Sample size calculations How many individuals do we need??? It - PowerPoint PPT Presentation

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f b i o s t a t i s t i c s Sample size calculations How many individuals do we need??? It depends on • the size of the effect we are looking for • how certain we want to be in finding the effect and the purpose of the investigation : • obtain a specific precision of an estimate • obtain a specific ( power ) of a test (most common). 1 / 5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f b i o s t a t i s t i c s Precision We want to estimate the mean SBP level µ for females with a precision of µ ± a . µ ± 1 . 96 SD 95% confidence interval (95%) : √ n . a = 1 . 96 SD Thus i.e. √ n n = 3 . 84 × SD 2 . a 2 We need to come up with a guess of SD. In the Framingham data we find SD=25. Example : a = 3 and SD=25 gives n = 3 . 84 × 25 2 = 266 . 67 → 267 . 3 2 2 / 5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f b i o s t a t i s t i c s Test of hypotheses A test of a hypothesis H 0 can give two types of error : Type I: Reject the hypothesis even though it is true. Type II: Accept the hypothesis even though it is wrong. Probability of type I error : α = level of significance. Probability of type II error : β . 1 − β = power. Truth Conclusion Hypothesis true Hypothesis wrong Accept Correkt conclusion Type II error 1 − α β Reject Type I fejl Correkt conclusion α 1 − β 3 / 5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f b i o s t a t i s t i c s Comparison of means in two groups We determine the number of individuals in each group . Determine µ 1 , µ 2 , SD, α an β . SD 2 ( µ 1 − µ 2 ) 2 × ( z 1 − α/ 2 + z 1 − β ) 2 . = n 2 We need ∆ = µ 1 − µ 2 and SD. The needed samples size depends on : • the level of significance ( α ) • the power ( 1 − β ) and • the difference between the groups : the larger difference the smaller the needed sample size • the variation (SD) : the larger , the larger the sample size 4 / 5

u n i v e r s i t y o f c o p e n h a g e n d e p a r t m e n t o f b i o s t a t i s t i c s Example Assume that we want to detect af difference in SBP of ∆ =10 mmHg for women randomized to placebo / treatment. We want to be 90% sure to detect the difference ( 1 − β = 0 . 90 ) when testing on the 5% significance level ( α = 0 . 05 ). In the Framingham data we find (SD=25). I.e. 2 ( SD ∆ ) 2 ( z 1 − 0 . 05 / 2 + z 0 . 90 ) 2 = n 2 ( SD ∆ ) 2 ( 1 . 96 + 1 . 28 ) 2 = 2 × ( 25 10 ) 2 × 10 . 5 = 131 . 25 . = We need 132 women in each group. 5 / 5