Integrated Sensitivity Analysis of a Growth Hormone Trial

From Dropout Patterns to a Treatment Conclusion

In a longitudinal growth hormone trial, the 12-month height increase may be missing for participants who stop attending visits. If discontinuation is related to unobserved growth, analysing only participants with a recorded 12-month measurement can misrepresent the randomized groups. A complete analysis therefore has two linked tasks: describe who is missing and when, then state how the unobserved outcomes are assumed to differ from predictions based on observed data.

The case below is entirely hypothetical. It uses change in height from baseline to month 12, measured in centimeters, with larger values indicating more growth. The target contrast is the mean 12-month change under assignment to growth hormone minus the mean change under assignment to control, for all randomized participants. This treatment-policy contrast retains participants in their assigned groups regardless of treatment discontinuation. It is not a comparison restricted to those who completed follow-up.

Hypothetical Trial and Missingness Summary

Suppose 120 children were randomized equally: 60 to growth hormone and 60 to control. Visit records show the following monotone discontinuation patterns. Each listed number is the number of children whose last attended scheduled visit was at the indicated point; those children have no recorded 12-month outcome.

Thus 15 of 60 growth-hormone participants (25%) and 12 of 60 control participants (20%) lack the endpoint. The timing matters: early withdrawal leaves less observed history with which to predict the missing outcome. The table also shows somewhat more endpoint missingness in the growth-hormone arm, but counts alone do not establish why participants withdrew or whether their unobserved growth differed from predictions.

Reference Model and Delta Adjustment

The reference analysis predicts missing 12-month outcomes using randomized arm, baseline characteristics such as age and baseline height, and the available longitudinal height measurements. A typical implementation uses multiple imputation from a model that includes relevant observed predictors and allows the mean growth trajectory to differ by arm. Under its MAR assumption, after conditioning on the included information, the distribution of the missing outcome is the same as the corresponding predictive distribution for participants with observed outcomes.

MAR cannot be verified from the observed data alone. The delta adjustment makes the departure from that reference assumption explicit. For this illustration, assume a location shift: within arm \(g\), the imputed missing outcomes have the reference predictive distribution shifted by \(\delta_g\), while observed outcomes are not shifted. A negative \(\delta_g\) means that participants missing the endpoint are assumed to have grown less than otherwise similar participants predicted under MAR. The shift is conditional on the observed information used by the reference model; it is not a claim that every missing child has exactly the same growth value.

If the shift is constant within each arm and \(q_g\) is that arm’s missing fraction, its effect on the arm mean is \(q_g\delta_g\). Consequently, for the growth-hormone-minus-control contrast,

Here \(T\) denotes growth hormone and \(C\) denotes control. This identity follows by separating each arm’s full mean into its observed and missing portions. If the reference-model mean is \(\hat{\mu}_{g,\mathrm{MAR}}\), shifting only the missing fraction changes it to \((1-q_g)\hat{\mu}_{g,\mathrm{observed}}+q_g(\hat{\mu}_{g,\mathrm{missing,MAR}}+\delta_g)\). Relative to the MAR mean, the change is exactly \(q_g\delta_g\). Subtracting the adjusted control mean from the adjusted growth-hormone mean gives the stated contrast.

With multiple imputation, this adjustment is applied to the imputed values and the treatment contrast is re-estimated in each completed data set. The final standard error and confidence interval should be obtained by pooling the completed-data estimates and their variances using Rubin’s rules. In general, uncertainty should be recalculated for each sensitivity scenario rather than assumed to be identical across scenarios.

Varying Assumptions and Assessing Robustness

A single MNAR assumption is not enough to show how dependent the conclusion is on untestable choices. Vary the delta values over a scientifically interpretable range, give particular attention to differential departures between arms, and report the estimate and interval for each scenario. For this illustration, the table gives hypothetical results from refitting and pooling the multiple-imputation analysis at each scenario. Intervals use the large-sample form \(\hat{\tau}\pm1.96SE\); their standard errors are scenario-specific.

The estimate in the third row follows directly from the shift identity: \(1.20+(0.25)(-1.5)-(0.20)(-0.2)=0.865\) cm, reported as 0.87 cm. Its interval is \(0.865\pm1.96(0.41)=0.865\pm0.804\), approximately 0.06 to 1.67 cm. In the fourth row, the estimate is \(1.20+(0.25)(-2.0)-(0.20)(-0.2)=0.74\) cm; its interval is \(0.74\pm1.96(0.42)\), approximately -0.08 to 1.56 cm.

For the first three scenarios, the 95% intervals exclude zero; in the most adverse scenario shown, the interval includes zero. Thus the positive estimated contrast persists, but the conclusion that the interval excludes zero is not robust to every considered assumption. The values in the table are hypothetical outputs, not a claim that the stated deltas are biologically plausible for any real trial. A real report should justify the range using clinical knowledge, observed growth before withdrawal, and input from investigators or participants.

Diagnostics, Alternatives, and Limitations

Before interpreting a sensitivity analysis, check that the reference model is suitable for the observed data. Inspect growth trajectories and residual patterns by arm, compare observed histories of participants who do and do not later discontinue, and examine whether the imputation model includes predictors of both growth and dropout. Check that the model produces plausible predictive distributions, including plausible ranges and variability, rather than only plausible means. Sparse early-dropout patterns, poor overlap in observed histories, or a small number of participants can make predictions especially dependent on model form.

A delta analysis also needs diagnostics of its own. State whether deltas are common across dropout times or vary by pattern, confirm that the direction and units of each shift are clear, and inspect the implied distributions for implausible values. A constant shift can be too restrictive if early and late discontinuers differ. If substantive knowledge supports different departures by dropout time, pattern-specific deltas can be considered, but the added parameters require more assumptions and should not be chosen solely to obtain a preferred conclusion.

The delta approach is a pattern-mixture sensitivity model: it adjusts the outcome distribution among participants with missing outcomes relative to a chosen reference model. A selection model instead specifies how the probability of observing an outcome depends on its value and observed information. These approaches express MNAR assumptions differently, and observed data generally cannot determine which untestable assumption is right. Copy-reference analysis is another possible sensitivity approach when its treatment-course assumption suits the question; it is not interchangeable with an arm-specific delta shift. The choice should follow the estimand and clinical rationale, not convenience or the direction of the result.

A useful report states the randomized denominators, endpoint availability and timing by arm, the estimand, the reference model, the delta interpretation and units, and the range of sensitivity assumptions. It presents both estimates and uncertainty, and distinguishes a positive point estimate from an interval that excludes zero. Sensitivity analysis does not repair biased randomization, a misspecified outcome model, or missing data that are not adequately represented by the chosen patterns and predictors. It also cannot prove that the treatment effect is robust beyond the assumptions actually examined.