I have a set of generalized linear models fit to 5 multiply imputed datasets. I am interested in testing the statistical significance of a set of predictors, coded from dummy variables.

Rubin's Rules makes conducting 1 degree of freedom Wald tests very easy, since the general form of the test statistic $\hat{\beta}/SE_{RR}(\hat{\beta})$ is available from basic arithmetic provided the model summaries for each multiply imputed dataset. I understand this test statistic also has good asymptotic properties, relative to the general Wald test. But I do not understand how to go about 2 degree-of-freedom (or higher) tests. Putting aside the issue of finding higher dimensional Rubin's Rules, is it tractable to just use the likelihoods for each model, and conduct a test by doing:

$$-2 \sum_{s=1}^5 \sum_{i=1}^n \left( \log \mathcal{L}(X_{s,i}, y_{s,i}, \hat{\beta}_s) - \log \mathcal{L} (X_{s,i}, y_{s,i}, \hat{\beta}_{s,0}) \right) \sim_d \chi^2_k$$

Where $\mathcal{H}_0: \beta = [\beta_1, \beta_2, \ldots, \beta_{p-k-1}, 0, 0, \ldots, 0]$ is a $p$-length vector having the last $k \le p$ entries equal to 0


Apparently, the mice package has an option pool.compare. The help file cites the following publication by Meng, Rubin "Performing Likelihood Ratio Tests with Multiply-Imputed Data Sets" in Biometrika 1992. I cannot access this with my limited research library. However, looking at the source code for pool.compare suggests that the implementation is a very clever trick indeed.

The outcome, of any scale, is compared to a binomial likelihood by creating $Z = (Y - \min(Y))/\text{range}(Y)$ and taking a logistic inverse transformation of the linear predictors $E[Z|X] = \mbox{logit}^{-1} (\mathbf{X}\hat{\beta})$. Two log likelihood ratios are defined for each dataset, the $D_L$ which uses the pooled coefficient estimate, $\bar{Q}$ and the $D_M$ which uses the coefficient estimates from each dataset. The $D_L$ and $D_M$ are averaged together, and some algebra is done to create the test-statistic. Namely, $r_m$ is defined as

$$r_m = \frac{m+1}{k (m-1)} (D_M - D_L)$$

$$T = D_L/(k (1+r_m))$$

which has a $\chi^2_{df}$ distribution with $df = 4(mk-4)(1+(1-2/(mk))^2)$ if $mk>4$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.