Some statistical software use the Wald statistic when reporting on the regression coefficients. As examples, R and Stata report Wald by default.
The logistic regression article on Wikipedia says, unfortunately without reference:
“Rather than the Wald method, the recommended method to calculate the p-value for logistic regression is the likelihood-ratio test (LRT)”
How are Wald and LRT calculated for a logistic regression coefficient (independent variable)? This is for a reminder on how they are calculated and to highlight their differences.
From the Wald test page on Wikipedia:
- [T]he Wald test is not invariant to a reparametrization, while the Likelihood ratio tests will give exactly the same answer whether we work with R, log R or any other monotonic transformation of R.
So in the context of logistic regression, if you logged a regressor its p-value would be different compared to if it was unlogged (is this correct). Would the p-value change if the LRT was used?
From the same Wald test page:
- The other reason is that the Wald test uses two approximations (that we know the standard error, and that the distribution is χ2), whereas the likelihood ratio test uses one approximation (that the distribution is χ2).
Although Wald and likelihood ratio are asymptotically equivalent, in the logistic regression we are usually in the pre-asymptote setting, so this is not a reason to view then as equivalent.
Thus it seems that the Wald test disadvantages outweigh the advantages in the logistic setting, and the likelihood ratio is better.
It is my guess that the Wald test is used by logistic regression software routines for its easier computational efficiency, which was more important in the past when software such as R and Stata were first created. Then, through backwards compatibility and not wanting to change the semantics of their logistic functions, the Wald statistic has remained the default. Is there any evidence for this being the case?
Should I be changing the default Wald to likelihood ratio? A lesser question, is there an easy was to do this in R?