Complete-Case Analysis With a Worked Example

What a Complete-Case Estimate Represents

In a multi-omics study, a subject may have transcriptomic measurements but lack proteomic measurements, or may be missing another modality needed for the planned integration. Complete-case analysis keeps only subjects with every required modality observed. It is easy to calculate, but its meaning depends on two choices: how the integrated quantity is defined and which population the retained subjects represent.

To make the target explicit, suppose each subject has \(M\) modalities and the analysis defines one integrated score \(Y_i\) from those modalities. For example, \(Y_i\) might be a prespecified weighted combination of modality-level summaries after the chosen preprocessing. The score, its transformations, and its weights must be defined consistently for every subject in the target population. Complete-case analysis cannot estimate a score that is undefined for subjects whose modalities are missing.

The population target considered here is \(\mu=E(Y_i)\), the mean integrated score in the study's target population. The complete-case mean directly describes \(E(Y_i\mid C_i=1)\), the mean among subjects who are complete. These two quantities need not be equal. When they differ, increasing the number of complete cases can make the estimate of the complete-case mean more precise without making it representative of the full target population.

Assumptions and a Justification

The calculation requires a meaningful score \(Y_i\) for each target-population subject, even if it cannot be computed from that subject's observed data. The complete-case estimate also requires at least one complete subject. For the usual standard error, assume subjects are independent draws from the relevant population and that the score has finite variance. If the goal is inference about the full target population, an additional selection condition is needed: a sufficient condition is that complete-case status is independent of the integrated score, \(C_i\perp Y_i\). This is the MCAR condition for complete-case selection with respect to this score.

Under that condition, the distribution of the score among complete subjects equals its distribution in the target population. In particular,

Conditional on observing \(n_{CC}=\sum_i C_i>0\) complete subjects, the complete-case mean is an average of \(n_{CC}\) scores with mean \(\mu\). Its conditional expectation is therefore \(\mu\), and its variance is \(\sigma_Y^2/n_{CC}\), where \(\sigma_Y^2\) is the target-population variance of the score. In practice, estimate that variance with the complete-case sample variance:

The standard error describes sampling variability under the assumed sampling and selection model. It does not account for bias caused by selective completeness. If \(C_i\) depends on \(Y_i\), the complete-case mean generally estimates \(E(Y_i\mid C_i=1)\), not \(\mu\). If completeness depends on observed covariates related to \(Y_i\), a raw complete-case mean can likewise differ from the marginal population mean, even when the missingness is MAR. The earlier material on missingness mechanisms helps distinguish these assumptions; the mechanism cannot generally be established from observed data alone.

How Selection Changes the Target

A complete-case analysis changes which subjects contribute to the estimate. If completion is related to the integrated score, or to measured characteristics that predict it, the retained subjects may have a different score distribution from the target population. The resulting difference is a change in estimand, and possibly selection bias if the analyst interprets the complete-case mean as the full-population mean.

Completeness is not necessarily related to the score itself to create this problem. For example, if subjects with a particular observed clinical characteristic are less likely to provide every modality, and that characteristic predicts the score, the complete cases may over- or under-represent score values. MAR does not automatically make the unadjusted complete-case mean representative: conditional independence given observed covariates is not the same as independence marginally. Appropriate adjustment may be possible under additional assumptions, but it is not part of the raw complete-case calculation.

Precision, Diagnostics, and Alternatives

Discarding incomplete subjects reduces the number contributing to the estimate. Under independent sampling and MCAR, the standard error is approximately \(s_{CC}/\sqrt{n_{CC}}\). If the score variance is similar to what it would be in the full sample, fewer complete cases mean less precision. However, the observed standard error need not always be larger: selection can also change the score variance among retained subjects. A narrow interval among a selected low-variability subset does not establish precise estimation of the full-population mean.

For a small sample, an interval for the complete-case mean can be based on a \(t\) distribution if the complete-case scores are independent and approximately normally distributed. For the five scores in the first example, a 95% interval under those assumptions is \(1.00\pm t_{0.975,4}(0.200)=1.00\pm2.776(0.200)\), or approximately \([0.445,\ 1.555]\) score units. This interval concerns the mean represented by the complete-case sampling model; interpreting it as an interval for the full target-population mean additionally requires valid selection assumptions. It is not a 95% probability statement about the fixed parameter.

Useful diagnostics include reporting the number and proportion of complete subjects, listing observed modality-availability patterns, and comparing available covariates between complete and incomplete subjects. Examine whether completeness varies by site, batch, clinical group, or other observed predictors of the integrated score. These checks can expose a plausible selection problem, but similar observed groups do not establish MCAR, and unobserved differences remain possible. Also verify that the integrated score was defined before restricting the analysis and that preprocessing or weights have not inadvertently changed across availability patterns.

Complete-case analysis is transparent and straightforward when the target is explicitly the population with all modalities observed, or when selection assumptions justify generalization to the full target population. It can be inefficient when many subjects have partial data, and misleading when complete cases are selected. Alternatives include observed-data likelihood, imputation, and inverse-probability weighting; each requires its own modeling assumptions and diagnostics. Pairwise calculations do not automatically solve the problem of defining a coherent integrated estimate.

Common Mistakes

• Calling the complete-case mean the cohort mean by default. It directly estimates the mean among complete subjects. Generalization to the target population requires justified selection assumptions.
• Treating MAR as sufficient for the raw mean. MAR conditional on observed covariates does not generally imply that complete and incomplete subjects have the same marginal score distribution.
• Equating a small standard error with lack of bias. The usual standard error quantifies sampling variability under its assumptions; it does not measure unmodeled selection bias.
• Assuming fewer cases must produce a larger observed standard error. The sample size tends to reduce precision, but selection can also alter the score variance. Report both \(n_{CC}\) and \(s_{CC}\).
• Defining the score only after seeing which subjects are complete. A score intended for a wider target population must have a consistent definition for that population, including subjects whose values are not observed.