Complete-Case Restrictions With Nonlinear Outcomes

From a Complete-Case Restriction to a Nonlinear Prediction

A restriction on conditional means can answer a question about average outcomes, but many clinical questions concern a nonlinear feature of the outcome distribution: the probability of response, the risk of a severe event, or the expected number of patients exceeding a count threshold. For these targets, transferring only a mean is generally insufficient. The complete-case restriction can instead be stated for the whole conditional distribution, after which the desired probability or other distributional quantity is calculated from that distribution.

Consider a two-visit study. Let $Y_1$ be the first-visit outcome and $Y_2$ the second-visit outcome. Let $R=1$ identify participants with observed outcomes at both visits and $R=0$ identify participants missing $Y_2$. Write $H$ for the observed history used to make the restriction, such as $Y_1$, treatment group, and baseline covariates. The same definition of $H$ must be used for completers and non-completers.

Definition: The complete-case distributional restriction states that, for histories $h$ represented among both completers and non-completers, the conditional distribution of the missing outcome among non-completers equals the conditional distribution of the observed outcome among completers: $$ P(Y_2=y\mid H=h,R=0)=P(Y_2=y\mid H=h,R=1). $$ For continuous outcomes, the equality is understood as equality of conditional distributions (or densities, when densities exist). It is a modeling restriction about outcomes that are unobserved for $R=0$, not a comparison that can be verified directly from those missing outcomes.

The restriction has two essential conditions. First, histories used for prediction must have positivity: there must be completers at the relevant histories, or a defensible conditional model that can predict there. Second, the conditional distribution among completers must be modeled adequately. Even if these conditions hold, the equality across $R$ is not established by the observed data alone; it is an identifying assumption.

Why the Whole Distribution Matters

If the outcome is binary, its conditional distribution is determined by the success probability. If it is a count, a mean alone does not determine probabilities such as $P(Y_2\geq 4\mid H=h)$. More generally, once a distribution is specified, an outcome feature can be calculated as an expectation of a function $g$:

E\{g(Y_2)\mid H=h,R=0\} = \sum_y g(y)P(Y_2=y\mid H=h,R=1),

for discrete outcomes, with an integral replacing the sum for continuous outcomes. This follows directly from the restriction: replace each non-completer conditional probability in the expectation by the corresponding completer probability. For a binary outcome, choosing $g(y)=y$ gives the success probability. For a count and $g(y)=I(y\geq k)$, it gives the probability of reaching a threshold $k$. These are distributional predictions, not claims that a particular missing value has been recovered.

Worked Example: Predicting Binary Responses by History

Hypothetical data. In a clinical study, define response at visit 2 as $Y_2=1$, and no response as $Y_2=0$. Let $H$ be a two-level observed baseline-history category. Among completers with $H=0$, 6 of 30 responded; among completers with $H=1$, 12 of 20 responded. There are 10 non-completers with $H=0$ and 5 with $H=1$.

Assume the complete-case restriction within each history category, adequate completer representation in both categories, and a Bernoulli conditional model. Here the saturated estimates of the completer response probabilities are the observed proportions:

\hat p_C(0)=\frac{6}{30}=0.20,\qquad \hat p_C(1)=\frac{12}{20}=0.60.

Under the restriction, these are also the estimated response probabilities for non-completers in their respective categories. The expected number of responses among the non-completers is therefore

$$ 10(0.20)+5(0.60)=2+3=5. $$

The predicted response proportion among the 15 non-completers is $5/15=0.3333$. It is a model-based expectation, not a statement that exactly five of these particular participants responded. If the target is the response proportion among all 65 participants, the observed completers contributed $6+12=18$ responses, so the plug-in estimate is $(18+5)/65=0.3538$. This last calculation combines observed responses with expected missing responses and should be described as an estimate under the restriction.

The assumptions are visible in the calculation: both history categories occur among completers, and the binary outcome is represented by a conditional Bernoulli probability. With only these aggregate counts, one cannot check whether finer histories would be needed or whether the restriction is true; those are modeling and sensitivity concerns.

Count Outcomes: Predicting a Threshold Probability

For count outcomes, a common conditional model is Poisson: $Y_2\mid H=h,R=1\sim\text{Poisson}(\lambda_h)$, where $\lambda_h$ is the conditional rate parameter. If the model is adequate and the complete-case restriction is imposed, the non-completer distribution at history $h$ is assigned that same Poisson distribution. A threshold probability then comes from the Poisson probability mass function, not just from its mean.

Worked Example: Counts at Least Four Among Non-Completers

Hypothetical data and fitted model. Suppose a count outcome is the number of symptom episodes by visit 2. There are 12 non-completers with history $H=A$ and 8 with history $H=B$. Among completers, conditional Poisson models give fitted rates $\hat\lambda_A=2$ and $\hat\lambda_B=3$ episodes. For this illustration, suppose conditional distributional checks show no material lack of fit or overdispersion, the fitted rates are supported by completers in both histories, and the complete-case restriction is adopted.

For a Poisson variable with rate $\lambda$, the probability of at least four episodes is one minus the probabilities of zero through three:

P(Y_2\geq4\mid H=h,R=0) = 1-e^{-\lambda_h}\left(1+\lambda_h+\frac{\lambda_h^2}{2}+\frac{\lambda_h^3}{6}\right).

For history A, substituting $\lambda_A=2$ gives

1-e^{-2}\left(1+2+\frac{2^2}{2}+\frac{2^3}{6}\right) =1-\frac{19}{3}e^{-2} \approx 0.1428765.

For history B, substituting $\lambda_B=3$ gives

1-e^{-3}\left(1+3+\frac{3^2}{2}+\frac{3^3}{6}\right) =1-13e^{-3} \approx 0.3527681.

The expected number of non-completers with at least four episodes is therefore

12(0.1428765)+8(0.3527681) \approx 1.714518+2.822145 =4.536663.

Thus the model predicts about 4.54 of the 20 non-completers to have counts of at least four. The decimal is an expected count, not a fractional observed person. Rounding the two probabilities earlier in the calculation changes the final digits slightly; using the unrounded probabilities gives approximately 4.5366634.

This calculation relies on more than the equality of distributions across completion patterns: it also relies on the conditional Poisson model. The stated fit checks and history overlap make that model a plausible working specification for the example, but they cannot verify the complete-case restriction for outcomes that are missing.

Conditional Predictions Must Be Standardized Correctly

A conditional prediction is not automatically the population prediction of interest. To predict the proportion above a threshold among all non-completers, weight each history-specific probability by the fraction of non-completers in that history. In the count example, the estimated proportion is $4.536663/20\approx0.2268$. For a target population with a different history composition, the weights must be changed accordingly. The conditional probabilities can remain the same while the marginal prediction changes.

Nonlinear summaries need particular care. Averaging conditional odds, rates, or other transformed quantities is generally not equivalent to first averaging probabilities and then transforming. A probability-scale calculation is often the clearest way to report an event risk.

Worked Example: Averaging Probabilities Before Converting to Odds

Hypothetical prediction. Suppose non-completers are equally divided between two histories. The restricted response probabilities are $0.20$ for $H=0$ and $0.60$ for $H=1$, as in the binary example. The marginal predicted response probability is the history-weighted average:

\hat p=\frac{1}{2}(0.20)+\frac{1}{2}(0.60)=0.40.

The odds corresponding to this standardized probability are $0.40/(1-0.40)=0.6667$. By contrast, the two conditional odds are $0.20/0.80=0.25$ and $0.60/0.40=1.50$, whose unweighted average is $0.875$. Converting that average odds to a probability gives $0.875/(1+0.875)=0.4667$, not $0.40$.

The histories have equal weights, so the discrepancy is not caused by unequal group sizes. It arises because odds are a nonlinear transformation. The relevant marginal event prediction is obtained by weighting conditional probabilities first. This example assumes the two conditional probabilities are valid under the restriction and that the stated history weights describe the non-completers being summarized.

Diagnostics, Alternatives, and Limitations

Diagnostics can assess the observed-data components of the procedure, but cannot test the equality of missing and observed outcome distributions within a history. Useful checks include:

History overlap: inspect whether each history represented among non-completers has completers. Predictions outside observed completer support depend on extrapolation.
Conditional distributional fit: for a binary model, examine calibration and residual patterns across histories. For a count model, inspect predicted versus observed frequencies, dispersion, excess zeros, and tail fit, rather than checking only whether the fitted mean is reasonable.
Model sensitivity: compare reasonable conditional distribution specifications and history definitions. A Poisson model can miss overdispersion; a logistic model can be too restrictive if important history variables or interactions are omitted.
Weighting and target: verify that predictions are weighted using the history distribution of the population being described. A prediction for non-completers is not automatically a prediction for the full study population.

A restriction stated for the conditional mean, as in the earlier material on applying complete-case restrictions to means, may suffice for a mean target but does not identify a binary event probability or a count tail probability. A conditional regression model can express how a mean varies with history, but a nonlinear target still requires a suitable conditional distribution or a direct model for that target. The distributional restriction makes this requirement explicit. Other identifying restrictions, such as available-case or neighboring-case restrictions, use different information and assumptions; they are not interchangeable with this complete-case restriction.

The main limitation is that missing outcomes supply no direct evidence about whether their conditional distribution matches that of completers. A good fit among completers, substantial overlap, and stable predictions do not establish that equality. The restriction can also be fragile when histories are sparse or the conditional distribution is misspecified. Report the target, the conditioning history, the model used, and the identifying restriction together; do not present a predicted probability or count as if it were directly observed.

A Compact Calculation Sequence

Define the target and history.
Specify the binary event or count feature, the outcome time, and the observed variables $H$ on which the restriction conditions.

State the restriction and check support.
Equate the non-completer conditional distribution to the completer distribution at each relevant history, and assess whether completers represent those histories.

Fit and assess the conditional distribution.
Choose a binary or count distributional model and examine its fit among completers; do not treat a fitted mean as a complete distribution.

Calculate and standardize the prediction.
Compute the desired event probability within each history, then apply the target population’s history weights. Label the result as an assumption-dependent prediction.

Common mistake: “The completers’ average count is 2, so the probability of at least four episodes can be read off from that mean.” This is incorrect: different count distributions can share the same mean but have different tail probabilities. Specify or estimate a conditional distribution, calculate the tail probability from it, and state that applying it to non-completers requires the complete-case restriction.

Check Your Understanding

Work through each question, then compare with the solutions below.

In one history stratum, 9 of 30 completers have a binary event. Under the complete-case restriction, what event probability is assigned to non-completers in that stratum?
A count model gives $Y_2\mid H=h,R=1\sim\text{Poisson}(1)$. Under the restriction, write the formula for the probability of at least two events among non-completers with that history.
There are 4 non-completers in a history with event probability $0.25$ and 6 in a history with event probability $0.50$. What is the expected number with the event, and what is the expected proportion?
Why does a good Poisson fit among completers fail to demonstrate the complete-case restriction?

Solutions. (1) The completer estimate is $9/30=0.30$, so the restriction assigns an estimated event probability of $0.30$ to non-completers in that stratum. (2) The required probability is $P(Y_2\geq2\mid H=h,R=0)=1-P(Y_2=0\mid h)-P(Y_2=1\mid h)=1-e^{-1}(1+1)=1-2e^{-1}\approx0.2642$. (3) The expected event count is $4(0.25)+6(0.50)=1+3=4$, and the expected proportion is $4/10=0.40$. (4) The fit assesses the conditional distribution where outcomes are observed, among completers. The restriction concerns whether that distribution also describes outcomes missing for non-completers, which cannot be checked directly from the missing values.