Instead of taking the effect size as the input and calculating the group sizes, this calculator takes the group sizes as inputs and calculates the effect size that the study has (1 - β) power to detect.

The effect size is calculated 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. Answer 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. S = Standard deviation of the outcome in the population N1 = Number of subjects in Group 1 N0 = Number of subjects in Group 0

Total group size = Ntotal = N1 + N0 =

Proportion of subjects in Group 1 = q1 = N1 / Ntotal =

Proportion of subjects in Group 0 = q0 = 1 - q1 =

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

Degrees of freedom = DoF = Ntotal - 2 =

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

k = √1/N1 + 1/N0 =

Non-centrality parameter = δ =

E/S = k * δ =

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

Standard normal deviate for α = Zα =

Standard normal deviate for β = Zβ =

A = (Zα + Zβ)2 =

B = 1/q1 + 1/q0 =

C = AB/Ntotal =

E/S = √C =

Because the formula used here is based on approximating the T statistic with a Z statistic, it will slightly underestimate the effect size for smaller values of Ntotal.

References:
Hulley SB, Cummings SR, Browner WS, Grady D, Newman TB. Designing clinical research : an epidemiologic approach. 4th ed. Philadelphia, PA: Lippincott Williams & Wilkins; 2013. Appendix 6A, page 73.
Chow S-C, Shao J, Wang H. Sample size calculations in clinical research. 2nd ed. Boca Raton: Chapman & Hall/CRC; 2008. Section 3.2.1, page 58.