Efficient Estimation in Marginal Partially Linear Models for Longitudinal/Clustered Data Using Splines

Abstract.  We consider marginal semiparametric partially linear models for longitudinal/clustered data and propose an estimation procedure based on a spline approximation of the non-parametric part of the model and an extension of the parametric marginal generalized estimating equations (GEE). Our estimates of both parametric part and non-parametric part of the model have properties parallel to those of parametric GEE, that is, the estimates are efficient if the covariance structure is correctly specified and they are still consistent and asymptotically normal even if the covariance structure is misspecified. By showing that our estimate achieves the semiparametric information bound, we actually establish the efficiency of estimating the parametric part of the model in a stronger sense than what is typically considered for GEE. The semiparametric efficiency of our estimate is obtained by assuming only conditional moment restrictions instead of the strict multivariate Gaussian error assumption.     

Comparing alternating logistic regressions to other approaches to modelling correlated binary data

Alternating logistic regressions (ALRs) seem to offer some of the advantages of marginal models estimated via generalized estimating equations (GEE) and generalized linear mixed models (GLMMs). Via simulation study we compared ALRs to marginal models estimated via GEE and subject-specific models estimated via GLMMs, with a focus on estimation of the correlation structure in three-level data sets (e.g. students in classes in schools). Data set size and structure, and amount of correlation in the data sets were varied. For simple correlation structures, ALRs performed well. For three-level correlation structures, all approaches, but especially ALRs, had difficulty assigning the correlation to the correct level, though sample sizes used were small. In addition, ALRs and GEEs had trouble attaching correct inference to the mean effects, though this improved as overall sample size improved. ALRs are a valuable addition to the data analyst's toolkit, though care should be taken when modelling data with three-level structures.     

Joint estimation of disease-specific sensitivities and specificities in reader-based multi-disease diagnostic studies of paired organs

Binocular data typically arise in ophthalmology where pairs of eyes are evaluated, through some diagnostic procedure, for the presence of certain diseases or pathologies. Treating eyes as independent and adopting the usual approach in estimating the sensitivity and specificity of a diagnostic test ignores the correlation between fellow eyes. This may consequently yield incorrect estimates, especially of the standard errors. The paper is concerned with diagnostic studies wherein several diagnostic tests, or the same test read by several readers, are administered to identify one or more diseases. A likelihood-based method of estimating disease-specific sensitivities and specificities via hierarchical generalized linear mixed models is proposed to meaningfully delineate the various correlations in the data. The efficiency of the estimates is assessed in a simulation study. Data from a study on diabetic retinopathy are analyzed to illustrate the methodology.     

Estimation of regression parameters in generalized linear models for cluster correlated data with measurement error

Liang and Zeger (1986) introduced a class of estimating equations that gives consistent estimates of regression parameters and of their asymptotic variances in the class of generalized linear models for cluster correlated data. When the independent variables or covariates in such models are subject to measurement errors, the parameter estimates obtained from these estimating equations are no longer consistent. To correct for the effect of measurement errors, an estimator with smaller asymptotic bias is constructed along the lines of Stefanski (1985), assuming that the measurement error variance is either known or estimable. The asymptotic distribution of the bias-corrected estimator and a consistent estimator of its asymptotic variance are also given. The special case of a binary logistic regression model is studied in detail. For this case, methods based on conditional scores and quasilikelihood are also extended to cluster correlated data. Results of a small simulation study on the performance of the proposed estimators and associated tests of hypotheses are reported.     

Semiparametric Likelihood Based Method for Goodness of Fit Tests and Estimation in Upgraded Mixture Models

We use Owen's (1988, 1990) empirical likelihood method in upgraded mixture models. Two groups of independent observations are available. One is z ₁, ..., z _n which is observed directly from a distribution F ( z ). The other one is x ₁, ..., x _m which is observed indirectly from F ( z ), where the x _i s have density ∫ p ( x | z ) dF ( z ) and p ( x | z ) is a conditional density function. We are interested in testing H ₀: p ( x | z ) = p ( x | z ; θ ), for some specified smooth density function. A semiparametric likelihood ratio based statistic is proposed and it is shown that it converges to a chi-squared distribution. This is a simple method for doing goodness of fit tests, especially when x is a discrete variable with finitely many values. In addition, we discuss estimation of θ and F ( z ) when H ₀ is true. The connection between upgraded mixture models and general estimating equations is pointed out.