Sparse Completer Patterns and Extrapolation

Why Sparse Completers Matter

A complete-case restriction uses the conditional outcome distribution among people observed at every visit to supply the distribution of later outcomes among people who dropped out. When the complete-case pattern is small, the restriction may require predictions at histories or covariate values that few, or no, completers have. In that situation, estimates for missing outcomes depend not only on the restriction but also on how the conditional distribution is modeled.

The central distinction is between the identifying restriction and the extrapolation used to implement it. The restriction equates conditional distributions across patterns. It does not, by itself, say that a fitted linear or logistic model is correct, nor does it guarantee that the complete cases provide adequate information for every history in the dropout pattern.

Restriction, Support, and Extrapolation

Consider a two-visit study. Let $Y_1$ and $Y_2$ be outcomes at visits 1 and 2, let $X$ denote baseline covariates, and let $R=2$ indicate that both outcomes are observed. Let $R=1$ indicate that $Y_1$ is observed but $Y_2$ is missing. Write $H=(X,Y_1)$ for the observed history available at dropout.

Definition: The two-visit complete-case restriction is $$ f(Y_2\mid H,R=1)=f(Y_2\mid H,R=2). $$ It asserts that, conditional on the same observed history $H$, the distribution of the missing visit-2 outcome among dropouts equals the distribution among complete cases. It is an identifying assumption, not a consequence of having complete cases.

The conditional distribution on the right can be estimated directly only for histories represented among completers. If a dropout has a history outside that support, the restriction refers to a conditional distribution that the observed complete cases do not directly reveal. A regression model, smoothing rule, or other extrapolation assumption must then connect observed histories to unsupported ones. Even within the observed range, very few completers near a dropout’s history can make the estimated conditional distribution unstable.

For a conditional mean analysis, define $m_2(h)=E(Y_2\mid H=h,R=2)$. Under the restriction, $m_2(h)$ is also the conditional mean for dropouts at history $h$. A fitted model for $m_2(h)$ may provide predictions where the data are sparse, but its form and the range of its predictions matter. A correct conditional-mean model is enough to reconstruct a mean; reconstructing the full distribution requires a model or estimate for that full conditional distribution.

How Extrapolation Enters the Reconstructed Mean

Let $\pi_r=P(R=r)$. The population mean at visit 2 can be written as a mixture over the two patterns. This is an identity from conditioning on $R$; using the complete-case restriction supplies the otherwise missing conditional mean for $R=1$.

E(Y_2)=\pi_2 E(Y_2\mid R=2)+\pi_1 E\{m_2(H)\mid R=1\}.

In a sample, the second term is typically estimated by predicting $Y_2$ for each dropout from the completer-based conditional model and averaging those predictions. If the model is evaluated at histories unsupported by completers, that average inherits the extrapolation. A small number of unsupported or high-leverage histories can have a substantial effect when many dropouts share them.

Describe the histories.
Compare the covariate and observed-outcome histories of dropouts with those of completers. Identify gaps and thin regions.

Fit the conditional model.
Estimate the distribution or mean of $Y_2$ among completers conditional on the histories specified by the restriction.

Predict and aggregate.
Apply the fitted conditional model to dropout histories, then weight predictions by the pattern sizes to reconstruct the target summary.

Examine dependence on extrapolation.
Check whether the result changes materially under plausible alternative model forms or restrictions on unsupported histories.

Worked Example: A Thin Covariate Stratum

Suppose a hypothetical study has 100 participants, with a binary baseline covariate $X$. Thirty completers have $X=0$, and their visit-2 mean is 10 units. Only two completers have $X=1$, with visit-2 outcomes 12 and 18 units, so their mean is $(12+18)/2=15$ units. There are 20 dropouts with $X=0$ and 48 with $X=1$. Assume the complete-case restriction conditional on $X$, and use the corresponding completer means to predict dropout outcomes.

The $X=0$ prediction for each dropout is 10 units; the $X=1$ prediction is 15 units. The estimated total of observed and reconstructed visit-2 outcomes is $$ 30(10)+2(15)+20(10)+48(15)=300+30+200+720=1250\text{ units}. $$ Thus the estimated overall mean is $$ \widehat{E}(Y_2)=\frac{1250}{100}=12.50\text{ units}. $$

The $X=1$ dropout prediction is based on only two completers. The calculation is arithmetically straightforward, but the estimated conditional mean of 15 units may be imprecise and sensitive to those two observations. If the complete-case restriction is not credible within $X$, or if those observations are unrepresentative, the reconstructed contribution from the 48 $X=1$ dropouts can be seriously affected. The pattern weight makes this cell particularly consequential.

Continuous Histories and Model-Based Extrapolation

With a continuous observed history, a dropout’s value may lie beyond the range observed among completers. A regression model then produces a prediction there by extending its fitted relationship. The complete-case restriction does not justify that extension: it equates distributions at a given history, while the model supplies the relationship across histories.

Worked Example: Predicting Beyond the Completer Range

In a hypothetical study, three completers have observed histories $Y_1=2,4,6$, with corresponding visit-2 outcomes $Y_2=5,6,7$ units. The fitted conditional-mean model is $m_2(Y_1)=a+bY_1$. Its ordinary least-squares slope is $$ \hat b=\frac{\sum_i(Y_{1i}-\bar Y_1)(Y_{2i}-\bar Y_2)} {\sum_i(Y_{1i}-\bar Y_1)^2} =\frac{(-2)(-1)+0(0)+2(1)}{(-2)^2+0^2+2^2} =\frac{4}{8}=0.5. $$

The means are $\bar Y_1=4$ and $\bar Y_2=6$, so $\hat a=\bar Y_2-\hat b\bar Y_1=6-(0.5)(4)=4$. The fitted model is therefore $\widehat m_2(Y_1)=4+0.5Y_1$. Suppose 12 dropouts each have $Y_1=10$, outside the completer range from 2 to 6. The restriction and this model give a predicted outcome of $4+0.5(10)=9$ units for each dropout.

The completers’ observed total is $5+6+7=18$ units. The reconstructed overall mean among the 15 participants is $$ \frac{18+12(9)}{15}=\frac{126}{15}=8.400\text{ units}. $$ For a simple sensitivity check, keep the intercept at 4 but use a slope of 0.3. The prediction at $Y_1=10$ becomes $4+0.3(10)=7$ units, and the reconstructed mean becomes $\{18+12(7)\}/15=102/15=6.800$ units. This comparison is not a second fitted estimate; it illustrates how an unsupported slope choice can shift the result by 1.600 units.

The restriction is an assumption in both calculations. The second calculation changes the extrapolation model, not the restriction. Since the dropout histories lie beyond the observed completer range, the data shown cannot determine which slope is more credible.

Nonlinear Outcomes and Sparse Support

For a binary $Y_2$, a model for the conditional event probability can avoid impossible predictions below zero or above one. It does not remove the extrapolation problem: a predicted probability far outside the completer histories may still be highly sensitive to the model’s shape and estimated coefficients.

Worked Example: Binary Outcomes at an Unsupported Covariate Value

Suppose completers in a hypothetical study have baseline values of $X$ only from 0 to 2. A logistic regression fitted to their binary visit-2 outcome gives $$ \operatorname{logit}\{\hat p(X)\}=-2+0.6X, \qquad \hat p(X)=\frac{\exp(-2+0.6X)}{1+\exp(-2+0.6X)}. $$ Assume the complete-case restriction conditional on $X$, and suppose 20 dropouts have $X=5$. This value is beyond the completer range, so applying the fitted model is extrapolation.

At $X=5$, the fitted log-odds are $-2+0.6(5)=1$, and the predicted event probability is $$ \hat p(5)=\frac{\exp(1)}{1+\exp(1)} =\frac{2.7183}{3.7183}=0.7311. $$ The model therefore predicts $20(0.7311)=14.62$ events in expectation among these 20 dropouts. This is an expected count, not a claim that a fractional number of events was observed.

If a sensitivity model instead uses slope 0.3 while retaining intercept $-2$, the log-odds at $X=5$ are $-0.5$, giving $\hat p(5)=\exp(-0.5)/\{1+\exp(-0.5)\}=0.3775$. The expected count is then $20(0.3775)=7.550$ events. The difference shows that a bounded prediction is not necessarily a well-supported prediction. The sparse complete-case data provide no direct observations at $X=5$.

Diagnostics, Alternatives, and Limitations

Diagnostics should focus on the histories where predictions are being made, not only on how well the model fits the completers overall. Useful checks include:

Compare the distributions and ranges of $X$ and $Y_1$ by pattern; flag dropout histories outside the completer range.
Tabulate completer counts in important covariate strata and inspect whether a small number of observations determine a conditional estimate.
Check residual patterns and influential observations among completers, while remembering that good in-sample fit does not validate extrapolation.
Recompute target summaries under plausible alternative conditional models and report whether conclusions materially change.
Track the proportion of dropout predictions that require extrapolation and the total pattern weight attached to them.

A complete-case restriction is one identifying choice among several. Available-case and neighboring-case restrictions use different pattern comparisons and therefore make different assumptions; neither is automatically justified by sparse complete cases. A more flexible model may reduce functional-form rigidity but can be especially unstable with few completers. Pooling or regularization may stabilize estimates, but it introduces additional modeling assumptions and does not create information in unsupported regions.

The key limitation is that observed data alone cannot verify the complete-case equality for outcomes missing among dropouts. Sparse support adds a second limitation: even conditional quantities among completers may be estimated poorly or may need model-based extension. Uncertainty intervals that account only for sampling variation under one fixed model do not capture all uncertainty about the identifying restriction or the extrapolation form.

Common Mistake

Common mistake: “The complete-case restriction says dropouts have the same visit-2 outcomes as completers, so I can use the overall completer mean for every dropout.” This ignores the conditioning history. The restriction equates distributions only among people with the same $H$; it does not generally equate the marginal distributions across patterns. The correction is to model or estimate the conditional distribution given $H$, inspect whether completers support the dropout histories, and state clearly when predictions rely on extrapolation.

A second mistake is to treat a numerically precise model prediction as evidence that the extrapolation is reliable. Precision conditional on a chosen model is not confirmation of that model beyond the observed range. Sparse completers make it especially important to distinguish a computed estimate from the assumptions that make it interpretable.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

State the two-visit complete-case restriction using $H$, $R=1$, and $R=2$. What does it assert, and what does it not assert?
There are 10 completers with $X=0$, whose mean $Y_2$ is 8 units, and one completer with $X=1$, whose $Y_2$ is 14 units. There are 30 dropouts with $X=0$ and 20 with $X=1$. Under the restriction conditional on $X$, calculate the reconstructed overall mean.
Why is predicting for dropouts with $Y_1=9$ extrapolation if all completers have $Y_1$ between 1 and 5, even when a linear regression gives a numerical prediction?
A logistic model predicts probability 0.4 for each of 25 dropout histories. What is the expected event count, and why is that not an observed count?

Solutions. (1) The restriction is $f(Y_2\mid H,R=1)=f(Y_2\mid H,R=2)$. It equates the conditional distributions at the same history; it does not assert equal marginal outcome distributions across patterns or establish that a fitted model extrapolates correctly. (2) The estimated total is $10(8)+1(14)+30(8)+20(14)=80+14+240+280=614$ units. Dividing by 61 participants gives $614/61=10.07$ units. The $X=1$ prediction is based on only one completer. (3) The value 9 lies outside the observed completer range, so the prediction extends the fitted relationship beyond direct support; its validity depends on the model’s form as well as the restriction. (4) The expected count is $25(0.4)=10$ events. It is an expectation under the fitted probabilities; the realized observed count is an integer and need not equal 10.