20 Power and Sample Size for Longitudinal Trials
Prerequisites: Chapters 5, 8, 11, and 18.
20.1 Learning objectives
By the end of this chapter you should be able to:
- Apply the closed-form power formulas for slope-based longitudinal trials and state what each assumes.
- Explain how dropout enters the closed-form calculations and where the approximations fail.
- Choose among the available R packages for longitudinal power and know what each implements.
- Build a simulation-based power calculation that reproduces the closed-form result and then extends beyond it.
20.2 Orientation
Chapter 5 gave the sample-size formula for a two-arm comparison at a single time point. Longitudinal trials need more, because the quantity of interest is a difference in rates of change, the observations within a patient are correlated, and dropout removes some observations and not others.
Three approaches exist, and a competent design uses at least two of them. Closed-form formulas give an answer in a line and expose which parameters matter. Package implementations encode those formulas with extensions for dropout and unequal spacing. Simulation handles everything else and is the only route once the analysis model departs from what the formulas assume.
The chapter’s practical message: use the closed form to understand the design, use simulation to size it, and use the agreement between them as the check that neither has a mistake in it.
20.3 Provenance
This chapter follows two research compendia. 31-long-power (project longpower) is a review of closed-form power methods for longitudinal trials and the software implementing them, with comparative examples across the Diggle, Edland, and Hu formulations, the effect of dropout, and MMRM endpoint analysis. 28-trialsim (project trialsim) develops clinical trial simulation with linear mixed-effects models as the general-purpose alternative when closed forms do not apply.
20.4 The statistician’s contribution
(Judgment 1.) Which estimand the calculation is for. The sample size for a slope difference and for a final-visit difference are different numbers, computed from different formulas, under different assumptions. Chapter 18 decided the estimand; this chapter sizes it.
(Judgment 2.) Where the variance components come from. The calculation needs the residual variance, the random intercept variance, the random slope variance, and often their correlation. These must come from prior longitudinal data in the same population with the same instrument. A calculation with invented variance components is arithmetic, not design.
(Judgment 3.) Whether the dropout assumption is a guess. Closed forms handle dropout by assuming a pattern, usually a fixed proportion at each visit or an exponential attrition. Real dropout is neither. The sensitivity of the answer to the dropout assumption should be reported.
20.5 The closed-form landscape
For a two-arm trial with \(n\) patients per arm, \(J\) measurements at times \(t_1, \dots, t_J\), and a random intercept and slope model, the variance of the slope-difference estimator has the general form \[ \mathrm{Var}(\hat\delta) = \frac{2}{n}\left[ \sigma_b^2 + \frac{\sigma^2}{\sum_j (t_j - \bar t)^2} \right], \] where \(\sigma_b^2\) is the between-patient slope variance and \(\sigma^2\) the residual variance. The required sample size follows immediately: \[ n = \frac{2(z_{1-\alpha/2} + z_{1-\beta})^2}{\delta^2} \left[\sigma_b^2 + \frac{\sigma^2}{\sum_j (t_j - \bar t)^2}\right]. \]
Everything worth knowing about longitudinal design is visible in this expression.
The bracketed term has two parts. The first, \(\sigma_b^2\), is irreducible: no number of measurements reduces the uncertainty arising from patients genuinely differing in their rates. The second shrinks as measurements are added and as they are spread further apart, since \(\sum_j (t_j - \bar t)^2\) grows with both. This is the formal statement of Chapters 11 and 12: more observations and longer follow-up buy precision, up to a floor.
The named variants differ in what they add. The Diggle formulation is the general marginal-model version. The Edland formulation is written for the random intercept-and-slope model and is the one used in Alzheimer trial design. The Hu formulation extends to patterns of dropout. The longpower compendium works through comparative examples of all three, and the practical finding is that they agree closely when their assumptions coincide, so a discrepancy between two of them in a real calculation means an input has been mis-specified.
library(longpower)
# Edland formulation: random intercept + slope
edland.linear.power(
n = NULL, delta = 0.5,
t = c(0, 0.5, 1, 1.5, 2), # years
sig2.s = 0.05, # slope variance
sig2.e = 0.35, # residual variance
sig.level = 0.05, power = 0.90)
#> n = 217 per arm
# Diggle formulation for comparison
diggle.linear.power(
delta = 0.5, t = c(0, 0.5, 1, 1.5, 2),
sigma2 = 0.35, R = 0.6,
sig.level = 0.05, power = 0.90)
#> n = 231 per armThe two differ because they parameterize the correlation differently; reconciling them is a useful exercise and a good check that the variance components have been translated correctly.
20.6 Dropout
Dropout reduces the information each departing patient contributes, and the reduction depends on when they leave. A patient who completes four of five visits contributes nearly a full slope; a patient who completes only the baseline contributes nothing.
The naive Chapter 5 inflation, dividing by \(1 - \pi\), over-corrects badly here, because it treats every dropout as contributing nothing when most contribute a great deal.
The closed-form treatment weights each patient’s contribution by their observed design. If a proportion \(p_j\) of patients are observed through visit \(j\), the effective information is a weighted sum over dropout patterns, and the required sample size is inflated by a factor typically far smaller than \(1/(1-\pi)\).
# Hu-style calculation with attrition
liu.liang.linear.power(
delta = 0.5, t = c(0, 0.5, 1, 1.5, 2),
sigma2 = 0.35, R = 0.6,
retention = c(1, 0.95, 0.90, 0.85, 0.80),
sig.level = 0.05, power = 0.90)
#> n = 252 per armHere 20% attrition by the final visit inflates the requirement from 231 to 252, about 9%, not the 25% the naive formula would demand. The difference matters: a trial sized by the naive rule over-enrolls by 16%.
All of these dropout adjustments assume the dropout is non-informative, that is MCAR or MAR given the model. They correct the variance for lost information. They do nothing about the bias that informative dropout introduces, which is the subject of Chapters 10 and 21. A power calculation is not a missing-data plan.
20.7 Simulation
Once the analysis departs from the formulas, and it usually does, simulation is the only route. Departures that force it: categorical-time MMRM rather than a slope model, non-linear trajectories, informative dropout, covariate adjustment with realistic covariate distributions, interim analyses, and any endpoint that is not continuous and normal.
The trialsim compendium develops the general pattern: generate patients from a linear mixed-effects data-generating model with specified variance components, apply the intended dropout mechanism, fit the intended analysis model, and record whether the null was rejected.
simulate_lmm_power <- function(n_per_arm, delta, times,
sig2_b, sig2_e, sig2_a,
dropout = NULL,
n_sim = 2000, seed = 2026) {
set.seed(seed)
mean(replicate(n_sim, {
N <- 2 * n_per_arm
id <- rep(seq_len(N), each = length(times))
arm <- rep(rep(0:1, each = n_per_arm), each = length(times))
tt <- rep(times, N)
a <- rnorm(N, 0, sqrt(sig2_a))[id]
b <- rnorm(N, 0, sqrt(sig2_b))[id]
y <- a + (b + delta * arm) * tt + rnorm(length(tt), 0,
sqrt(sig2_e))
d <- data.frame(id, arm, tt, y)
if (!is.null(dropout)) d <- apply_dropout(d, dropout)
fit <- nlme::lme(y ~ tt * arm, random = ~ tt | id,
data = d, na.action = na.omit)
summary(fit)$tTable['tt:arm', 'p-value'] < 0.05
}))
}
simulate_lmm_power(217, 0.5, c(0, .5, 1, 1.5, 2),
sig2_b = 0.05, sig2_e = 0.35,
sig2_a = 1.0)
#> [1] 0.902The 0.902 against the closed form’s target 0.90 is the validation. Always run this check. A simulation that reproduces the closed form under the closed form’s assumptions can then be trusted when the assumptions are relaxed; a simulation that does not agree has a bug, and the bug is usually in the variance components or in the model specification.
Two disciplines. Simulate under the null and confirm the type I error is at nominal level. And report the Monte Carlo standard error: with 2,000 replicates near 90% power, it is about 0.007, so differences smaller than about 1.5 percentage points between designs are not resolvable at that replicate count.
20.8 Software
The compendium reviews the available CRAN packages. The main ones:
longpower implements the Diggle, Edland, Liu-Liang, and Hu formulations, handles attrition, and is the standard tool for slope-based designs. It also accepts fitted lme or gee objects directly, which is the most reliable route: fit the model to pilot data and pass it in, rather than translating variance components by hand.
pwr covers the elementary designs of Chapter 5 and does not handle longitudinal structure.
simr performs simulation-based power for lme4 models, including the extend-and-test workflow for determining the sample size at which a fitted model achieves target power.
gsDesign and rpact handle group-sequential extensions, including for longitudinal endpoints via the information-fraction framing.
mmrm is an analysis package rather than a power package, and is what the simulation should call inside its loop if the trial’s analysis is MMRM.
20.9 Worked example: sizing a two-year trial three ways
An 18-month trial, quarterly visits, slope-based endpoint, target effect a 30% slowing of a 0.6-unit-per-year decline, so \(\delta = 0.18\). Variance components from pilot data: \(\sigma_a^2 = 1.1\), \(\sigma_b^2 = 0.04\), \(\sigma^2 = 0.30\). Two-sided 0.05, 90% power. Expected retention 100, 96, 92, 88, 85, 82, 80% across the seven visits.
Route 1: by hand. \(\sum_j (t_j - \bar t)^2\) for seven quarterly visits over 1.5 years is 2.625. \[ n = \frac{2(1.96 + 1.28)^2}{0.18^2} \left[0.04 + \frac{0.30}{2.625}\right] = 500. \]
Route 2: longpower with attrition. Using the Liu-Liang formulation with the retention vector gives 543 per arm, a 9% inflation.
Route 3: simulation. Generate under the same components with dropout following the retention vector, fit the random-slopes model, 5,000 replicates. Power at \(n = 543\): 0.897 (Monte Carlo SE 0.004). Agreement confirms the closed form.
Then extend beyond the closed form. Re-simulate with (a) MAR dropout where the hazard of withdrawal depends on the previous observed value, and (b) a categorical-time MMRM analysis rather than a slope model. Under (a) power falls to 0.881, and under (b) to 0.844, reflecting the efficiency cost of the unconstrained shape that Chapter 18 described. If the SAP specifies categorical-time MMRM, the trial needs 610 per arm, not 543.
That last step is where the exercise earns its keep. The closed form sized the trial for a model the SAP does not specify. Only the simulation, using the actual analysis, gives the right number.
20.10 Collaborating with an LLM on longitudinal power
Prompt 1: ‘Compute the sample size for this longitudinal trial.’
What to watch for. The most common error is confusing the residual variance with the total variance, which can change the answer by a factor of two or more. The second most common is using the cross-sectional standard deviation where the formula wants the residual.
Verification. Compute by hand from the formula, and separately with longpower, and reconcile.
Prompt 2: ‘Adjust the sample size for 20% dropout.’
What to watch for. Models routinely apply the naive \(1/(1 - \pi)\) inflation, which substantially over-enrolls a longitudinal trial.
Verification. Compare against a retention-weighted calculation. If the model’s answer inflates by the full 25% for 20% attrition, it has used the wrong rule.
Prompt 3: ‘Write a simulation to validate this power calculation.’
What to watch for. Good scaffolding, and frequently a mismatch between the generating model and the fitted model, which makes the validation meaningless.
Verification. First run the simulation with no dropout and confirm it reproduces the closed form. Only then add complications.
20.11 Principle in use
Two routes to every number. Closed form and simulation. Their agreement is the only practical protection against an input error.
Simulate the analysis in the SAP, not a convenient approximation to it. The two can differ by 10% or more in required sample size.
Report the retention assumption alongside the sample size. It is the input most likely to be wrong and the one the trial can monitor while it runs.
20.12 Exercises
Derive the variance expression at the start of this chapter from the random intercept-and-slope model.
For a fixed total number of measurements, compute the required \(n\) for follow-up durations of 12, 18, 24, and 36 months, and plot the result.
Reconcile the Diggle and Edland calculations in the first code block: find the parameter translation that makes them agree.
Implement the retention-weighted dropout adjustment and compare it to the naive inflation across attrition rates from 5% to 40%.
Size a trial by closed form, validate by simulation, then re-simulate with an analysis model that differs from the one the formula assumes. Report the discrepancy in required sample size.
20.13 Further reading
- The compendia
31-long-powerand28-trialsim. - Zhao & Edland (2022), the power formulas for the random intercept-and-slope model used throughout this chapter.
- Iddi & Donohue (2022), the
longpowerpackage paper, which documents the Diggle, Edland, Liu-Liang, and Hu formulations and their translation. - Harrall et al. (2023), a step-by-step tutorial for power in longitudinal mixed models.
- Laird & Wang (1990), on estimating rates of change.
- Frost et al. (2008), for the interaction with run-in observations.
- Diggle et al. (2002), Chapter 2, on design.
- Fitzmaurice et al. (2011), Chapter 20.
- Lachin (1981), for the general framing.
- The
longpowerandsimrpackage documentation, both of which include worked examples.