Handling Missing Periods in Reference-Scaled Analyses

Why Missing Periods Matter

A replicate design estimates within-subject reference variability by observing reference treatment more than once in some participants. When a period is missing, the central question is not simply how many measurements remain. It is which within-subject comparisons remain available. A participant with one observed reference period contributes information about that observation, but not the same direct reference-repeat contrast as a participant with two observed reference periods.

This distinction matters in reference-scaled analyses because the reference variance is part of the scaling calculation. Missing periods may reduce precision, remove particular contrasts, or make a variance component unidentifiable under the chosen model. The consequences depend on the design, which treatment and period are missing, the covariance model, and the assumptions used to handle missingness.

The examples use log-transformed exposure measurements. Period and treatment effects are understood to have been handled in the stated model; where a simple paired contrast is calculated, the values are explicitly described as adjusted for those fixed effects. The introductory material on replicate designs establishes why repeated reference observations are needed; here the focus is on what remains when some of those observations are absent.

Which Reference Contrasts Survive?

Suppose the adjusted log reference measurements for subject \(i\) are \(Z_{iR1}\) and \(Z_{iR2}\). If both are observed, define their difference as \(D_i=Z_{iR1}-Z_{iR2}\). Under the simple model that the two reference occasion errors are independent, have mean zero, and share variance \(\sigma_{WR}^2\), the subject effect cancels in the difference and \(D_i\sim N(0,2\sigma_{WR}^2)\). Thus, when \(m\) subjects have usable paired reference measurements, a simple moment estimate is \(s_D^2/2\), where \(s_D^2\) is the sample variance of their differences. The degrees of freedom for this sample variance are \(m-1\), provided the differences are independent and the stated model applies.

A subject missing one of the two scheduled reference measurements has no such paired difference. The subject may still contribute to estimation of treatment or period means, and to other covariance information, but the missing value cannot be replaced by a contrast. A subject with two observed reference measurements retains the contrast even if a different treatment period is missing.

This formula is a simple contrast-based estimate, not a universal substitute for the model-based REML procedures used in reference-scaled analyses. It requires measurements on the same subject, appropriate adjustment for fixed effects, independent occasion errors with a common reference variance, and enough paired contrasts to estimate their variance. If period effects are not removed or modeled, a difference between periods can include a period effect as well as within-subject variation.

What a Model Can Use Beyond Complete Pairs

A mixed model can retain incomplete subjects rather than discarding every subject who lacks a complete set of periods. For illustration, write an observed log endpoint as \(Y_{ij}=\mu+\tau_{T_{ij}}+\pi_j+b_i+e_{ij}\), where \(\tau_{T_{ij}}\) is the treatment effect, \(\pi_j\) is the period effect, \(b_i\) is a subject effect, and \(e_{ij}\) is an occasion error. Assume \(b_i\sim N(0,\sigma_b^2)\); conditional on \(b_i\), occasion errors are independent and normally distributed, with variance \(\sigma_{WR}^2\) for reference observations. The treatment and period structure must match the actual design. A model may include additional terms where required, but those terms must be specified rather than assumed away.

If a subject has only one observed reference measurement, that observation informs the fitted mean and the marginal variation, but on its own it does not separate the subject variance from the reference occasion variance. A subject with two reference measurements provides a within-subject covariance pattern and a difference that removes the shared subject effect. REML combines the information in all observed patterns according to the specified covariance model. It can use incomplete records, but it cannot create the covariance information that no observed pattern contains.

Model-based use of incomplete records also depends on why the data are missing. A common working condition is that, conditional on observed outcomes and included covariates, the probability of a missing measurement does not depend on its unobserved value (an ignorable missingness assumption under the model). If missingness depends on the unobserved exposure after conditioning on those observed quantities, the missingness mechanism may be informative; the observed-data likelihood alone may then yield biased estimates. This assumption is not established merely because a mixed model converges.

When the Reference Variance Is Not Identifiable

A model's ability to retain an incomplete record is different from its ability to estimate every variance component. In the random-intercept model above, one observed reference measurement has marginal variance \(\sigma_b^2+\sigma_{WR}^2\). If every subject contributes at most one reference measurement, there are no within-subject reference pairs to reveal the reference covariance pattern. Without additional design information or restrictions that identify the components, \(\sigma_b^2\) and \(\sigma_{WR}^2\) cannot be separated from those observations alone.

The same issue can occur less starkly when very few paired reference observations remain: the component may technically be estimable under the specified model, yet be weakly informed and imprecise. Whether a particular incomplete design still identifies a component depends on all observed treatment-period patterns and the covariance structure, not only on the total number of recorded endpoints. The earlier material on degrees of freedom for scaled variance estimates is relevant to the uncertainty calculation once an estimate is available; missing periods can reduce the information supporting that calculation.

Checks, Alternatives, and Common Pitfalls

Before interpreting a scaled estimate, make a period-by-period inventory of observed treatments and subjects. Confirm that the retained paired reference contrasts correspond to the intended replicate design, that period effects are represented, and that the model's covariance assumptions are defensible. Inspect the observed patterns for whether reference repeats are present and whether missingness is concentrated in a treatment, period, sequence, or subject subgroup. Such patterns do not prove informative missingness, but they are reasons to investigate how data became unavailable. Formal residual diagnostics are a separate step.

The main alternatives answer different questions. A complete-pair calculation is transparent and directly shows the information in observed within-subject differences, but it discards incomplete records and can be inefficient. A mixed-model REML analysis can use the observed parts of incomplete records, subject to its mean, covariance, and missingness assumptions. Complete-case analysis is not automatically unbiased: if completion is associated with outcomes or relevant covariates, the retained subjects may not represent the intended analysis population. Imputation is not an automatic remedy either; filling in a missing period with a mean or a predicted value as though it were observed can distort both means and within-subject variance.

A common mistake is to count every observed reference value as if it contributed equally to reference repeatability. For example, one subject with two reference observations and another with one do not provide two and one independent reference differences. The correction is to identify which subjects have the required paired observations, then state separately how a model uses the other records. A related mistake is to infer that REML must estimate a reference variance because software returned a number. Convergence does not establish identifiability, adequate information, or a plausible missingness assumption.