtain a product with a given ratio of components, for example, A:B=3:. One of the possible options for solving such problems is to express the ratio as a single number (.5) and use the value of deviations (or squared deviations) from this number as an optimization parameter. The third requirement related to the quantitative nature of the optimization parameter is uniqueness in the statistical sense. A given set of factor values must correspond to one optimization parameter value, while the opposite is not true: different sets of factor values can correspond to the same parameter value. The fourth, most important requirement, the requirement for optimization parameters is its ability to really effective evaluation functioning of the system. The idea of the object does not remain constant during the study. It
7 changes as information accumulates and depending on the results achieved. This leads to a consistent approach when choosing an optimization parameter. So, for example, in the first stages of the study of technological processes, the product yield is often used as an optimization parameter. However, in the future, when the possibility of increasing the yield is exhausted, they begin to be interested in such parameters as cost, product purity, etc. Evaluation of the effectiveness of the functioning of the system can be carried out both for the entire system as a whole, and by evaluating the effectiveness of a number of subsystems that make up this system. But at the same time, it is necessary to take into account the possibility that the optimality of each of the subsystems in terms of its optimization parameter "does not exclude the possibility of the death of the system as a whole." This means that an attempt to achieve the optimum, taking into account some local or intermediate optimization parameter, may turn out to be inefficient or even lead to marriage. The fifth requirement for the optimization parameter is the requirement of universality or completeness. The universality of the optimization parameter is understood as its ability to comprehensively characterize the object. In particular, technological parameters are not universal enough: they do not take into account the economy. Universality is possessed, for example, by generalized optimization parameters, which are constructed as functions of several particular parameters. The sixth requirement: it is desirable that the optimization parameter has a physical meaning, be simple and easy to calculate. The requirement of physical meaning is connected with the subsequent interpretation of the results of the experiment. It is not difficult to explain what the maximum extraction means, the maximum content of a valuable component. These and similar technological optimization parameters have a clear physical meaning, but sometimes they may not be satisfied, for example, the requirement of statistical efficiency. Then it is recommended to proceed to the transformation of the optimization parameter. A transformation, such as arcsn y, can make the optimization parameter statistically efficient (for example, the variances become uniform), but it remains unclear: what does it mean to reach an extremum of this value? The second requirement, i.e. simplicity and easy computability are also essential. For separation processes, the thermodynamic optimization parameters are more universal. However, they are rarely used in practice: their calculation is quite difficult. Of these two requirements, the first is more significant, because it is often possible to find the ideal characteristic of the system and compare it with the real characteristic... PROBLEMS WITH MULTIPLE OUTPUT PARAMETERS obvious benefits. But in practice, it is often necessary to take into account several output parameters. Sometimes their number is quite large. For example, in the production of rubber and plastic products it is necessary to take into account physical, mechanical, technological, economic, artistic and aesthetic and other parameters. Mathematical models can be built for each of the parameters, but it is impossible to optimize several functions at the same time. Usually, one function that is most important from the point of view of research is optimized, under the restrictions imposed by other functions. Therefore, out of many output parameters, one is chosen as the optimization parameter, and the rest serve as constraints. It is always a good idea to explore the possibility of reducing the number of outputs. To do this, you can use correlation analysis.
8 In this case, between all possible pairs of parameters, it is necessary to calculate the coefficient of pair correlation, which is a generally accepted characteristic in mathematical statistics of the relationship between two random variables. If we designate one parameter as y and the other as y, and the number of experiments in which they will be measured, through so that u=, where u is the current number of the experiment, then the pair correlation coefficient r is calculated by the formula 88 Here ryyy = u= ( y)(yy) u (y y) (yy) uu= u= uuu = and y = u= y are the arithmetic means for y and y, respectively. The values of the pair correlation coefficient can range from to. If with the growth of the value of one parameter the value of another increases, the coefficient will have a plus sign, and if it decreases, then a minus sign. The closer the found value of r y y is to one, the stronger the value of one parameter depends on what value the other takes, i.e. there is a linear relationship between such parameters, and when studying the process, only one of them can be considered. It must be remembered that the coefficient of pair correlation as a measure of the tightness of the relationship has a clear mathematical meaning only with a linear relationship between the parameters and in the case of their normal distribution. To check the significance of the pair correlation coefficient, you need to compare its value with the tabular (critical) value of r, which is given in App. 6. To use this table, you need to know the number of degrees of freedom f = and choose a certain significance level, for example, equal to 0.05. This value of the significance level corresponds to the probability of a correct answer when testing the hypothesis p = a = 0.05 = 0.95, or 95%. This means that, on average, only 5% of cases can lead to an error in testing the hypothesis. If the experimentally found value of r is greater than or equal to the critical one, then the hypothesis of a correlation linear relationship is confirmed, and if it is less, then there is no reason to believe that there is a close linear relationship between the parameters. With a high significance of the correlation coefficient, any of the two analyzed parameters can be excluded from consideration as not containing additional information about the object of study. You can exclude the parameter that is more difficult to measure, or the one whose physical meaning is less clear. u= y 3. GENERALIZED OPTIMIZATION PARAMETER The path to a single optimization parameter often lies through generalization. It has already been pointed out that of the many responses that define an object, it is difficult to choose one that is the most important. If this is possible, then they fall into the situation described in the previous chapter. In this chapter, more complex situations are considered, where it is necessary to generalize many responses into a single quantitative attribute. There are a number of difficulties associated with such a generalization. Each response has its own physical meaning and its own dimension. To combine different responses, first of all, it is necessary to introduce some dimensionless scale for each of them. The scale must be of the same type for all combined responses u.
9 makes them comparable. Choosing a scale is not an easy task, depending on a priori information about the responses, as well as on the accuracy with which the generalized feature is determined. After constructing a dimensionless scale for each response, the following difficulty arises: choosing the rule for combining the initial partial responses into a generalized indicator. There is no single rule. Here you can go in different ways, and the choice of the path is not formalized. Consider several ways to construct a generalized indicator. 3.. SIMPLE WAYS OF CONSTRUCTING A GENERALIZED RESPONSE Let the object under study be characterized by n partial responses y u (u,..., n) = and each of these responses is measured in experiments. Then u has the value of the uth response in the th experiment (=,...,). Each of the responses of u has its own physical meaning and, most often, a different dimension. We introduce the simplest transformation: we put the data set for each in accordance with the simplest standard analog scale, on which there are only two values: 0 defective, unsatisfactory quality, good product, satisfactory quality. Having thus standardized the scale of partial responses, we proceed to the second stage of their generalization. In a situation where each transformed partial response takes only two values of 0, and it is desirable that the generalized response also takes one of these two possible values, and so that the value takes place if all partial responses in this experiment take on a value. And if at least one of the responses turned to 0, then the generalized response will be zero. With such reasoning, to construct a generalized response, it is convenient to use the formula where Y is a generalized response in the 1st experiment; n u= Y = n n y u u= y y,..., y,. product of partial responses n The root is introduced in order to connect this formula with another, more complex one, which will be discussed later. In this case, nothing will change if we write n Y = y u. The disadvantage of this approach is its rudeness and rigidity. Let's consider another way to obtain a generalized response, which can be used in cases where for each of the particular responses an "ideal" is known to strive for. There are many ways to introduce a metric that defines "closeness to the ideal". Here the concept of "introduce a metric" means to specify a rule for determining the distance between any pairs of objects from the set of interest to us. u= Let's supplement the previous designation with one more:, u u has the best (“ideal”) value of the uth response. Then уu у уо can be considered as some measure of proximity to the ideal. However, it is impossible to use the difference when constructing a generalized response for two reasons. It has the dimension of the corresponding response, and each of the responses can have its own dimension, which prevents them from being combined. Negative or
10 positive sign difference also creates inconvenience. To pass to dimensionless values, it is enough to divide the difference by the desired value: 88 y u y y If in some experiment all partial responses coincide with the ideal, then Y will become equal to zero. This is the value you should aim for. The closer to zero, the better. Here it is necessary to agree on what to consider as the lower limit if the upper one is equal to zero. Among the shortcomings of such an assessment, the leveling of private responses stands out. All of them are included in the generalized response to equal rights. In practice, the various indicators are far from equal. This drawback can be eliminated by introducing some weight a u and u u= a = u > 0 a. u Y uo u u u a u u= uuo = To rank the responses in order of importance and find the appropriate weights, you can use expert judgment. We have considered the simplest ways of constructing a generalized indicator. For the transition and more complex methods, you need to learn to capture more subtle differences on the response transformation scale. Here one has to rely mainly on the experience of the experimenter. But in order for this experience to be reasonably used within the framework of formal procedures, it must also be formalized. The most natural way of such formalization is to introduce the system of preferences of the experimenter on the set of values of each partial response, obtain a standard scale, and then generalize the results. Using the preference system, you can get a more meaningful scale instead of a classification scale with two classes. An example of constructing such a scale is discussed in the next subsection. The construction of this generalized function is based on the idea of transforming the natural values of partial responses into a dimensionless scale of desirability or preference. The desirability scale refers to psychophysical scales. Its purpose is to establish a correspondence between physical and psychological parameters. Here, physical parameters are understood as all kinds of responses that characterize the functioning of the object under study. Among them may be aesthetic and even statistical parameters, while psychological parameters are understood as purely subjective assessments of the experimenter of the desirability of a particular response value. To obtain a desirability scale, it is convenient to use ready-made tables correspondence between preference relations in the empirical and numerical systems (Table 3..). Table 3. Standard Desirability Scale Scores Desirability Desirability Scale Scores Very Good,000.80,
11 Good 0.800.63 Satisfactory 0.630.37 Poor 0.370.0 Very bad 0.00.00 Figure 3.. shows the numbers corresponding to some points of the curve (Fig. 3.), e y which is given by the equation d = e or d = exp[exp(y)], where exp accepted designation exhibitors. d Desirability function 0, Fig. 3.. Desirability values are plotted on the y-axis, varying from 0 to. The abscissa shows the response values written on a conditional scale. The value corresponding to the desirability of 0.37 is chosen for the origin of 0 along this axis. The choice of this particular point is due to the fact that it is the inflection point of the curve, which in turn creates certain convenience in calculations. The desirability curve is usually used as a nomogram. Example. Let among the responses be the reaction output y, the natural boundaries of which are between 0% and 00%. Suppose that 00% corresponds to one on the desirability scale, and 0% to zero, then we get two points on the x-axis: 0 and 00 (Fig. 3.). The choice of other points depends on a number of circumstances, such as the situation at the initial moment, the requirements for the result, and the capabilities of the experimenter. In this case, the area of good results (0.80 0.63 on the scale of desirability) is enclosed within the boundaries of 5055%. 50% gives the lower bound. Example. A different picture emerges when it comes to the synthesis of a new substance, which has not yet been obtained in quantities sufficient for identification. At less than % yield, there is no way to identify the product. Any yield above 0% is excellent (Figure 3.). Here, the output is denoted by y. In our examples, the same reaction yield responses with measurement limits from 0% to 00% are considered. However, this is not always the case. It is worth including such responses as the quality of the material, and the boundaries become vague. In these cases, limits are set on the allowable values for private responses, and the limits can be one-sided in the form y y y y. Here it must be borne in mind that y y,% y,% y u mn and bilateral in the form mn u max ymn corresponds to the mark on the scale of desirability
12 d u = 0.37, and the researcher's max value. y is established on the basis of experience and situation 3.3. GENERALIZED DESILIABILITY FUNCTION After choosing a scale of desirability and converting particular responses into particular desirability functions, we proceed to the construction of a generalized desirability function. Generalize according to the formula n D = n d u u= where D is the generalized desirability; d u private desirability. The way to specify the generalized desirability function is such that if at least one desirability d u = 0, then the generalized function will be equal to zero. On the other hand, D= only when d u =. The generalized function is very sensitive to small values of partial desirability. Example: when determining the suitability of a material with a given set of properties for use under specified conditions, if at least one particular response does not meet the requirements, then the material is considered unsuitable. For example, if at certain temperatures a material becomes brittle and breaks down, then no matter how good other properties are, this material cannot be used for its intended purpose. The method for setting the base marks of the desirability scale, presented in Table 3, is the same for both private and generalized desirability. The generalized desirability function is some abstract construction, but it has such important properties as adequacy, statistical sensitivity, efficiency, and these properties are not lower than those for any technological indicator corresponding to them. The generalized desirability function is a quantitative, unambiguous, unified and universal indicator of the quality of the object under study and has such properties as adequacy, efficiency, statistical sensitivity, and therefore can be used as an optimization criterion. 4. FACTORS After choosing the object of study and the optimization parameter, it is necessary to consider all the factors that can influence the process. If any significant factor turns out to be unaccounted for and takes on arbitrary values that are not controlled by the experimenter, then this will significantly increase the error of the experiment. By maintaining this factor at a certain level, a false idea of the optimum can be obtained, because. there is no guarantee that the resulting level is optimal. On the other hand, a large number of factors increases the number of experiments and the dimension of the factor space. The section states that the number of experiments is equal to p k, where p is the number of levels and k is the number of factors. The question arises of reducing the number of experiments. Recommendations for solving this problem are given in Section 7. Thus, the choice of factors is very significant, since the success of the optimization depends on it. 4.. CHARACTERISTICS OF FACTORS, 88
13 A factor is a measured variable that takes on a certain value at some point in time and affects the object of study. Factors must have a domain of definition, within which its specific values are set. The domain of definition can be continuous or discrete. When planning an experiment, the values of the factors are assumed to be discrete, which is associated with the levels of the factors. IN practical tasks the domains of determining the factors have limitations that are either of a fundamental or technical nature. Factors are divided into quantitative and qualitative. Quantitative factors include those factors that can be measured, weighed, etc. Qualitative factors are various substances, technological methods, devices, performers, etc. Although the numerical scale does not correspond to qualitative factors, but when planning an experiment, a conditional ordinal scale is applied to them in accordance with the levels, i.e. encoding is performed. The order of levels here is arbitrary, but after encoding it is fixed. 4.. REQUIREMENTS FOR FACTORS Factors must be controllable, which means that the selected desired value of the factor can be maintained constant throughout the experiment. An experiment can be planned only if the levels of factors obey the will of the experimenter. For example, the experimental setup is mounted in an open area. Here, we cannot control the air temperature, it can only be controlled, and therefore, when performing experiments, we cannot take into account temperature as a factor. To accurately determine the factor, you need to specify the sequence of actions (operations) by which its specific values are set. Such a definition is called operational. So, if the factor is the pressure in some apparatus, then it is absolutely necessary to indicate at what point and with what instrument it is measured and how it is established. The introduction of an operational definition provides an unambiguous understanding of the factor. The accuracy of measurements of factors should be as high as possible. The degree of accuracy is determined by the range of factors. In long processes, measured in many hours, minutes can be ignored, and in fast processes, fractions of a second must be taken into account. The study becomes much more complicated if the factor is measured with a large error or the values of the factors are difficult to maintain at the selected level (the level of the factor “floats”), then special research methods have to be used, for example, confluent analysis. The factors must be unambiguous. It is difficult to manage a factor that is a function of other factors. But other factors may be involved in planning, such as the ratios between the components, their logarithms, and so on. The need to introduce complex factors arises when you want to present the dynamic features of an object in a static form. For example, it is required to find the optimal regime for raising the temperature in the reactor. If it is known with respect to temperature that it should increase linearly, then instead of a function (linear in this case), one can use the slope tangent as a factor, i.e. gradient. When planning an experiment, several factors are simultaneously changed, so it is necessary to know the requirements for a combination of factors. First of all, the requirement of compatibility is put forward. Compatibility of factors means that all their combinations are feasible and safe.
14 Incompatibility of factors is observed at the boundaries of their areas of definition. You can get rid of it by reducing the areas. The situation becomes more complicated if the incompatibility appears within the domains of definition. One of the possible solutions is splitting into subdomains and solving two separate problems. When planning an experiment, the independence of factors is important, i.e. the possibility of establishing a factor at any level, regardless of the levels of other factors. If this condition is impracticable, then it is impossible to plan an experiment. CHOICE OF LEVELS OF VARIATION OF FACTORS AND THE BASIC LEVEL A factor is considered given if its name and domain of definition are indicated. In the chosen domain of definition, it can have several values that correspond to the number of its different states. The quantitative or qualitative states of the factor chosen for the experiment are called factor levels. In the design of an experiment, the values of factors corresponding to certain levels of their variation are expressed in coded values. Under the factor variation interval is meant the difference between its two values, taken as a unit during coding. When choosing the area for determining factors, special attention is paid to the choice of a zero point, or a zero (main) level. The choice of the zero point is equivalent to the determination of the initial state of the object of study. Optimization is related to the improvement of the state of the object compared to the state at the zero point. Therefore, it is desirable that this point be in the optimum region or as close as possible to it, then the search for optimal solutions is accelerated. If the experiment was preceded by other studies on the issue under consideration, then such a point is taken as zero, which corresponds to the best value of the optimization parameter established as a result of the formalization of a priori information. In this case, the zero levels of the factors are those values of the latter whose combinations correspond to the coordinates of the zero point. Often, when setting a problem, the domain of definition of factors is given, being a localized domain of the factor space. Then the center of this region is taken as the zero point. Suppose, in some task, the factor (temperature) could vary from 40 to 80 ° C. Naturally, the average value of the factor corresponding to 60 ° C was taken as the zero level. After establishing the zero point, the intervals of factor variation are selected. This is due to the determination of such factor values that correspond to and in the coded values. Variation intervals are chosen taking into account the fact that the values of the factors corresponding to the levels and must be sufficiently different from the value corresponding to the zero level. Therefore, in all cases, the value of the variation interval should be greater than twice the quadratic error of fixing this factor. On the other hand, an excessive increase in the value of the variation intervals is undesirable, because this can lead to a decrease in the efficiency of the search for the optimum. A very small interval of variation reduces the area of the experiment, which slows down the search for the optimum. When choosing a variation interval, it is advisable to take into account, if possible, the number of levels of variation of factors in the area of the experiment. The amount of experiment and optimization efficiency depend on the number of levels. In general, the dependence of the number of experiments on the number of levels of factors has the form where the number of experiments; p number of factor levels; k number of factors. k = p,
15 The minimum number of levels usually applied in the first stage of work is equal to. These are the upper and lower levels, denoted in coded coordinates by and. The variation of factors at two levels is used in screening experiments, at the stage of movement towards the optimum region, and in describing the object of study by linear models. But such a number of levels is not enough to build second-order models (after all, the factor takes only two values, and through two points you can draw many lines of different curvature). With an increase in the number of levels, the sensitivity of the experiment increases, but at the same time the number of experiments increases. When constructing second-order models, 3, 4, or 5 levels are required, and here the presence of odd levels indicates that experiments are being carried out at zero (basic) levels. In each individual case, the number of levels is chosen taking into account the conditions of the problem and the proposed methods for planning the experiment. Here it is necessary to take into account the presence of qualitative and discrete factors. In experiments related to the construction of linear models, the presence of these factors, as a rule, does not cause additional difficulties. When planning the second order, qualitative factors are not applicable, because they have no clear physical meaning for the zero level. For discrete factors, measurement scale transformations are often used to ensure that factor values are fixed at all levels. 5. CHOICE OF MODELS As already mentioned in the section, a model is a response function of the form y = f (x, x,..., x k). To choose a model means to choose the form of this function, to write down its equation. Then it remains to plan and conduct an experiment to estimate the numerical values of the constants (coefficients) of this equation. A visual, convenient, perceptible idea of the response function is given by its geometric analogue, the response surface. In the case of many factors, geometric clarity is lost, because passes into an abstract multidimensional space, where most researchers do not have the skill of orientation. We have to switch to the language of algebra. Therefore, consider simple examples of cases with two factors. The space in which the response surface is constructed is called the factor space. It is set by coordinate axes, along which the values of the factors and the optimization parameter are plotted (Fig. 5.). Y X X Fig. 5 .. For two factors, you can not go to three-dimensional space, but limit yourself to a plane. To do this, it is enough to make sections of the surface with planes parallel to the x ox plane (Fig. 5.) and project the lines obtained in the sections onto this plane. Here each line corresponds to a constant value of the parameter
16 optimizations. Such a line is called an equal response line. X X 88 Fig. 5. Having received some idea of the model, consider the requirements for them. The main requirement for the model is the ability to predict the direction of further experiments, and to predict with the required accuracy. This means that the response value predicted by the model does not differ from the actual value by more than some predetermined value. A model that meets this requirement is called adequate. The verification of the feasibility of this requirement is called the verification of the adequacy of the model and it is performed using special statistical methods, which will be discussed later. The next requirement is the simplicity of the model. But simplicity is a relative thing, it must first be formulated. When planning an experiment, it is assumed that algebraic polynomials are simple. The following polynomials are most commonly used. First degree polynomial: y = in o k in x Second degree polynomial: y = in o kkk in jxx in x in j xxj Third degree polynomials: y = in k o kk in x in j xxj in j xxj in jj xxjk in x 3. jkk in x Here in these equations: y is the criterion value; into linear coefficients; in j are the double interaction coefficients; x coded factor values. Experiments when planning an experiment are needed to determine the numerical values of the coefficients. The more coefficients, the more experiments are needed. And we are trying to reduce their number. Therefore, you need to find a polynomial that contains as few coefficients as possible, but satisfies the requirements for the model. Polynomials of the first degree have the least number of coefficients, other than that they are.
17 allow predicting the direction of the fastest improvement of the optimization parameter. But polynomials of the first degree are not effective in the region close to the optimum. Therefore, when planning an experiment at the first stage of the study, polynomials of the first degree are used, and when they become ineffective, they switch to polynomials of higher degrees. 6. COMPLETE FACTOR EXPERIMENT Work on planning an experiment begins with the collection of a priori information. Analysis of this information allows you to get an idea about the optimization parameter, about the factors, about best conditions research, the nature of the response surface, etc. A priori information can be obtained from literary sources, from a survey of specialists, by performing one-factor experiments. The latter, unfortunately, is not always possible to implement, because. the possibility of their implementation is limited by the cost of experiments, their duration. Based on the analysis of a priori information, the choice of the experimental area of the factor space is made, which consists in choosing the main (zero) level and intervals of factor variation. The main level is the starting point for constructing an experiment plan, and the variation intervals determine the distance along the coordinate axes from the upper and lower levels to the main level. When planning an experiment, the values of the factors are encoded by a linear transformation of the coordinates of the factor space with the transfer of the origin to the zero point and the choice of scales along the axes in units of the intervals of variation of the factors. The ratio x c c ε o = is used here, where x is the encoded value of the factor (dimensionless value); c natural values of the factor (respectively, the current value and c o zero level); ε is the natural value of the interval of variation of factors (C). The values of the factors equal to (upper level) and (lower level) are obtained. The location of the experimental points in the factor space for the full factorial experiment at k= and k=3 is shown in fig. 6.. As you can see, the points of the plan are set by the coordinates of the vertices of the square, and the points of the plan are 3 by the coordinates of the vertices of the cube. The experimental points at k>3 are located according to a similar principle. C X C X C C C a) k= c) k=3
18 Fig COMPLETE FACTOR TYPE k EXPERIMENT The first stage of experiment design to obtain a linear model is based on variation at two levels. In this case, with a known number of factors, one can find the number of experiments necessary to implement all possible combinations of factor levels. The formula for calculating the number of experiments was given in the section and in this case it looks like = k. An experiment in which all possible combinations of factor levels are implemented is called a full factorial experiment (FFE). If the number of factor levels is two, then we have a PFE of type k. It is convenient to write the conditions of the experiment in the form of a table, which is called the experiment planning matrix. Experiment planning matrix Table 6. Experiment number x x y 3 4 y y 3 y 4 6.. When filling in the planning matrix, the values of the levels of factors, for the sake of simplicity, are indicated by the corresponding signs, and the number is omitted. Taking into account the interaction of factors x and x, Table 6. can be rewritten as follows: Planning matrix Table 6. Experience number 3 4 x x x x y y y 3 y 4 Each column in the planning matrix is called a column vector, and each row is called a row vector. Thus, in table. 6.. we have two column vectors of independent variables and one column vector of the optimization parameter. What is written in algebraic form can be represented graphically. A point corresponding to the main level is located in the area of determining the factors, and new coordinate axes are drawn through it, parallel to the axes of the natural values of the factors. Next, the scales along the new axes are chosen so that the variation interval for each factor is equal to one. Then the conditions for conducting experiments will correspond to the vertices of the square, at k=, and the vertices of the cube, at k=3. The centers of these figures is the main level, and each side is equal to two intervals (Fig. 6.). The numbers of the vertices of the square and cube correspond to the numbers of experiments in the planning matrix. The area bounded by these figures is called the experiment area. The experimental points at k>3 are located according to a similar principle. 88
19 Location of points in the factor space for PFE at k= and k=3 С Х С Х C C С С a) k= c) k=3 6.. If for two factors all possible combinations of levels can be easily found by enumeration, then with an increase in the number of factors, it becomes necessary to use some technique for constructing matrices. Three methods are usually used, based on the transition from matrices of a smaller dimension to matrices of a larger dimension. Let's consider the first approach. When a new factor is added, each combination of levels of the original factor occurs twice, in combination with the top and bottom levels of the new factor. From here, a technique naturally appears: write down the initial plan for one level of a new factor, and then repeat it for another level. This technique can be applied to matrices of any dimension. In the second technique, the rule of multiplication of matrix columns is introduced. When multiplying the levels of the original matrix by row, we obtain an additional column of the product x x, then we repeat the original plan, and the signs of the column of products are reversed. This technique is applicable to constructing matrices of any dimension, but it is more complicated than the first one. The third technique is based on the alternation of signs. In the first column, the signs change alternately, in the second column they alternate two times, in the third after four, in the fourth after eight, and so on. by powers of two. An example of constructing planning matrices p 3, see table. 6. Table 6.3 Experiment design matrix 3 Experiment number 3 4 x x x 3 y y y 3 y 4
20 y 5 y 6 y 7 y PROPERTIES OF THE COMPLETE FACTORY EXPERIMENT OF TYPE k Full factorial experiment is one of the designs that are the most effective in building linear models. Efficiency, otherwise optimality, of a complete factorial experiment is achieved due to its properties listed below. Two properties follow directly from the construction of the matrix. The first of them, symmetry with respect to the center of the experiment, is formulated as follows: the algebraic sum of the elements of the column vector of each factor is equal to zero, or j= x j = 0, where =, k is the number of the factor, the number of experiments. The second property, the so-called normalization condition, is formulated as follows: the sum of the squares of the elements of each column is equal to the number of experiments, or j= This is a consequence of the fact that the values of the factors in the matrix are given by and. We have considered the properties of individual columns of the planning matrix. Consider the properties of a set of columns. The sum of the termwise products of any two matrix column vectors is equal to zero, or x j uj = 0 j= x j = x for u, and also, u = 0,..., k. This important property is called planning matrix orthogonality. The last, fourth property is called rotatability, i.e. the points in the planning matrix are selected so that the accuracy of the optimization parameter values predictions is the same at equal distances from the center of the experiment and does not depend on the direction. The fulfillment of these conditions ensures the minimum dispersion of the regression coefficients, but also the equality of the dispersion. This facilitates the statistical analysis of the results of the experiment. CALCULATION OF REGRESSION COEFFICIENTS After constructing the planning matrix, the experiment is carried out. Having received the experimental data, the values of the regression coefficients are calculated. The value of the free term (in o) is taken as the arithmetic mean of all values of the optimization parameter in the matrix: where in o y u. =, u y values of the optimization parameter in the uth experiment; the number of experiences in the matrix.
21 Linear regression coefficients are calculated by the formula in x y u u = = xu where хu is the coded value of the x factor in uth trial. The regression coefficients characterizing the pair interaction of factors are found by the formula in x x y u ju u j = = xu Consider an example of calculating the regression coefficients for planning, the planning matrix of which is given in Table. 6. y y y3 y4 in o = ; 4 y y y3 y4 c = ; 4 y y y3 y4 c = ; 4 y y y3 y4 in =. 4 Consider the regression equation for k=3. y \u003d b0 in in in in3x3 in in xx in3xx3 in 3 x x3 in3 xx x3, where in0 is a free member; c, c c linear coefficients;, 3, c3, c3 c coefficients of double interaction; in 3 triple interaction coefficient. The total number of all possible regression coefficients, including at 0, linear coefficients and interaction coefficients of all orders, is equal to the number of trials of the full factorial experiment. To find the number of interactions of a certain order, you can use the formula for the number of combinations C m k x x k! m!(km)! u u =, where k is the number of factors; m is the number of elements in the interaction. So, for plan 4, the number of pair interactions is six 4! C 4 == 6.!! This shows that with an increase in the number of factors, the number of possible interactions quickly y x u ju, y u.
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1 Plans for one independent variable
The design of a "true" experimental study differs from others in the following ways: the most important features:
1) using one of the strategies for creating equivalent groups, most often - randomization;
2) the presence of an experimental and at least one control group;
3) completion of the experiment by testing and comparing the behavior of the group that received the experimental exposure (X1) with the group that did not receive the exposure X0.
The classic version of the plan is the plan for 2 independent groups. In psychology, experiment planning has been used since the first decades of the 20th century.
There are three main versions of this plan. When describing them, we will use the symbolization proposed by Campbell.
Table 5.1
Here R-randomization, X-exposure, O1 - testing of the first group, O2 - testing of the second group.
1) Plan for two randomized groups with post-exposure testing. Its author is the famous biologist and statistician R. A. Fisher. The structure of the plan is shown in Table. 5.1.
The equality of the experimental and control groups is an absolutely necessary condition for the application of this plan. Most often, to achieve group equivalence, a randomization procedure is used (see Chapter 4). This plan is recommended for use when it is not possible or necessary to conduct preliminary testing of subjects. If the randomization is done well, then this plan is the best one, it allows you to control most of the sources of artifacts; in addition, various variants of analysis of variance are applicable to it.
After randomization or other group equalization procedure, an experimental impact is carried out. In the simplest version, only two gradations of the independent variable are used: there is an impact, there is no impact.
If it is necessary to use more than 1 level of exposure, then plans with several experimental groups (according to the number of exposure levels) and one control group are used.
If it is necessary to control the influence of one of the additional variables, then a plan with 2 control groups and 1 experimental group is used. Behavior measurement provides material for comparing the 2 groups. Data processing is reduced to the use of estimates traditional for mathematical statistics. Consider the case when the measurement is carried out with an interval scale. Student's t-test is used to assess differences in mean group scores. Evaluation of differences in the variation of the measured parameter between the experimental and control groups is carried out using the F criterion. The corresponding procedures are discussed in detail in the textbooks of mathematical statistics for psychologists.
The use of a 2-group randomized design with post-exposure testing controls for the main sources of intrinsic invalidity (as defined by Campbell). Since there is no preliminary testing, the effect of the interaction between the testing procedure and the content of the experimental impact and the testing effect itself are excluded. The plan allows you to control the influence of the composition of groups, spontaneous dropout, the influence of the background and natural development, the interaction of the composition of the group with other factors, it also allows you to exclude the effect of regression due to randomization and comparison of data from the experimental and control groups. However, when conducting most pedagogical and socio-psychological experiments, it is necessary to strictly control the initial level of the dependent variable, whether it be intelligence, anxiety, knowledge, or the status of the individual in the group. Randomization is the best possible procedure, but it does not give an absolute guarantee of the correct choice. When there is doubt about the results of randomization, a plan with pre-testing is used.
Table 5.2
2) Plan for two randomized groups with pre-test and post-test. Consider the structure of this plan (Table 5.2).
The pretest plan is popular with psychologists. Biologists have more confidence in the randomization procedure. The psychologist is well aware that each person is unique and different from others, and subconsciously seeks to catch these differences with the help of tests, not trusting the mechanical procedure of randomization. However, the hypothesis of most psychological research, especially in the field of developmental psychology (“formative experiment”), contains a prediction of a certain change in the property of an individual under the influence of an external factor. Therefore, the test-exposure-retest plan using randomization and a control group is very common.
In the absence of a group equalization procedure, this plan is converted into a quasi-experimental one (it will be discussed in Section 5.2).
The main source of artifacts that violate the external validity of the procedure is the interaction of testing with experimental influence. For example, testing the level of knowledge in a particular subject before conducting an experiment on memorizing material can lead to updating the initial knowledge and to an overall increase in memorization productivity. This is achieved by updating the mnemonic abilities and creating a setting for memorization.
However, other external variables can be controlled with this plan. The “history” (“background”) factor is controlled, since in the interval between the first and second testing, both groups are exposed to the same (“background”) influences. At the same time, Campbell notes the need to control for "intra-group events", as well as the effect of testing non-simultaneity in both groups. In reality, it is impossible to ensure that the test and retest are carried out in them at the same time. The plan becomes quasi-experimental, for example:
Usually, testing non-simultaneity control is carried out by two experimenters testing two groups at the same time. The procedure of randomization of the order of testing is considered optimal: testing of members of the experimental and control groups is performed in random order. The same is done with the presentation - not the presentation of experimental influence. Of course, such a procedure requires the presence of a significant number of subjects in the experimental and control samples (at least 30-35 people in each).
Natural development and test effects are controlled by the fact that they are equally manifested in the experimental and control groups, and the effects of group composition and regression [Campbell, 1980] are controlled using a randomization procedure.
The results of applying the "test-impact-retest" plan are presented in the table.
When processing data, parametric criteria t and F are usually used (for data on an interval scale). Three values of t are calculated: comparison 1) O1 and O2; 2) O3 and O4; 3) O2 and O4. The hypothesis of a significant influence of the independent variable on the dependent variable can be accepted if two conditions are met: a) the differences between O1 and O2 are significant, and between O3 and O4 are insignificant, and b) the differences between O2 and O4 are significant. It is much more convenient to compare not absolute values, but the magnitude of the increase in indicators from the first test to the second (δ(i)). δ(i12) and δ(i34) are calculated and compared by Student's t-test. In the case of significant differences, an experimental hypothesis is accepted about the influence of the independent variable on the dependent one (Table 5.3).
It is also recommended to use Fisher's analysis of covariance. In this case, the indicators of preliminary testing are taken as an additional variable, and the subjects are divided into subgroups depending on the indicators of preliminary testing. Thus, the following table is obtained for processing data using the MANOVA method (Table 5.4).
The use of the “test-impact-retest” plan allows you to control the influence of “side” variables that violate the internal validity of the experiment.
External validity is related to the ability to transfer data to a real situation. The main point that distinguishes the experimental situation from the real one is the introduction of preliminary testing. As we have already noted, the “test-exposure-retest” plan does not allow controlling the effect of the interaction of testing and experimental exposure: the pre-tested subject “sensitizes” - becomes more sensitive to the impact, since we measure in the experiment exactly the dependent variable that we are going to influence by varying the independent variable.
Table 5.5
To control external validity, the plan of R. L. Solomon, which was proposed by him in 1949, is used.
3) Solomon's plan is used when conducting an experiment on four groups:
1. Experiment 1: R O1 X O2
2. Control 1: R O3 O4
3. Experiment 2: R X O5
4. Control 2: R O6
The design includes a study of two experimental and two control groups and is essentially a multigroup (2 x 2 type), but for convenience of presentation it is discussed in this section.
Solomon's plan is a combination of two previously discussed plans: the first, when no pre-testing is performed, and the second - "test-impact-retest". With the help of the "first part" of the plan, it is possible to control the effect of the interaction of the first test and experimental exposure. Solomon, with the help of his plan, reveals the effect of experimental exposure in four different ways: when comparing 1) O2 - O1; 2) O2 - O4; 3) O5 - O6 and 4) O5 - O3.
If we compare O6 with O1 and O3, then we can identify the combined influence of the effects of natural development and "history" (background effects) on the dependent variable.
Campbell, criticizing Solomon's data-processing schemes, suggests ignoring pre-testing and reducing the data to a 2 x 2 scheme suitable for applying analysis of variance (Table 5.5).
Comparison of the averages by columns makes it possible to reveal the effect of the experimental influence - the influence of the independent variable on the dependent one. Row averages show the effect of pre-testing. Comparison of cell means characterizes the interaction of the test effect and experimental exposure, which indicates the degree of violation of external validity.
In the case when the effects of preliminary testing and interaction can be neglected, one proceeds to comparing O4 and O2 by the method of covariance analysis. As an additional variable, pre-testing data is taken according to the scheme given for the “test-impact-retest” plan.
Finally, in some cases it is necessary to check the persistence of the effect of the influence of the independent variable on the dependent one over time: for example, to find out whether a new teaching method leads to long-term memorization of the material. For these purposes, they use next plan:
1 Experiment 1 R O1 X O2
2 Control 1 R O3 O4
3 Experiment 2 R O5 X O6
4 Control 2 R O7 O8
2. Designs for one independent variable and several groups
Sometimes comparing two groups is not enough to confirm or refute an experimental hypothesis. Such a problem arises in two cases: a) when it is necessary to control external variables; b) if necessary, to identify quantitative relationships between two variables.
Various variants of the factorial experimental design are used to control external variables. As for the identification of a quantitative relationship between two variables, the need to establish it arises when testing an "exact" experimental hypothesis. In a two-group experiment, at best, a causal relationship can be established between the independent and dependent variables. But an infinite number of curves can be drawn between two points. In order to verify that there is a linear relationship between two variables, you must have at least three points corresponding to three levels of the independent variable. Therefore, the experimenter must select several randomized groups and put them in different experimental conditions. The simplest option is to design for three groups and three levels of the independent variable:
Experiment 1: R X1 O1
Experiment 2: R X2 O2
Control: R O3
The control group in this case is the third experimental group for which the level of the variable X = 0.
When implementing this plan, each group is presented with only one level of the independent variable. It is also possible to increase the number of experimental groups according to the number of levels of the independent variable. To process the data obtained using such a plan, the same statistical methods are used as listed above.
Simple "systemic experimental designs" are, surprisingly, very rarely used in modern experimental research. Maybe the researchers are "embarrassed" to put forward simple hypotheses, remembering the "complexity and multidimensionality" of psychic reality? The inclination to use designs with many independent variables, moreover, to conduct multivariate experiments, does not necessarily contribute to a better explanation of the causes of human behavior. As you know, "the smart one strikes with the depth of the idea, and the fool - with the scope of construction." It's best to prefer a simple explanation over a complex one, although regression equations where everything equals everything and intricate correlation graphs can impress some dissertation councils.
3 Factorial designs
Factorial experiments are used when it is necessary to test complex hypotheses about the relationships between variables. The general form of such a hypothesis is: "If A1, A2,..., An, then B." Such hypotheses are called complex, combined, etc. At the same time, there can be various relationships between independent variables: conjunctions, disjunctions, linear independence, additive or multiplicative, etc. Factor experiments are a special case of a multidimensional study, during which they try to establish relationships between several independent variables. and several dependent variables. In a factorial experiment, as a rule, two types of hypotheses are tested simultaneously:
1) hypotheses about the separate influence of each of the independent variables;
2) hypotheses about the interaction of variables, namely, how the presence of one of the independent variables affects the effect of the impact on the other.
The factorial experiment is built according to the factorial plan. Factorial design of the experiment is to ensure that all levels of independent variables are combined with each other. The number of experimental groups is equal to the number of combinations of levels of all independent variables.
Today, factorial plans are the most common in psychology, since simple relationships between two variables are practically not found in it.
There are many variants of factorial plans, but not all of them are used in practice. Most often, factorial designs are used for two independent variables and two levels of the 2x2 type. To draw up a plan, the principle of balancing is applied. The 2x2 design is used to identify the effect of two independent variables on one dependent variable. The experimenter manipulates the possible combinations of variables and levels. The data are given in a simple table (Table 5.6).
Less frequently, four independent randomized groups are used. Fisher's analysis of variance is used to process the results.
Other versions of the factorial design are also rarely used, namely: 3x2 or 3x3. The 3x2 plan is used in cases where it is necessary to establish the type of dependence of one dependent variable on one independent variable, and one of the independent variables is represented by a dichotomous parameter. An example of such a plan is an experiment to identify the impact of external observation on the success of solving intellectual problems. The first independent variable varies simply: there is an observer, there is no observer. The second independent variable is the difficulty levels of the task. In this case, we get a 3x2 plan (Table 5.7).
A variant of the 3x3 plan is used if both independent variables have several levels and it is possible to identify the types of relationship between the dependent variable and the independent ones. This plan allows you to identify the effect of reinforcement on the success of completing a task of varying difficulty (Table 5.8).
IN general case the design for two independent variables looks like N x M. The applicability of such plans is limited only by the need to recruit a large number of randomized groups. The amount of experimental work increases exorbitantly with the addition of each level of any independent variable.
Designs used to investigate the effects of more than two independent variables are rarely used. For three variables, they have the general form L x M x N.
The most commonly used plans are 2x2x2: "three independent variables - two levels." Obviously, adding each new variable increases the number of groups. Their total number is 2, where n is the number of variables in the case of two intensity levels and K in the case of a K-level intensity (we assume that the number of levels is the same for all independent variables). An example of this plan may be the development of the previous one. In the case when we are interested in the success of the experimental series of tasks, which depends not only on the general stimulation, which is produced in the form of punishment - electric shock, but also on the ratio of reward and punishment, we apply the 3x3x3 plan.
A simplification of a complete plan with three independent variables of the form L x M x N is planning according to the “Latin square” method. The "Latin square" is used when it is necessary to investigate the simultaneous influence of three variables that have two or more levels. The principle of the "Latin square" is that two levels of different variables occur in the experimental plan only once. This greatly simplifies the procedure, not to mention the fact that the experimenter gets rid of the need to work with huge samples.
Suppose we have three independent variables, each with three levels:
The plan according to the "Latin square" method is presented in Table. 5.9.
The same technique is used to control external variables (counterbalancing). It is easy to see that the levels of the third variable N (A, B, C,) occur in each row and in each column once. By combining the results across rows, columns, and levels, it is possible to identify the influence of each of the independent variables on the dependent variable, as well as the degree of pairwise interaction of the variables.
"Latin Square" allows you to significantly reduce the number of groups. In particular, the 2x2x2 plan turns into a simple table (Table 5.10).
The use of Latin letters in cells to indicate the levels of the 3rd variable (A - yes, B - no) is traditional, so the method is called "Latin square".
A more complex plan according to the "Greco-Latin square" method is used very rarely. It can be used to investigate the effect of four independent variables on the dependent variable. Its essence is as follows: to each Latin group of a plan with three variables, a Greek letter is attached, denoting the levels of the fourth variable.
Consider an example. We have four variables, each with three levels of intensity. The plan according to the method of "Greco-Latin square" will take this form (Table 5.11).
For data processing, the method of variance analysis according to Fisher is used. The methods of the "Latin" and "Greco-Latin" square came to psychology from agrobiology, but were not widely used. The exceptions are some experiments in psychophysics and psychology of perception.
The main problem that can be solved in a factorial experiment and cannot be solved by applying several ordinary experiments with one independent variable is determining the interaction of two variables.
Consider the possible results of the simplest factorial experiment 2x2 from the standpoint of the interactions of variables. To do this, we need to present the results of experiments on a graph, where the values of the first independent variable are plotted along the abscissa axis, and the values of the dependent variable are plotted along the ordinate axis. Each of the two straight lines connecting the values of the dependent variable at different values of the first independent variable (A) characterizes one of the levels of the second independent variable (B). Let us apply for simplicity the results of not an experimental, but a correlation study. Let us agree that we have investigated the dependence of the child's status in the group on the state of his health and level of intelligence. Consider options for possible relationships between variables.
The first option: the lines are parallel - there is no interaction of variables (Fig. 5.1).
Sick children have a lower status than healthy children, regardless of the level of intelligence. Intellectuals always have a higher status (regardless of health).
The second option: physical health with a high level of intelligence increases the chance of getting a higher status in the group (Figure 5.2).
In this case, the effect of divergent interaction of two independent variables is obtained. The second variable amplifies the effect of the first on the dependent variable.
The third option: converging interaction - physical health reduces the chance of an intellectual to acquire a higher status in the group. The variable "health" reduces the influence of the variable "intelligence" on the dependent variable. There are other cases of this interaction option:
variables interact in such a way that an increase in the value of the first leads to a decrease in the influence of the second with a change in the sign of the dependence (Fig. 5.3).
In sick children with a high level of intelligence, there is less chance of getting a high status than in sick children with low intelligence, and in healthy children, the relationship between intelligence and status is positive.
It is theoretically possible to imagine that sick children will have a greater chance of achieving high status with a high level of intelligence than their healthy low-intellect peers.
The last, fourth, possible variant of the relationships between independent variables observed in the studies: the case when there is an intersecting interaction between them, presented in the last graph (Fig. 5.4).
So, the following interactions of variables are possible: zero; divergent (with different dependency signs); intersecting.
The magnitude of the interaction is estimated using analysis of variance, and Student's t-test is used to assess the significance of differences in group X.
In all the considered options for planning an experiment, a balancing method is used: different groups of subjects are placed in different experimental conditions. The procedure for equalizing the composition of groups allows you to compare the results.
However, in many cases it is required to plan the experiment so that all its participants receive all options for the influence of independent variables. Then the technique of counterbalancing comes to the rescue.
Plans that embody the strategy "all subjects - all exposures" McCall calls rotational experiments, and Campbell - "balanced plans". To avoid confusion between the concepts of "balancing" and "counter-balancing", we will use the term "rotational plan".
Rotational plans are built according to the “Latin square” method, but, unlike the example considered above, the groups of subjects are indicated in the rows, and not the levels of the variable, in the columns - the levels of exposure to the first independent variable (or variables), in the cells of the table - the levels of exposure to the second independent variable.
An example of an experimental design for 3 groups (A, B, C) and 2 independent variables (X, Y) with 3 intensity levels (1st, 2nd, 3rd) is given below. It is easy to see that this plan can also be rewritten so that the levels of the variable Y are in the cells (Table 5.12).
Campbell includes this design as a quasi-experimental design on the basis that it is not known whether it controls external validity. Indeed, it is unlikely real life the subject can receive a series of such influences, as in the experiment.
As for the interaction of group composition with other external variables, sources of artifacts, the randomization of groups, according to Campbell, should minimize the influence of this factor.
Column sums in a rotational design indicate differences in effect levels for different values of one independent variable (X or Y), while row sums should characterize differences between groups. If the groups are successfully randomized, then there should be no between-group differences. If the composition of the group is an additional variable, it becomes possible to control it. The counterbalancing scheme does not allow one to avoid the effect of training, although the data of numerous experiments using the "Latin square" do not allow such a conclusion to be drawn.
Summing up the consideration of various variants of experimental plans, we propose their classification. Experimental plans differ on the following grounds:
1. Number of independent variables: one or more. Depending on their number, either a simple or a factorial design is used.
2. The number of levels of independent variables: at 2 levels, we are talking about establishing a qualitative relationship, at 3 or more - a quantitative relationship.
3. Who gets the exposure. If the scheme “each group has its own combination” is applied, then we are talking about an intergroup plan. If the “all groups - all impacts” scheme is used, then we are talking about a rotational plan. Gottsdanker calls it cross-individual comparison.
The experimental design scheme can be homogeneous or heterogeneous (depending on whether the number of independent variables is equal or not equal to the number of levels of their change).
4 Experimental plans for one subject
Experiments on samples with variable control is a situation that has been widely used in psychology since the 1910s-1920s. Experimental studies on equalized groups became especially widespread after the creation by the outstanding biologist and mathematician R. A. Fisher of the theory of planning experiments and processing their results (variance and covariance analyses). But psychologists were using experiment long before the advent of sampling design theory. The first experimental studies were carried out with the participation of one subject - he was the experimenter himself or his assistant. Beginning with G. Fechner (1860), the technique of experimentation came into psychology to test theoretical quantitative hypotheses.
The classic experimental study of one subject was the work of G. Ebbinghaus, which was carried out in 1913. Ebbinghaus investigated the phenomenon of forgetting by memorizing meaningless syllables (invented by him). He memorized a series of syllables, and then tried to reproduce them after a certain time. As a result, a classic forgetting curve was obtained: the dependence of the volume of stored material on the time elapsed from the moment of memorization (Fig. 5.5).
In empirical scientific psychology, three research paradigms interact and struggle. Representatives of one of them, traditionally coming from a natural-science experiment, consider only that which is obtained in experiments on equivalent and representative samples as the only reliable knowledge. The main argument of the supporters of this position is the need to control external variables and leveling individual differences to find common patterns.
Representatives of the methodology of "experimental analysis of behavior" criticize the supporters of statistical analysis and design of experiments on samples. In their opinion, it is necessary to conduct studies with the participation of one subject and using certain strategies that will allow reducing the sources of artifacts during the experiment. Supporters of this methodology are such well-known researchers as B.F. Skinner, G.A. Murrayidr.
Finally, classical idiographic research is opposed to both single-subject experiments and plans that study behavior in representative samples. Idiographic research involves the study of individual cases: biographies or behavioral patterns of individuals. Examples are Luria's wonderful works The Lost and Returned World and A Little Book of Great Memory.
In many cases, single-subject studies are the only option. The methodology for the study of one subject was developed in the 1970s-1980s. by many authors: A. Kezdan, T. Kratochwill, B.F. Skinner, F.-J. McGuigan and others.
During the experiment, two sources of artifacts are identified: a) errors in the planning strategy and in the conduct of the study; b) individual differences.
If you create a "correct" strategy for conducting an experiment with one subject, then the whole problem will be reduced to accounting for individual differences. An experiment with one subject is possible when: a) individual differences can be neglected in relation to the variables studied in the experiment, all subjects are considered equivalent, so data can be transferred to each member of the population; b) the subject is unique, and the problem of direct data transfer is irrelevant.
The single-subject experimentation strategy was developed by Skinner to investigate the learning process. Data during the study are presented in the form of "learning curves" in the coordinate system "time" - "total number of responses" (cumulative curve). The learning curve is initially analyzed visually; its changes over time are considered. If the function describing the curve changes when the influence A changes to B, then this may indicate the presence of a causal dependence of behavior on external influences (A or B).
Single-subject research is also called time series planning. The main indicator of the influence of the independent variable on the dependent variable in the implementation of such a plan is the change in the nature of the responses of the subject from the impact on him of changing the conditions of the experiment over time. There are a number of basic schemes for applying this paradigm. The simplest strategy is the A-B scheme. The subject initially performs the activity in conditions A, and then - in conditions B (see Fig. 5.8).
When using this plan, a natural question arises: would the response curve have retained its previous form if there had been no impact? Simply put, this regimen does not control for the placebo effect. In addition, it is not clear what led to the effect: perhaps it was not the variable B that had the effect, but some other variable that was not taken into account in the experiment.
Therefore, another scheme is more often used: A-B-A. Initially, the behavior of the subject under conditions A is recorded, then the conditions change (B), and at the third stage, the previous conditions return (A). We study the change in the functional relationship between the independent and dependent variables. If, when conditions change at the third stage, the former form of the functional relationship between the dependent and dependent variables is restored, then the independent variable is considered a cause that can modify the behavior of the subject (Fig. 5.9).
However, both the first and the second time series planning options do not allow taking into account the impact cumulation factor. Perhaps a combination - a sequence of conditions (A and B) - leads to an effect. It is also not obvious that after returning to situation B, the curve will take the same form as it was when conditions B were first presented.
An example of a design that reproduces the same experimental effect twice is the A-B-A-B scheme. If during the 2nd transition from conditions A to conditions B a change in the functional dependence of the subject's responses on time is reproduced, then this will become evidence of the experimental hypothesis: the independent variable (A, B) affects the behavior of the subject.
Let's consider the simplest case. As a dependent variable, we choose the total amount of knowledge of the student. As an independent - physical education in the morning (for example, wushu gymnastics). Let us assume that the wushu complex has a positive effect on the general mental state of the student and contributes to better memorization (Fig. 5.10).
Obviously, gymnastics had a positive effect on learning.
There are various options for time series planning. There are schemes of regular alternation of series (AB-AB), series of stochastic sequences and positional adjustment schemes (example: ABBA). Modifications of the A-B-A-B scheme are scheme A-B-A-B-A or longer: A-B-A-B-A-B-A.
The use of "longer" time plans increases the guarantee of detection of the effect, but leads to fatigue of the subject and other cumulative effects.
In addition, the A-B-A-B plan and its various modifications do not remove three major problems:
1. What would happen to the subject if there was no effect (placebo effect)?
2. Is the sequence impacts A-B by itself another impact (side variable)?
3. What reason led to the effect: if there were no impact in place B, would the effect be repeated?
To control the placebo effect in series A-B-A-B include conditions that “simulate” either action A or action B. Consider the solution last problem. But first, let's analyze such a case: let's say a student constantly practices wushu. But periodically at the stadium or in gym a pretty girl appears (just a spectator) - impact B. Plan A-B-A-B revealed an increase in the effectiveness of the student's studies during the periods of the appearance of variable B. What is the reason: the presence of a spectator as such or a specific pretty girl? To test the hypothesis about the presence of a specific cause, the experiment is built according to the following scheme: A-B-A-C-A. For example, in the fourth time period, another girl or a bored pensioner comes to the stadium. If the effectiveness of classes decreases significantly (not the same motivation), then this will indicate a specific reason for the deterioration in learning. It is also possible to check the impact of condition A (wushu classes without spectators). To do this, you need to apply the plan A-B-C-B. Let the student stop classes for some time in the absence of the girl. If her reappearance at the stadium will lead to the same effect as the first time, then the reason for the increase in academic performance is in her, and not only in wushu classes (Fig. 5.11).
Please do not take this example seriously. In reality, just the opposite is happening: the passion for girls sharply reduces the academic performance of students.
There are many methods of conducting research with the participation of a single subject. An example of the development of the A-B plan is the "alternative impact plan". Exposures A and B are randomly distributed over time, for example by day of the week, if we are talking about different ways to get rid of smoking. Then all the moments when there was an impact A are determined; a curve is constructed connecting the corresponding consecutive points. All moments of time when there was an "alternative" impact B are selected, and in order of sequence in time they are also connected; the second curve is drawn. Then both curves are compared and it is revealed which effect is more effective. Efficiency is determined by the magnitude of the rise or fall of the curve (Fig. 5.12).
Synonyms for the term “alternative impact plan” are: “series comparison plan”, “synchronized impact plan”, “multiple schedule plan”, etc.
Another option is a reverse plan. It is used to explore two alternative forms of behavior. Initially, the basic level of manifestation of both forms of behavior is recorded. The first behavior can be actualized with the help of a specific impact, and the second, incompatible with it, is simultaneously provoked by another type of impact. The effect of two exposures is evaluated. After a certain time, the combination of influences is reversed so that the first form of behavior receives the influence that initiated the second form of behavior, and the second - the influence relevant to the first form of behavior. Such a plan is used, for example, in the study of the behavior of young children (Figure 5.13).
In the psychology of learning, the method of changing criteria, or the “plan of increasing criteria,” is used. Its essence lies in the fact that a change in the behavior of the subject is recorded in response to an increase (phase) of exposure. An increase in the registered parameter of behavior is fixed, and the next impact is carried out only after the subject reaches the specified level of the criterion. After stabilization of the level of performance, the subject is presented with the following gradation of exposure. The curve of a successful experiment (confirming the hypothesis) resembles a staircase knocked down by heels, where the beginning of the step coincides with the beginning of the exposure level, and its end coincides with the test subject's exit to the next criterion.
The way to level the "sequence effect" is the inversion of the sequence of influences - plan A-B-B-A. Sequence effects are related to the influence of the previous impact on the subsequent one (another name is order effects, or transfer effects). The transfer can be positive or negative, symmetrical or asymmetric. The A-B-B-A sequence is called a positionally equalized scheme. As Gottsdanker points out, the effects of variables A and B are due to early or late carryover effects. Impact A is associated with late transfer, and B - with early transfer. In addition, if there is a cumulative effect, then two successive exposures B can affect the subject as a single cumulative exposure. An experiment can only be successful if these effects are negligible. The variants of plans discussed above with regular alternation or with random sequences are most often very long, so they are difficult to implement.
To summarize briefly, we can say that the exposure schemes are applied depending on the possibilities that the experimenter has.
A random sequence of exposures is obtained by randomizing tasks. It is used in experiments requiring a large number of samples. Random alternation of exposures ensures that sequence effects do not occur.
For a small number of samples, a regular alternation scheme of the A-B-A-B type is recommended. Attention should be paid to the periodicity of background influences, which may coincide with the action of the independent variable. For example, if you give one test for intelligence in the morning, and the second - always in the evening, then under the influence of fatigue, the effectiveness of the second test will decrease.
A positionally equalized sequence can be suitable only when the number of actions (tasks) is small and the influence of early and late transfer is insignificant.
But none of the schemes excludes the manifestation of differentiated asymmetric transfer, when the influence of the previous influence A on the effect of influence B is greater than the influence of the previous influence B on the effect of influence A (or vice versa).
A variety of plans for one subject were summarized by D. Barlow and M. Hersen in the monograph "Experimental designs for single cases" (Single case experimental designs, 1984) (Table 5.13).
Table 5.13
Major artefacts in a single-subject study are virtually unremovable. It is difficult to imagine how the effects associated with the irreversibility of events can be eliminated. If the effects of order or the interaction of variables are to some extent controllable, then the already mentioned effect of asymmetry (differential transfer) cannot be eliminated.
No less problems arise when establishing from entry level the intensity of the recorded behavior (the level of the dependent variable). The initial level of aggressiveness that we recorded in a child in a laboratory experiment may not be typical for him, since it was caused by recent previous events, such as a quarrel in the family, suppression of his activity by peers or kindergarten teachers.
The main problem is the possibility of transferring the results of a study of one subject to each of the representatives of the population. We are talking about taking into account individual differences that are significant for the study. Theoretically, the following move is possible: presentation of individual data in a "dimensionless" form; in this case, the individual values of the parameter are normalized to a value equal to the spread of values in the population.
Consider an example. In the early 1960s in the laboratory of B. N. Teplov, a problem arose: why are all the graphs describing changes in reaction time depending on the intensity of the stimulus different for the subjects? V. D. Nebylitsyn [Nebylitsyn V. D. units of physical intensity, but in units of previously measured individual absolute threshold (“one threshold”, “two thresholds”, etc.). The results of the experiment brilliantly confirmed Nebylitsyn's hypothesis: the dependence curves of the reaction time on the level of exposure, measured in units of the individual absolute threshold, turned out to be identical in all subjects.
A similar scheme is applied to data interpretation. At the Institute of Psychology of the Russian Academy of Sciences A. V. Drynkov conducted research on the formation of simple artificial concepts. The learning curves showed the dependence of success on time. They turned out to be different for all subjects: they were described by power functions. Drynkov suggested that normalizing individual indicators by the value of the initial level of learning (along the Y axis) and by the individual time to achieve the criterion (along the X axis) makes it possible to obtain a functional dependence of success on time, which is the same for all subjects. This was confirmed: the indicators of changes in the individual results of the subjects, presented in a "dimensionless" form, obeyed a power square law.
Therefore, the identification of a general pattern by leveling individual differences is decided each time on the basis of a meaningful hypothesis about the influence of an additional variable on the interindividual variation in the results of the experiment.
Let us dwell once again on one feature of the experiments with the participation of one subject. The results of these experiments are very dependent on the experimenter's prejudices and the relationship that develops between him and the subject. When conducting a long series of successive exposures, the experimenter can unconsciously or consciously act so that the subject actualizes behavior that confirms the experimental hypothesis. That is why in this kind of research it is recommended to use "blind experiments" and "double-blind experiments". In the first option, the experimenter knows, but the subject does not know, when the latter receives a placebo, and when - the effect. A "double-blind" experiment is when the experiment is conducted by a researcher who is unfamiliar with the hypothesis and does not know when the subject is receiving a placebo or treatment.
Experiments involving one subject play important role in psychophysiology, psychophysics, psychology of learning, cognitive psychology. The methodology of such experiments has penetrated the psychology of programmed learning and social management, in clinical psychology, especially in behavioral therapy, the main propagandist of which is Eysenck [Eysenk G. Yu., 1999].
Experimental design arose in the 1920s from the need to eliminate or at least reduce bias in agricultural research by randomizing experimental conditions. The planning procedure turned out to be aimed not only at reducing the variance of the estimated parameters, but also at randomization with respect to concomitant, spontaneously changing and uncontrolled variables. As a result, we managed to get rid of the bias in the estimates.
Since 1918, R. Fisher began his well-known series of works at the Rochemsted agrobiological station in England. In 1935, his monograph "Design of Experiments" appeared, which gave the name to the whole direction. In 1942, A. Kishen reviewed the design of an experiment on Latin cubes, which was a further development of the theory of Latin squares. Then R. Fischer independently published information about orthogonal hyper-Greek-Latin cubes and hyper-cubes. Shortly thereafter, in 1946, R. Rao considered their combinatorial properties. The works of H. Mann (1947-1950) are devoted to the further development of the theory of Latin squares.
The first deep mathematical study of the flowchart was carried out by R. Bowes in 1939. First, the theory of balanced incomplete block plans (BIB-schemes) was developed. Then R. Bowes, K. Ner and R. Rao generalized these plans and developed the theory of partially balanced incomplete block plans (PBIB schemes). Since then, much attention has been paid to the study of block diagrams both by specialists in experiment planning (F. Yates, G. Cox, V. Cochren, W. Federer, K. Gulden, O. Kempthorn and others), and by specialists on combinatorial analysis (Bose, F. Shimamoto, V. Klatsworthy, S. Shrikhande, A. Hoffman and others).
R. Fisher's research marks the beginning of the first stage in the development of experiment planning methods. Fisher developed the factor planning method. Yates proposed a simple computational scheme for this method. Factor planning has become widespread. A feature of the factorial experiment is the need to set up a large number of experiments at once.
In 1945, D. Finney introduced fractional replicas from a factorial experiment. This made it possible to reduce the number of experiments and opened the way for technical planning applications. Another possibility of reducing the required number of experiments was shown in 1946 by R. Plakett and D. Berman, who introduced rich factorial designs.
G. Hotelling in 1941 proposed to find the extremum from experimental data using power expansions and a gradient. The next important step was the introduction of the principle of sequential stepwise experimentation. This principle, expressed in 1947 by M. Friedman and L. Savage, made it possible to extend the experimental definition of the extremum - iteration.
To build modern theory planning the experiment, one link was missing - the formalization of the object of study. This link appeared in 1947 after N. Wiener created the theory of cybernetics. The cybernetic concept of "black box" plays an important role in planning.
In 1951, the work of American scientists J. Box and C. Wilson began new stage development of experiment planning. It formulated and brought to practical recommendations the idea of a consistent experimental determination of the optimal conditions for conducting processes using the estimation of the coefficients of power expansions by the least squares method, moving along a gradient and finding an interpolation polynomial in the region of the extremum of the response function (almost stationary region).
In 1954-1955. J. Box, and then P. Yule. It was shown that the planning of an experiment can be used in the study of physical and chemical processes, if one or several possible hypotheses are stated a priori. The direction was developed in the works of N. P. Klepikov, S. N. Sokolov and V. V. Fedorov in nuclear physics.
The third stage in the development of the theory of experimental design began in 1957, when Box applied his method to industry. This method became known as "evolutionary planning". In 1958, G. Schiffe proposed a new method for planning an experiment for studying physicochemical composition diagrams - a property called "simplex lattice".
The development of the theory of experiment planning in the USSR is reflected in the works of V. V. Nalimov, Yu. P. Adler, Yu. V. Granovsky, E. V. Markova, V. B. Tikhomirov
Experiment planning steps
Experiment planning methods allow minimizing the number of necessary tests, establishing a rational procedure and conditions for conducting research, depending on their type and the required accuracy of the results. If, for some reason, the number of tests is already limited, then the methods give an estimate of the accuracy with which the results will be obtained in this case. The methods take into account the random nature of the dispersion of the properties of the tested objects and the characteristics of the equipment used. They are based on the methods of probability theory and mathematical statistics.
Planning an experiment involves a number of steps.
1. Setting the purpose of the experiment(determination of characteristics, properties, etc.) and its type (determinative, control, comparative, research).
2. Refinement of the conditions for the experiment(existing or available equipment, terms of work, financial resources, number and staffing of employees, etc.). Selection of the type of tests (normal, accelerated, reduced in laboratory conditions, on the stand, field, full-scale or operational).
3. Identification and selection of input and output parameters based on the collection and analysis of preliminary (a priori) information. Input parameters (factors) can be deterministic, that is, registered and controlled (depending on the observer), and random, that is, registered, but unmanaged. Along with them, the state of the object under study can be influenced by unregistered and uncontrolled parameters that introduce a systematic or random error into the measurement results. These are mistakes measuring equipment, change in the properties of the object under study during the period of the experiment, for example, due to aging of the material or its wear, exposure to personnel, etc.
4. Establishing the required accuracy of measurement results(output parameters), areas of possible change in input parameters, clarification of the types of impacts. The type of samples or objects under study is selected, taking into account the degree of their compliance with the real product in terms of condition, device, shape, size and other characteristics.
The purpose of the degree of accuracy is influenced by the conditions of manufacture and operation of the object, the creation of which will use these experimental data. Manufacturing conditions, ie manufacturing possibilities, limit the highest realistically achievable accuracy. The operating conditions, that is, the conditions for ensuring the normal operation of the object, determine the minimum requirements for accuracy.
The accuracy of experimental data also significantly depends on the volume (number) of tests - the more tests, the higher the reliability of the results (under the same conditions).
For a number of cases (with a small number of factors and a known law of their distribution), it is possible to calculate in advance the minimum required number of tests, which will allow obtaining results with the required accuracy.
0. Drawing up a plan and conducting an experiment- the number and order of tests, the method of collecting, storing and documenting data.
The order of testing is important if the input parameters (factors) in the study of the same object during one experiment take different meanings. For example, when testing for fatigue with a step change in the level of load, the endurance limit depends on the sequence of loading, since the accumulation of damage occurs in different ways, and, therefore, there will be a different value of the endurance limit.
In some cases, when systematic parameters are difficult to take into account and control, they are converted into random ones, specifically providing for a random order of testing (randomization of the experiment). This makes it possible to apply the methods of the mathematical theory of statistics to the analysis of the results.
The order of testing is also important in the process of exploratory research: depending on the chosen sequence of actions in the experimental search for the optimal ratio of the parameters of an object or some process, more or less experiments may be required. These experimental problems are similar to mathematical problems of numerical search for optimal solutions. The most well-developed methods are one-dimensional searches (one-factor one-criteria problems), such as the Fibonacci method, the golden section method.
0. Statistical processing of the results of the experiment, construction of a mathematical model of the behavior of the studied characteristics.
The need for processing is due to the fact that a selective analysis of individual data, out of touch with the rest of the results, or their incorrect processing can not only reduce the value of practical recommendations, but also lead to erroneous conclusions. Results processing includes:
· definition confidence interval the mean value and variance (or standard deviation) of the values of the output parameters (experimental data) for a given statistical reliability;
Checking for the absence of erroneous values (outliers), in order to exclude questionable results from further analysis. It is carried out for compliance with one of the special criteria, the choice of which depends on the distribution law of the random variable and the type of outlier;
Checking the compliance of the experimental data with the previously introduced distribution law. Depending on this, the chosen experimental plan and methods for processing the results are confirmed, and the choice of the mathematical model is specified.
The construction of a mathematical model is performed in cases where quantitative characteristics of interrelated input and output parameters under study should be obtained. These are problems of approximation, that is, the choice of a mathematical dependence that best suits the experimental data. For these purposes, regression models are used, which are based on the expansion of the desired function in a series with the retention of one (linear dependence, regression line) or several (non-linear dependences) expansion members (Fourier, Taylor series). One of the methods for fitting the regression line is the widely used least squares method.
To assess the degree of interrelatedness of factors or output parameters, a correlation analysis of test results is carried out. As a measure of interconnectedness, the correlation coefficient is used: for independent or non-linearly dependent random variables, it is equal to or close to zero, and its proximity to unity indicates the complete interconnectedness of the variables and the presence of a linear relationship between them.
When processing or using experimental data presented in tabular form, there is a need to obtain intermediate values. For this, the methods of linear and non-linear (polynomial) interpolation (determination of intermediate values) and extrapolation (determination of values that lie outside the interval of data change) are used.
0. Explanation of the results and formulating recommendations for their use, clarifying the methodology for conducting the experiment.
Reduction of labor intensity and reduction of testing time is achieved by using automated experimental complexes. Such a complex includes test benches with automated setting of modes (allows you to simulate real operating modes), automatically processes the results, conducts statistical analysis and documents the research. But the responsibility of the engineer in these studies is also great: clearly defined test objectives and a correctly made decision allow you to accurately find the weak point of the product, reduce the cost of fine-tuning and the iteration of the design process.
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1) Meaningful and formal business planningstealmental studies
Planning a psychological experiment
Experiment planning- one of the most important stages in the organization of psychological research, at which the researcher tries to design the most optimal model (that is, plan) of the experiment for implementation in practice.
A well-designed research scheme, plan, allows you to achieve optimal values of validity, reliability and accuracy in the study, to provide for nuances that are difficult to keep track of in everyday "spontaneous experimentation". Often, in order to adjust the plan, experimenters conduct a so-called pilot study, or trial study, which can be considered as a “draft” of a future scientific experiment.
Key questions answered by the pilot plan
The pilot plan is designed to answer basic questions about:
the number of independent variables that are used in the experiment (one or more?);
the number of levels of the independent variable (does the independent variable change or remain constant?);
Methods for controlling additional, or perturbing, variables (which ones are necessary and appropriate to apply?):
o direct control method (direct exclusion of a known additional variable),
o alignment method (take into account a known additional variable if it is impossible to exclude it),
o randomization method (random selection of groups in case of unknown additional variable) .
One of the most important questions that an experimental design must answer is to determine in what sequence the change in the considered stimuli (independent variables) should occur, affecting the dependent variable. Such exposure can vary from a simple "A 1 --A 2 " scheme, where A 1 is the first stimulus value, A 2 is the second stimulus value, to more complex ones, such as "A 1 --A 2 --A 1 --A 2 ”, etc. The sequence of presentation of stimuli is a very important issue, directly related to compliance with the validity of the study: for example, if you constantly present the same stimulus to a person, he may become less receptive to it.
Planning steps
Planning includes two stages:
o Determination of a number of theoretical and experimental provisions that form the theoretical basis of the study.
o Formulation of theoretical and experimental hypotheses of the study.
o Selecting the required experimental method.
o Decision of the issue of sampling of subjects:
§ Determining the composition of the sample.
§ Determining the sample size.
§ Determination of the sampling method.
2. Formal design of the experiment:
o Achieving the ability to compare results.
o Achieving the possibility of discussing the received data.
o Ensuring cost-effective conduct of the study.
The main goal of formal planning is considered to be the elimination of the maximum possible number of reasons for distorting the results.
Types of plans
1. Simple (one-factor) plans
o Experiments with reproducible conditions
o Experiments involving two independent groups (experimental and control)
2. Comprehensive plans
o Plans for multilevel experiments
o Factorial plans
3. Quasi-experimental plans
o Plans ex post facto
o Small N Experiment Plans
4. Plans for correlation studies
simple plans, or univariate, involve the study of the effect on the dependent variable of only one independent variable. The advantage of such plans is their effectiveness in establishing the influence of the independent variable, as well as the ease of analysis and interpretation of the results. The disadvantage is the inability to draw a conclusion about the functional relationship between the independent and dependent variables.
Experiments with reproducible conditions
Compared to experiments involving two independent groups, such plans require fewer participants. The plan does not include different groups(for example, experimental and control). The purpose of such experiments is to establish the effect of one factor on one variable.
Experiments involving two independent groups- experimental and control - experiments in which only the experimental group is subjected to experimental influence, while the control group continues to do what it usually does. The purpose of such experiments is to test the action of one independent variable.
Comprehensive plans
Comprehensive plans are compiled for experiments that study either the effects of several independent variables (factorial designs) or the sequential effects of different gradations of one independent variable (multilevel designs).
Plans for multilevel experiments
If experiments use one independent variable, the situation when only two of its values are studied is considered the exception rather than the rule. Most univariate studies use three or more values of the independent variable—these designs are often referred to as single-factor multilevel. Such designs can be used both to investigate non-linear effects (that is, cases where the independent variable takes on more than two values) and to test alternative hypotheses. The advantage of such plans is the ability to determine the type of functional relationship between the independent and dependent variables. The disadvantage, however, is that it takes a lot of time and also the need to attract more participants.
Factorial plans
Factorial plans involve the use of more than one independent variable. There can be any number of such variables, or factors, but usually they are limited to using two, three, less often four.
Factorial designs are described using a numbering system showing the number of independent variables and the number of values (levels) taken by each variable. For example, a factorial design 2x3 ("two by three") has two independent variables (factors), the first of which takes two values ("2"), and the second - three values ("3"); 3x4x5 factorial design has three independent variables, respectively, taking "3", "4" and "5" values, respectively.
In a 2x2 factorial design experiment, let's say one factor, A, can take two values, A 1 and A 2 , and the other factor, B, can take values B 1 and B 2 . During the experiment, according to the 2x2 plan, four experiments should be carried out:
The sequence of experiments may be different depending on the expediency, determined by the tasks and conditions of each particular experiment.
Quasi-experimental plans- plans for experiments in which, due to incomplete control over variables, it is impossible to draw conclusions about the existence of a causal relationship. The concept of a quasi-experimental design was introduced by Campbell and Stanley in Experimental and quasi-experimental designs for research (Cambell, D. T. & Stanley, J. C., 1966). This was done in order to overcome some of the problems faced by psychologists who wished to conduct research in a less rigorous environment than the laboratory. Quasi-experimental plans are often used in applied psychology.
Types of quasi-experimental plans:
1. Experimental plans for non-equivalent groups
2. Plans of discrete time series.
1. Experiment according to the time series plan
2. Plan of series of time samples
3. Plan of series of equivalent impacts
4. Plan with non-equivalent control group
5. Balanced plans.
ex post facto plans. Studies in which the collection and analysis of data is carried out after the event has already taken place, called studies ex post facto , many experts refer to quasi-experimental. Such research is often carried out in sociology, pedagogy, clinical psychology and neuropsychology. The essence of the study ex post facto consists in the fact that the experimenter himself does not influence the subjects: some real event from their life acts as an influence.
In neuropsychology, for example, for a long time (and even today) research has been based on the paradigm of localizationism, which is expressed in the "locus - function" approach and claims that lesions of certain structures make it possible to reveal the localization of mental functions - the specific material substrate in which they "are" in the brain [see. A. R. Luria, “Brain lesions and cerebral localization of higher functions”]; such studies can be classified as studies ex post facto.
When planning a study ex post facto the design of a rigorous experiment is simulated with equalization or randomization of groups and testing after exposure.
Low N plans also called "single-subject plans" because each subject's behavior is considered individually. One of the main reasons for the use of small N experiments is the impossibility in some cases to apply the results obtained from generalizations on large groups of people to any of the participants individually (which, therefore, leads to a violation of individual validity).
Psychologist B. F. Skinner is considered the most famous advocate of this line of research: in his opinion, the researcher should “study one rat for a thousand hours,<…>not a thousand rats for an hour each, or a hundred rats for ten hours each. Ebbinghaus's introspective studies can also be attributed to experiments with small N (only the subject he studied was himself).
A plan with a single subject must meet at least three conditions:
1. The target behavior must be precisely defined in terms of events that are easy to capture.
2. It is necessary to establish a baseline response level.
3. It is necessary to influence the subject and fix his behavior.
Correlation study-- a study conducted to confirm or refute the hypothesis of a statistical relationship (correlation) between several (two or more) variables. The plan of such a study differs from the quasi-experimental plan in that it does not have a controlled effect on the object of study.
In a correlation study, a scientist hypothesizes that there is a statistical relationship between several mental properties of an individual or between certain external levels and mental states, while assumptions about causal dependence are not discussed. Subjects must be in equivalent, unchanging conditions. In general terms, the design of such a study can be described as PxO ("subjects" x "measurements").
Types of correlation studies
Comparison of two groups
One-dimensional study
Correlation study of pairwise equivalent groups
Multivariate correlation study
Structural correlation study
· Longitudinal correlation study *
* Longitudinal studies are considered an intermediate option between a quasi-experiment and a correlation study.
Experiment (psychology)
Psychological experiment-- held in special conditions experience for obtaining new scientific knowledge through the targeted intervention of the researcher in the life of the subject.
Various authors interpret the concept of "psychological experiment" ambiguously; often, under the experiment in psychology, a complex of different independent empirical methods is considered ( actual experiment, observation, questioning, testing). However, traditionally in experimental psychology, the experiment is considered an independent method.
The main stages of the experiment
Stage 1 - Preparatory:
1.1 Define the research topic
Preliminary acquaintance with the object of study
Determine the purpose and objectives of the study
Refine object
Determine and select research methods and techniques.
2. Stage - the stage of collecting research data:
2.1 Conducting a pilot study.
2.2 Direct interaction with the object of study
3. Stage - Final:
3.1 Processing of received data
3.2 Analysis of the received data
3.3 Hypothesis testing
4. Stage - Interpretation:
4.1 Conclusions.
2 )
Polls are an indispensable method of obtaining information about the subjective world of people, their inclinations, motives, and opinions.
Polling is an almost universal method. With proper precautions, information can be obtained no less reliable than in the study of documents or observation. And this information can be about anything. Even about things you can't see or read.
For the first time, official polls appeared in England at the end of the 18th century, and at the beginning of the 19th century in the United States. In France and Germany, the first surveys were conducted in 1848, in Belgium - in 1868-1869. And then they began to actively spread.
The art of using this method is to know what to ask, how to ask, what questions to ask, and finally, how to make sure that the answers you receive can be trusted.
For the researcher, first of all, it is necessary to understand that not the “average respondent” is participating in the survey, but a living, real person endowed with consciousness and self-awareness, who affects the sociologist in the same way as the sociologist affects him.
Respondents are not impartial registrars of their knowledge of opinions, but living people who are not alien to some kind of sympathy, preferences, fears, etc. Therefore, perceiving questions, they cannot answer some of them due to lack of knowledge, and they do not want to answer others or answer insincerely.
Types of polls
There are two large classes of survey methods: interviews and questionnaires.
An interview is a conversation conducted according to a certain plan, involving direct contact between the interviewer and the respondent (interviewee), and the latter's answers are recorded either by the interviewer (his assistant) or mechanically (on film).
There are many types of interviews.
2) According to the technique of conducting - they are divided into free, non-standardized and formalized (as well as semi-standardized) interviews.
Free - a long conversation (several hours) without a strict specification of questions, but according to a general program ("interview guide"). Such interviews are appropriate at the exploration stage in the formulational research plan.
Standardized interviews, like formalized observation, require a detailed development of the entire procedure, including overall plan conversations, the sequence and design of questions, options for possible answers.
3) Depending on the specifics of the procedure, the interview can be intensive (“clinical”, i.e. deep, sometimes lasting for hours) and focused on identifying a fairly narrow range of respondent's reactions. The purpose of a clinical interview is to obtain information about the internal motives, motives, inclinations of the respondent, and a focused interview is to extract information about the subject's reactions to a given impact. With its help, they study, for example, to what extent a person reacts to individual components of information (from the mass press, lectures, etc.). Moreover, the text of information is pre-processed by content analysis. In a focused interview, they try to determine which semantic units of text analysis are at the center of the attention of the respondents, which are on the periphery, and which are not left in the memory at all.
4) The so-called undirected interviews are "therapeutic" in nature. The initiative for the course of the conversation belongs here to the respondent himself, the interviewer only helps him “pour out his soul”.
5) according to the method of organizing interviews, they are divided into group and individual. The former are used relatively rarely; this is a planned conversation, during which the researcher seeks to provoke a discussion in the group. the methodology for conducting reader's conferences resembles this procedure. Telephone interviews are used to quickly solicit opinions.
Questionnaire survey
This method assumes a rigidly fixed order, content and form of questions, a clear indication of the methods of answer, and they are registered by the respondent either alone (correspondence survey) or in the presence of the questionnaire (direct survey).
Questionnaires are classified primarily by the content and design of the questions asked. Distinguish between open polls, when respondents speak freely. In a closed questionnaire, all answers are provided in advance. Semi-closed questionnaires combine both procedures. Probing or express polling is used in public opinion surveys and contains only 3-4 items of basic information plus a few items related to the demographic and social characteristics of the respondents. Such questionnaires are reminiscent of sheets of popular referendums. A survey by mail is distinguished from a survey on the spot: in the first case, the return of the questionnaire is expected at a prepaid postal item, in the second - the questionnaire itself collects the completed sheets.
Group surveys are different from individual surveys. In the first case, up to 30-40 people are questioned at once: the questionnaire gathers the respondents, instructs them and leaves them to fill out the questionnaires, in the second case, he addresses each respondent individually.
The organization of “distributing” surveys, including surveys at the place of residence, is naturally more laborious than, for example, surveys through the press, which are also widely used in our and foreign practice. However, the latter are not representative of many groups of the population, so they can rather be attributed to the methods of studying the public opinion of readers of these publications.
Finally, when classifying questionnaires, numerous criteria related to the topic of the surveys are also used: event questionnaires, questionnaires for clarifying value orientations, statistical questionnaires (in population censuses), timing of daily time budgets, etc.
When conducting surveys, one should not forget that they reveal subjective opinions and assessments that are subject to fluctuations, the influence of the conditions of the survey and other circumstances.
To minimize the data distortion associated with these factors, any kind of survey methods should be carried out in a short time. It is impossible to stretch the survey for a long time, since by the end of the survey external circumstances may change, and information about its conduct will be transmitted by the respondents to each other with any comments, and these judgments will affect the nature of the answers of those who later fall into the composition of the respondents.
Whether we use interviews or questionnaires, most of the problems associated with the reliability of information are common to them.
In order for the questionnaire survey to be more effective, it is necessary to follow a number of rules that help to correctly set the course of the survey and reduce the number of errors in the study.
Questions addressed to the respondents are not isolated - they are links of one chain, and as links, each of them is connected with the previous and subsequent ones (L.S. Vygodsky called this relationship “the influence of meanings”). The questionnaire is not a mechanical sequence of questions that can be placed in it in any way or as convenient for the researcher, but a special whole. It has its own properties, which are not reducible to a simple sum of the properties of the individual questions that make it up.
At the very beginning, they ask simple questions, and not according to the logic of the researcher contained in the program, so as not to bring down serious questions on the answerer right away, but to let him get used to the questionnaire and gradually move from simple to more complex (funnel rule).
The radiation effect - when all questions are logically interconnected and logically narrow the topic, the respondent has a certain attitude according to which he will answer them - this influence of the question is called the radiation effect or the echo effect and it manifests itself in the fact that the previous question or questions direct the train of thought respondents in a certain direction, create some mini-coordinate system within which a very specific answer is formed or selected.
Sometimes there are problems related to the sequence of questions. Differences in answers to the same question should not be due to their different sequence.
So, for example, if a low-paid worker is asked the question “Do you intend to quit your job in the near future? this enterprise?” after a question about wages, the likelihood of receiving an affirmative answer increases. And if the same question is put after finding out, say, the prospects for growth in wages, the probability of getting a negative answer increases.
The fact of conjugation of answers to different questions is taken into account when compiling the questionnaire. For this, for example, buffer questions are introduced.
So far, one can only assume that with the help of the questionnaire, a greater isolation of answers to each question is achieved than with direct communication with the interviewer. The respondent does not need to care about his image in the eyes of the communication partner (of course, on condition of anonymity), as during the interview. Therefore, apparently, the nature of contingency of responses is less pronounced here. However, this has not been proven.
General and private questions. The questionnaire begins with the most specific questions and gradually concretizes them (funnel rule). This allows you to gradually introduce the respondent into the situation. But the general solution does not always presuppose a specific one, while the latter strongly influences the general one (people are more willing to generalize particulars than do deductions).
Example: General self-assessment questions about interest in politics and religion, posed before and after specific questions about the political and religious behavior of respondents, received an uneven number of “votes”. In the second case, the respondents expressed their interest much more often. At the same time, general assessments of the economic and energy situation turned out to be very slightly influenced by the formulation of particular questions about the respondent's income and energy sources before and after them. This suggests that general and particular issues affect each other ambiguously. The distribution of answers to general questions depends on the previous formulation of a particular question on the same topic more than vice versa. In addition, this dependence is also due to the content of the phenomenon under discussion.
Applying filter questions
The purpose of the filters is to influence the answers to subsequent questions. These questions make it possible to single out a group of people whose answers turn out to be based not only on general ideas, but also on personal experience:
“Does your child attend a children's music school?
If so, who usually accompanies him there?
Any of the parents
Grandma, grandpa, etc.”
These questions save the time of those to whom the question following the filter is not addressed.
Using filters results in missed responses.
These gaps are caused not only by the conscious transition of some of the respondents to questions that they can answer, bypassing those that are not related to them, but also by some other factors. For example, it turned out that the use of a series of filtering questions (“If you higher education, then...?"; “If you have a higher humanitarian education, then...?”; “If you have a higher education in the humanities and you have had an internship in high school, then...?”), although it is a very convenient way of arranging questions for a sociologist, it extremely complicates the perception of the questionnaire by the respondents. Sometimes this results in such a significant number of missed responses that the entire study is in jeopardy.
preamble question
A question about facts, like any other, can be perceived as an evaluative characteristic of the respondent, therefore it is advisable in some cases to ask it in a form that somewhat weakens its evaluative nature. For example: “Some people clean the apartment every day, others do it from time to time. What do you do most of the time?”
Table questions
Table questions are very convenient for the researcher. These are difficult questions in which the respondent has to make a number of efforts to answer them.
In such questions, we are talking about things that can be answered only when the knowledge and mental abilities of the respondents are used. After such questions, it is desirable to move on to simpler ones.
Such questions should not be repeated often, because respondents experience fatigue, distraction of attention, and a radiation effect occurs.
For example, in one study, respondents were offered a list of the same topics. In the first case, it was required to evaluate their effectiveness, in the second - efficiency, in the third - the completeness of coverage of problems. The presentation of this list in the second, and even more so in the third, caused the respondents to feel that not only the list was repeated, but also the evaluation criteria. Many survey participants, looking at the third table, said: “I already answered you”, “It already happened”, etc., skipped it, left it unanswered.
The uniformity of filling tables leads to the fact that the danger of getting mechanical fillings, thoughtless answers increases.
Having once chosen the rating “3” for the answer, the respondent can fix it throughout the entire table, regardless of what the actual rating is and even regardless of the content of the question.
Mo problemnotation
To a large extent, the influence of uniform questions on the answers of respondents is also related to the radiation effect. As in the cases with tables, and in many others, especially when respondents are asked several questions formulated according to the same syntactic scheme, the questionnaire turns out to be monotonous. This leads to an increase in the proportion of ill-conceived answers or their omission. In order to overcome monotony, the following techniques are recommended:
“dilute” tables and questions, and data in the same syntactic form, with other questions; vary the categories for the answer (in the first case, ask the respondent to express agreement or disagreement, in the second - to evaluate, in the third - to decide whether this or that statement is true or false, in the fourth to formulate the answer yourself, etc.); make wider use of a variety of functional-psychological questions, “extinguishing the mutual influence of answers”; diversify the design of the questionnaire.
Functional-psychological OPdew
In order to create and maintain interest in the questionnaire, relieve the tension that arises, and transfer the respondent from one topic to another, special questions are used in the questionnaire, called functional-psychological questions.
These questions serve not so much to collect information as to provide a relationship of communication between the researcher and the respondents.
These questions serve not only as an incentive to answer, they contain a variety of information: explanations and justifications for the sociologist's statements addressed to the respondents, some comments perceived as signs of a more symmetrical communication, a more equal exchange of information.
Functional-psychological issues include contact issues and buffer issues.
Contact questions
Any communication begins with an adaptation phase. This phase provides for the perception of communication with the respondents, acquaintance with the purpose of the study and instructions for filling out the questionnaire.
The first question of the questionnaire is contact. It can be expected that due to the interconnection of all questions of the questionnaire, if a person answers the first question, then he can answer all the others.
A number of requirementsovations to the first question of the questionnaire
1) The contact question should be very simple. Here, questions of a purely eventual nature are often used - for example, work experience, area of residence, habits, interest in problems.
2) The contact question should be very general, ie. apply to all respondents. Therefore, it is undesirable to start the questionnaire with a filter.
3) It is recommended to make a contact question so broad that any respondent can answer it. Answering, a person begins to believe in his competence, to feel confident. He has a desire to develop his thoughts further, to speak more fully. Therefore, it is better to start the questionnaire with what is accepted by everyone, which is the most understandable.
It is not necessary that contact questions contain the information you are looking for. Their main function is to facilitate interaction. Answers to contact questions are not necessarily involved in scientific analysis in connection with substantive problems. On the other hand, in methodological plan these answers are of great importance: depending on their content, one can determine the attitude of the respondents to the survey, its influence on their conscientiousness, sincerity, etc.
buffer questions
Quite rarely, the questionnaire is devoted to any one topic. But even within the same topic, various aspects are discussed. Abrupt and unexpected transitions from one topic to another can produce an unfavorable impression on respondents.
Buffer questions are designed to mitigate the mutual influence of questions in the questionnaire. First, as already mentioned, they play the role of a kind of “bridges” when moving from topic to topic. For example, after discussing a series production problems the following wording is given:
“Free time is not only the time we need to restore the energy expended at work. First of all, it is an opportunity for the comprehensive development of the individual. Therefore, we ask you to answer a series of questions about activities outside of work.”
With the help of a buffer question (this function was not the question itself, but the preamble to it), the researcher explains to the respondents the course of his thoughts.
With the help of such “buffers”, the researcher not only invites respondents to switch their attention to another topic, but also explains why this is necessary. For example, after a question about leisure, the following wording is given: “A person spends most of his life at work. Sorrows and joys, successes and failures in work are not indifferent to us. Therefore, it is not surprising that we want to talk to you about work.”
Second, buffer questions are designed to neutralize the effect of radiation. In this case, any substantive questions that are not directly related to the subject that is discussed in the questions, the mutual influence of which the sociologist assumes, can act as buffers.
Concluding the discussion of the significance of functional-psychological questions in the design of the questionnaire, we note that, like any other, their wording may not be indifferent to the respondents and, therefore, affect the content and availability of their answers. The sociologist's knowledge that this or that question appears as a functional-psychological one does not yet ensure that he will fulfill his role as expected. In order for the sociologist's assumptions to be justified, it is necessary to conduct special methodological experiments in this area.
The situation of the questionnaire
A very important role is played by the way the environment for conducting a questionnaire survey is set. First of all, it is necessary to make it clear to the respondents that all their answers are absolutely anonymous. This will provide more reliable information in the responses. Respondents are also affected by the presence of strangers. To create a more favorable atmosphere during the survey, it is necessary to take measures for the presence of people directly related to the questionnaire (researcher, respondents). The location of the survey also plays a role. It must be familiar to the respondent. It is important that he feels free in such a place. The space should not be too formal (the manager's office) or too informal (the locker room). A lot depends on what the questions are about.
If the questionnaire asks questions about the enterprise where the survey is being conducted, the answers are likely to be insincere. It is necessary to pay attention to the timing of the survey. It should not last too long, so as not to tire the respondents (they have more important things to do).
List of literature sources
1) Meaningful and formal planningexpertmental research
1. ^ Experimental psychology: textbook. -- M.: Prospekt, 2005. S. 80--81.
2. ^ See ibid.
3. ^ See ibid. pp. 82--83.
4. ^ Research in psychology: methods and planning / J. Goodwin. - St. Petersburg: Piter, 2004. S. 248.
5. ^ Zarochentsev K. D., Khudyakov A. I. Experimental psychology. pp. 82--83.
6. ^ Research in psychology: methods and planning / J. Goodwin. pp. 258--261.
7. ^ See ibid. S. 275.
8. ^ See ibid.
9. ^ See ibid. S. 353.
10. ^ Solso R. L., Johnson H. H., Beal M. C. Experimental psychology: a practical course. St. Petersburg: prime-EVROZNAK, 2001. S. 103.
11. ^ See ibid.
12. ^ Druzhinin V. N. Experimental psychology. St. Petersburg: Piter, 2002. S. 138.
13. ^ Research in psychology: methods and planning / J. Goodwin. pp. 388--392.
14. ^ See ibid.
15. ^ Druzhinin V. N. Experimental psychology. S. 140.
16. ^ See ibid.
17. ^ See ibid. S. 142
18. Research in psychology: methods and planning / J. Goodwin. -- 3rd ed. - St. Petersburg: Peter, 2004.
19. Solso R. L., Johnson H. H., Beal M. K. Experimental psychology: a practical course. St. Petersburg: prime-EVROZNAK, 2001.
20. Robert Gottsdanker "Fundamentals of Psychological Experiment": Moscow University Press 1982
2) General characteristics of survey methods
1. Butenko I.A. “Questionnaire survey as a method of communication between a sociologist and a respondent”, Moscow, 1989
2. Noel E. “Mass polls. Introduction to the methodology of demoscopy”, M., 1987.
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Above all speculative knowledge and arts is the ability to make experiments, and this science is the queen of sciences.
R. Bacon
Experiment planning- this is the process of choosing conditions, procedures and methods for conducting experiments, their number and conditions necessary and sufficient to solve the problem with the required accuracy.
Requirements for planning an experiment:
- 1) the number of experiments should be minimal so as not to complicate the procedure of the experiment and not increase its cost, but not to the detriment of the accuracy of the result;
- 2) it is necessary to determine the set of factors influencing the results of the experiment, rank them, identify the main ones, and insignificant variables can be excluded;
- 3) the condition for the correctness of the experiment should be considered the simultaneous variation of all variables (factors) that have a mutual influence on the process under study;
- 4) a number of actions in the experiment can be replaced by their models (primarily mathematical ones), while the adequacy of the models must be checked and evaluated;
- 5) it is necessary to develop an experiment strategy and an algorithm for its implementation: the series of experiments should be analyzed after the completion of each of them before moving on to the next series.
Experiment plan should include the following sections:
- 1. Name of the research topic.
- 2. Purpose and objectives of the experiment.
- 3. Experimental conditions: optimization parameter and variable factors.
- 4. Research methodology.
- 5. Justification of the number of experiments (experiment volume).
- 6. Means and methods of measurements.
- 7. Material support of the experiment (list of equipment).
- 8. Method of processing and analysis of experimental data.
- 9. Calendar plan for conducting tests, which indicates the timing of their implementation, the performers, the data of the experiment presented.
- 10. Cost estimate.
Purpose and objectives of the experiment- the starting point of the plan. They are formulated on the basis of an analysis of a scientific hypothesis, theoretical results of their own research or research by other authors.
The goal determines the final result of the experiment, i.e. what the researcher should get as a result. For example, to confirm correct scientific hypotheses; to check in practice the adequacy, efficiency and practical suitability of models, methods; determine optimal conditions technological process etc.
In different conditions, the goals require different costs, means and methods of measurement, the time of the experiment, and are reflected in the methodology for conducting it. These points of the plan will be different, for example, in the conditions of laboratory, field and production experiments.
The objectives of the experiment determine the particular goals with which the final goal or ways to achieve it can be achieved. For example, determining the optimal temperature and pressure indicators in the manufacture of fullerene nanotubes; establishing the optimal ratio of starting materials; substantiation of the speed of the technological process, etc.
Particular tasks of the experiment in its planning can be:
- - verification of theoretical positions in order to confirm their truth;
- - verification (clarification) of the constants of mathematical or other models;
- - search for optimal (permissible) conditions of any process;
- - construction of interpolation analytical dependencies.
Particular tasks of the experiment can have several levels, i.e., a tree-like form. It is recommended to formulate 2-4 complex problems and 10-15 easier problems.
Formulation experiment conditions- optimization parameter and variable factors.
The value describing the result of the experiment is called optimization parameter(response) of the system to the impact. The set of values that an optimization parameter takes is called its domain of definition.
The optimization parameter must be quantitative, given by a number and be measurable for any fixed set of factor levels. It must be characterized unambiguously, i.e., a given set of factor levels must correspond, with the accuracy of the experimental error, to one value of the optimization parameter. The optimization parameter must comprehensively characterize the object of study, satisfy the requirement of universality and completeness. It must have a physical meaning to ensure the subsequent interpretation of the results of the experiment, be simple and easy to calculate.
The optimization parameter (response) depends on the factors influencing the experiment. Factor(lat .factor - producing) - the cause of any process, phenomenon, which determines its influence on the object of study, its character or individual features. This is a measurable quantity, and each value that a factor can take on is called the level of the factor.
Each factor in the experiment can take one of several values. A fixed set of levels of several factors will determine some specific conditions for the experiment. A change in at least one of the factors leads to a change in the conditions, and, as a result, to a change in the value of the optimization parameter.
Variable factors in a multivariate experiment determine the goals and conditions of the study. For example, factors in an experiment to find optimal conditions for the production of nanomaterials can be: temperature, exposure time, amount of oxide, etc.
A large number of factors makes the experiment very complex and time consuming. Therefore, it is very important when planning an experiment to reduce the number of factors and choose the most significant ones. In this case, one can be guided by the Pareto principle, according to which 20 % factors determine 80% of the properties of the system.
The significance of factors can be determined empirically or analytically. In the first case, a limited experiment is carried out. In this case, one factor changes, while the rest do not, etc. The ranking of "significant" factors is carried out according to the strength of their influence on the result of the experiment. Those factors, the change of which has a stronger effect on the final result, are considered more important. "Insignificant" factors can be neglected.
If there are many factors, this path is inefficient, then an analytical path based on factor analysis methods is used.
If the factors are dependent, they can be calculated using the topological decomposition method and structure by their influence on ultimate goal. The task of determining the ranks of factors is to identify the most connected part of the graph. It is being solved in stages.
First, "reachable sets" are determined for each vertex of the graph (for each factor). Then "counterreachable sets" are defined, each of which includes all vertices that have a path to the vertex. Finally, the most significant vertices of the graph that make up the strongly connected graph are determined. The most significant factors are left, the rest are discarded.
The most important requirement of the experiment is the controllability of factors, and the experimenter must be able to choose the desired value of the factor and maintain it constant throughout the experiment. The factor must also be operational so that it can be specified by the sequence of operations required to set a particular value.
When formalizing the conditions for conducting an experiment, it is also important to determine the area of its conduct. To do this, it is necessary to estimate the boundaries of the areas of determination of the factors. Several types of restrictions are possible here: which cannot be violated under any conditions (for example, the temperature may not be below absolute zero); technical and economic constraints (for example, the cost of equipment or the duration of the process under study); specific process conditions.
Under experiment model usually understand the black box model, which uses a response function that establishes the relationship between the optimization parameter and the factors: y = f(x y X2 > ...,Jc n).
To choose a model means to choose the form of this function and write down its equation. Then it remains only to conduct an experiment to calculate the numerical coefficients of this model. The main requirement for an experiment model is the ability to predict the future direction of experiments with the required accuracy. Among all possible adequate models, it is necessary to choose the one that seems to be the simplest.
Most often, in planning an experiment, polynomial models of the first (linear) or second degree are chosen:
Experiment Method is a key part of the experimental design. It includes:
- - the sequence of actions of the researcher;
- - basic techniques and rules for the implementation of each stage, the use of instruments and equipment;
- - the procedure for measuring, fixing the results and methods for their processing;
- - the procedure for analyzing the results of the experiment and formulating conclusions.
When developing a methodology, it is important to correctly justify the number of experiments,
which guarantees the required accuracy of the result, and on the other hand, does not lead to unjustified cost and time overruns for redundant tests.
More than ten tests justification for the number of experiments can be carried out on the basis of the Chebyshev inequality:
where X- the average value of a randomly measured quantity; M(x)- mathematical expectation of the value; e is the required accuracy of the result; D(x) - quantity variance X, calculated from the results N conducted experiments.
The inequality can be formulated as follows: “the probability that the difference between the mathematical expectation and the average value of a random variable X does not exceed the required accuracy of the result - e, is equal to the difference between unity and the ratio D(x): Ne 2".
There are three unknowns in the inequality: N and statistical characteristics depending on N. Therefore, the calculation process N is iterative.
If the inequality is satisfied, then the number of experiments is sufficient. Otherwise, the number of experiences increases.
A sufficient number of observations (experiments) can be determined using a table of sufficiently large numbers (Table 8.1). It shows that a sufficient number of observations depends on the degree of confidence in the results of the experiment (confidence probability), the size of the allowable error (confidence interval). In other words, the degree of confidence is determined by the magnitude of the probability with which the corresponding conclusion is made.
Concerning the choice of the probability value R there is no general solution that is the same for all studies. The closer to unity the value of the considered probability is, the more reliable the conclusion will be. In the practice of scientific research, the confidence probability is usually taken P = 0.9-0.99. The required accuracy in research is established depending on the nature of the phenomenon under study. In most cases, the required accuracy is assumed to be e = 0.01-0.05.
For example, if the value of the confidence probability is taken equal to R= 0.95, and the allowable error is e = 0.05, then the sufficient number of observations during the experiment will be 384.
Another important part of the experimental design is substantiation of funds and measurement techniques. It involves the choice of measuring instruments, apparatus and equipment, allows you to record the data of the experiment; convert them to a convenient form; store, provide issuance upon request, etc.
The measurement system should be formed taking into account the requirements of metrology, the science of methods and means of measurement, the choice of units, scales and measurement systems; measurement accuracy problems. Measurement methods that can be applied in various experiments are discussed in the previous chapter.
These measurement methods can be divided into two groups: direct (the desired value is measured directly in the course of the experiment) and indirect measurements (the desired value obtained from the results of direct measurements). In addition, on the basis of units of measurement, there are absolute measurements carried out in units of the quantity under study, and relative measurements, which involve fixing the ratio of the measured quantity to its certain limit value.
The considered fundamentals of organizing and conducting an experiment are only an overview, and the essence, content, conditions for applying the above recommendations and the sequence of using one or another method of conducting an experiment require a more detailed study. In addition, it should be clearly understood that each method of conducting an experiment will have its own characteristics depending on the object of study.
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