Missing data

Missing data occur when some intended measurements are unobserved. Whether valid inference can ignore the missingness mechanism depends on how the probability of observation relates to observed and unobserved values.

Item nonresponse leaves a particular measurement unobserved while retaining other measurements on that individual. Unit nonresponse leaves all survey measurements unavailable. Adjusting for an observed auxiliary variable can address item nonresponse under missing at random, but does not identify the auxiliary-variable distribution among unit nonresponders. Extrapolation to them needs further assumptions or external information.

Vary the unobserved outcome probabilities for item and unit nonresponders, and any unknown auxiliary-variable mixture, over scientifically plausible ranges. Recompute the population estimate and identify values at which substantive conclusions change. This probes untestable transport or missing at random assumptions; it is distinct from sampling uncertainty represented by a confidence interval. Letting each missing-outcome probability vary from zero to one gives worst-case identification bounds.

The mechanism assigns probabilities to observation patterns $R$ conditional on complete measurements $D$: $p(R\mid D)$. Whether it depends on missing values distinguishes missing completely at random, missing at random and missing not at random.

Missingness depends only on fully observed covariates $X$, not on the response after conditioning on them. A responder-only marginal estimate may still be biased when response rates and outcome probabilities both vary across covariate strata. MAR standardization over a fully observed covariate weights within-stratum respondent estimates using covariate frequencies from the whole target cohort.

Suppose outcome collection stops at the first success or after $m$ unsuccessful attempts, and let $A$ count unsuccessful attempts. Conditional on the outcome $y$, put $q_j(y)=\mathbb{P}(A\ge j\mid A\ge j-1,Y=y)$. For $k<m$, the joint density of $A=k$ and an observed $Y=y$ isFor $A=m$, the unobserved outcome is integrated out, giving $\int f_{\beta }(y){\prod }_{j=1}^m{q}_j(y)dy$. These probabilities sum to one by telescoping and integration. Dependence of $q_j$ on $y$ models missing not at random. If all $q_j$ are independent of $y$ conditional on the regression predictors, the outcome density among responders remains $f_{\beta }$, as required for complete-case analysis of that conditional model.

Data are missing completely at random when the missingness indicator is independent of both observed and missing data.

With a fully observed covariate $X$, response $Y$, and an independent response indicator with probability $\rho$, let $m(X)=E(Y\mid X)$. A covariate-standardized mean has asymptotic variance $[\mathrm{Var}(m)+E\{\mathrm{Var}(Y\mid X)\}/\rho ]/N$, whereas the complete-case mean has variance $\mathrm{Var}(Y)/(N\rho )$. The difference is $(1-\rho )\mathrm{Var}(m)/(N\rho )$. Thus missing completely at random does not justify discarding predictive observed covariates on efficiency grounds.

Data are missing at random when the probability of missingness, conditional on observed data, does not depend further on the missing values.

For each fixed observation pattern, the probability of that pattern may depend on observed response components and fully observed covariates, but not on the response components missing under that pattern. Conditional missing completely at random is the special case with no dependence on observed response components either.

When a response $Y$ is missing at random conditional on a fully observed category $X$, estimate its conditional probabilities using respondents within each category and average them with the category frequencies from all patients. For a binary response, $\hat{P}(Y=1)={\sum }_x\hat{P}(X=x)\hat{P}(Y=1\mid X=x,R=1)$. The weighting uses the full cohort rather than only complete cases.

A missingness mechanism is ignorable for likelihood or Bayesian inference about outcome-model parameters when the missingness-model factor can be omitted without changing that inference. Missing at random together with distinct outcome and missingness parameters is a standard sufficient condition.

If a regression response $Y$ is observed but covariate $X$ is sometimes missing, likelihood inference requires a joint model: for example $f_{\beta }(y\mid x)g_{\eta }(x)$. An incomplete record contributes $\int f_{\beta }(y\mid x)g_{\eta }(x)dx$ to the observed-data likelihood. Missing at random with distinct parameters allows the missingness mechanism to be omitted; it does not remove the need to integrate over $X$.

Outcome and missingness parameters are distinct when their joint parameter space is the Cartesian product of their separate parameter spaces, so variation-independent choices can be made for the two models.

Data are missing not at random when the probability of missingness still depends on an unobserved value after conditioning on the observed data.

Model the joint distribution by multiplying a response model by a sequential dropout model. For example, the logit of dropout before visit $j$, conditional on remaining observed, can depend on the currently unobserved $Y_{ij}$ as well as previous responses and baseline covariates. The observed-data likelihood sums over the missing binary responses. Such a dependence parameter is usually weakly identified or unidentified from observed responses alone; sensitivity analysis must vary it over scientifically justified values.

A response model and a dropout model share latent random effects. Conditional on those effects and observed predictors, the two processes may be independent, yet marginally dropout is missing not at random because the shared effects also predict unseen outcomes. Joint likelihood fitting integrates the product of both models over the shared effects; integrating the outcome model alone is insufficient. This is an explicit identifying model, and its latent-dependence assumptions should be assessed by sensitivity analysis.

Repeated measurements have a monotone missing-data pattern when every measurement after a subject's first missing value is also missing.

A complete-case analysis uses only observations for which every variable required by the analysis is observed. It can be biased when completeness selects observations according to their outcomes or other relevant variables.

Let $R$ indicate observation of a regression response $Y$, with fully observed predictors $X$. If $Y\perp R\mid X$ and $\mathbb{P}(R=1\mid X)>0$ on the target support, the observed conditional density equals the full conditional density. Thus a correctly specified regression model can be fitted to observed responses under conditional missing at random. This does not justify the unadjusted complete-case mean when observation rates and outcome means both vary with $X$, nor does it imply missing completely at random.

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

A recorded variable included in imputation models to help predict missing values or explain response patterns, even if it is absent from the final substantive model. Useful auxiliaries can improve precision and make missing at random more plausible; they cannot establish that assumption from the observed data.

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.

Use the pooled estimate and total variance from Rubin's rules. A basic large-complete-sample scalar interval is $\overline{Q}\pm t_{\nu ,1-\alpha /2}\sqrt{T}$, with $\nu =(M-1)[1+\overline{U}/((1+M^{-1})B){]}^2$ for $B>0$. Small complete-data samples need an appropriate degrees-of-freedom adjustment; the normal limit applies as the available degrees of freedom become large.

Specify a conditional distribution for each incomplete variable given the others, cycle through parameter draws and missing-value draws, and retain multiple completed datasets after suitable convergence checks. Preserve observed entries. Adequate models, appropriate support and the intended analysis structure matter; arbitrary conditional models need not be compatible with one joint distribution. Standard applications assume missing at random given included predictors. Use Rubin's rules to pool estimates and uncertainty.