Parametric Statistics and Levels of Measurement

If Y is a continuous, ordinal measure of a latent variable θ and Y has a normal distribution with equal variances in several groups, then t tests and one-way analyses of variance on Y can be used to test hypotheses about population mean differences on θ in the corresponding groups. If X and Y are continuous, ordinal measures of latent variables θ and Φ, and if X and Y have a bivariate normal distribution, then a test of the null hypothesis that the population correlation between X and Y is zero is also a test of the hypothesis that θ and Φ are independent.