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 calculation to allow comparison with other calculators which use the Z approximation.)
Instructions: Enter parameters in the green cells. Answers will appear in the blue box 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.