Traditionally, asymptotic tests are studied and applied under local alternative (Aivazian, et al., 1985). There exists a widespread opinion that the Wald, likelihood ratio, and score tests are asymptotically equivalent. We dispel this myth by showing that These tests have different statistical power in the presence of nuisance parameters. The local properties of the tests are described in terms of the first and second derivative evaluated at the null hypothesis. The comparison of the tests are illustrated with two popular regression models: linear regression with random predictor and logistic regression with binary covariate. We study the aberrant behavior of the tests when the distance between the null and alternative does not vanish with the sample size. We demonstrate that these tests have different asymptotic power. In particular, the score test is generally asymptotically biased but slightly superior for linear regression in a close neighborhood of the null. The power approximations are confirmed through simulations.
Keywords: Effective sample size; GLM; Linear regression; Local alternative; Logistic regression; Sample size determination.