Generalized Functional Linear Regression Models With Functional and Scalar Covariates Prone to Measurement Error

Most methods for adjusting for biases due to measurement errors in covariates in generalized linear regression models focus on scalar covariates. Less work exists to correct for biases due to measurement error in a mixture of functional and scalar covariates. To address this limitation, we develop joint functional simulation extrapolation (FSIMEX) and mixed effects model-based (MEM) approaches in a mixture of functional and scalar covariates prone to classical measurement errors in generalized functional linear regression. The proposed methods were compared with the Oracle estimator, which served as a benchmark, principal components analysis through the conditional expectation (PACE), Naiveₐve, and Naiveₒne methods through extensive simulations to assess their finite sample performance. Across a range of conditions, the joint FSIMEX estimator generally had low biases, followed by the MEM estimator, similar to those of the Oracle estimator. The SIMEX and MEM methods for measurement error adjustment performed notably better than the Naiveₒne estimators that did not adjust for measurement error. Although the Naiveₐve and PACE estimators adjusted for some of the measurement errors, they did not account for the heteroscedastic covariance structures associated with the functional covariate and the PACE approach was not designed for modeling scalar predictors. We applied our methods to data from the 2011-2014 cycles of the National Health and Examination Survey (NHANES) to assess the relationships physical activity, total caloric intake and demographic characteristics have with type 2 diabetes status of community dwelling adults living in the United States. We treated the device-based measure of physical activity as a functional covariate prone to complex arbitrary heteroscedastic errors and total caloric intake as a scalar covariate prone to error. Estimation with the joint FSIMEX and MEM estimators reduced the associations for some covariates but increased them for others, particularly physical activity, in comparison to estimation with the estimators that did not adjust for biases due to measurement error. Overall, we observed that failing to account for measurement error due to biases can lead to biased estimations of the functional and scalar covariates, which are prone to errors in the generalized functional linear regression models.