Why examine availability patterns?
An integrated analysis can use subjects with different subsets of omics modalities. For example, one group may have both transcriptomic and proteomic measurements, while another has only transcriptomic measurements. A single reported estimate can conceal how much each group contributes. If a small group has an unusual estimate, the pooled result may depend noticeably on only a few subjects. Conversely, estimates may be similar across patterns, suggesting that no one pattern dominates the result descriptively.
The purpose of an availability-pattern diagnostic is to make that dependence visible. It is not, by itself, a test of why measurements are missing, a correction for missingness, or evidence that an association is causal. It asks a narrower question: how do the estimate and its supporting summaries change across the observed groups defined by available modalities?
For \(n\) subjects, pattern \(r\) has sample fraction \(w_r=n_r/n\), with \(\sum_r w_r=1\). For a subject-level quantity \(Y_i\), such as a prespecified integrated score measured on the same scale for all subjects, write \(\bar{Y}_r\) for its mean within pattern \(r\). These summaries are meaningful only when the quantity being compared is defined comparably across patterns. If a pattern-specific analysis estimates a model coefficient instead, the same model specification and outcome scale should be used for each subset, and the coefficient must be re-estimated on that subset.
Decomposing a pooled mean
For a mean, the pooled estimate is exactly a weighted average of the pattern-specific means. This identity is useful because it separates two sources of influence: how different a pattern's mean is from the others, and how many subjects have that pattern.
The identity follows by grouping the subject-level sum by pattern. Within pattern \(r\), \(\sum_{i:R_i=r}Y_i=n_r\bar{Y}_r\). Summing these group totals and dividing by \(n\) gives the displayed expression. It applies to an ordinary pooled mean; it does not imply that a regression coefficient or another nonlinear estimator is the weighted average of its pattern-specific estimates.
Worked Example: A Small Pattern with a High Integrated Score
Suppose a hypothetical study has 100 subjects. An integrated score, in standardized score units, is defined for each subject on the same scale. The observed pattern summaries are:
| Availability pattern | Subjects \(n_r\) | Pattern mean \(\bar{Y}_r\) (score units) |
|---|---|---|
| Transcriptomic and proteomic | 60 | 0.20 |
| Transcriptomic only | 35 | 0.35 |
| Proteomic only | 5 | 1.10 |
The fractions are \(60/100=0.60\), \(35/100=0.35\), and \(5/100=0.05\). In particular, the proteomic-only pattern accounts for 0.05, or 5%, of the subjects. The pooled mean is
Thus the pooled estimate is 0.2975 standardized score units. After excluding the five proteomic-only subjects, the mean among the remaining 95 subjects is
Exclusion changes the estimate by \(0.2553-0.2975=-0.0422\) score units, approximately. The rare pattern has a high mean and raises the pooled mean, despite comprising only 5% of the subjects. This is a descriptive influence finding; it does not establish why those subjects have higher scores or whether the full-sample or restricted estimate better targets a scientific question.
Quantifying exclusion influence
For a mean, the change after removing one pattern can be expressed directly in terms of that pattern's size and mean. Let \(\bar{Y}_{-r}\) denote the mean after excluding pattern \(r\), and let the full-sample mean be \(\bar{Y}\). Since the sum of the values outside \(r\) is \(n\bar{Y}-n_r\bar{Y}_r\), the reduced-sample mean is
The second equality follows by subtracting \(\bar{Y}\) from the first expression and simplifying the numerator: \(n\bar{Y}-n_r\bar{Y}_r-(n-n_r)\bar{Y} =n_r(\bar{Y}-\bar{Y}_r)\). If the pattern mean exceeds the full mean, its exclusion lowers the estimate; if it is below the full mean, exclusion raises it. The change depends on both the discrepancy and the pattern fraction. A very unusual but extremely small pattern can have limited influence on a mean, while a moderately different pattern with many subjects can matter substantially.
Worked Example: Checking an Exclusion Change Algebraically
Consider a second hypothetical integrated-score summary with \(n=80\) subjects. Seventy-six subjects have an integrated score mean of 2.10 units, and four subjects in a rare pattern have mean 8.10 units. First calculate the full mean:
The rare-pattern fraction is \(4/80=0.05\). Excluding it leaves the mean \(2.10\) units, so the change is \(2.10-2.40=-0.30\) units. The formula gives the same result:
This agreement is a useful arithmetic check. It also shows why reporting only the rare pattern's percentage would be inadequate: the mean's shift depends on how far that pattern's summary lies from the pooled estimate.
Pattern-specific estimates for model parameters
For a model parameter, such as an exposure coefficient, compute the estimate on each pattern and on the full set using the same outcome definition, covariates, coding, and fitting method. The pattern-specific estimates describe associations within the observed subsets. Their differences may reflect different subject composition, different precision, or different relationships in those subsets. They should not be interpreted automatically as effects of modality availability.
A pooled regression coefficient generally is not a weighted average of coefficients from separate pattern-specific regressions. Even in a simple unadjusted two-group comparison, the full-sample difference in group means depends on the numbers in each group and on their means within each pattern. Here is a numerical illustration.
Worked Example: An Exposure Difference Across Three Patterns
In a hypothetical study, an integrated outcome is measured in score units. Exposure is coded 0 or 1. The table gives the number of subjects and mean outcome within each exposure group and availability pattern. Each pattern contains both exposure groups, so the within-pattern differences are defined.
| Pattern | Exposure 0: \(n\), mean | Exposure 1: \(n\), mean | Within-pattern difference (score units) |
|---|---|---|---|
| A | 20, 0.10 | 20, 0.30 | 0.20 |
| B | 15, 0.20 | 15, 0.50 | 0.30 |
| Rare pattern C | 3, 0.90 | 3, 1.50 | 0.60 |
For the full sample, the exposure-0 total is \(20(0.10)+15(0.20)+3(0.90)=2+3+2.7=7.7\), across \(20+15+3=38\) subjects. Its mean is \(7.7/38\approx0.2026\) score units. The exposure-1 total is \(20(0.30)+15(0.50)+3(1.50)=6+7.5+4.5=18\), also across 38 subjects, giving a mean of \(18/38\approx0.4737\). The full-sample unadjusted difference is therefore
After excluding pattern C, the exposure-0 mean is \((2+3)/35=5/35\approx0.1429\); the exposure-1 mean is \((6+7.5)/35=13.5/35\approx0.3857\). The restricted difference is
The estimate decreases by about \(0.2429-0.2711=-0.0282\) score units. Pattern C has the largest within-pattern difference, 0.60 units, but only six subjects. The full-sample difference is not the simple average of 0.20, 0.30, and 0.60: it is formed from the exposure-group means in the full sample. This example also shows why pattern-specific estimates and a pooled estimate should all be reported when the scientific question warrants them.
Assumptions, diagnostics, and failure modes
Pattern stratification is primarily a descriptive diagnostic, so it does not require a missing-at-random assumption to calculate. Its interpretation does require care. Define patterns from the modality-availability indicators before examining the estimates; ensure the patterns are mutually exclusive and exhaustive; and use a consistent outcome scale and estimator wherever estimates are compared. For a pattern-specific model, check that the model is estimable in every subset and that relevant groups or covariate levels are represented. If the estimate is based on imputed or weighted data, state which estimate is being summarized and apply a consistent procedure across patterns.
Useful diagnostics include a table of pattern counts and fractions, pattern-specific summaries, the full estimate, and leave-one-pattern-out estimates. For model coefficients, report the model specification and the number of subjects contributing to each fit. Plotting estimates against pattern sizes can make a small but unusual group visible. Where uncertainty estimates are available, show them as well: a large difference between point estimates may be imprecise, and overlapping intervals do not establish equality. Estimates across subsets are often dependent because the subsets overlap with the full sample.
- Sparse patterns: A small denominator can make a mean or model coefficient unstable. Report its sample count, not just its percentage, and avoid treating a noisy pattern estimate as definitive.
- Changed composition: Excluding a pattern changes the subjects represented. A changed estimate may reflect different covariate or exposure distributions, not a change in a biological relationship.
- Empty cells or non-estimability: If a pattern lacks one exposure group, a within-pattern group contrast cannot be computed from that pattern. Do not invent a value or silently omit the limitation.
- Too many patterns: With many modalities, possible patterns proliferate and several may contain very few subjects. Summaries can become unreadable or misleading if the pattern definitions are chosen after inspecting results.
- Scale mismatch: Comparing summaries that use different score definitions or model specifications does not isolate availability-pattern influence.
How this diagnostic differs from alternatives
Pattern-stratified summaries complement, rather than replace, other approaches to incomplete modalities. A complete-case analysis restricts attention to subjects with all required modalities; it answers a restricted-data question and may discard substantial information. Inverse-probability weighting changes the contribution of observed subjects according to modeled availability probabilities. Imputation supplies values for missing blocks under an imputation model, while pooled estimates account for variation across imputations. Pattern diagnostics instead display how estimates vary across the observed availability groups and under selected exclusions. They do not adjust for pattern differences or recover information absent from a pattern.
A stable estimate across patterns is reassuring only in a limited sense: it says the chosen summaries did not change much across these observed subsets. It does not prove that missingness is ignorable, that unobserved values resemble observed ones, or that the estimator is unbiased. A marked change is a reason to investigate the estimand, pattern composition, model assumptions, and uncertainty—not a reason to automatically discard the rare pattern.
Check Your Understanding
Answer each question, then compare your work with the solutions.
- A pattern has 12 of 240 subjects. What is its sample fraction, and what percentage is that?
- Two patterns have means 1.0 and 3.0 score units, with 30 and 10 subjects respectively. Calculate the pooled mean.
- For a pooled mean \(\bar{Y}=4.0\), a pattern has \(n_r=8\), \(\bar{Y}_r=7.0\), and the total sample size is \(n=48\). Find the mean after excluding that pattern and the change from the full mean.
- Why is the pooled regression coefficient not generally the weighted average of pattern-specific regression coefficients?
- A pattern-specific exposure contrast cannot be estimated because the pattern contains no unexposed subjects. What should be reported?
Solutions. (1) \(12/240=0.05\), or 5%. (2) \([30(1.0)+10(3.0)]/40=(30+30)/40=1.50\) score units. (3) The full total is \(48(4.0)=192\). After exclusion it is \(192-8(7.0)=136\), over \(48-8=40\) subjects, so \(\bar{Y}_{-r}=136/40=3.40\). The change is \(3.40-4.0=-0.60\) units. (4) A regression coefficient is obtained by fitting a model to the joint data; it generally depends nonlinearly on the design and outcome values, so it need not equal a weighted average of separately fitted coefficients. (5) Report the pattern count and that the contrast is not estimable in that subset; do not fabricate an estimate or conceal the empty group.