Making logits behave

For the multivariate analysis of binary (or n-ary) dependent variables, common statistical analysis systems offer models with non-homogeneous error variance (LSN-models), and optionally transformations (“response functions”) of the relative frequencies, above all the logit function. Using the simplest possible example—two groups with a relative frequency in each—it is shown that significance tests based on transformed and non-transformed variables can yield substantially diverging results, and even an example of flagrantly inadequate behavior of logit as a response function in significance tests is presented. Dispensing with LSN models can remedy the latter problem; this recommendation is in line with other known objections against using LSN with binary dependents. Furthermore, an improved approximation for the variance of (any) response function makes tests based on transformed and non-transformed variables equivalent, so that problems of discrepant results depending on the response function used are completely removed. In the Appendix a sketch is given of a method to extend the approach to more general situations.