21 Sample-Size Adjustment for Anticipated Dropout
Prerequisites: Chapters 5, 10, and 20.
21.1 Learning objectives
By the end of this chapter you should be able to:
- Enumerate the common dropout-adjustment methods and state what each assumes.
- Explain why the standard \(N_0/(1-d)\) inflation over-corrects for MMRM analyses and under-corrects at high dropout rates.
- Choose an adjustment method matched to the planned analysis.
- Explain how the timing of dropout, not just its amount, changes the correct adjustment.
- Distinguish adjusting for lost information from addressing dropout bias.
21.2 Orientation
Every longitudinal trial inflates its sample size for anticipated dropout, and almost every one does it by dividing by \(1 - d\). That rule was derived for a complete-case comparison at a single endpoint, and it is applied routinely to trials analyzed by MMRM, where patients who drop out still contribute substantial information through their observed visits.
The consequence is systematic mis-sizing, in a direction that depends on the dropout rate and the analysis. The menu of alternatives is large and largely undocumented comparatively: heuristic inflation factors, replacement strategies, and analytical adjustments derived from the likelihood of the planned analysis.
This chapter compares them. Its practical conclusion is that the adjustment should match the planned analysis, and that when the analysis is MMRM the analytical adjustment derived for MMRM is available, unbiased, and better than any of the heuristics.
21.3 Provenance
This chapter follows the research compendium 14-dropout-adjustment (project dropout-adjustment), which formalizes ten adjustment methods and evaluates each in an ADEMP-compliant Monte Carlo program. The data-generating model is a linear mixed trajectory with Gaussian random intercepts and slopes and compound-symmetric covariance; three dropout-hazard patterns (constant, early-heavy, late-heavy) are crossed with dropout rates from 0% to 40%, at 1,000 replicates per cell, reporting bias, empirical standard error, mean squared error, and 95% interval coverage with Monte Carlo standard errors.
21.4 The statistician’s contribution
(Judgment 1.) Matching the adjustment to the analysis. An inflation factor derived for complete-case analysis applied to an MMRM design is a category error. The question is always: given the planned analysis, how much information does a patient who leaves at visit \(j\) contribute?
(Judgment 2.) Estimating the dropout pattern, not just the rate. Early-heavy and late-heavy attrition at the same total rate require different corrections and make replacement strategies work or fail. The pattern comes from prior trials in the same population.
(Judgment 3.) Keeping the two problems separate. Adjusting the sample size handles the loss of precision. It does nothing about the bias from informative dropout, which Chapter 10 addresses. A protocol that inflates \(N\) and says nothing about the missing-data mechanism has solved the easier half.
21.5 The methods
The compendium formalizes ten.
No adjustment. The baseline, showing what dropout costs when nothing is done.
Simple inflation, \(N_0 / (1 - d)\). The standard rule. Derived for a complete-case comparison in which a dropout contributes nothing.
Squared inflation, \(N_0 / (1 - d)^2\). The most conservative of the common rules.
Overall’s additive rule, \(N_0 (1 + d)\). Derived specifically for slope-based longitudinal analyses, and intermediate between the two above.
Full replacement. Keep the original target and enroll a replacement for each dropout, who contributes a partial trajectory from their later start.
Half replacement. Replace half of them.
MMRM analytical adjustment. Derive the required \(n\) from the information matrix of the planned MMRM under the assumed dropout pattern.
Random-slopes adjustment. The analogous derivation for a random-slopes analysis.
Two-stage slope-summary adjustment. Compute each patient’s slope from their observed data, then compare slopes, with the sample size derived from the resulting variance under the dropout pattern.
Piecewise linear growth adjustment. For trajectories with a change in slope.
21.6 What the simulation shows
No adjustment. Even at 10 to 20% dropout, MMRM analyses retain substantial power, because partial trajectories contribute information. This is the finding that undermines the naive rule: the loss is much smaller than the dropout rate suggests. At 30 to 40% the loss becomes pronounced.
Simple inflation performs adequately at moderate rates and tends toward slight conservatism for MMRM analyses, because it was derived for complete-case comparisons. At high rates it under-corrects, since the relationship between dropout and information loss is not linear in \(d\).
Squared inflation produces the largest samples and over-enrolls considerably for MMRM analyses of longitudinal data. It may be appropriate when both arms experience independent dropout and the analysis is endpoint-only, which describes few modern longitudinal trials.
Overall’s additive rule falls between the two and was derived for slope-based analyses, where it offers the best balance of efficiency and robustness among the heuristics.
Full replacement is sensitive to dropout timing. Under late-heavy patterns there is little calendar time left for replacements to accumulate follow-up, so the strategy adds patients who contribute almost nothing. Under early-heavy patterns, replacements enroll early enough to deliver meaningful trajectories and the strategy works.
Half replacement is the pragmatic middle: less costly than full replacement, meaningful power recovery over none.
MMRM analytical adjustment directly accounts for the information contributed by partial trajectories and therefore avoids the heuristics’ conservatism. It is the recommended method when the planned analysis is MMRM with compound symmetry.
Two-stage slope-summary adjustment performs comparably to the MMRM analytical method when the between-subject slope variance is non-trivial, which connects back to the equivalence conditions of Chapter 18.
21.7 Timing matters
Three patterns at the same 25% total dropout:
Constant hazard. Withdrawals spread evenly across follow-up. The average dropout has completed about half the visits.
Early-heavy. Most withdrawals in the first months, often reflecting tolerability. The average dropout contributes little, so the information loss is closer to the naive rule, and replacement strategies work because replacements have time to accumulate data.
Late-heavy. Most withdrawals near the end, often reflecting study fatigue or a competing event. The average dropout has contributed most of their trajectory, so the information loss is small and the naive rule badly over-enrolls, while replacement strategies contribute almost nothing.
The design implication: obtain the dropout curve from prior trials, not just the final rate. A Kaplan-Meier plot of retention from a previous trial in the same population is the input that makes the adjustment defensible.
# Retention-informed adjustment: compute the information
# under the assumed dropout pattern rather than inflating.
information_ratio <- function(retention, corr_matrix) {
# relative information of the final-visit contrast under
# the observed-data patterns implied by `retention`
J <- length(retention)
patterns <- diff(c(retention, 0)) * -1 # prop. dropping at j
contribs <- sapply(seq_len(J), function(j) {
obs <- seq_len(j)
S <- corr_matrix[obs, obs, drop = FALSE]
# information about the visit-J mean from visits 1..j
if (j == J) 1 else
corr_matrix[j, J]^2 # simplified illustration
})
sum(patterns * contribs) + retention[J] * 1
}The illustration is deliberately simplified; the compendium’s implementation uses the full information matrix of the fitted model. The point of showing it is that the calculation is mechanical once the retention curve and the correlation structure are specified, and there is no reason to substitute a rule of thumb for it.
21.8 Worked example: adjusting a 12-month MMRM trial
A trial with visits at months 3, 6, 9, and 12, primary contrast at month 12, MMRM with compound-symmetric covariance and within-patient correlation 0.65. Unadjusted requirement: 180 per arm. Expected dropout 25%, with a roughly constant hazard.
Naive simple inflation. \(180 / 0.75 = 240\) per arm.
Squared inflation. \(180 / 0.75^2 = 320\) per arm. Clearly excessive for this analysis.
Overall’s rule. \(180 \times 1.25 = 225\) per arm.
MMRM analytical adjustment, computing the information under the constant-hazard retention curve and the assumed correlation: 207 per arm.
Simulation check. Generate under the assumed model and dropout pattern, fit the MMRM, 5,000 replicates. At \(n = 207\), power is 0.898 (Monte Carlo SE 0.004). At the naive 240, power is 0.932, confirming the over-enrollment: 33 extra patients per arm buying 3 points of power that were not required.
Sensitivity to the pattern. Repeating with an early-heavy pattern at the same 25% total, the analytical requirement rises to 228 per arm; with a late-heavy pattern it falls to 193. The spread across patterns, 35 patients per arm, is larger than the difference between several of the heuristic methods, which is the argument for obtaining the retention curve.
What is not addressed. All of the above assumes non-informative dropout. If patients withdraw because they are deteriorating, the estimate is biased regardless of sample size, and the protocol needs the Chapter 10 apparatus: retention plan, follow-up after discontinuation, reference-based sensitivity analyses, and a tipping point.
21.9 Collaborating with an LLM on dropout adjustment
Prompt 1: ‘Adjust our sample size for 25% dropout.’
What to watch for. Nearly universal use of \(N_0/(1-d)\) with no reference to the planned analysis. This is the single most reliable failure mode in this area.
Verification. Ask what the analysis is and whether the proposed adjustment accounts for partial trajectories.
Prompt 2: ‘Should we replace dropouts?’
What to watch for. Generic discussion without reference to timing, which is the factor that determines the answer.
Verification. Supply the expected retention curve and the enrollment window, and ask how much follow-up a replacement enrolled at month \(t\) would accumulate.
Prompt 3: ‘Simulate power under this dropout pattern.’
What to watch for. Dropout implemented as MCAR when the intent was MAR, or as deletion of the final visit only rather than of all subsequent visits.
Verification. Check that dropout is monotone: once a patient leaves, all later visits are missing.
21.10 Principle in use
Match the adjustment to the analysis. MMRM analytical adjustment when the analysis is MMRM; Overall’s rule for slope-based analyses when the analytical route is unavailable; simple inflation only for endpoint-only analyses.
Get the retention curve, not the retention rate. The pattern changes the answer by more than the choice among heuristics does.
Say explicitly that the adjustment addresses precision and not bias. They are separate problems with separate remedies, and conflating them is how protocols end up with a large trial and no missing-data plan.
21.11 Exercises
For a four-visit MMRM design with correlation 0.65, compute the information contributed by patients dropping out after visits 1, 2, and 3, relative to a completer.
Implement the ten adjustment methods for a design of your choosing and tabulate the resulting sample sizes at dropout rates of 10, 20, 30, and 40%.
Simulate the achieved power of each method at each rate and identify which methods hit the 90% target most closely.
Repeat under early-heavy and late-heavy dropout patterns and describe how the ranking changes.
Add informative dropout, where the withdrawal hazard depends on the current unobserved value, and report both the achieved power and the bias for the method that performed best under MAR.
21.12 Further reading
- The compendium
14-dropout-adjustment. - Overall et al. (2006), the additive rule-of-thumb derived for two-stage analyses of repeated measurements, which is the best-performing heuristic for slope-based designs.
- Iddi & Donohue (2022), for the retention-weighted calculations implemented in
longpower. - National Research Council (2010), on prevention and treatment of missing data.
- Mallinckrodt et al. (2008), on MMRM under dropout.
- Diggle & Kenward (1994), on informative dropout.
- Friede & Kieser (2006), on sample-size recalculation as an alternative to guessing the dropout rate in advance.
- The
longpowerpackage’s attrition functions.