Omitting empty cells in regression analysis introduces sample selection bias.
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Available studies partially cover the relationship between missing data and regression bias, with some indicating that omitting cases can result in selection bias or coefficient distortion while others note conditions where complete-case analysis remains unbiased.
Recent research on fair regression focused on developing new fairness notions and approximation methods as target variables and even the sensitive attribute are continuous in the regression setting. However, all previous fair regression research assumed the training data and testing data are drawn from the same distributions. This assumption is often violated in real world due to the sample selection bias between the training and testing data. In this paper, we develop a framework for fair regression under sample selection bias when dependent variable values of a set of samples from the training data are missing as a result of another hidden process. Our framework adopts the classic Heckman model for bias correction and the Lagrange duality to achieve fairness in regression based on a variety of fairness notions. Heckman model describes the sample selection process and uses a derived variable called the Inverse Mills Ratio (IMR) to correct sample selection bias. We use fairness inequality and equality constraints to describe a variety of fairness notions and apply the Lagrange duality theory to transform the primal problem into the dual convex optimization. For the two popular fairness notions, mean difference and mean squared error difference, we derive explicit formulas without iterative optimization, and for Pearson correlation, we derive its conditions of achieving strong duality. We conduct experiments on three real-world datasets and the experimental results demonstrate the approach’s effectiveness in terms of both utility and fairness metrics.
Indicator and Stratification Methods for Missing Explanatory Variables in Multiple Linear Regression: Journal of the American Statistical Association: Vol 91, No 433
## Abstract
The statistical literature and folklore contain many methods for handling missing explanatory variable data in multiple linear regression. One such approach is to incorporate into the regression model an indicator variable for whether an explanatory variable is observed. Another approach is to stratify the model based on the range of values for an explanatory variable, with a separate stratum for those individuals in which the explanatory variable is missing. For a least squares regression analysis using either of these two missing-data approaches, the exact biases of the estimators for the regression coefficients and the residual variance are derived and reported. The complete-case analysis, in which individuals with any missing data are omitted, is also investigated theoretically and is found to be free of bias in many situations, though often wasteful of information. A numerical evaluation of the bias of two missing-indicator methods and the complete-case analysis is reported. The missing-indicator met
Sample sizes in cross-country growth regressions vary greatly, depending on data availability. But if the selected samples are not representative of the underlying population of nations in the world, ordinary least squares coefficients (OLS) may be biased. This paper re-examines the determinants of economic growth in cross-sectional samples of countries utilizing econometric techniques that take into account the selective nature of the samples. The regression results of three major contributions to the empirical growth literature by Mankiw-Romer-Weil (1992), Barro (1991) and Mauro (1995), are considered and re-estimated using a bivariate selectivity model. Our analysis suggests that sample selection bias could significantly change the results of empirical growth analysis, depending on the specific sample utilized. In the case of the Mankiw-Romer-Weil paper, the value and statistical significance of some of the estimated coefficients change drastically when adjusted for sample selectivity. But the results obtained by Barro and Mauro are robust to sample selection bias.
When estimating a population parameter by a nonprobability sample, that is, a sample without a known sampling mechanism, the estimate may suffer from sample selection bias. To correct selection bias, one of the often-used methods is assigning a set of unit weights to the nonprobability sample, and estimating the target parameter by a weighted sum. Such weights are often obtained with classification methods. However, a tailor-made framework to evaluate the quality of the assigned weights is missing in the literature, and the evaluation framework for prediction may not be suitable for population parameter estimation by weighting. We try to fill in the gap by discussing several promising performance measures, which are inspired by classical calibration and measures of selection bias. In this paper, we assume that the population parameter of interest is the population mean of a target variable. A simulation study and real data examples show that some performance measures have a strong positive relationship with the mean squared error and/or error of the estimated population mean. These performance measures may be helpful for model selection when constructing weights by logistic regression or machine learning algorithms.
for inference about relationships of scientific interest based on samples without well-defined probability sampling mechanisms. Motivated by the potential for selection bias in: (a) estimated relationships of polygenic scores (PGSs) with phenotypes in genetic studies of volunteers and (b) estimated differences in subgroup means in surveys of smartphone users, we derive novel measures of selection bias for estimates of the coefficients in linear and probit regression models fitted to nonprobability samples, when aggregate-level auxiliary data are available for the selected sample and the target population. The measures arise from normal pattern-mixture models that allow analysts to examine the sensitivity of their inferences to assumptions about nonignorable selection in these samples. We examine the effectiveness of the proposed measures in a simulation study and then use them to quantify the selection bias in: (a) estimated PGS-phenotype relationships in a large study of volunteers recruited via Facebook and (b) estimated subgroup differences in mean past-year employment duration in a nonprobability sample of low-educated smartphone users. We evaluate the performance of the measures in these applications using benchmark estimates from large probability samples.
Keywords: Linear regression, probit regression, nonprobability samples, selection bias, polygenic scores, National Survey of Family Growth 1. Introduction.
The random selection of elements from a finite population of interest into a probability sample, where all population elements have a known nonzero probability of selection, ensures that elements included in the sample, appropriately weighted if necessary, mirror the target population in expectation. That is, for all variables of interest the mechanism of selection of a subset of elements into the sample is ignorable, following the theoretical framework for missing data mechanisms originally introduced by Rubin (1976) .
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