Available-Case Restriction at a Single Visit

The Available-Case Distribution at One Visit

In a pattern-mixture model, the distribution of an outcome may be described separately for groups defined by their missingness patterns. At a particular visit, some participants have an observed outcome and others do not. The available-case restriction is a way to fill in the otherwise unspecified distribution for a pattern whose outcome is missing: it sets that pattern-specific distribution equal to the distribution among all participants whose outcome at that visit is observed.

This is an identifying assumption, not a consequence of having observed data. It can be useful because the available cases provide an empirical basis for specifying a missing-outcome distribution. But the unobserved outcomes cannot verify that the assumption is true.

Definition: Fix a visit \(t\). Let \(Y_t\) be the outcome at that visit, \(R_t=1\) indicate that \(Y_t\) is observed, and \(R_t=0\) indicate that it is missing. Let \(G\) denote the participant’s missingness pattern. The available-case distribution at visit \(t\) is the population distribution \(F_{t,\mathrm{av}}(y)=P(Y_t\le y\mid R_t=1)\). For every pattern \(g\) with \(P(G=g)>0\) and \(R_t=0\) in that pattern, the available-case restriction is \(P(Y_t\le y\mid G=g)=F_{t,\mathrm{av}}(y)\) for every \(y\). Thus, within each pattern missing at visit \(t\), the marginal distribution of \(Y_t\) is assumed to match the distribution among all available cases at that visit.

The equality is a statement about entire distributions, not just their averages. It applies to the specified visit and the stated, unstratified conditioning: it does not additionally condition on baseline covariates or past outcomes. The definition presumes that the available-case group has positive probability, so its distribution is defined, and that the pattern under discussion is a meaningful population group.

What the Restriction Implies

A useful way to understand a distributional assumption is to ask what it implies for summaries that matter in an analysis. If the available-case restriction holds for pattern \(g\), then any summary defined by integrating a function of \(Y_t\) has the same value in that pattern and among available cases, provided the expectation exists.

F_{t,g}(y)=F_{t,\mathrm{av}}(y)\ \text{for every }y \quad\Longrightarrow\quad E\{h(Y_t)\mid G=g\}=E\{h(Y_t)\mid R_t=1\}

Here \(F_{t,g}\) denotes the conditional distribution given \(G=g\), and \(h\) is any measurable function for which the expectations exist. The justification is that equal probability distributions integrate the same function to the same value. For \(h(y)=y\), the pattern-specific mean equals the available-case mean, when finite. For \(h(y)=y^2\), the second moments also agree; hence the variances agree when finite. For an indicator such as \(h(y)=I(y>c)\), the probability of exceeding the threshold \(c\) agrees. These consequences are not separate assumptions: they follow from equality of distributions.

Specify the visit and pattern.
Name the outcome \(Y_t\), identify \(R_t\), and choose a pattern \(g\) in which \(Y_t\) is missing.

Define the reference distribution.
Use the distribution of \(Y_t\) among all participants with \(R_t=1\), not only among participants observed at every visit.

Impose the equality.
Set the distribution in pattern \(g\) equal to the available-case distribution. Derive means, probabilities, or other summaries only after stating this assumption.

Worked Example: A Discrete Available-Case Distribution

Consider a hypothetical study with one outcome visit of interest. Of 120 participants, 80 have an observed outcome at that visit. Their empirical outcome distribution has 20 values equal to 2, 40 equal to 4, and 20 equal to 6 points. The remaining participants belong to two patterns in which the outcome is missing: pattern A has 25 participants and pattern B has 15. Apply the available-case restriction to each missing pattern.

The available-case proportions are \(20/80=0.25\) at 2 points, \(40/80=0.50\) at 4 points, and \(20/80=0.25\) at 6 points. The corresponding available-case mean is

\bar{x}_{\mathrm{av}} =\frac{20(2)+40(4)+20(6)}{80} =\frac{320}{80} =4\ \text{points}.

Under the restriction, the population distribution of the outcome in pattern A is set equal to this available-case distribution, and the same equality is imposed separately for pattern B. Consequently, each pattern has mean 4 points under the assumed distribution. Each also has probability \(0.25\) of an outcome of 2 points, \(0.50\) of 4 points, and \(0.25\) of 6 points. The restriction does not assert that every missing participant has outcome 4; it specifies a distribution, not individual values. The observed counts estimate the reference distribution in an analysis, but they do not prove that the missing-pattern distributions match it.

Worked Example: A Continuous Outcome Model

Suppose, hypothetically, that a continuous outcome at visit \(t\) is modeled among available cases by \(N(10,9)\), where the second argument is the variance. Assume the analysis imposes the available-case restriction for a pattern with missing \(Y_t\). State the restricted distribution and calculate the probability of an outcome above 13.

The assumed distribution in the missing pattern is also \(N(10,9)\). Its standard deviation is \(\sqrt{9}=3\) outcome units. Standardizing the threshold gives

P(Y_t>13\mid G=g) =1-\Phi\left(\frac{13-10}{3}\right) =1-\Phi(1) \approx 1-0.8413 =0.1587.

Thus, under this model and restriction, the probability of an outcome above 13 units is about 15.9% in the specified missing pattern. This is a model-based implication, not a probability learned directly from the unobserved outcomes. The distributional form \(N(10,9)\) is an additional modeling choice for the available cases; the available-case restriction transfers that distribution to the missing pattern but does not itself require normality.

Worked Example: A Binary Outcome

In a hypothetical study, 60 participants have an observed binary outcome at the visit, and 12 experience the event. A missingness pattern contains 30 participants whose outcome at that visit is missing. Under the available-case restriction, what event probability is assigned to that pattern, and what is the expected event count?

The available-case event proportion is \(12/60=0.20\). A binary distribution is determined by its event probability, so the restriction assigns event probability 0.20 to the missing pattern and no-event probability \(1-0.20=0.80\). The expected number of events among 30 participants is

30(0.20)=6\ \text{events}.

Six is the expected count under the assumed distribution; it is not a claim that exactly six of the 30 participants experienced the event. The restriction supplies a probability for the unobserved outcomes, rather than recovering their individual values.

Assumptions, Interpretation, and Diagnostics

The restriction is substantive. It assumes that, at the selected visit, the outcome distribution in a group whose outcome is missing is the same as the distribution in the group whose outcome is observed. It does not follow merely from defining a pattern, from the observed sample sizes, or from the fact that both groups came from the same study. In particular, missingness related to the unobserved outcome can make this equality implausible.

The available-case distribution is itself a population quantity. In practice it is estimated using observed outcomes, possibly with a statistical model. Sampling error in that estimate is distinct from uncertainty about the identifying assumption. A large available-case sample may estimate its own distribution precisely while providing no direct evidence that the missing-pattern distribution equals it.

Diagnostics therefore have limits. Analysts can check whether the reference distribution is estimated stably, whether its model fits the observed available cases, and whether the assumed distribution gives sensible values for the outcome. They can also examine observed outcomes across groups that differ in their observation patterns, where such comparisons are available. But the equality for outcomes missing in pattern \(g\) cannot be directly tested from the observed data: those outcomes are precisely the values unavailable for comparison. Apparent agreement among observed groups is informative context, not proof of equality for the missing group.

A practical analysis should make the assumption visible and consider whether conclusions change under plausible alternatives. The introductory material on the identification gap explains why observed data alone do not generally determine distributions for outcomes that are missing. The available-case restriction closes that gap by assumption; it does not remove the uncertainty that motivated the assumption.

How It Differs From Other Choices

Choice	Reference or specification	Main implication
Available-case restriction	All participants with the outcome observed at the selected visit	Each pattern missing that outcome is assigned the available-case distribution at that visit.
Complete-case restriction	Participants observed at every required visit	The reference group is narrower when some participants have the selected outcome but are missing another required outcome.
No identifying restriction	No equality is imposed for the missing-pattern distribution	The missing-pattern distribution remains unspecified without further assumptions or information.

The distinction between available cases and complete cases matters whenever some participants are observed at the selected visit but are not complete cases for the full study. The two reference distributions need not be the same. Choosing the available-case restriction is not a claim that it is universally more credible; it is a choice of reference group and an equality assumption that should be justified for the scientific setting.

Common Mistakes and Limitations

Confusing the restriction with an observed fact. The observed data estimate the available-case distribution. They do not reveal whether the missing-pattern distribution equals it.
Using “available cases” to mean “complete cases.” Available cases have \(R_t=1\) at the specified visit. They may still be missing outcomes at other visits.
Restricting only the mean without saying so. The available-case restriction equates the full distributions. Equal means alone are weaker and do not imply equal variances, quantiles, or event probabilities.
Treating an expected count as an actual count. A probability of 0.20 in 30 people gives an expected count of 6, not a guarantee of exactly 6 events.

The restriction is marginal as stated here. If outcome distributions differ systematically across baseline or history groups, a single marginal equality may obscure those differences. Conversely, adding conditioning variables changes the restriction and requires its own specification and justification. The present equality should therefore be read exactly as written: at one visit, for the specified missingness pattern, the outcome distribution matches that among all cases observed at that visit.

Key takeaway: At a single visit, the available-case restriction assigns each pattern missing that visit’s outcome the same distribution as the outcome distribution among all participants observed there. It is a clear, strong identifying assumption with direct implications for distributional summaries, but it cannot be verified for the missing outcomes from observed data alone.

Check Your Understanding

Answer each question using the single-visit definition. Solutions follow the questions.

At visit \(t\), what population defines the available-case distribution?
If the available-case mean is 7 units and the restriction holds for pattern \(g\), what mean does the restriction imply for that pattern? Does it imply that every missing value is 7?
There are 50 available cases, of whom 15 have a binary event. Under the restriction, what event probability and expected event count are assigned to a missing pattern of size 20?
Why can observed data not directly test the equality between the available-case distribution and the distribution in a pattern missing \(Y_t\)?

Solutions. (1) It is the distribution of \(Y_t\) among all participants with \(R_t=1\), meaning the outcome is observed at that visit; they need not be complete cases across the study. (2) The pattern-specific mean is 7 units, provided the mean exists. The restriction concerns the distribution and does not set every individual outcome to 7. (3) The available-case event proportion is \(15/50=0.30\), so the missing pattern is assigned event probability 0.30. Its expected event count is \(20(0.30)=6\) events, not a guaranteed realized count of six. (4) The outcomes in that pattern are missing at visit \(t\), so their distribution cannot be compared directly with the observed available-case distribution. The equality is an identifying assumption rather than an empirically testable equality for those unobserved outcomes.