Excellent question. You've correctly identified a key challenge in frequentist inference after multiple imputation (MI) and are asking about the standard, and most accepted, solution.

Yes, a generalization for multi-parameter tests (often called "chunk tests" or omnibus tests) exists, and it's a crucial part of the MI inference framework developed by Rubin and his colleagues. The resulting test is an F-test, which is a natural generalization of the t-test.

Summary of the Single-Parameter Case (Rubin's Rules)

Let's quickly recap the single-parameter case to set the stage. For a single scalar parameter :

1. Point Estimate: (the average of the estimates from imputations).
2. Within-Imputation Variance: (the average of the squared standard errors).
3. Between-Imputation Variance: .
4. Total Variance: .
5. Test Statistic: The statistic for is .
6. Reference Distribution: This statistic is compared to a -distribution with degrees of freedom, where:
   • is the "relative increase in variance due to nonresponse" or an estimate of the fraction of missing information.

The key insight is that the uncertainty from the missing data (captured by ) reduces the effective degrees of freedom from infinity (for a Normal/Z distribution) to a finite number .

Generalization to Multi-Parameter Tests (The F-Test)

The generalization for testing a set of parameters simultaneously (e.g., ) was developed by Li, Raghunathan, and Rubin (1991) and Meng and Rubin (1992). It leads to a multi-parameter Wald test that follows an approximate F-distribution.

Let be a vector of parameters of interest.

1. Pooled Point Estimate: (a vector of the averaged estimates).
2. Pooled Within-Imputation Covariance Matrix: (the average of the covariance matrices from each imputed dataset).
3. Pooled Between-Imputation Covariance Matrix: .
4. Total Covariance Matrix: . This is the same formula as the scalar case, but now with matrices.

The Wald test statistic for the null hypothesis is:

If there were no missing data, would be compared to an distribution, or would be compared to a distribution for large samples. With MI, we use a scaled F-test.

Test Statistic and Reference Distribution:

The test statistic is defined as:

This statistic is compared to an F-distribution , where is the number of parameters being tested and is the generalized degrees of freedom. The formula for is a multivariate extension of the single-parameter case:

• Degrees of Freedom:
• Multivariate Relative Increase in Variance:

Here, is the trace (sum of the diagonal elements) of the matrix product . This trace essentially provides an "average" fraction of missing information across the parameters.

The p-value for the test is then .

Note: There are alternative, more complex formulas for the denominator degrees of freedom that may perform better when is small or the fractions of missing information are highly variable across parameters (e.g., Reiter, 2007). However, the formula above is the most commonly implemented and cited.

Adjusting Covariance Matrices and Confidence Regions

You asked about adjusting the covariance matrix so that standard distributions could be used. The F-test approach is the direct analogue of this. Instead of adjusting the matrix , we adjust the critical value used to form confidence regions.

A confidence region for the vector is the set of all points satisfying:

where is the critical value from the F-distribution. This defines a confidence ellipsoid whose size is appropriately "inflated" (compared to one based on a critical value) to account for the missing data uncertainty, thereby achieving better coverage probability. This is the multi-parameter version of "bumping up the standard error."

Practical Implementation

This procedure is standard in major statistical software:

• R: The mice package is the cornerstone.
  • To compare nested models, use pool.compare() or the D1() function (which implements this Wald test).
  • For example, to test if x2 and x3 are jointly zero:

    library(mice)
    imp <- mice(nhanes, m = 20, seed = 123)
    fit_full <- with(imp, lm(bmi ~ age + hyp + chl))
    fit_reduced <- with(imp, lm(bmi ~ age))
    
    # D1() performs the multivariate Wald test on the difference in coefficients
    # It correctly calculates the F-statistic and degrees of freedom
    D1(fit_full, fit_reduced) 
    

• Stata: The mi estimate prefix automatically handles this. After running an MI estimation, the test command will correctly use the F-test with the adjusted degrees of freedom.

  mi estimate: reg y x1 x2 x3
  test x1 x2 // This automatically uses the correct F-test
  

• SAS: PROC MIANALYZE with the MULT statement performs this multivariate test.

Your Follow-up on Bayesian Methods

Your follow-up is spot on. The complexities of Rubin's rules (which are based on a frequentist approximation to a Bayesian posterior) are elegantly resolved by a fully Bayesian approach.

When you use MCMC to fit a model, you can treat the missing data as just another set of unknown parameters to be estimated. At each step of the MCMC sampler, you draw a value for the model parameters and a value for each missing data point.

By marginalizing (i.e., integrating out) the imputed data values across all posterior draws, you get the posterior distribution for your parameters of interest. This posterior automatically and correctly incorporates the uncertainty from the missing data. The tails of the posterior distributions will naturally be heavier, reflecting this extra uncertainty, without any need for special t or F distribution adjustments.

"Posterior stacking" is a pragmatic way to combine results if you run MCMC on each of the completed datasets separately (as described in your link to the RMS course notes). It's an excellent and robust method that bridges the gap between traditional MI and a fully integrated Bayesian analysis.

In summary, the path you're on is correct:

1. For frequentist analysis, generalize the t-test to the multi-parameter F-test using the degrees of freedom from Li, Raghunathan, and Rubin.
2. For a more integrated and often theoretically cleaner solution, use a fully Bayesian MCMC model that samples both parameters and missing data, or use posterior stacking/pooling as a powerful alternative.

References

1. Li, K. H., Raghunathan, T. E., & Rubin, D. B. (1991). Large-Sample Significance Levels from Multiply Imputed Data Using Moment-Based Statistics and an F Reference Distribution. Journal of the American Statistical Association, 86(416), 1065–1073.
2. Meng, X. L., & Rubin, D. B. (1992). Performing Likelihood Ratio Tests with Multiply-Imputed Data Sets. Biometrika, 79(1), 103–111. (This paper proposes an alternative, the D3 statistic, based on likelihood ratios).
3. Reiter, J. P. (2007). Small-sample degrees of freedom for multi-component significance tests with multiple imputation. Statistical Neerlandica, 61(4), 502-514.