Generalization of degrees of freedom for t distribution for coefficients after multiple imputation
Generalization of degrees of freedom for t distribution for coefficients after multiple imputation
Donald Rubin has shown that regression coefficient estimates have fatter tails after multiple imputation and has provided a formula for the degrees of freedom to use as a t-distribution approximation to the coefficient estimates resulting from Rubin's rule for combining multiple imputations. I would like a generalization of this approach that handles more than single degree of freedom tests, e.g., an adjustment to the multiple degree of freedom
Ultimately it would be nice to have a way to convert covariance matrices arising from Rubin's rule so that normal and
Follow-up
As I've gotten more into Bayesian modeling I came to know about posterior stacking for incorporating multiple imputation into Bayesian analysis using MCMC posterior sampling. This makes the tails of the posterior distribution properly heavier, automatically, and avoids complexities of the approximate Bayesian method of multiple inputation requiring the use of Rubin's rule. An example is in the RMS course notes.
K2-Think
To address the generalization of degrees of freedom adjustments for chunk tests (multivariate hypothesis tests) after multiple imputation (MI), we build on Rubin's original framework and extensions proposed in the statistical literature. Here's a structured approach:
1. Rubin's Framework for Single Parameters
Rubin's rules combine estimates from ( m ) imputed datasets. For a single coef...
GPT-5
Yes. The scalar t-approximation that Rubin derived generalizes to “chunk” (multi-df) tests via an F-approximation, with a denominator degrees-of-freedom (df) that depends on the between- and within-imputation variability for the joint hypothesis. This is standard in the multiple-imputation (MI) literature and is implemented in major software (SAS PROC MIANALYZE, Stata mi test, R mice/mitools/Ameli...
Claude Opus 4
This is an excellent question about extending Rubin's degrees of freedom adjustment from univariate to multivariate tests after multiple imputation.
The Challenge
Rubin's original work provided the adjusted degrees of freedom for univariate t-tests:
where
o3 Pro
Yes – the single-degree-of-freedom (“univariate”) t–approximation that Rubin (1987) derived has been generalised to any set of linear hypotheses.
The generalisations were worked out in a series of papers by Xiao-Ling Meng, Don Rubin and collaborators and are implemented in standard MI software (e.g. mice::pool, mitools, SAS PROC MIANALYZE, Stata mi test, …).
Key references
- Meng, X.-L. & Rub...
o3
Yes – the univariate “fraction‐of‐missing‐information’’ argument that leads to the scalar t–distribution can be extended to any linear hypothesis that involves several regression coefficients simultaneously. Exactly the same quantities that are used in Rubin’s combining rules (the within– and between–imputation covariance matrices) are needed, one only has to form them for the linear combination o...
o4 mini
After you do your m imputations and fit the same regression model to each one you end up (via Rubin’s‐rules) with
Q_i = the p×1 vector of estimates from imputation i,
U_i = the p×p “within‐imputation” covariance for Q_i.
You form
Q̄ = (1/m)∑{i=1}^m Q_i
Ū = (1/m)∑{i=1}^m U_i
B = (1/(m−1))∑_{i=1}^m (Q_i−Q̄)(Q_i−Q̄)′
and the total covariance
T = Ū + (1+1/m) B.
Un...
Gemini 2.5 Pro
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 i...
DeepSeek R1
The original question is about extending Rubin's method for combining multiple ...
Claude 3.7 Sonnet
Generalizing Degrees of Freedom After Multiple Imputation
Your question addresses an important nuance in multiple imputation (MI) methodology, particularly when moving from single-parameter inference to multivariate or "chunk" tests. Let me address this systematically.
Background: Rubin's Rules and the t-Distribution
As you noted, Rubin showed that after multiple imputation, the total vari...
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