Alternatives To The Wald Test
The likelihood-ratio test can also be used to test whether an effect exists or not. Usually the Wald test and the likelihood ratio test give very similar conclusions (as they are asymptotically equivalent), but very rarely, they disagree enough to lead to different conclusions: the researcher finds him/herself asking, or being asked, why the p-value is significant when the confidence interval includes 0, or why the p-value is not significant when the confidence interval excludes 0. In this situation, first remember that statistical significance is always somewhat arbitrary, as it depends on an arbitrarily chosen significance level.
There are several reasons to prefer the likelihood ratio test to the Wald test. One is that the Wald test can give different answers to the same question, depending on how the question is phrased. For example, asking whether R = 1 is the same as asking whether log R = 0; but the Wald statistic for R = 1 is not the same as the Wald statistic for log R = 0 (because there is in general no neat relationship between the standard errors of R and log R). Likelihood ratio tests will give exactly the same answer whether we work with R, log R or any other monotonic transformation of R. The other reason is that the Wald test uses two approximations (that we know the standard error, and that the distribution is chi-squared), whereas the likelihood ratio test uses one approximation (that the distribution is chi-squared).
Yet another alternative is the score test, which has the advantage that it can be formulated in situations where the variability is difficult to estimate; e.g. the Cochran–Mantel–Haenzel test is a score test.
Read more about this topic: Wald Test
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