Traditionally, asymptotic tests are studied and applied under local alternative. 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.
ChandraT. K. & JoshiS. N. (1983). Comparison of the likelihood ratio, Rao’s and Wald’s tests and a conjecture of C. R. Rao. Sankhyā Ser. A 45, 226-246.
5.
ChandraT. K. & MukerjeeR. (1984). On the optimality of the Rao’s statistic. Communications in Statistics. Theory and Methods13, 1507-1515.
6.
CohenJ. (1994). The earth is round (p<0.05). American psychologist49, 997-1003.
7.
CoxD. R. & HinkleyD. V. (1974). Theoretical statistics. Boca Raton, FL: Chapman & Hall.
8.
DemidenkoE. (2013). Mixed models: Theory and applications. 2d. ed. Hoboken, NJ: Wiley.
9.
DemidenkoE. (2020). Advanced Statistics with Applications in R. Hoboken, NJ: Wiley. www.eugened.org
10.
DemidenkoE. (2007). Sample size determination for logistic regression revisited. Statistics in Medicine26, 3385-3394.
11.
DemidenkoE. (2008). Sample size and optimal design for logistic regression with binary interaction. Statistics in Medicine27, 36-46.
12.
DuchateauL.Mc DermottB., & RowlandsG. R. (1998). Power evaluation of small drug and vaccine experiments with binary outcomes. Statistics in Medicine17, 111-120.
13.
FearsT. R.BenicouJ., & GailM. H. (1996). A reminder of the fallibility of the Wald statistics, The American Statistician,50, 226-227.
14.
GaudermanW. J. (2002). Sample size requirements for association studies of gene-gene interaction. American Journal of Epidemiology155, 478-484.
15.
HauckW. W. & DonnerA. (1977). Wald’s test as applied to hypotheses in logit analysis. Journal of American Statistical Association72, 851-853.
16.
KulinskayaE.MorgenthalerS., & StaudteR. G. (2008). Meta analysis. A guide to calibrating and combining statistical evidence. Chichester: Wiley.
17.
LehmannE. L. & RomanoJ. P. (2005). Testing statistical hypotheses, 3d edition. New York: Springer.
18.
LemonteA. J. & FerrariS. L. P. (2012). Local power and size properties of the LR, Wald, score and gradient tests in dispersion models. Statistical Methodology 9, 537-554.
19.
MadanskyA. (1989). A comparison of the likelihood ratio, Wald, and Rao tests. In: Contributions to Probability and Statistics, (Eds: GleserL. J. et al.). New York: Springer-Verlag.
20.
Mc CullaghP. (1987). Tensor Methods in Statistics. London: Chapman & Hall.
21.
PawitanY. (2001). In all likelihood. Statistical modeling and inference using likelihood. Oxford: Clarendon Press.
22.
RaoC. R. (1948). Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation. Proc. Camb. Phil. Soc.44, 50-57.
23.
RaoC. R. (1973). Linear Statistical Inference and Its Applications. 2d ed. New York: Wiley.
24.
RaynerJ. C. (1997). The asymptotically optimal tests, The Statistician46, 337-346.
25.
SchervishM. J. (1995). Theory of statistics. New York: Springer–Verlag.
26.
SelfG. & MautitzenR. H. (1988). Power/sample size calculations for generalized linear models. Biometrics44, 79-86.
27.
SelfG.MautitzenR. H., & OharaJ. (1992). Power calculations for likelihood ratio tests in generalized linear models. Biometrics48, 31-39.
28.
SerflingR. J. (1980). Approximation theorems of mathematical statistics. New York: Wiley.
29.
SevereniT. A. (2000). Likelihood methods in statistics. Oxford: Oxford University Press.
30.
ShaoJ. (2010). Mathematical statistics. 2d ed. Springer: New York.
31.
ShiehG. (2000). On power and sample size calculations for likelihood ratio tests in generalized linear models. Biometrics56, 1192-1196.
32.
ShiehG. (2005). On power and sample size calculations for Wald tests in generalized linear models. Journal of Statistical Planning and Inference 128, 43-59.
33.
Væth, M. (1985). On the use of Wald’s test in exponential families. International Statistical Review53, 199-214.
34.
WhittemoreA. S. (1981). Sample size for logistic regression with small response probability. Journal of the American Statistical Association76, 27-32.
35.
YoungG. A. & SmithR. L. (2005). Essentials of statistical inference. Cambridge: Cambridge University Press.
