# Statistical control based on covariance analysis procedures...

## Statistical control based on covariance analysis procedures

The statistical method of covariance analysis ANCOVA is a mix of methods of variance and regression analysis. This method as a method of indirect experimental control is used every time when the researcher in the intergroup experiment lacks the ability to select equivalent groups based on standard primary control schemes, or in situations when the application of such schemes is undesirable and may threaten the external validity of the experiment.

Because the consequence of the rejection of the equivalence of experimental groups may be the emergence of a systematic blending of the investigated independent variable with any secondary variable, the value of the dependent variable, measured during the experiment, must be corrected based on the previously changed values ​​of this variable. In the covariance analysis, the value of the dependent variable is denoted as a varia, or a criterion, and the value of the secondary variable, on the basis of which the variations are made, as covariates. or a predictor.

The scheme of the experiment using this control is given in Table. 14.1.

Table 14.1

Quasi-experimental plan using indirect, statistical control

 Independent variable levels l ... j k Covariates (X) Variatia (Y) Covariates (X) Variatia (Y) Covariates (X) Variatia ( Y ) X l1 l1 X lj lj X lk lk ... ... ... ... ... X n1 Y n1 X nj Y nj X nk Y nk

As you can see, for each level of the independent variable, two measurements of the dependent variable are made. One measurement is made before the experimental effect is produced. It gives the value of covariates. The second measurement is made after the implementation of the experimental treatment. It gives the value of the variations.

Imagine that the researcher's task is to compare the effectiveness of various teaching methods within the same academic discipline, say, a foreign language. In the standard case of a true experimental plan, it would be necessary to form two homogeneous groups of subjects, the level of initial knowledge in which in this discipline would be approximately equal for all subjects in the average group.

Suppose, however, that in a real situation the researcher is forced to deal with already formed training groups. For example, a psychologist conducting research in an ordinary secondary school wants to compare the success of mastering a foreign language in two parallel classes, in which the teaching methods for this subject matter differ. It is clear that in such a situation the researcher hardly has the opportunity to demand the reformatting of the classrooms recruited several years before the beginning of his research. Therefore, not being able to equalize the classes, the experimenter, even before the beginning of the study, assesses the initial level of possession of all the subjects in this discipline. This gives the value of the covariates, which will be used as a control condition when assessing the overall effectiveness of training.

It is clear that if the differences in the mean values ​​of the covariates between the experimental groups are initially found, then this fact clearly indicates the nonequivalence of the experimental groups themselves. Consequently, the differences in the mean values ​​of the variat,

if they are marked with the results of an experimental study, they can be related either to the effect of an independent variable (by teaching the discipline) or to the original nonequivalence of the experimental groups under study.

Therefore, the covariance analysis assumes the calculation of the adjusted values ​​of the variable based on the values ​​of covariates. For this, the logic of simple linear regression is used. The adjusted values ​​are then compared in accordance with the rules of dispersion analysis already known to us.

Let's consider this procedure in more detail. For data within an arbitrarily chosen experimental group, the adjusted values ​​of the dependent variable, the variations Y ' , can be given by the following linear equation:

where In 'is the coefficient of a simple linear regression.

The total square for the residual variance, reflecting the effect of the experimental error, for this regression equation will be described by the following relationship:

This statistic has ( to (n - 1) - 1) the degree of freedom. Thus, the corrected mean square (variance) for the effect of an experimental error expressing the effect of possible side variables can be expressed by the formula:

The linear regression equation for an arbitrary Y criterion pair and the covariates X i of the test subject in j -th group will look like this:

The total square of the remainder for this regression equation will obviously have the following form:

This expression corresponds to the total variance of the adjusted data. If we have k groups of subjects by n observations, the statistics will have kn - 2 degrees of freedom.

The variance (mean square) of the effects of the experimental impact is calculated on the basis of the difference D and D :

The statistics constructed in this way are used in a standard way to construct a F-ratio, reflecting the relationship between the variance of the experimental action and the variance of the error. It has a to -1 degrees of freedom for the numerator and to (n - 1) - 1 degrees of freedom for the denominator . The statistical significance of this ratio (p-level) is estimated on the basis of the standard F-distribution. The logic of such an analysis does not differ from the logic of the usual ANOVA analysis ANOVA.

The method of covariance analysis allows us to evaluate not only the overall effect of an independent variable, corrected based on the values ​​of covariates, but also any contrasts, both paired and multiple. There is also a factorial version of the covariance analysis, which makes it possible to carry out statistical control in more complex quasi-experimental plans that investigate the effects of not one but several independent variables.

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