Measurement Error Analysis from Independent to Longitudinal Setup

In a generalized linear models (GLMs) setup, when scalar responses along with multidimensional covariates are collected from a selected sample of independent individuals, there are situations where it is realized that the observed covariates differ from the corresponding true covariates by some measurement error, but it is of interest to find the regression effects of the true covariates on the scalar responses. Further it may happen that the true covariates may be fixed but unknown or they may be random. It is understandable that when observed covariates are used for either likelihood or quasi-likelihood-based inferences, the naive regression estimates would be biased and hence inconsistent for the true regression parameters. Over the last three decades there have been a significant number of studies dealing with this bias correction problem for the regression estimation due to the presence of measurement error. In general these bias correction inferences are relatively easier for the linear and count response models, whereas the inferences are complex for the logistic binary models. In the first part of the paper, we review some of the widely used bias correction inferences in the GLMs setup and highlight their advantages and drawbacks where appropriate. As opposed to the independent setup, the bias correction inferences for clustered (longitudinal) data are, however, not adequately addressed in the literature. To be a bit more specific, some attention has been given to deal with bias correction in linear longitudinal setup (also called panel data setup) only. Bias corrected generalized method of moments (BCGMM) and bias corrected generalized quasi-likelihood (BCGQL) approaches are introduced and discussed. In the second part of this paper, we review these BCGMM and BCGQL approaches along with their advantages and drawbacks. The bias correction inferences for count and binary data are, however, more complex, because of the fact that apart from the mean functions, the variance and covariance functions of the clustered responses also involve time-dependent covariates. This makes the bias correction difficult. However, following some recent works, in the second part of the paper, we also discuss a BCGQL approach for longitudinal models for count data subject to measurement error in covariates. Developing a similar bias correction approach for longitudinal binary data appears to be difficult and it requires further in-depth investigations.