Achieved power for comparing rates of change

When to use it

Assess a trial whose question is whether one group’s trajectory is steeper, such as a slower decline in lung function.

Not for

Spreading the same visits over a longer period helps as much as adding visits; a short follow-up cannot estimate a slope well.

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.
$s_t^2$: Variance of the visit times across the follow-up period.

Formulas

Variance of a participant’s slope

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

A participant’s own slope ignores where they started, so only within-person variance is left [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 in slopes, in outcome units per unit of time.

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.

Follow-up period $T$

Values: Positive

From the first to the last visit, in the slope’s time unit.

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 30 per group, 5 visits over 4 years, σ = 2, ρ = 0.5, and a slope difference of 0.3 per year at α = 0.05: s_t² = 2, so V = 4 × 0.5/(5 × 2) = 0.2.

Needed for analysis: 30 · To approach: 60

Open this example in the calculator

Validation

At the size the rates-of-change 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 rates of change. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-longitudinal-slopes/

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.