Sample size is fixed. What is the minimum detectable effect size? Measure a continuous outcome y in each subject at the start and end of the study period. For each subject, calculate the change Δ = y_end - y_start. Compare the mean value of Δ to 0. This requires the standard deviationS(Δ). The estimate of S(Δ) should be based on data from other subjects who were followed for similar time periods. However, it can be estimated as S(Δ) = S_y√2(1-ρ) where S_y is the standard deviation of the underlying outcome measurement, assumed to be the same at both time points, and ρ is the within-subject correlation of the outcome.

This calculator derives the effect size in two different ways: first using the T statistic (with a non-centrality parameter), then using the Z statistic. The Z statistic approximates the T statistic, but provides an effect size that is slightly too small. (We provide the Z statistic calcultion to allow comparison with other calculators which use the Z approximation.)

Instructions: Enter parameters in the red cells. Answers will appear in blue below.

α (two-tailed) =		Threshold probability for rejecting the null hypothesis. Type I error rate.
β =		Probability of failing to reject the null hypothesis under the alternative hypothesis. Type II error rate.
N =		Sample size
S(Δ) =		Standard Deviation of the CHANGE in the outcome. (If you don't know S(Δ), click here to calculate it.)

If you don't know S(Δ) for your study:
What is the standard deviation of the outcome in the population?	S =
What is the within-subject correlation of the outcome?	rwithin =
	Recalculate S(Δ)
Calculated Standard Deviation of change: S(2(1-rwithin))1/2	S(Δ) =

Calculate

1. Calculation using the T statistic and non-centrality parameter:

Degrees of freedom = DoF = N_total - 1 =

The standard T value (with DoF degrees of freedom) corresponding to α = T_α =

Non-centrality parameter = δ =

E/S(Δ) = δ/√N =

2. Normal approximation using the Z statistic instead of the T statistic:

The standard normal deviate for α = Z_α =

The standard normal deviate for β = Z_β =

A =

B = (Z_α+Z_β)² =

AB/N =

E/S(Δ) = √AB/N =

References:
Rosner B. Fundamentals of Biostatistics. 4th ed. Duxbury Press; 1995. Page 221.
Chow S-C, Shao J, Wang H. Sample size calculations in clinical research. 2nd ed. Boca Raton: Chapman & Hall/CRC; 2008. Section 3.1.1, page 50.