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Donald Rubin's work connects Causal inference, missing data and Bayesian statistics. His Harvard profile lists Causal inference in experiments and observational studies, nonresponse in sample surveys, and Bayesian and empirical Bayesian methods as research interests.^[ref]

The potential outcomes framework makes a causal comparison explicit by distinguishing the outcomes that would occur under alternative treatments. Only the outcome under the received treatment is observed for each individual, so identifying an average treatment effect requires design or modelling assumptions. In missing-data analysis, multiple imputation propagates uncertainty by analyzing several completed datasets and pooling their results with Rubin's rules. These ideas explain why filling in one convenient value and then treating it as observed generally understates uncertainty.

Homepage: statistics.fas.harvard.edu/people/donald-b-rubin

John Carlin develops biostatistics for health and medical research, including methods for missing data and causal inference. His institutional profiles describe work on multiple imputation and the need to specify the scientific purpose of an analysis before choosing a regression model.^[ref][ref]

With multiple imputation, several plausible completed datasets represent uncertainty about unobserved values. Their analyses must account both for uncertainty within a completed dataset and for variation between completed datasets. This makes the assumptions about the missing values visible in the analysis. Carlin is a coauthor of Bayesian Data Analysis by Andrew Gelman and others.^[ref]

Missing data are intended measurements that were not observed. Write $R=1$ when an outcome $Y$ is observed and let $X$ denote fully observed covariates. The observation mechanism matters because observed cases can differ systematically from the target population.

Missing completely at random means $R$ is independent of both $Y$ and $X$. Missing at random allows observation to depend on $X$, but requires $R$ and $Y$ to be conditionally independent given $X$. Missing not at random permits additional dependence on the unobserved $Y$. For example, follow-up determined only by recorded age can be missing at random after conditioning on age; follow-up depending on unrecorded current symptoms can be missing not at random. The missing-at-random assumption generally cannot be established from the observed data alone.

An observed-case analysis need not estimate the population mean even under missing at random, because the distribution of $X$ among observed cases can be different. Regression, Inverse probability weighting or Multiple imputation can address this under suitable assumptions, including a positive probability of observation throughout the target population. Stating those assumptions is part of the statistical model rather than a consequence of an imputation algorithm.

Multiple imputation repeatedly draws missing values from fitted predictive distributions, including uncertainty about fitted parameters, analyzes each completed dataset, and combines the resulting estimates and uncertainty.

Choose candidate variables from subject-matter knowledge, measurement quality and the target question before screening on outcomes. For an adjusted-effect analysis, retain prespecified confounders and essential design variables even when their individual $P$ values are large. For prediction, consider reliable predictors available at the intended prediction time; do not include future information. Handle missingness transparently, using a justified multiple imputation strategy where appropriate rather than changing the analyzed population unpredictably as variables enter.

Encode a categorical variable with $K-1$ indicators relative to a stated reference category, and assess it as a group. Check multicollinearity, sparse categories and plausible interactions. The relevant information is chiefly the number and distribution of events, not merely the number of enrolled subjects: hundreds of correlated or rarely varying predictors cannot be supported by a small event count.

Compare prespecified nested models by a likelihood-ratio test based on the partial log-likelihood and the change in parameter dimension. Penalized methods such as ridge regression or Lasso regression applied to the Cox likelihood can stabilize many-variable models; choose tuning by suitable validation and account for mandatory variables. The Akaike information criterion or a limited, documented model-selection procedure can help balance fit and complexity. Unrestricted stepwise searches and repeated univariable screening invite unstable choices, omit joint confounding effects, and make naive post-selection intervals misleading. Prefer a defensible, validated covariate set to a collection chosen only for small $P$ values.

The monthly nurse assessments are auxiliary longitudinal outcomes observed even after clinic dropout. Investigators can include their histories, treatment, and earlier QoL in a multiple imputation model for missing 12-month QoL, or in a joint longitudinal model. Because the assessments predict disability and QoL, conditioning on them reduces residual outcome variance and makes each imputed value more informative, improving precision of the treatment-effect estimate. A prespecified covariate-adjusted or augmented inverse-probability estimator could use the same information.

For $Z=0$, the fitted conditional success probabilities for $W$ areAmong all 250 subjects with $Z=0$, multiple imputation asymptotically assigns $100$ to $Y=0$ and $150$ to $Y=1$. HenceUsing the supplied limit for ${\hat{\psi }}_1$ gives

The distribution of a new or missing quantity integrates its model over the Bayesian posterior: $p(y^{\ast }\mid y)=\int p(y^{\ast }\mid \varphi ,y)p(\varphi \mid y)d\varphi$. Both parameter and residual uncertainty should enter predictive draws used for multiple imputation.

Under proper Multiple imputation with compatible imputation and analysis models, for $M>1$ completed-data estimates $Q_m$ with variances $U_m$, pool using $\overline{Q}=M^{-1}{\sum }_m{Q}_m$, $\overline{U}=M^{-1}{\sum }_m{U}_m$, $B=(M-1{)}^{-1}{\sum }_m(Q_m-\overline{Q}{)}^2$, and $T=\overline{U}+(1+M^{-1})B$. The total variance includes within-imputation uncertainty, between-imputation uncertainty and finite-$M$ simulation uncertainty.