Estimation of the nonresponse models for categorical data.

Abstract

For categorical outcomes subject to nonignorable nonresponses, log-linear models may be used to adjust for the nonresponse. These models are used to fit the data in a frequency table in which one index corresponds to whether or not the subject is respondent. The likelihood is maximized over the expected cell frequencies with respect to the log-linear model using the EM algorithm (Baker and Laird, 1988). However, maximum likelihood (ML) estimation may provide boundary estimates for the frequencies of the nonrespondents. The occurrence of boundary estimates depends on the relationship of the marginal distribution of the nonrespondents to those of the respondents conditional on the value of the response. As alternative methods to ML estimation, we propose two approaches; restricted maximum likelihood (RML) estimation and adjusted maximum likelihood (AML) estimation. RML estimation imposes restrictions on the ML estimators of cell frequencies by bounding the range of the odds ratio $\alpha$ which represents the nonresponse mechanism. AML estimation adjusts the E-step of the EM algorithm by defining the pseudo-observed cell frequency as a weighted sum of the estimate determined at the M-step (with weight w) and the fixed constant (with weight 1-w), where the constant is proportional to the cell frequency of the respondents. The two proposed estimation methods are compared with ML estimation by simulation in 2 x 2 x 2 tables and 4 x 4 x 2 tables. Biases and mean square errors (MSE) of the ML estimates, RML estimates, and AML estimates of the cell frequencies are used to compare the methods. For a reasonable choice of the boundary conditions, RML estimation provided smaller MSE's than those of ML estimation, while it yielded slightly larger biases. Similarly, for w's between 0.9 and 0.95, AML estimation yielded estimates which had smaller MSE's than those of ML estimation, while it increased biases slightly. Since a reasonable choice of the boundary condition for RML is not always feasible, AML with w = 0.95 is recommended because it provides estimates which have small MSE's without a large increase in bias.