Achieved power for a 2 × 2 crossover trial

When to use it

Assess a crossover in a stable condition, where every participant receives both treatments and the first washes out before the second.

Not for

Carryover is assumed absent. If the first treatment can affect the second period, a crossover is the wrong design, not a smaller one.

Notation

$n$: Participants in each sequence; 2n in total.
$\Delta$: Treatment difference.
$\sigma_w$: Within-participant standard deviation.
$\nu$: Degrees of freedom, 2n − 2.

Formulas

Within-participant noncentrality

\lambda=\frac{|\Delta|\sqrt n}{\sigma_w}, \qquad \nu=2n-2

Each participant is their own control, so only the within-participant variance enters [1, 2].

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

Participants per sequence $n$

Values: Integer, at least 2

Half the participants take each order, AB and BA.

Treatment difference $\Delta$

Values: Any non-zero value in outcome units

The expected difference between the treatments.

Within-participant SD $\sigma_w$

Values: Positive

From earlier crossover data, or σ√(1 − ρ) from a total SD σ and the correlation ρ between periods.

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

No carryover, no treatment-by-period interaction, and approximately normal within-participant differences.
One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Worked examples

A worked input set

With 16 participants per sequence, a difference of 10, and σ_w = 15 at α = 0.05: λ = 10 × √16/15 = 2.67 on 30 degrees of freedom.

Needed for analysis: 16 · To approach: 32

Open this example in the calculator

Validation

At the size the crossover 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 a 2 × 2 crossover trial. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-crossover/

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

References

1.Senn S. Cross-over Trials in Clinical Research. 2nd ed. Chichester: John Wiley & Sons; 2002.
2.Chow SC, Shao J, Wang H. Sample Size Calculations in Clinical Research. 2nd ed. Boca Raton: Chapman & Hall/CRC; 2008.