Mixed linear nonlinear aggregate level models for small area estimation from surveys of binary counts - ARCHIVED

We propose an aggregate level generalized linear model with additive random components (GLMARC) for binary count data from surveys. It has both linear (for random effects) and nonlinear (for fixed effects) parts in modeling the mean function and hence belongs to a class termed as mixed linear non-linear models. The model allows for linear mixed model (LMM)-type approach to small area estimation (SAE) somewhat similar to the well-known Fay-Herriot (1979) method and thus takes full account of the sampling design. Unlike the alternative hierarchical Bayes (HB) approach of You and Rao (2002), the proposed method gives rise to easily interpretable SAEs and frequentist diagnostics as well as self-benchmarking to reliable large area direct estimates. The usual LMM methodology is not appropriate for the problem with count data because of lack of range restrictions on the mean function and the possibility of unrealistic (e.g. zero in the context of SAE) estimates of the variance component as the model does not allow the random effect part of the conditional mean function to depend on the marginal mean. The proposed method is an improvement of the earlier method due to Vonesh and Carter (1992) which also uses mixed linear nonlinear models but the variance-mean relationship was not accounted for although typically done via range restrictions on the random effect. Also the implications of survey design were not considered as well as the estimation of random effects. In our application for SAE, however, it is important to obtain suitable estimates of both fixed and random effects. It may be noted that unlike the generalized linear mixed model (GLMM), GLMARC like LMM offers considerable simplicity in model fitting. This was made possible by replacing the original fixed and random effects of GLMM with a new set of parameters of GLMARC with quite a different interpretation as the random effect is no longer inside the nonlinear predictor function. However, this is of no consequence for SAE because the small area parameters correspond to the overall conditional means and not on individual model parameters. We propose a method of iterative BLUP for parameters estimation which allows for self-benchmarking after a suitable model enlargement. The problem of small areas with small or no sample sizes or zero direct estimates is addressed by collapsing domains only for the stage of parameter estimation. Application to the 2000-01 Canadian Community Health Survey for estimation of the proportion of daily smokers in subpopulations defined by provincial health regions by age-sex groups is presented as an illustration.