Complete-Case Restriction for Two-Wave Dropout

Why a Complete-Case Restriction Is Needed

In a two-wave study, suppose the first outcome $Y_1$ is recorded for everyone, but the second outcome $Y_2$ is recorded only for participants who remain in the study through wave 2. The complete cases reveal how $Y_2$ is distributed among participants with a recorded second outcome. They do not, by themselves, reveal how $Y_2$ is distributed among those who dropped out after wave 1. A restriction is needed to connect these two distributions.

The complete-case restriction makes that connection by comparing participants at the same value of the observed first outcome. It is an assumption about distributions, not a procedure that discards incomplete records. Its role is to supply the missing conditional distribution of $Y_2$ using the corresponding distribution among completers.

Definition: Let $R=1$ indicate that a participant has an observed $Y_2$ and is therefore a complete case; let $R=0$ indicate dropout after wave 1, so $Y_2$ is missing. Assume $Y_1$ is observed for everyone. The complete-case restriction is $$ \mathcal{L}(Y_2\mid Y_1=y_1,R=0) = \mathcal{L}(Y_2\mid Y_1=y_1,R=1) $$ for values $y_1$ relevant to the dropout group. Here $\mathcal{L}$ denotes a probability distribution. Equivalently, for every measurable set $A$ of possible $Y_2$ values, the conditional probabilities of $Y_2\in A$ are equal for dropouts and completers at that $Y_1$.

The equality is conditional: it compares the two dropout patterns among participants with the same $Y_1$. It does not require the two patterns to have the same distribution of $Y_1$. Where conditional distributions are described by densities, equality means equality of those densities almost everywhere; the distribution formulation also applies when outcomes are discrete or mixed.

What the Restriction Says—and Does Not Say

At a fixed $Y_1$, the restriction says that knowing whether a participant dropped out provides no further information about the distribution of $Y_2$. In conditional-independence notation, this is $Y_2\perp R\mid Y_1$, over the relevant support. To see the equivalence, let $A$ be any set of possible second-outcome values. Equality of the conditional distributions gives the same conditional probability for $Y_2\in A$ in each dropout pattern. That common probability is also the probability conditional on $Y_1$ alone, since it is the same for both values of $R$. Thus $$ P(Y_2\in A\mid Y_1,R)=P(Y_2\in A\mid Y_1). $$

This statement does not assert that dropout is unrelated to $Y_1$. Dropout rates can vary with the first outcome. Nor does it say that dropouts and completers have the same unconditional distribution of $Y_2$: their $Y_1$ distributions may differ, and the conditional distribution of $Y_2$ can vary with $Y_1$. The restriction concerns the relationship between $Y_2$ and dropout status after conditioning on $Y_1$.

The restriction provides an extrapolation: among completers, $Y_2$ is observed, so its conditional distribution given $Y_1$ can be estimated. Under the restriction, that same conditional distribution describes $Y_2$ among dropouts with the corresponding $Y_1$. Combining it with the observed distribution of $Y_1$ among dropouts characterizes their joint distribution of $(Y_1,Y_2)$. This conclusion relies on the restriction; it is not learned from the missing second outcomes themselves.

Worked Example: Checking Conditional Distribution Equality

Consider a hypothetical study in which $Y_1$ has two categories, lower and higher. Suppose the assumed conditional distributions of a binary $Y_2$ are as follows.

$Y_1$ category	$P(Y_2=1\mid Y_1,R=1)$	$P(Y_2=1\mid Y_1,R=0)$
Lower	0.30	0.30
Higher	0.70	0.70

Because $Y_2$ is binary, each listed probability also determines the probability of $Y_2=0$. For the lower category, both patterns have $P(Y_2=0\mid Y_1)=1-0.30=0.70$; for the higher category, both have $1-0.70=0.30$. Thus the full conditional distributions agree in each category, and the restriction holds in this hypothetical specification.

The values differ across $Y_1$ categories, which is allowed: the assumed probability of $Y_2=1$ changes from 0.30 to 0.70 as $Y_1$ changes. The restriction requires equality between patterns within each category, not a constant $Y_2$ distribution across all participants.

Using Completers to Describe the Missing Distribution

In observed data, the conditional distribution for $R=1$ can be estimated from participants with both outcomes recorded. The conditional distribution for $R=0$ cannot be directly checked or estimated from their $Y_2$ values, because those values are missing. Under the complete-case restriction, the completer distribution is used as the distributional description for dropouts at the same $Y_1$. In practice, that description is only as credible as the assumption and the available information at each $Y_1$.

Worked Example: Translating a Completer Distribution to Dropouts

In a hypothetical study, consider participants with $Y_1=4$ units. Among 40 completers in this group, the observed $Y_2$ values are 2 units for 10 participants, 4 units for 20, and 6 units for 10. There are 12 dropouts with $Y_1=4$, but their $Y_2$ values are unobserved. Use the complete-case restriction to describe the distribution for those dropouts.

First calculate the completer proportions: $$ \begin{aligned} \widehat P(Y_2=2\mid Y_1=4,R=1)&=\frac{10}{40}=0.25,\\ \widehat P(Y_2=4\mid Y_1=4,R=1)&=\frac{20}{40}=0.50,\\ \widehat P(Y_2=6\mid Y_1=4,R=1)&=\frac{10}{40}=0.25. \end{aligned} $$ Under the restriction, these proportions are used for the conditional distribution among dropouts with $Y_1=4$. Applied to the 12 dropouts, the corresponding expected counts are $12(0.25)=3$, $12(0.50)=6$, and $12(0.25)=3$.

The counts 3, 6, and 3 describe the distribution implied by the assumption; they do not reveal which particular dropout had which $Y_2$, nor do they claim that the missing values are known. They also use estimated completer proportions as plug-in values, so the uncertainty in those proportions would matter in an actual analysis.

A basic condition for this extrapolation is overlap: for dropout-relevant values of $Y_1$, there must be completers with those values, or a justified model must extend the completer distribution into regions without observed completers. If no completer has a particular $Y_1$ value seen among dropouts, the observed data provide no direct estimate of the needed conditional distribution there. Stating the restriction does not create information in such a region.

Distributional Equality Is Not Mean Equality

The restriction concerns the entire conditional distribution. Equality of conditional means is not enough: distributions can have the same mean and still differ in spread, skewness, or the probabilities assigned to individual values. Conversely, a comparison of overall distributions can differ across patterns even when the conditional distributions agree at every $Y_1$.

Worked Example: Equal Conditional Means but Different Distributions

At one fixed value of $Y_1$, suppose the hypothetical completer distribution assigns probability 0.50 to $Y_2=0$ and 0.50 to $Y_2=2$. Suppose the dropout distribution assigns probability 1 to $Y_2=1$. The conditional mean for completers is $$ 0(0.50)+2(0.50)=1, $$ and the conditional mean for dropouts is $1(1)=1$. The means are equal.

However, at the set $A=\{0\}$, the conditional probabilities are $P(Y_2\in A\mid Y_1,R=1)=0.50$ and $P(Y_2\in A\mid Y_1,R=0)=0$. Since these probabilities differ, the conditional distributions are not equal and the complete-case restriction fails. Matching means would not justify using the completer distribution for the dropouts.

Worked Example: Equal Conditional Distributions but Different Marginal Distributions

Let $Y_1$ be either group A or group B, and let $Y_2$ be binary. Suppose the conditional probability of $Y_2=1$ is 0.20 in group A and 0.80 in group B, for both completers and dropouts. The complete-case restriction holds within each $Y_1$ group.

Now suppose 80% of completers are in group A and 20% are in group B. Their marginal probability of $Y_2=1$ is $$ 0.80(0.20)+0.20(0.80)=0.16+0.16=0.32. $$ Suppose instead that 20% of dropouts are in group A and 80% are in group B. Their marginal probability is $$ 0.20(0.20)+0.80(0.80)=0.04+0.64=0.68. $$ The marginal probabilities, 0.32 and 0.68, differ because the patterns have different $Y_1$ compositions. That difference does not contradict the conditional restriction: within each group, the conditional probability is the same for the two patterns.

Diagnostics, Comparisons, and Limitations

The central diagnostic limitation is that $Y_2$ is unobserved for $R=0$. Therefore, the defining conditional equality cannot be directly tested by comparing observed $Y_2$ distributions across the two patterns. This is an identifying assumption, not a claim that the observed data can verify it. Observed differences in the distribution of $Y_1$ by $R$ are possible and do not, by themselves, refute the restriction.

Some checks can still reveal practical problems. Examine whether completers are present across the $Y_1$ values seen among dropouts, and whether the estimated completer conditional distributions are unstable in sparsely represented regions. Assess how sensitive the resulting conclusions are to plausible changes in the assumed dropout distribution. These checks can expose weak support or dependence on extrapolation; they cannot establish that the unobserved conditional equality is true.

The restriction can be contrasted with two broad alternatives. One is to leave the dropout distribution of $Y_2$ unrestricted; then the observed data alone do not determine that missing distribution. Another is to impose a different assumption linking missing outcomes to observed information. These choices are not interchangeable: each supplies a different identifying claim. The complete-case restriction specifically uses the completer distribution conditional on $Y_1$, rather than asserting equality of marginal distributions or merely matching a summary such as the mean.

A common mistake is to read “complete-case restriction” as “analyze only complete cases,” or to say that it assumes the two patterns have identical $Y_2$ distributions overall. The correction is that the restriction is a conditional distributional equality at fixed $Y_1$. Completers provide the reference distribution for dropouts only after conditioning on the first outcome. A second mistake is to treat the equality as empirically confirmed because completer data fit a chosen distribution well. Good fit among completers does not test the unobserved dropout distribution.

The restriction is useful because it turns an otherwise unidentified conditional distribution into one that can be represented using observed completer data, subject to overlap and estimation uncertainty. Its limitation is equally important: if dropouts would have a different $Y_2$ distribution than completers with the same $Y_1$, the extrapolation is wrong, and the observed data cannot diagnose that discrepancy directly.

Key takeaway: In a two-wave study with $Y_1$ observed for all, the complete-case restriction equates the full conditional distribution of $Y_2$ for dropouts and completers at the same $Y_1$. It is a substantive, generally untestable assumption; it is neither unconditional distribution equality nor a complete-case-only analysis.

Check Your Understanding

Answer each question, then compare with the solutions.

State the complete-case restriction using $Y_1$, $Y_2$, and $R$, with $R=1$ for completers.
At a fixed $Y_1$, completers have $P(Y_2=1)=0.40$, while dropouts are assumed to have $P(Y_2=1)=0.40$. If $Y_2$ is binary, what are the probabilities of $Y_2=0$ in the two patterns?
Can the complete-case restriction hold if the marginal distributions of $Y_2$ differ between completers and dropouts? Explain.
There are dropouts at a value of $Y_1$ for which no completers are observed. What problem does this create?

Solutions. (1) For every relevant $y_1$ and measurable set $A$, $P(Y_2\in A\mid Y_1=y_1,R=0)=P(Y_2\in A\mid Y_1=y_1,R=1)$. (2) Each probability is $1-0.40=0.60$, so the full binary conditional distributions agree at that $Y_1$. (3) Yes. The $Y_1$ distributions can differ across patterns, so averaging the same within-$Y_1$ distributions over different $Y_1$ compositions can produce different marginals. (4) There is no direct completer-based estimate of the needed conditional distribution there. Applying the restriction requires an additional justified extrapolation, and conclusions may depend on it.