Parametric mode regression for bounded responses

Abstract

This article has earned an open data badge "Reproducible Research" for making publicly available the code necessary to reproduce the reported results. The results reported in this article were reproduced partially for data confidentiality reasons and due to their computational complexity.

Parametric modal regression with error in covariates

“…For the beta modal regression model without error in covariates, Zhou and Huang (2020) propose a score-based test statistic defined below for the purpose of model diagnostics,…”

“…Abundant data collected from this line of research are available in the Alzheimer's Disease Neuroimaging Initiative (ADNI) database (http://adni.loni.usc.edu/). Zhou and Huang (2020) analyzed a data set relating to 245 individuals diagnosed with mild cognitive impairment from this database. The goal is to study roles that an individual's volumetric measure of entorhinal cortex (ERC) and that of hippocampus (HPC) play in predicting one's risk of developing Alzheimer's disease.…”

“…For the beta modal regression model without error in covariates, Zhou and Huang (2020) propose a score‐based test statistic defined below for the purpose of model diagnostics, $$$$\begin{equation} Q(\hat{\mathbf {\Omega }}_0;\mathcal {D}) = \frac{n-2}{2(n-1)}\overline{\mathbf {S}}^{ \mathrm{\scriptscriptstyle T} }\hat{\mathbf {\Sigma }}^{-1}\overline{\mathbf {S}}, \end{equation}$$$$where $$\hat{\mbox{\boldmath $\Omega $}}_0$$ is the maximum likelihood estimator of $$\mbox{\boldmath $\Omega $}$$, $$\overline{\mathbf {S}} =n^{-1} \sum _{j=1}^{n} \mathbf {S}(\hat{\mathbf {\Omega }}_0; Y_j, \bm {X}_j)$$, and $$\hat{\bm{\Sigma }} =\lbrace n(n-1)\rbrace ^{-1} \sum _{j=1}^{n}\lbrace \mathbf {S}(\hat{\bm{\Omega }}_0; Y_j, \bm {X}_j...$…”

“…By construction, a larger value of the nonnegative $$Q(\hat{\mbox{\boldmath $\Omega $}}_0; \mathcal {D})$$ provides stronger evidence indicating model misspecification. A parametric bootstrap procedure is developed in Zhou and Huang (2020) to estimate the null distribution of $$Q(\hat{\mbox{\boldmath $\Omega $}}_0; \mathcal {D})$$, from which one onbtains an estimated p ‐value for the test.…”

“…But the next hurdle emerges, that is the design of a parametric bootstrap procedure for estimating the null distribution of $$\tilde{Q}(\hat{\mbox{\boldmath $\Omega $}}; \mathcal {D}^*)$$. Traditional parametric bootstrap in the regression setting, such as the procedure in Zhou and Huang (2020), involves generating response data from the primary regression model that again requires evaluating an estimated regression function at the true covariates that are partly unobserved in the current context. We overcome this hurdle by “estimating” unobserved true covariate data, as implemented in the method of regression calibration (Chapter 4, Carroll et al., 2006) that takes on the structural viewpoint of measurement error models.…”