# Power and Sample Size

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Power and Sample SizeBoulder 2006Benjamin Neale

To Be AccomplishedIntroduce concept of power via correlation coefficient () exampleIdentify relevant factors contributing to powerPractical:Empirical power analysis for univariate twin model (simulation)How to use mx for power

Simple exampleInvestigate the linear relationship (r)between two random variables X and Y: r=0 vs. r0 (correlation coefficient).

draw a sample, measure X,Y calculate the measure of association r (Pearson product moment corr. coeff.) test whether r 0.

How to Test r 0assumed the data are normally distributeddefined a null-hypothesis (r = 0)chosen a level (usually .05)utilized the (null) distribution of the test statistic associated with r=0t=r [(N-2)/(1-r2)]

How to Test r 0Sample N=40r=.303, t=1.867, df=38, p=.06 =.05As p > , we fail to reject r = 0

have we drawn the correct conclusion?

= type I error rate probability of deciding r 0(while in truth r=0) a is often chosen to equal .05...why?

DOGMA

N=40, r=0, nrep=1000 central t(38), a=0.05 (critical value 2.04)

Observed non-null distribution (r=.2) and null distribution

In 23% of tests of r=0, |t|>2.024 (a=0.05), and thus draw the correct conclusion that of rejecting r = 0.

The probability of rejecting the null-hypothesis (r=0) correctly is 1-b, or the power, when a true effect exists

Hypothesis TestingCorrelation Coefficient hypotheses:ho (null hypothesis) is =0 ha (alternative hypothesis) is 0Two-sided test, where > 0 or < 0 are one-sidedNull hypothesis usually assumes no effectAlternative hypothesis is the idea being tested

Summary of Possible ResultsH-0 trueH-0 falseaccept H-01-abreject H-0 a1-b

a=type 1 error rateb=type 2 error rate1-b=statistical power

Rejection of H0Non-rejection of H0H0 trueHA true

PowerThe probability of rejection of a false null-hypothesis depends on: the significance criterion ()the sample size (N) the effect size ()The probability of detecting a given effect size in a population from a sample of size N, using significance criterion

Talpha 0.05Sampling distribution if HA were trueSampling distribution if H0 were truePOWER: 1 - Standard CaseNon-centrality parameter

Talpha 0.05Sampling distribution if HA were trueSampling distribution if H0 were truePOWER: 1 - Increased effect sizeNon-centrality parameter

Impact of more conservative Talpha 0.01Sampling distribution if HA were trueSampling distribution if H0 were truePOWER: 1 - Non-centrality parameter

Impact of less conservative Talpha 0.10Sampling distribution if HA were trueSampling distribution if H0 were truePOWER: 1 - Non-centrality parameter

Talpha 0.05Sampling distribution if HA were trueSampling distribution if H0 were truePOWER: 1 - Increased sample sizeNon-centrality parameter

Effects on Power RecapLarger Effect SizeLarger Sample SizeAlpha Level shifts Type of Data:Binary, Ordinal, ContinuousMultivariate analysisEmpirical significance/permutation

When To Do Power Calculations?Generally study planning stages of studyOccasionally with negative resultNo need if significance is achievedComputed to determine chances of success