Multiple imputation and functional methods in the presence of measurement error and missingness in explanatory variables

Abstract

In many applications involving regression analysis, explanatory variables (or covariates) may be imprecisely measured or may contain missing values. Although there exists a vast literature on measurement error modeling to account for errors-in-variables, and on missing data methodology to handle missingness, very few methods have been developed to simultaneously address both. In this paper, we consider likelihood-based multiple imputation to handle missing data, and combine this with two well-known functional measurement error methods: simulation-extrapolation and corrected score. This unified approach has several appealing characteristics: the model fitting procedure is easy to understand and off-the-shelf software can be incorporated into the modeling framework; no calibration data or a validation subset is required in the model fitting procedure; and the missing data component of the proposed approach is likelihood-based which allows standard likelihood machinery. We demonstrate our methods on simulated datasets and apply them to daily ozone pollution measurements in Los Angeles where observed covariates consist of missing data and imprecise measurements. We conclude that the proposed methods substantially reduce bias and mean squared errors in regression coefficients, in comparison to methods that ignore either measurement error or missingness in covariates.

Supplementary material

The following plots, Figs. 8 and 9, show the results of the simulation study in Sect. 3.1 in terms of the magnitute of bias when $n=50$ and $n=1000$. Figures 8, 9, 10 and 11 show the accuracy of comparing methods in parameter estimation when $X \sim N(0,1)$ and $X \sim U(-1,1)$, respectively, for each n. Figures 8 and 9 are consistent with Fig. 1 (see Sect. 3.1) when $n=100$ and $X \sim N(0,1)$. They show that MI-CS and MI-SIMEX outperform the other comparing methods in terms of accuracy as the missingness and $\sigma ^2_u$ are increased. Also, Figs. 8 and 9 are comparable with Fig. 2 in Sect. 3.1 when $n=100$ and $X \sim U(-1,1)$ in a similar way.

Fig. 8

Boxplot of the slope coefficient estimates for different methods when $X \sim N(0,1)$ and $n=50$. Different methods are compared with the original dataset (complete, in mint) based on the accuracy of their estimators as the missing proportion and the error variance increase from the top-left corner to the bottom-right. Note that the true value of the slope coefficient is 1

Fig. 9

Boxplot of the slope coefficient estimates for different methods when $X \sim N(0,1)$ and $n=1000$. Different methods are compared with the original dataset (complete, in mint) based on the accuracy of their estimators as the missing proportion and the error variance increase from the top-left corner to the bottom-right. Note that the true value of the slope coefficient is 1

Fig. 10

Boxplot of the slope coefficient estimates for different methods when $X \sim U(-1,1)$ and $n=50$. Different methods are compared with the original dataset (complete, in mint) based on the accuracy of their estimators as the missing proportion and the error variance increase from the top-left corner to the bottom-right. Note that the true value of the slope coefficient is 1

Fig. 11

Boxplot of the slope coefficient estimates for different methods when $X \sim U(-1,1)$ and $n=1000$. Different methods are compared with the original dataset (complete, in mint) based on the accuracy of their estimators as the missing proportion and the error variance increase from the top-left corner to the bottom-right. Note that the true value of the slope coefficient is 1

The following plots, Figs. 12 and 13, show the results of the simulation study in Sect. 3.1 in terms of MSE when $n=50$ and $n=1000$. Figures 12 and 13 show the predictive performances of the comparing methods when $n=50$ and $n=1000$, respectively. Comparing Fig. 3 (see Sect. 3.1) with these two plots suggests that MI-CS had the best predictive performance as the missingness and $\sigma ^2_u$ increased. MI-SIMEX showed relatively high MSE for $n=50$, however, the predictive performance of MI-CS improved substantially and equaled MI-SIMEX as n increased. The predictive performances of the comparing methods were similar when $X \sim N(0,1)$ and $X \sim U(-1,1)$.

Fig. 12

Mean squared error of the slope coefficient estimates for different methods with $n=50$ when a$X \sim N(0,1)$ and b$X \sim U(-1,1)$. Different methods are compared with the original dataset (complete, in solid mint line) for different values of error variance as the missing proportion increases

Fig. 13

Mean squared error of the slope coefficient estimates for different methods with $n=1000$ when a$X \sim N(0,1)$ and b$X \sim U(-1,1)$. Different methods are compared with the original dataset (complete, in solid mint line) for different values of error variance as the missing proportion increases

The following plot, Fig. 14, shows the results of the Poisson log-linear simulation study in Sect. 3.2 when $X \sim U(-1,1)$. Figure 14 is consistent with Fig. 4 (see Sect. 3.2) for when $X \sim N(0,1)$. It shows that MI-CS and MI-SIMEX outperform the other comparing methods in terms of accuracy as the missingness and $\sigma ^2_u$ increased.

Fig. 14

Boxplot of the slope coefficient estimates for different methods for Poisson log-linear model when $X \sim U(-1,1)$. Different methods are compared with the original dataset (complete, in mint) based on the accuracy of their estimators as the missing proportion and the error variance increase from the top-left corner to the bottom-right. Note that the true value of the slope coefficient is 1