Testing Covariance Matrix Hypotheses: An Example From The Measurement Of Musical Abilities

In using tests of statistical significance, the null hypothesis will almost certainly be rejected if there is a large number of subjects. This statement also applies to the testing of hypotheses about covariance matrices. The problem can be solved within a decision theoretic framework of hypothesis testing using Monte Carlo procedures. The covariance structure of theoretical interest is specified in the alternative hypothesis, while the null hypothesis is identical in all secondary aspects to the alternative hypothesis but differs clearly from the alternative hypothesis in primary theoretical aspects. Using reasonable values for the parameters in both hypotheses, the relations between the number of subjects, the significance level, and the power of the test are studied with simulated data. From the Monte Carlo study an optimal and feasible combination of number of subjects, significance level, and power is determined for the empirical study. The procedure is demonstrated in detail with an example from the measurement of musical abilities. It is emphasized that the power of the test for the actual data should be investigated again in a simulation study, especially if the data differ in important ways from the planning of the experiment.