Table 1:
Summary of some existing and proposed imputation and data analysis strategies. Proposed methods highlighted in gray.
| Standard MICE | Bartlett et al. (2014) | Stacked, 1/M weighted | Stacked, f(Y|X) weighted | |
|---|---|---|---|---|
| Covariate Imputation | f(Xp|X−p, Y), specified as regression model | f(Xp|X−p, Y)∝ f(Y|X)f(Xp|X−p), where f(Xp|X−p) is a regression model | Often, same as MICE. Could also apply other imputation methods. | f(Xp|X−p), specified as regression model |
| Point Estimation | Fit model to each imputed dataset separately | Fit model to each imputed dataset separately | Fit single weighted model to stacked imputations.* | Fit single weighted model to stacked imputations. Weights ∝f(Y|X) |
| Standard Errors | Rubin’s rules | Rubin’s rules | Previously, unclear how to estimate.** We propose new approach in Eq. 3. | We propose new approach in Eq. 3. |
| Comments | ↳ Easy to implement ↳ Tricky to specify imputation regressions |
↳ Limited outcome models supported by current software ↳ Easy to implement for supported models ↳ Outcome model built into imputation |
↳ Inherits properties of imputation approach chosen ↳ Different data analysis ↳ Proposed new standard errors |
↳ Imputation ignores Y. Easy to implement. ↳ Imputation and analysis separated. Easy to compare outcome models. |
| R Packages | mice | smcfcs | mice, StackImpute † | mice, StackImpute † |
Tall stack corresponds to stack of M imputed datasets, with complete cases listed M times. All rows given weight 1/M. Short stack corresponds to stack with complete cases listed only once. Imputed rows given weight 1/M and complete cases given weight 1.
Sandwich estimator applied to weighted, stacked data known to under-estimate standard errors. Wood et al. (2008) proposed largely untested ad hoc correction method for stacked analysis standard errors. Bootstrap methods for estimating standard errors are computationally expensive.
R package for estimating standard errors using Eq. 3. Development version available at https://github.com/lbeesleyBIOSTAT/StackImpute. Can be implemented for additional outcome models using custom software. See Web Appendix Section 3 for details.