PLANNING AND STATISTICAL ANALYSIS OF FACTOR EXPERIMENTS WITH REDIAL MEASUREMENTS
As a result of studying this chapter, the student must:
know
• general principles for planning and analyzing factor experiments with repeated measurements on one or more independent variables based on statistical models;
• the main provisions and principles of twoway variance analysis with repeated measurements of one variable;
• the main provisions and principles of threefactor analysis of variance with a repetition of one or two independent variables;
• the possibility of generalizing the basic principles of analyzing multifactor plans with repeated measurements;
• methods of applying standard multivariate analysis of variance analysis for connected samples in the processing of experimental data;
be able to
• it is correct to build factorial intrasubject and mixed experimental plans on the basis of statistical models of variance analysis and choose an adequate model of statistical processing of experimental data;
• correctly interpret the main results of multivariate analysis of variance with repeated measurements and formulate on their basis adequate statistical conclusions;
own
• the basic theoreticmethodological apparatus of multivariate analysis of variance with repeated measurements;
• methods of statistical data processing by the method of multivariate analysis of variance with repeated measurements using statistical programs.
The factorial plans considered in Ch. 5, in which each group of subjects deals only with one combination of levels of independent variables studied in the experiment, are not always economical enough. So, if the experimental tests are simple enough, there is no sense to limit the subject to only one of them. It makes sense to offer him other tests in which the value of one or more independent variables will be changed. And this proves to be a fairly common case, allowing to attract to the study a smaller number of subjects, which undoubtedly saves the experimenter's time and effort.
Factor experimental plans in which one or more independent variables are considered as intrasubjects are called plans with repeated measurements. The principles of their statistical analysis differ somewhat from those that were considered in Ch. 5. In this chapter, we first consider the general statistical principles for planning and analyzing experiments with repeated measurements. Then we will consider concrete examples of how the variance analysis of twofactor experiments with repeated measurements is carried out according to one of the factors. After this, we will try to generalize these approaches first with respect to threefactor plans with repeated measurements on one or two factors, and then with respect to an arbitrarily given factorial plan with any number of independent variables.
Twofactorial plan with repeated measurements for one of the factors
First, consider the situation where we have two or more groups of subjects divided according to the levels of one independent variable, and each subject in the group is exposed to the second independent variable. For example, it could be an experiment investigating the positional effects of memorization (see subparagraph 4.4.2) in a situation of immediate or delayed playback. In this case, the experimenter will be interested in how the subjects reproduce the initial, middle and final part of the list of words or meaningless syllables, which they are asked to read and remember. In this case, according to the experimental plan, one group of subjects reproduces the list immediately after its presentation or memorization, the second  after, say, several minutes, and the third  after several hours. Thus, we will have a twofactor plan of 3 x 3 with repeated measurements by the reproduction time factor.
Table 6.1 summarizes the structure of a similar twofactor plan for an arbitrary number of levels of two independent variables A and B. p groups of subjects. Each of these groups is exposed to only one level of the A, factor, but all groups pass through all the levels of the В.
Table 6.1
The structure of the factorial plan p x q with repeated measurements by the factor In
Factor Levels 

... 

... 


Group 1 
Group 1 
Group 1 

... 


Group i 
Group i 
Group i 

... 


Group p 
Group p 
Group p 
The structure of an arbitrarily chosen experimental group i in this plan is presented in Table. 6.2. If we consider the subject's factor as a separate independent variable, then it can be noted that this factor turns out to be embedded into the levels of the factor A. This is indicated by the index of the subject, where the level of the factor A, is given in brackets for example k ( i ). Such an investment, or nesting, in the theory of experimental planning is called nesting (TV Kornilova [11]) .
Table 6.2
Group structure in the factor plan p × times. q with repeated measurements by the factor In

Subject 

... 

... 



... 

... 


... 
... 
... 
... 
... 
... 



... 

... 


... 
... 
... 
... 
... 
... 



... 

... 

The structural model of such an experimental plan assumes that each value of the dependent variable at an arbitrarily chosen level i of the factor A and the arbitrarily chosen level j of the factor In of the subject k ( i ) should include seven additive parts:
• population mean μ;
• the effect of the factor A  αi;
• the effect of the subject  pk (i);
the effect of the factor In  βj;
• the effect of the interaction of the factors A and In  αβij;
• the interaction effect of the factor In with the subject's factor  βπjk (i);
• the effect of the experimental error is εk (ij).
In other words, it is assumed that the value of the dependent variable can be decomposed into the following additive components:
The population mean μ is a constant, and therefore is not a source of variance. The effect of A varies between groups of subjects, and the effect of the subject is a source of both intergroup and intragroup variance. The remaining effects are intrasubject, except for the experimental error factor, which varies both between the subjects and within them. Eliminating the error factor, we find five sources of variance of the dependent variable, two of which provide variance between the subjects, and three of which are within the test subjects (Figure 6.1).
Fig. 6.1. General principles of twoway analysis of variance with repeated measurements by one of the factors
The mean square for the subject's factor within the group can be estimated as follows:
Similarly, the interaction of the factor B with the factor of the subjects within the groups can be estimated:
The remaining effects are evaluated in the same way as in the analysis of intergroup plans.
In Table. 6.3 shows the theoretically expected values of the mean squares and the number of degrees of freedom for each dispersion source in the structural model of twofactor analysis of variance ( ANOVA ) we are considering, with repeated measurements for one of the factors in this in the case of the factor B.
Table 6.3
Degrees of freedom and theoretically expected mean square values for different dispersion sources in a twofactor analysis of variance with repeated measurements for one of the factors
Dispersion Source 
df 
E ( MS ) 
Between subjects 


The A 


Subjects within a group 


Inside the subjects 


Factor In 


Interaction A B 


Interaction In & x; Subjects within a group 


The statistical hypotheses for this factorial plan are obviously identical to those that we put forward in the case of intergroup analysis. They assume a zero variance for the expected effects of both factors and their interactions:
Evaluating the effect of the A, factor in constructing the F relations in the numerator, the average square of the factor of the subject within the group . Thus, the theoretically expected value of F would look like this:
If the first of the statistical hypotheses put forward by us is true, then this expression will take the following form:
In this case, the constructed statistics will be described by F distribution with p  1 degree of freedom in the numerator and p ( n  1) the degree of freedom in the denominator. As usual, the calculated statistics are estimated using tables (see Appendix 4). In case of using special statistical packages, the results of such an evaluation are displayed in the analysis results. Once again, it is worth paying attention to the fact that, since it is theoretically impossible to get the value F, smaller than unity, then in practice this result automatically assumes a null hypothesis that affirms the equality of all effects of the factor under investigation at all its levels.
Similarly, arguments are constructed about the effect of the intrasubject factor B, as well as the interactions of A and In. In this case, when constructing the F statistics, the mean square for the interaction of the In with the factor of the subjects within the groups.
Thus, the theoretically expected value of the F statistics in evaluating the effect of the B factor, provided the null hypothesis is true, assuming the equality of all the effects of this factor at all levels, will look like this:
Theoretically, the expected value of the statistics F in the evaluation of the interaction effect of the factors and under the condition of the null hypothesis validating the equality of all interactions at all levels will be:
Note that the model we are considering is based on the following assumptions:
• First, it assumes homogeneity of the data dispersion for the subjects within the groups. This assumption is not strict and, in principle, rather tolerant of exceptions.
• Secondly, it is assumed that the variance of the data is homogeneous for the interaction of the factor by which the repeated measurements are performed, i.e. factor In with the factor of the subjects within the groups.
• Thirdly, the homogeneity of the variationalcovariance matrix q x is assumed. q. This assumption is very strict and requires mandatory pretests.
As in the case of ordinary, singlefactor, experimental plans with repeated measurements, the hypothesis of the homogeneity of the variationalcovariance matrix can be verified using the Moochley sphericity test. If this test gives a negative result, it may be worthwhile to convert the degrees of freedom of the numerator and denominator in order to make the statistical conclusion more conservative. The strategies for such a transformation are analogous to what we already know by discussing conventional singlefactor plans with repeated measurements. Similarly, the problem of nonsphericity can be solved by using multivariate tests and estimating polynomial contrasts.
thematic pictures
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