How Can the Score Test Be Inconsistent?

Abstract

The score test can be inconsistent because-at the MLE under the null hypothesis-the observed information matrix generates negative variance estimates. The test can also be inconsistent if the expected likelihood equation has spurious roots.KEY WORDS: Maximum likelihood, score test inconsistent, observed information, multiple roots for likelihood equationTo appear in The American Statistician vol. 61 (2007) pp. 291-295 The short answerAfter a sketch of likelihood theory, this paper will answer the question in the title. In brief, suppose we use Rao's score test, normalized by observed information rather than expected information. Furthermore, we compute observed information atθ S , the parameter value maximizing the log likelihood over the null hypothesis. This is a restricted maximum. At a restricted maximum, observed information can generate negative variance estimates-which makes inconsistency possible.At the unrestricted maximum, observed information will typically be positive definite. Thus, if observed information is computed at the unrestricted MLE, consistency should be restored. The "estimated expected" information is also a good option, when it can be obtained in closed form. (Similar considerations apply to the Wald test.) However, the score test may have limited power if the "expected likelihood equation" has spurious roots: details are in section 8 below.The discussion provides some context for the example in Morgan, Palmer, and Ridout (2007), and may clarify some of the inferential issues. Tedious complications are avoided by requiring "suitable regularity conditions" throughout: in essence, densities are positive, smooth functions that decay rapidly at infinity, and with some degree of uniformity. Mathematical depths can be fathomed another day.David A. Freedman is Professor, Department of Statistics, University of California Berkeley, CA 94720-3860 (E-mail: freedman@stat.berkeley.edu). Peter Westfall (Texas Tech) made many helpful comments, as did Morgan, Palmer, and Ridout.

Model equivalence tests in a parametric framework

“…Reeds (1985) illustrates it for a Cauchy model. Freedman (2007) gives an example for a discrete distribution and points out that this yields inconsistency of the classical score significance test. The nonlinear regression Example 2 of Dominguez and Lobato (2004), where Y = θ 2 0 X + θ 0 X 2 + ε, can be recast in a maximum likelihood setup assuming ε ∼ N (0, σ 2 ) to give another illustration.…”

Score Test for Conditional Independence Between Longitudinal Outcome and Time to Event Given the Classes in the Joint Latent Class Model

“…As the score statistic is a sum over the subjects , Freedman (2007) recommended to estimate using the empirical covariance matrix of the summands that may be more robust than the asymptotic estimate. More precisely: This variance estimate is easy to compute and accounts for the uncertainty due to parameter estimation, but it is an estimate of the variance of U at the true value η, whereas Var as ( U ) is computed under H 0 .…”

“…Neglecting uncertainty due to the estimation of parameters θ, under the null hypothesis, the test statistic U (0,θ) T Var(U ) −1 U (0,θ) follows asymptotically a χ 2 m distribution, where m is the size of η. 3.4 Robust Variance of the Score Statistic As the score statistic is a sum over the subjects U (0,θ) = N i =1 U i (0,θ), Freedman (2007) recommended to estimate Var(U (0,θ)) using the empirical covariance matrix of the summands that may be more robust than the asymptotic estimate. More precisely:…”

The score test for the two‐sample occupancy model

“…(e.g. Freedman , p. 293). There are reservations on the use of the score statistic based on the observed information matrix.…”