Achieved power for comparing average levels over time

When to use it

Assess a trial whose question is whether one group sits higher throughout follow-up, measured several times per participant.

Not for

Extra measurements help less as their correlation rises: with ρ near 1 they add almost nothing.

Notation

$n_1$: Group 1 size; group 2 has ⌈r n₁⌉.
$m$: Measurements per participant.
$\sigma$: Standard deviation of one measurement.
$\rho$: Correlation between a participant’s measurements.
$V$: Variance one participant contributes to the comparison.

Formulas

Variance of a participant’s average

V=\frac{\sigma^2[1+(m-1)\rho]}{m}, \quad \lambda=\frac{|\Delta|}{\sqrt{V(1/n_1+1/n_2)}}, \quad \nu=n_1+n_2-2

The design effect [1 + (m − 1)ρ]/m is never above 1 [1].

Power from the rejection region

\text{Power}=P(T_{\nu,\lambda}>t_{1-\alpha/\text{tails},\nu}) + I_{\text{tails}=2}P(T_{\nu,\lambda}<-t_{1-\alpha/\text{tails},\nu})

One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Parameters

Group 1 size $n_1$

Values: Integer, at least 2

Participants available in group 1; group 2 follows from the allocation ratio.

Allocation ratio $r$ Optional

Values: Positive; 1 for equal groups

Group 2 has n₂ = ⌈r n₁⌉ participants.

Difference $\Delta$

Values: Any non-zero value

Difference between the groups’ average levels.

Standard deviation $\sigma$

Values: Positive

The SD of a single measurement.

Measurements per participant $m$

Values: Integer, at least 2

Equally spaced visits, including baseline.

Correlation between measurements $\rho$

Values: At least 0 and below 1

Compound symmetry: every pair of visits correlates equally.

Significance level $\alpha$

Values: Between 0 and 1

The type I error rate for the specified test.

Test direction $s$

Values: 1 or 2

One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Assumptions

Measurements are approximately normal with compound symmetry; other correlation structures can change power in either direction.
One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Worked examples

A worked input set

With 300 per group, 5 measurements, σ = 2, ρ = 0.5, and a difference of 0.3 at α = 0.05: V = 4 × (1 + 4 × 0.5)/5 = 2.4.

Needed for analysis: 300 · To approach: 600

Open this example in the calculator

Validation

At the size the average-level calculator returns, this power reaches that calculator’s target, and one fewer falls short; it is checked against an independent calculation in R.

How to cite

StatWisely. Achieved power for comparing average levels over time. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-longitudinal-mean-level/

Cite the original references for the formulas your design uses as well.

References

1.Diggle PJ, Heagerty P, Liang KY, Zeger SL. Analysis of Longitudinal Data. 2nd ed. Oxford: Oxford University Press; 2002.