A linear mixed model with a smooth random effects density is proposed. A similar approach to P-spline smoothing of Eilers and Marx (1996, Statistical Science 11, 89-121) is applied to yield a more flexible estimate of the random effects density. Our approach differs from theirs in that the B-spline basis functions are replaced by approximating Gaussian densities. Fitting the model involves maximizing a penalized marginal likelihood. The best penalty parameters minimize Akaike's Information Criterion employing Gray's (1992, Journal of the American Statistical Association 87, 942-951) results. Although our method is applicable to any dimensions of the random effects structure, in this article the two-dimensional case is explored. Our methodology is conceptually simple, and it is relatively easy to fit in practice and is applied to the cholesterol data first analyzed by Zhang and Davidian (2001, Biometrics 57, 795-802). A simulation study shows that our approach yields almost unbiased estimates of the regression and the smoothing parameters in small sample settings. Consistency of the estimates is shown in a particular case.
Objective. Recent data suggest that a geneenvironment interaction between smoking and the HLA shared epitope alleles plays a role in shaping the autoimmune reaction to specific citrullinated antigens. This study was undertaken to determine the effects of HLA shared epitope alleles and tobacco exposure on the immune response against various citrullinated antigens. These associations were analyzed in the anticitrullinated protein antibody (ACPA)-positive stratum to control for the possibility that the associations found are explained by the known interaction between HLA shared epitope alleles and tobacco exposure on ACPA status.Methods. In 661 patients with rheumatoid arthritis, reactivity against several citrullinated antigens from vimentin, fibrinogen, enolase, and myelin basic protein was determined by enzyme-linked immunosorbent assay. The effects of the HLA shared epitope alleles and tobacco exposure were assessed by logistic regression analysis. Biologic interaction was analyzed by investigating whether the effects of the risk factors combined exhibited departure from additivity. Results. A significant interaction between tobacco exposure and HLA shared epitope alleles was found for the presence of ACPA as reported previously. When these interaction effects were studied for several ACPA "fine specificities," significant interactions were noted for several citrullinated peptides. However, these interactions were not present after stratification for ACPA status, indicating that the interaction between tobacco exposure and HLA shared epitope alleles influences autoimmunity not to specific citrullinated antigens, but rather to ACPA development.Conclusion. Our data indicate that the geneenvironment interaction between HLA shared epitope alleles and smoking does not appear to shape the reactivity of the ACPA response. These data suggest that smoking promotes nonspecific citrullination rather than citrullination of specific antigens.
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