Powerofahypothesistest

1、Power of a hypothesis testP(reject Ho| Hiis true)1 -P(type II error)1 -BThat is, the power of a hypothesis test is theprobability that it will reject when it to.PowerP N(0,1)|/20IPowers SIExample: Post-hoc power calc for Gum studyFor the group-A gum data our testdid not reject the null hypothesis of

2、 nomean cha nge in DMFS.? WE KNOW: our test does not furnish us good evide nee of a cha nge inDMFS.? WE DON T KNOW: Whether or not the DMFS truly cha nges because thelack of evide nee could be the result of either:a. the mean DMFS truly doesn t change, orb. the test was n t powerful eno ugh to provi

3、deevide nee of cha nge.We can assess b. by estimati ng the power of our test.WE KNOW: n=25,=0.05ASSUME:? true average change in the population is 1 DMFS? true populati on sta ndard deviati on isa =5.37.P N(0,1)1.96|1 0|P(Z 1.03) P(Z 1.03).1525.37/J25Only 15% power. This implies the truth could be a.

4、 or b.Example: Post-hoc power calc (con ti nu ed)Suppose now that we had see n the same result (did not reject n ullhypothesis of no mean cha nge), but the sample size had bee nn=250.The power from this hypothesis test would have bee n:P N(0,1)1.96110|P(Z 0.98).8365.37/J250This says that if there tr

5、uly were a change in dmfs of1 or greater, our test probably (83.6% cha nee) would have rejectedthe n ull hypothesis.This yields more evidenee for us to“ accept ” thhypothesis.Importa nt Poi nt:En suri ng reas on ably high power of your test will not only increase your cha nee of rejecti ng null hypo

6、thesis, it will alsofacilitate interpretation of your result should your test fail toreject.iFactors that affect the power of a test forPoweriPowerTas|o-gi|TPower Tasn TPower Tas0Power TasaSample Size CalculationWe can invert the power formula to find the mini mum n thatwill give a specified power.T

7、o have power 1- p to reject for a test with sig nifica nee levelto reject the null hypothesis Ho:卩=o, (in favor of H1:卩 玮then the sample size should be at least/2Example:In our previous example, to have 80% power to detect adiffere nee of 1 DMFS, the sample size should be at least02so should en roll

8、 227 kids.To have 80% power to find a differe nee of2 DMFS, thesample size should be5.3721.960 0.842225.3721.960 0.842226.4n256.6Comp onents in to a sample-size calculati on1. the desired powerl-卩2. the significanee levela3. the population standard deviationa4. thedifferenee in the means |o-训卩1. The

9、 desired powerl-卩：Comm on“i ndustry sta ndard”is mini mum 80%.Tests attempti ng to dem on strate evide nee of equality (instead of differe nces) will sometimes specify higher powers(95%)2. Significanee level :aUsual choices area= 0.05 or a = 0.01.Sometimes adjustme nts for multiple testi ng will lea

10、d tospecify ing other levels fora.3. Population standard deviationThe populati on sta ndard deviati on will not be known, andmust be estimated from previous studies. These estimatesshould be con servative (err on the high side).Example: gum dataWe estimated ausing s from a sample of size n=25.The 95

11、% con fide nee in terval for ain this case*would be (4.19, 7.47), so we see usinga= 5.37 may be an underestimate of the true populati on SD.Suppose we assumecb= 5.37, so used a sample of 227 inhopes to have 80% power to detect a differe nee of 1 dmfs.BUT, say that areally was 6.00. The n our true po

12、wer wouldbe onlyA con servative method is to use the upper 80%con fide nee limit for a as an estimate fora which(T :P N(0,1)1.96|1 0|6.00 . 227P(Z 0.55).709see Rosner, section 6.7 for details of calculationis give n by(n1)s2/n 1,0.204. Differenee in the means |o-训 卩Specifyi ng the alter native hypot

13、hesis mea n is themost tricky part of the calculati on, and the choice cangreatly affect the power estimates (and thussample-size estimates).Ideally one should specify1to be themi ni mal eli nicallysig nifica nt differe nee.Your study will have reas on able power to find a differenee of the size you specify in1. However, if卩the true issmaller than1yourjistudy has a g