Why the Outcome Model Matters
When a regression outcome is missing, the imputation model must describe how that outcome varies with the covariates used in the analysis. Imputing an outcome from its marginal distribution alone discards those relationships. For example, if the analysis estimates how treatment and baseline severity relate to a continuous follow-up score, the imputed scores should be generated conditional on treatment and baseline severity, not from the overall average score.
The key distinction is which variable is being imputed. When a predictor is missing, its imputation model should account for the outcome relationship needed by the analysis. When the outcome is missing, the natural model instead specifies the outcome conditional on the analysis covariates. The conditional model should represent the analysis relationship, while also respecting the outcome's scale and support.
Let \(R_i=1\) when person \(i\)'s outcome is observed and \(R_i=0\) when it is missing. If the covariates are complete, an outcome imputation model is fitted using the cases with \(R_i=1\), and the fitted conditional distribution is used to generate values for cases with \(R_i=0\). If predictors are also missing, their current imputed values can be used when updating the outcome model within MICE; predictor-imputation steps must still account for the outcome as required by the analysis. The model used at each update therefore depends on which variable is being imputed.
Assumptions and the Main Principle
A defensible outcome imputation model requires more than choosing a familiar regression family. The observed outcomes must provide information about the conditional relationship in the covariate patterns where outcomes are missing. This requires adequate overlap: the cases with observed outcomes should cover the relevant range and combinations of analysis covariates. The conditional model must also be correctly specified well enough for the intended analysis, including its outcome distribution, link function, covariates, and important nonlinear terms or interactions when those are part of the analysis.
For likelihood-based inference, a common justification for fitting the outcome model to observed outcomes is that the missingness mechanism is ignorable: conditional on the observed information used in the model, whether the outcome is missing does not depend further on its unobserved value, and the missingness and outcome-model parameters are distinct. This is the usual MAR setting. If the probability of a missing outcome depends on its unseen value even after conditioning on the observed information, the observed outcomes alone do not identify that dependence. An MNAR analysis then requires additional assumptions and sensitivity analysis.
The uncertainty requirement is important. A fitted mean or fitted probability is not, by itself, a proper imputation. For a continuous outcome, an imputation includes residual variation as well as uncertainty about regression parameters. For a categorical outcome, an imputation is a random category drawn using the conditional category probabilities, not simply the most likely category. Repeating this process across multiple completed data sets represents uncertainty that a single filled-in data set would hide.
Continuous Outcomes: Conditional Regression Imputation
For a continuous outcome in a linear regression analysis, a common model is \(Y_i\mid X_i\sim N(\beta_0+X_i^\mathsf{T}\beta,\sigma^2)\), where \(X_i\) contains the analysis covariates, \(\beta_0\) is the intercept, \(\beta\) is the vector of regression coefficients, and \(\sigma\) is the residual standard deviation. If the analysis uses categorical predictors, they should be represented by the corresponding indicator variables. If the analysis includes a nonlinear term, that term must be represented consistently in the imputation model; later material treats such specifications in more detail.
A proper draw first reflects uncertainty in \(\beta_0\), \(\beta\), and \(\sigma\), and then generates an outcome around the resulting conditional mean. In a Bayesian implementation, this can be expressed as drawing parameters from their posterior distribution and then drawing a new outcome conditional on those parameters. Other valid implementations can use suitable large-sample or bootstrap methods to represent parameter uncertainty. Simply substituting the fitted coefficients and imputing the fitted mean for everyone omits both sources of uncertainty.
Worked Example: Imputing a Missing Continuous Outcome
Suppose a hypothetical study measures a continuous follow-up score \(Y\) for participants with treatment indicator \(T\) and baseline score \(B\). The planned analysis is an additive linear regression of follow-up score on treatment and baseline score. In the observed cases, the fitted model is \(Y_i=48+5T_i+0.4B_i+\varepsilon_i\). For one participant with a missing follow-up score, \(T_i=1\) and \(B_i=60\). Suppose a parameter draw for one imputation gives coefficients \(50\), \(4\), and \(0.35\), and \(\sigma=5\).
The conditional mean under this parameter draw is \(50+4(1)+0.35(60)=50+4+21=75\) score points. Suppose the predictive residual draw for this imputation is \(z=0.4\) standard deviations. The imputed outcome is \(75+0.4(5)=77\) score points.
This is one plausible draw, not a claim that the participant's outcome is known to be 77. Another imputation can draw different parameter values and a different residual, producing another plausible score. Repeating the procedure creates completed data sets that preserve the modeled relationship with treatment and baseline score while reflecting uncertainty.
The normal model is not automatically suitable for every continuous outcome. Strong skewness, bounds, or heteroscedasticity may make its predictive draws implausible. Diagnostics should compare observed and imputed outcomes on their original scale and, crucially, examine whether their conditional distributions across covariate patterns are plausible. A model that produces reasonable overall means can still misrepresent the regression relationship.
Binary and Ordinal Outcomes
For a binary outcome, use a conditional model that generates only the two permitted values. For example, with \(Y_i\in\{0,1\}\), logistic regression specifies \(\operatorname{logit}\{P(Y_i=1\mid X_i)\}=\beta_0+X_i^\mathsf{T}\beta\), where \(\operatorname{logit}(p)=\log\{p/(1-p)\}\). After drawing the model parameters, calculate the conditional probability and draw \(Y_i\) as a Bernoulli random variable with that probability. This preserves the outcome support and represents the modeled regression association.
Worked Example: Imputing a Missing Binary Outcome
Consider a hypothetical binary outcome \(Y\), where 1 indicates an event and 0 indicates no event. The analysis is a logistic regression on treatment \(T\) and baseline risk score \(B\). Suppose one parameter draw gives \(\operatorname{logit}\{P(Y=1\mid T,B)\}=-2.4+0.8T+0.03B\). A participant with a missing outcome has \(T=1\) and \(B=20\).
The linear predictor is \(-2.4+0.8(1)+0.03(20)=-2.4+0.8+0.6=-1.0\). Thus the event probability is \(1/(1+\exp(1.0))=0.2689\), to four significant figures. Suppose the random-number draw used for this imputation is \(U=0.20\), where \(U\) is uniform between 0 and 1. Because \(0.20<0.2689\), draw \(Y=1\). If \(U\) had been at least \(0.2689\), the draw would have been \(Y=0\).
The probability is conditional on the participant's treatment and risk score. Imputing the overall event proportion instead would ignore those predictors and could weaken or distort the relationship estimated by the logistic regression.
For an ordinal outcome, a proportional-odds model is one option when its assumptions are reasonable. For three ordered categories, one common parameterization is \(\operatorname{logit}\{P(Y_i\leq k\mid X_i)\}=\theta_k-X_i^\mathsf{T}\beta\), for cutpoints \(k=1,2\) with \(\theta_1<\theta_2\). The model assumes the same slope vector across cumulative cutpoints. Imputed categories are drawn from the implied category probabilities, not assigned using only a cumulative probability or a modal category.
Worked Example: Imputing a Missing Ordinal Outcome
Suppose a hypothetical three-category ordinal outcome is analyzed using the proportional-odds model \(\operatorname{logit}\{P(Y\leq k\mid X)\}=\theta_k-\beta X\). For one parameter draw, let \(\theta_1=-1\), \(\theta_2=1\), and \(\beta=0.7\). A participant with missing outcome has \(X=1\).
The cumulative probabilities are \(P(Y\leq1\mid X=1)=1/\{1+\exp(1.7)\}=0.1545\) and \(P(Y\leq2\mid X=1)=1/\{1+\exp(-0.3)\}=0.5744\). The category probabilities are therefore \(0.1545\) for category 1, \(0.5744-0.1545=0.4199\) for category 2, and \(1-0.5744=0.4256\) for category 3. They sum to 1.0000, up to rounding.
Using a uniform draw \(U=0.50\), assign category 1 if \(U\leq0.1545\), category 2 if \(0.1545<U\leq0.5744\), and category 3 otherwise. Here \(0.1545<0.50\leq0.5744\), so the imputed category is 2. This draw respects the ordering and preserves the proportional-odds relationship specified by the model.
A Practical Workflow and How to Diagnose Problems
The model choice should follow the analysis, rather than being selected only for computational convenience. The following sequence makes the essential checks explicit.
Identify the outcome scale, analysis covariates, regression family, and functional form that define the target relationship.
Use a conditional distribution appropriate to the outcome and include the analysis covariates in a representation consistent with the analysis model.
Represent parameter uncertainty, then draw each missing outcome from its individual conditional distribution, preserving its possible values.
Check plausible ranges and conditional patterns, look for fitting or convergence problems, and verify that the completed-data analysis uses the prespecified model.
Useful diagnostics include comparing observed and imputed outcome distributions, stratifying those comparisons by important analysis covariates, and checking whether imputed values are plausible for the covariate patterns where they occur. For a binary outcome, check fitted probabilities for values near zero or one and inspect sparse groups or separation. For an ordinal outcome, check that category probabilities are valid and that the proportional-odds assumption is tenable if that model is used. For a continuous outcome, look for impossible values, severe departures from the assumed residual behavior, and poor fit across the covariate range.
A lack of observed outcomes in covariate regions where outcomes are missing is a particularly serious limitation: the conditional relationship there is determined mainly by extrapolation. Other warning signs include unstable coefficients, rare categories, convergence failures, and very different observed and imputed conditional patterns. More imputations do not repair a misspecified conditional model, lack of overlap, or a missingness process that is not ignorable.
Alternatives and Common Pitfalls
Complete-case analysis uses only records with observed outcomes (and any other required analysis variables observed). It can be appropriate under suitable assumptions, but it discards incomplete records and may change the target relationship or reduce precision. A conditional outcome imputation model can retain those records under its assumptions. Mean imputation for a continuous outcome, or assigning the most likely category for a categorical outcome, is not a sound substitute: it suppresses variation and understates uncertainty. Drawing from an unconditional outcome distribution is also inadequate when the outcome is related to analysis covariates.
| Approach | What it does | Main concern |
|---|---|---|
| Conditional outcome imputation | Draws missing outcomes given analysis covariates | Depends on a suitable model, adequate overlap, and an appropriate missingness assumption |
| Complete-case analysis | Analyzes records with observed outcomes | May lose precision or be biased, depending on the missingness and analysis conditions |
| Mean or modal substitution | Replaces a missing value with a single summary | Understates variation and fails to represent imputation uncertainty |
| Unconditional outcome imputation | Draws outcomes without conditioning on the analysis covariates | Can fail to preserve the outcome–covariate relationship |
The central limitation is that imputation cannot recover information unsupported by the observed data and assumptions. A plausible model does not prove MAR, and good-looking marginal diagnostics cannot establish that the missing outcomes follow the modeled conditional distribution. Where departures from the assumptions are scientifically plausible, sensitivity analyses should be considered rather than treating a single imputation model as definitive.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A linear regression analysis relates a continuous outcome to treatment and baseline score. What should the outcome imputation model condition on, and what two sources of uncertainty should its draws represent?
- For a binary outcome, a participant's imputed event probability is 0.31 and the uniform draw is 0.72. What outcome is drawn?
- An ordinal model gives cumulative probabilities \(P(Y\leq1)=0.20\) and \(P(Y\leq2)=0.65\). What are the three category probabilities, and which category is drawn for \(U=0.50\)?
- Why can mean substitution distort a regression analysis even if the substituted values are close to the average observed outcome?
- What does a lack of observed outcomes among people with a particular covariate pattern imply for conditional outcome imputation?
Solutions. (1) Condition on treatment and baseline score, represented consistently with the analysis model. Draws should reflect uncertainty in the fitted parameters and residual variation in individual outcomes. (2) Draw \(Y=0\), because \(0.72>0.31\); an event is drawn only when \(U<0.31\). (3) The probabilities are \(0.20\), \(0.65-0.20=0.45\), and \(1-0.65=0.35\). Since \(0.20<0.50\leq0.65\), draw category 2. (4) Substitution assigns the same value to cases rather than representing their conditional variation, and it omits uncertainty about the missing values; the completed data can therefore have distorted associations and understated uncertainty. (5) The conditional relationship in that region is weakly informed or unsupported by observed outcomes, so imputations rely on extrapolation and should be treated cautiously.