# Statistical Hypotheses in Multivariate Correlation-Regression...

## Statistical hypotheses in multivariative correlation/regression analysis

Evaluation of the statistical reliability of the coefficient of determination R 2 involves testing the null hypothesis about its equality to zero. The general strategy of the statistical analysis used here is not fundamentally different from what we already know.

All the variance of the dependent variable Y can be decomposed into two additive parts: 1) the part that is determined by the effects of all independent variables, and 2) the part that is independent of them - this part, as we know, is called Residual variance. The scheme for estimating these parts of the dispersion is not fundamentally different from the one considered in Ch. 7, devoted to the bivariate correlation (in Table 7.4 specific details of such analysis are reflected).

The main difference between the variance analysis of estimating multivariate correlation plans is the estimation of the degrees of freedom of statistics F. If in the case of one independent variable we have one degree of freedom in the numerator , then using the k independent variables, the number of degrees of freedom for the numerator increases to k. Thus, the mean square for regression can be estimated as follows:

Similarly, the number of degrees of freedom for the residual variance in the case of k independent variables should be adjusted. This time their number decreases by the number of independent variables. The residual mean square can be estimated as follows:

Therefore, the estimated F -statistics will look like this:

If the theoretical value of the coefficient of determination R 2 turns out to be zero, i.e. 2 = 0, this F-statistics is distributed according to the law F -distributions with k degrees of freedom in the numerator and n - k - 1 degrees of freedom in the denominator.

In practice, however, statistical estimation of the coefficient of determination, as a rule, is not enough. The researcher is usually more interested in not the total percentage of variance of the dependent variable that explains the causal model used, but the statistical contribution of each of the independent variables to the variance of the dependent variable. In this case, we are already talking about the statistical reliability of the regression coefficients and the associated partial correlation coefficients. As a null hypothesis, a hypothesis is advanced about the equality of these coefficients to the zero value. It is clear that if any of these coefficients, for example the correlation value of a part, turns out to be zero, the remaining partial coefficients also take a zero value. Therefore, in practice, there is no need to test several hypotheses concerning these communication parameters at once. It suffices to verify only one hypothesis. Statistical programs usually give an estimate of the standardized regression coefficient, calculating for t-statistics with and - k - 1 degrees of freedom. In manual it is easier to use the following formula, which also calculates t -statistics:

If a part correlation is assumed to be zero, this statistics is distributed according to the t -distribution of the Student with p - k - 1 degrees of freedom. If the value of the ί statistic computed in this way is reliably different from zero, the hypothesis of the zero value of the partial coefficients is rejected and a conclusion is made about the statistically reliable contribution of the investigated independent variable.

VE Vysokova [21] investigated possible predictors of the reaction time of the subject in the problems of the semantic solution.

The subject was given pairs of words. The first word in the pair was one example of a different level of typicality of one of the ten common semantic categories, such as, for example, furniture, vegetable, clothing, fruit. A total of nine words of each category. The second word either denoted the category to which the word could relate, or represented the designation of the appropriate category of the thematic context, such as room, lunch, wardrobe, garden. Thus , two lists of 180 pairs of words were compiled. One list was built on a thematic principle, the second - on a categorical basis. Each word in the pair was presented twice: once with the appropriate category or thematic context, the second time with an inappropriate category or context. The order of presentation of pairs of words was random.

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