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In mixed linear models with nonnormal data, the Gaussian Fisher information matrix is called a quasi-information matrix (QUIM). The QUIM plays an important role in evaluating the asymptotic covariance matrix of the estimators of the model parameters, including the variance components. Traditionally, there are two ways to estimate the information matrix: the estimated information matrix and the observed one. Because the analytic form of the QUIM involves parameters other than the variance components, for example, the third and fourth moments of the random effects, the estimated QUIM is not available. On the other hand, because of the dependence and nonnormality of the data, the observed QUIM is inconsistent. We propose an estimator of the QUIM that consists partially of an observed form and partially of an estimated one. We show that this estimator is consistent and computationally very easy to operate. The method is used to derive large sample tests of statistical hypotheses that involve the variance components in a non-Gaussian mixed linear model. Finite sample performance of the test is studied by simulations and compared with the delete-group jackknife method that applies to a special case of non-Gaussian mixed linear models., Comment: Published at http://dx.doi.org/10.1214/009053605000000543 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The term ``empirical predictor'' refers to a two-stage predictor of a linear combination of fixed and random effects. In the first stage, a predictor is obtained but it involves unknown parameters; thus, in the second stage, the unknown parameters are replaced by their estimators. In this paper, we consider mean squared errors (MSE) of empirical predictors under a general setup, where ML or REML estimators are used for the second stage. We obtain second-order approximation to the MSE as well as an estimator of the MSE correct to the same order. The general results are applied to mixed linear models to obtain a second-order approximation to the MSE of the empirical best linear unbiased predictor (EBLUP) of a linear mixed effect and an estimator of the MSE of EBLUP whose bias is correct to second order. The general mixed linear model includes the mixed ANOVA model and the longitudinal model as special cases.

This paper demonstrates that the various College Board examinations, most importantly the Scholastic Aptitude Tests (SATs), make predictions of grade point averages at the University of California at Berkeley that are biased against women. This finding persists even when one has made corrections for differences in fields that women and men study and for selection bias. Because women in fact are better students than the SATs predict, they are underrepresented both proportionately and relative to their merit in Berkeley's freshman classes (as they would also be in scholarship competitions and at other selective colleges). The differences in predicted grades are small, but if unbiased tests were used, Berkeley would have at least 5% more women in its freshmen classes (200 to 300 per year). Various solutions to the bias in the SATs are explored. Simple mathematical "fixes" by admissions officers will not work well. The best solution would be for the Educational Testing Service to correct the bias by altering the mix of questions in its tests. Failing this, college admissions staff should significantly expand the numbers of women's applications that are evaluated qualitatively. Five tables and six figures present analysis results. (Contains 18 references.) (Author/SLD)