A Two-Visit Workshop on Restriction-Based Identification

From a Two-Visit Pattern Table to an Overall Mean

With two visits, each participant belongs to one of four response patterns. A pattern table records how many participants have each observation history, and which visit outcomes are available within it. This makes restriction-based identification a direct calculation: observed means supply some pattern-specific quantities, a stated restriction supplies the missing pattern-specific means, and the pattern proportions determine how those means contribute to the overall mean.

The key is to keep the steps separate. First identify which outcomes are observed in each pattern. Then define the reference mean required by the restriction. Finally, weight every pattern-specific mean by that pattern's proportion. A restriction does not change the observed data or the pattern proportions; it specifies values for outcome means that the data alone do not reveal.

The four proportions satisfy \(p_{11}+p_{10}+p_{01}+p_{00}=1\). Some pattern-specific means are observed directly: \(Y_1\) is observed in patterns 11 and 10, while \(Y_2\) is observed in patterns 11 and 01. The remaining pattern-specific means are missing from the observed data. For example, the full-population mean at visit 1 can be written as

The first two terms involve observed means; the last two do not. A restriction-based calculation supplies values for those two missing means. The analogous decomposition applies to \(\mu_2\).

Two Restrictions and Their Reference Means

A complete-case restriction uses pattern 11 as the reference for a visit outcome that is missing in another pattern. An available-case restriction instead uses the mean among everyone whose outcome at that visit is observed, pooling across the patterns where it is observed. These are marginal mean restrictions: they do not automatically adjust for covariates or other history.

Both restrictions require the relevant reference mean to be defined. For a complete-case restriction, \(p_{11}>0\) is needed. For the available-case restriction at visit 1, \(p_{11}+p_{10}>0\) is needed; at visit 2, \(p_{11}+p_{01}>0\) is needed. These are positivity conditions, not evidence that the restrictions are substantively plausible. In a sample calculation, the corresponding sample pattern counts must also be nonzero.

The available-case reference is a weighted mean, not an unweighted average of the pattern means. If one observed pattern is much larger than another, it contributes more to the mean of all available outcomes. Likewise, an overall mean is not an average of four pattern means unless the four pattern proportions are equal.

Why the Available-Case Calculation Reproduces Its Reference Mean

The equality between the available-case reference mean and the resulting overall mean is not a coincidence. It follows directly from the restriction and the fact that all missing patterns for a given visit are assigned the same available-case mean.

The same argument applies to visit 2, with \(q_2=p_{11}+p_{01}\). In a finite sample, the available-case overall mean is therefore exactly the mean among participants with that outcome observed, apart from numerical rounding. This identity is also a useful arithmetic check. The complete-case overall mean need not equal the complete-case mean: its observed-pattern contributions can include a different mean, as pattern 10 does for \(Y_1\) in the example.

Diagnostics, Comparisons, and Limits

A careful calculation begins with an observation map: for each visit, mark the patterns where its outcome is observed. Then verify that the pattern proportions sum to one, that the reference group has positive size, and that each observed mean is paired with the correct pattern proportion. A useful final check is to separate the observed and missing contributions and confirm that their weights sum to one.

The two restrictions answer different assumptions about the unobserved pattern means. Complete-case restrictions borrow the pattern-11 mean, even when an outcome is observed in another incomplete pattern. Available-case restrictions borrow the pooled observed mean, which reflects the observed pattern composition for that visit. Neither rule is established by having a large sample, and neither can generally be verified from the observed pattern table: the missing outcome means are precisely the quantities not observed.

These are marginal restrictions. If patterns differ in baseline covariates or other measured history, equality of marginal means does not imply equality within levels of those variables. Pooling can then conceal differences in composition. Conversely, replacing a marginal rule with a conditional rule changes the assumption and the calculation; it is not a harmless arithmetic refinement. The earlier material on covariate imbalance across patterns and on marginal versus conditional equality is relevant when deciding whether this simple table-level restriction is scientifically defensible.

The calculation identifies a marginal mean only under the stated restriction and the pattern definitions used. It does not identify the full distribution of missing outcomes, their variances, or their correlations with outcomes at the other visit. Nor does equality of means imply equality of distributions. If the intended estimand is conditional, or concerns a nonlinear quantity such as a threshold probability, these mean restrictions alone may be insufficient.