Influential Subjects in Crossover ANOVA

Why Subject-Level Influence Matters

In a crossover study, each participant receives multiple formulations, so the treatment comparison is informed by within-participant data. A participant may have unusually high log exposure in both periods yet contribute little to the treatment contrast. Conversely, a participant whose treatment-minus-reference difference is far from the others can noticeably shift that contrast. These are different diagnostic questions: how unusual is the participant’s model residual, and how much does the fitted treatment estimate change when that participant is omitted?

Influence analysis answers the second question directly. It checks the sensitivity of a specified model’s treatment estimate to individual subjects; it does not establish that an observation is erroneous or that the subject should be excluded. The model, treatment contrast, and analysis population must remain clearly defined throughout the assessment.

Subject-Level Contrasts and Residuals

Let \(y_{ij}\) denote the log-transformed exposure for subject \(i\) in period \(j\), with treatment and period recorded for each observation. A typical two-period crossover model includes treatment, period, and sequence effects, along with a subject effect and residual error. Its interpretation depends on the design and covariance assumptions: for example, the treatment contrast is interpreted under the specified additive model, and subjects are assumed independent of one another. The residual distribution and covariance structure must be adequate for the fitted model and its standard errors.

For a complete, balanced two-treatment, two-period crossover, define \(d_i\) as subject \(i\)’s log exposure on treatment minus that subject’s log exposure on reference. With equal representation of the two treatment sequences, the average of these signed differences estimates the treatment contrast under the additive model. The period contribution has opposite signs in the two sequences and cancels in the balanced average. If sequences are not equally represented, periods are missing, or the design is more complex, this simple average need not be the fitted treatment estimate; use the specified crossover model instead.

The contrast residual \(e_i\) is directly relevant to the treatment comparison: it says how far subject \(i\)’s treatment-minus-reference difference lies from the fitted average. This is not necessarily the same as an individual observation’s residual from the full model. For instance, the two period observations for a subject may both be high because of a subject-level shift, while their difference is close to the average treatment contrast.

In a more general mixed model, let \(\hat{\boldsymbol{\beta}}\) be the fitted fixed-effect coefficients and let \(c\) specify the treatment contrast. Then the reported contrast is \(c^\mathsf{T}\hat{\boldsymbol{\beta}}\). Subject-level response residuals can be inspected as a vector, with scaling informed by the fitted covariance model; the treatment contrast residual is not a substitute for that fuller diagnostic. Most importantly for influence, remove one subject’s entire set of observations and refit the same model, then compare the resulting \(c^\mathsf{T}\hat{\boldsymbol{\beta}}_{(-i)}\) with the full-data estimate.

Why Deleting One Subject Changes the Estimate

In the balanced example, deleting a subject changes the average of the remaining \(n-1\) contrasts. The following identity makes the connection between residual and influence precise:

The first equality follows because the full-data sum is \(n\hat{\theta}\), and removing subject \(i\) subtracts \(d_i\). Subtracting \(\hat{\theta}\) from that remaining-sample average gives the second expression. Thus, in this particular setting, a more extreme contrast residual produces a larger absolute deletion change, with the change also depending on sample size. This identity does not apply in this form to an unbalanced design or a general mixed model; there, deletion requires a model refit.

Assessing the Change in Context

A deletion change should be read alongside the full-data estimate, its standard error, and the study’s scientific scale. One descriptive summary is the signed change in treatment contrast, \(\hat{\theta}_{(-i)}-\hat{\theta}\); its absolute value ranks the size of the shift but hides its direction. Dividing by the full-data \(SE\) expresses the shift in units of that estimate’s standard error. This ratio is a diagnostic scale, not a hypothesis test or a universal cutoff for influence.

For log outcomes, exponentiating a treatment contrast gives the corresponding treatment-to-reference geometric mean ratio. The difference between two log-scale estimates does not translate into the same additive difference between their ratios, so report the log-scale changes and, when useful, the exponentiated estimates. A shift should be interpreted in context: its size, direction, precision, and possible impact on the scientific conclusion all matter.

For a complete balanced crossover, the subject-level differences also allow a simple standard-error comparison. If they are independent and adequately modeled with a common variance, the usual standard error of their mean is \(s_d/\sqrt{n}\), where \(s_d\) is the sample standard deviation of the \(d_i\). This estimate of uncertainty complements, but does not replace, the deletion comparison. In a more complex mixed model, use the standard error from the refitted model rather than assuming the paired-difference formula still applies.

Leave-One-Subject-Out Refitting in General Models

When the data are incomplete or the crossover is not a simple balanced two-period design, use the same model specification for the full-data and deletion fits. Retain the same response scale, treatment contrast, fixed effects, and covariance structure. Remove all records for one subject, not merely one period, and refit the model. Record the new treatment estimate, its standard error, and the deletion change. If the model cannot be fitted after a deletion, report that limitation rather than silently changing the model.

Diagnostics, Alternatives, and Limitations

Residual diagnostics and deletion refits answer related but distinct questions. A residual describes disagreement between the observed data and the fitted model. A deletion refit describes how sensitive a particular estimate is to one subject’s data. In the balanced paired-contrast setting, these quantities are mathematically linked by the identity above; in general mixed models, the relationship also depends on the design, covariance structure, and estimation method. Do not assume that the most extreme raw observation must have the greatest effect on the treatment estimate.

Potential warning signs include a large change in the treatment estimate after a single-subject deletion, a change in the estimate’s direction, marked changes in its standard error, or a refit that fails or behaves differently from the full-data fit. These findings prompt checks of data transcription, treatment and period labels, protocol deviations, missingness, model convergence, and the suitability of the fitted covariance assumptions. They do not identify the cause of the sensitivity automatically.

The balanced paired-difference calculation is transparent and useful when its design conditions hold. A full mixed-model refit is the appropriate comparison when the analysis includes missing periods or other structure not represented by that simple average. Neither approach replaces model-based residual assessment, and neither provides a universal threshold for declaring a subject influential. Deletion diagnostics can also be unstable in small studies: removing one participant changes both the estimate and the information available for estimating variability.

A common mistake is to label a participant influential solely because one log exposure is extreme, or to remove that participant because the deletion estimate seems more plausible. The correction is to assess the subject’s contribution to the specified treatment contrast, examine the model and data, and report sensitivity transparently. Any exclusion requires a defensible, prespecified or scientifically justified reason independent of the desired result; influence analysis alone is not that reason.