Developing and evaluating methods to impute race/ethnicity in an incomplete dataset

Abstract

The availability of race data is essential for identifying and addressing racial/ethnic disparities in the health care system; however, patient self-reported racial/ethnic information is often missing. Indirect methods for estimating race have been developed, but they usually only consider geocoded and surname data as predictors, may perform poorly among racial minorities, they do not adjust for possible errors for specific datasets, and are unable to provide race estimates for subjects missing some of this information. The objective of this study was to address these limitations by developing novel methods for imputing race/ethnicity when this information is partially missing. By viewing the unobserved race as missing data, we explored different multiple imputation methods for imputing race/ethnicity, and we applied these methods to a subset of Rhode Island Medicaid beneficiaries. Current race imputation methods and newly developed ones were compared using area under the ROC curve statistics and racial composition estimates to identify methods and sets of predictors that yield superior race imputations. Family race was identified as an important predictor and should be included in race estimation models when possible. Bayesian regression models (BRM) provide better race estimates than previously proposed methods. Missing race was multiply imputed using joint modeling and fully conditional specification. Post-imputation analyses showed that fully conditional specification with a BRM is superior to joint modeling for race imputation. The proposed fully conditional specification method is a flexible, effective way of estimating race/ethnicity that allows for propagation of imputation error and ease of interpretation in further analyses.

Appendices

Appendix 1

Pseudocode for multiple imputation with the FCS algorithm is presented below.

• 1. Populate incomplete dataset with initial starting values

• 2. Draw $\theta_{R}$ from $P(\theta_{R} |R^{obs} ,Y_{ - R} ,M,A)$

• 3. Draw $R^{miss}$ from $P(R|R^{obs} ,Y_{ - R} ,M,A,\theta_{R} )$ and substitute this into the dataset

• 4. Draw $\theta_{{G^{\prime}}}$ from $P(\theta_{{G^{\prime}}} |G^{'obs} ,Y_{{ - G^{\prime}}} ,M,A)$

• 5. Draw $G^{'miss}$ from $P(G^{\prime}|G^{'obs} ,Y_{{ - G^{\prime}}} ,M,A,\theta_{{G^{\prime}}} )$ and substitute this into the dataset

• 6. Draw $\theta_{{S^{\prime}}}$ from $P(\theta_{{S^{\prime}}} |S^{'obs} ,Y_{{ - S^{\prime}}} ,M,A)$

• 7. Draw $S^{'miss}$ from $P(S^{\prime}|S^{'obs} ,Y_{{ - S^{\prime}}} ,M,A,\theta_{{S^{\prime}}} )$ and substitute this into the dataset

• 8. Repeat Steps 2–7 until the cycle reaches convergence. The current draws are the set of imputed values.

• 9. Repeat Steps 1–8 $m$ times to obtain $m$ imputed datasets

Appendix 2

To determine the specification for $P({\mathbf{R}}|{\mathbf{Y}}_{{ - {\mathbf{R}}}} ,{\mathbf{M}},{\mathbf{A}},{\varvec{\uptheta}}_{{\mathbf{R}}} )$ and to compare various multiple imputation methods, the observed race for $n_{e}$ of the $n$ = 6087 individuals with fully observed data was set to missing. The remaining set of $n_{f}$ individuals were used to obtain parameter estimates for the BRMs. Using the final sample of parameter estimates, probabilities of individual $i$ belonging to race $r \in \varUpsilon$ for each of the $n_{e}$ individuals, given by $p_{ir} = \frac{{exp({\mathbf{x}}_{{\mathbf{i}}}^{{\mathbf{T}}} {\varvec{\upbeta}}_{{\mathbf{r}}} )}}{{\sum\nolimits_{r \in \varUpsilon } {(exp({\mathbf{x}}_{{\mathbf{i}}}^{{\mathbf{T}}} {\varvec{\upbeta}}_{{\mathbf{r}}} ))} }}$, were computed. The probabilities ${\mathbf{p}}_{{\mathbf{i}}} = (p_{iWhite} ,p_{iBlack} ,p_{iAI} ,p_{iHispanic} ,p_{iAPI} )$ for $i \in \{ 1,2, \ldots ,n_{e} \}$ were compared to the observed races for these individuals using AUC and racial composition.

Individuals in the testing set were those whose race has been set equal to missing. Thus, when determining which set each of these $n$ individuals will be placed in, it is important to specify whether race is missing completely at random (MCAR), missing at random (MAR) or missing not at random (MNAR). We considered all three in our analysis; implementation details for each are available below.

2.1 MCAR

The race data is MCAR if the missingness of race is unrelated to any of the study variables; that is, the pattern of race missingness is independent of observable variables (Little and Rubin 2002). In practice, this is a very strong assumption. We implement this assumption by randomly selecting $n_{e}$ individuals from the group of $n$. We set $n_{e}$ = 2891 and $n_{f}$ = 3196 such that the percentage of individuals with missing race is 47.5% because this is the percentage of missing race in the full RI dataset.

2.2 MAR

The race data is MAR if the missingness can be explained by variables for which there is complete information. Using the individuals in rows V and VI of Fig. 1, we fit a logistic regression model where the dependent variable is an indicator for whether race is missing and the independent variables are the fully observed geocoded and surname probabilities, family race indicators, language, and age. Using the estimated parameters for this model (intercept, geocoded, surname, family race indicators, language, and age in Table 8), we estimate the probability that race is missing for each of the $n$ individuals with fully observed data. These probabilities are then used to determine whether an individual belongs in the training set or the testing set. In our analysis, $n_{f}$ = 3195 while $n_{e}$ = 2892; hence, race is set to missing for 47.5% of the $n$ beneficiaries considered for this simulation.

Table 8 Parameter values used to simulate MAR

2.3 MNAR

The race data is MNAR when the missing entries for race depends on the racial groups, even after controlling for other variables with complete information. To simulate MNAR, we fit the same logistic regression model described in the MAR section and also incorporate parameters for the observed race (Table 9). Using this combined set of parameters, we compute the probability that race is missing for each of the $n$ beneficiaries with completely observed data. Similar to before, these are used to determine whether an individual will be placed in the training set or the testing set. In our analysis, $n_{f}$ = 3197 while $n_{e}$ = 2890; the percentage of individuals whose race is set to missing is 47.5%.

Table 9 Parameter values used to simulate MNAR

Note: The intercepts reported in Tables 8 and 9 were not the intercepts estimated from the logistic regression model. Rather, these were modified so that the percentage of individuals in the test was 47.5% across all missing data mechanisms.