Multivariate analysis of variances
The advantage of multivariate analysis methods in evaluating the results of experimental effects in plans with repeated measurements is that they do not require uniformity of the variational-covariance matrix at all. In contrast to the usual tests, the multivariate analysis of variance ( MANOVA ) does not investigate the hypothesis that the effects of the independent variable are equal to/e levels, and the hypothesis of the equality k - 1 difference of the mean of the zero value. One such hypothesis, as an example, can be as follows:
Such hypotheses suggest transformation of initial results of the experiment in the values of the differences between the levels of the independent variable X ij - X ij . Thus, a matrix of differences is obtained for all subjects. Then, based on the weighting coefficients, the values of the differences obtained are calculated for each subject:
The weight coefficients themselves are calculated on the basis of analysis of group matrices using vector algebra procedures. The main task of such an analysis is to maximize the value of the statistics t, which is constructed as follows:
The square of the obtained value t -statistics is usually denoted as T 2 Hotelling. To obtain F -statistics, the value of T 2 is multiplied by the value ( n - k + 1 ) / ( k - 1). This statistic will be distributed according to the F -distribution rule with k - 1 degrees of freedom in the numerator and n - k + 1 degrees of freedom in the denominator. Therefore, in order not to receive a negative value of the degrees of freedom F, it is necessary that the number of subjects participating in the experiment always exceed the number of levels of the independent variable. This is the restriction of multivariate tests. We also pay attention to the fact that if the number of levels of an independent variable turns out to be equal to two, the multivariate value F turns out to be equivalent to its usual, univariant value, and thus the test results ANOVA and MANOVA are the same.
Statistical programs, as a rule, provide the researcher with several variants of statistics for multivariate analysis. However, in the case of the experimental plan under consideration, they all give the same value F, and therefore the differences between them are not significant.
It should be noted that multivariate tests have less power compared to conventional tests, so it makes sense to use them only with the expressed heterogeneity of the variational-covariance matrix. The optimal results of such tests can be obtained if the following conditions are met: 1) k & lt; 4; n & gt; k + 15; ε & lt; 0.90 or 2) 5 & lt; k & lt; 8; n & gt; k + 30; ε & lt; 0.85.
If we are interested in contrasts, and this is, in the final analysis, the goal of almost any statistical analysis in multi-level plans, the problem of sphericity proves to be insignificant. The fact is that contrasting sums have only one degree of freedom in the numerator. The problem here is only what statistics to use as a measure of experimental error. By default, in the statistical packages only the variance of the error that is directly related to the estimated average is estimated. In fact, this means using the mean t -test for connected samples (see formula (2.16)), although the statistical program itself can produce F, which in this case is the value of t 2.
If our independent variable is quantitative, we may be interested in more complex, polynomial contrasts. We already know that the method of a priori contrasts makes it possible not only to estimate the differences between these or those experimental conditions, but also to reveal the quantitative relationships between the independent and dependent variables. In Ch. 3, an example of a linear correspondence between the independent and dependent variables was considered (see Figure 3.2). However, if the number of levels of the independent variable turns out to be sufficiently large, then along with the linear dependence one can also evaluate the nonlinear dependence of different order.
So, the three levels of the independent variable give us the opportunity to evaluate the linear and quadratic, parabolic, dependencies. The four levels additionally enable the evaluation of cubic, S -like dependence. The more the number of levels of the independent variable is investigated in the experiment, the more opportunities are given to the experimenter. Such dependencies, as we already know, are called polynomial. The maximum value of the degree of a polynomial is defined as k - 1. And this means that the method of polynomial contrasts allows the entire variance of the dependent variable to be decomposed into k - 1 additive parts, each of which will have only one degree of freedom.
The value of the total and correspondingly average squares for polynomial contrasts is estimated in the standard way, i.e. by the formula (3.9).
The variance value of the error is also the result of the decomposition of the total variance of the experimental error into additive parts, each of which corresponds to a separately taken contrast. Thus, in the general case, the number of degrees of freedom for the residual variance of each polynomial is n - 1, which should also rule out the problem of homogeneity of variance. The mean squares themselves can be estimated as follows:
where - variance estimate for contrasts evaluated separately for each subject.
Since the calculation of this metric can be tedious at manual calculations, then instead of the error value calculated by formula (4.7), the usual value of the residual dispersion can be used. However, such an option does not solve the sphericity problem. In this case, corrections to the estimates of the mean squares for the denominator can be used. In more detail, such corrections are considered in Section 4.4, which is devoted to practical examples of applying a single-factor analysis of variance with repeated measurements.
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