An Approach to Addressing Multiple Imputation Model Uncertainty Using Bayesian Model Averaging

Abstract

This paper considers the problem of imputation model uncertainty in the context of missing data problems. We argue that so-called “Bayesianly proper” approaches to multiple imputation, although correctly accounting for uncertainty in imputation model parameters, ignore the uncertainty in the imputation model itself. We address imputation model uncertainty by implementing Bayesian model averaging as part of the imputation process. Bayesian model averaging accounts for both model and parameter uncertainty, and thus we argue is fully Bayesianly proper. We apply Bayesian model averaging to multiple imputation under the fully conditional specification approach. An extensive simulation study is conducted comparing our Bayesian model averaging approach against normal theory-based Bayesian imputation not accounting for model uncertainty. Across almost all conditions of the simulation study, the results reveal the extent of model uncertainty in multiple imputation and a consistent advantage to our Bayesian model averaging approach over normal-theory multiple imputation under missing-at-random and missing-completely-at random in terms of Kullback-Liebler divergence and mean squared prediction error. A small case study is also presented. Directions for future research are discussed.

Keywords:

• Missing data
• Bayesian model averaging
• Multiple imputation

Notes

1 Note: footnote ours. This paper was eventually published as Rubin (1996).

2 Note that the unit information prior is equivalent to Zellner’s g-prior (Zellner et al. 1986), where g = 1/N…, and where N is the sample size. See also Fernández et al. (2001).

3 The Laplace method of integrals is based on a Taylor expansion of a function f(u) of a q-dimensional vector u. The approximation is $\int e^{f(u)}du\simeq 2{(\pi )}^{q/2}|A{|}^{1/2}\hspace{0.17em}\text{exp}\hspace{0.17em}\{f(u^{*})\}$, where $u^{*}$ is the value of u at which f attains its maximum, and A is minus the inverse of the Hessian of f evaluated at $u^{*}$. Following Raftery (1996, pg. 253), when the Laplace method is applied to equation (5), we obtain the approximation $p(y|M_k)\simeq {(2\pi )}^{q_k}|A_k{|}^{1/2}p(y|{\tilde{\theta }}_k,M_k)p({\tilde{\theta }}_k,M_k)$, where q_k is the dimension of θ_k, ${\tilde{\theta }}_k$ is the posterior mode of θ_k, and A_k is minus the inverse of the Hessian of $\text{log}\hspace{0.17em}\{p(y|{\theta }_k,M_k)p({\theta }_k|M_k)\}$, evaluated at the posterior mode ${\tilde{\theta }}_k.$

4 Note that the “mice” program is flexible enough to allow different imputation methods to be chosen for different scales of variables. We address this issue and its implications for missing data with Bayesian model averaging in the Discussion section.

5 We thank an anonymous reviewer for bringing this paper to our attention.

6 The default number of models in the “BMA” package is 150.

7 In the interest of space, we only show the results for MCAR1, MCAR3, and MCAR5. The results for MCAR2 and MCAR4 results are very similar under all conditions.

8 We recognize that when using plausible values in the analysis of large-scale educational assessments, it is more appropriate to use all plausible values and combine them using Rubin’s (1987) rules. However, extending our miBMA approach to the plausible value framework is beyond the scope of this paper.