Statistical Analysis with Missing Data

I describe a Bayesian method to account for measurement errors in linear regression of astronomical data. The method allows for heteroscedastic and possibly correlated measurement errors and intrinsic scatter in the regression relationship. The method is based on deriving a likelihood function for the measured data, and I focus on the case when the intrinsic distribution of the independent variables can be approximated using a mixture of Gaussian functions. I generalize the method to incorporate multiple independent variables, nondetections, and selection effects (e.g., Malmquist bias). A Gibbs sampler is described for simulating random draws from the probability distribution of the parameters, given the observed data. I use simulation to compare the method with other common estimators. The simulations illustrate that the Gaussian mixture model outperforms other common estimators and can effectively give constraints on the regression parameters, even when the measurement errors dominate the observed scatter, source detection fraction is low, or the intrinsic distribution of the independent variables is not a mixture of Gaussian functions. I conclude by using this method to fit the X-ray spectral slope as a function of Eddington ratio using a sample of 39 z 0.8 radio-quiet quasars. I confirm the correlation seen by other authors between the radio-quiet quasar X-ray spectral slope and the Eddington ratio, where the X-ray spectral slope softens as the Eddington ratio increases. IDL routines are made available for performing the regression.

https://iopscience.iop.org/article/10.1086/519947/pdf

Some Aspects of Measurement Error in Linear Regression of Astronomical Data

: Scientific learning is an iterative process employing Criticism and Estimation. Correspondingly the formulated model factors into two complimentary parts - a predictive part allowing model criticism, and a Bayes posterior part allowing estimation. Implications for significance tests, the theory of precise measurement, and for ridge estimates are considered. Predictive checking functions for transformation, serial correlation, bad values, and their relation with Bayesian options are considered. Robustness is seen from a Bayesian viewpoint and examples are given. For the bad value problem a comparison with M estimators is made. (Author)

Sampling and Bayes' inference in scientific modelling and robustness

A wide variety of effect sizes (ESs) has been used in the single-case design literature. Several researchers have “stress tested” these ESs by subjecting them to various degrees of problem data (e.g., autocorrelation, slope), resulting in the conditions by which different ESs can be considered valid. However, on the back end, few researchers have considered how prevalent and severe these problems are in extant data and as a result, how concerned applied researchers should be. The current study extracted and aggregated indicators of violations of normality and independence across four domains of educational study. Significant violations were found in total and across fields, including low levels of autocorrelation and moderate levels of absolute trend. These violations affect the selection and interpretation of ESs at the individual study level and for meta-analysis. Implications and recommendations are discussed.

Violations of Assumptions in School-Based Single-Case Data: Implications for the Selection and Interpretation of Effect Sizes.

Response misclassification of counted data biases and understates the uncertainty of parameter estimators in Poisson regression models. To correct these problems, researchers have devised classical procedures that rely on asymptotic distribution results and supplemental validation data in order to estimate unknown misclassification parameters. We derive a new Bayesian Poisson regression procedure that accounts and corrects for misclassification for a count variable with two categories. Under the Bayesian paradigm, one can use validation data, expert opinion, or a combination of these two approaches to correct for the consequences of misclassification. The Bayesian procedure proposed here yields an operationally effective way to correct and account for misclassification effects in Poisson count regression models. We demonstrate the performance of the model in a simulation study. Additionally, we analyze two real-data examples and compare our new Bayesian inference method that adjusts for misclassification with a similar analysis that ignores misclassification.

A Bayesian approach to adjust for diagnostic misclassification between two mortality causes in Poisson regression.

The article considers regression models for binary response in a situation when the response is subject to classification error. It is also assumed that some of the covariates are unobservable, but measurements on its surrogates are available. Likelihood based analysis is developed to fit the model. A sensitivity analysis is also carried out through simulation to ascertain the effect of ignoring classification error and/or measurement error on the estimation of regression parameters. At the end, the methodology developed in this paper is illustrated through an example.

Measurement error model for misclassified binary responses

A multivariate probit model for correlated binary responses given the predictors of interest has been considered. Some of the responses are subject to classification errors and hence are not directly observable. Also measurements on some of the predictors are not available; instead the measurements on its surrogate are available. However, the conditional distribution of the unobservable predictors given the surrogate is completely specified. Models are proposed taking into account either or both of these sources of errors. Likelihood-based methodologies are proposed to fit these models. To ascertain the effect of ignoring classification errors and /or measurement error on the estimates of the regression and correlation parameters, a sensitivity study is carried out through simulation. Finally, the proposed methodology is illustrated through an example.

Analysis of misclassified correlated binary data using a multivariate probit model when covariates are subject to measurement error.

This paper focuses on the question of specification of measurement error distribution and the distribution of true predictors in generalized linear models when the predictors are subject to measurement errors. The standard measurement error model typically assumes that the measurement error distribution and the distribution of covariates unobservable in the main study are normal. To make the model flexible enough we, instead, assume that the measurement error distribution is multivariate t and the distribution of true covariates is a finite mixture of normal densities. Likelihood–based method is developed to estimate the regression parameters. However, direct maximization of the marginal likelihood is numerically difficult. Thus as an alternative to it we apply the EM algorithm. This makes the computation of likelihood estimates feasible. The performance of the proposed model is investigated by simulation study.

A Flexible Model for Generalized Linear Regression with Measurement Error

We consider the estimation of the prevalence of a rare disease, and the log-odds ratio for two specified groups of individuals from group testing data. For a low-prevalence disease, the maximum likelihood estimate of the log-odds ratio is severely biased. However, Firth correction to the score function leads to a considerable improvement of the estimator. Also, for a low-prevalence disease, if the diagnostic test is imperfect, the group testing is found to yield more precise estimate of the log-odds ratio than the individual testing.

Surupa Roy

Papers

Measurement error model for misclassified binary responses

Analysis of misclassified correlated binary data using a multivariate probit model when covariates are subject to measurement error.

A Flexible Model for Generalized Linear Regression with Measurement Error

Estimation of log-odds ratio from group testing data using Firth correction.

Analysis of ordinal longitudinal data under nonignorable missingness and misreporting: An application to Alzheimer’s disease study