Sample size calculations. $ available $ per sample

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1 Saple size calculatios = $ available $ per saple Too few aials A total waste Too ay aials A partial waste Power X 1,...,X iid oralµ A, A Y 1,...,Y iid oralµ B, B Test H 0 : µ A = µ B vs H a : µ A µ B at α = Test statistic: T = X Ȳ ŜD X Ȳ. Critical value: C such that Pr T > C µ A = µ B = α. Power: Pr T > C µ A µ B Power C 0 C

2 Power depeds o... The desig of your experiet What test you re doig Chose sigificace level, α Saple size True differece, µ A µ B Populatio SD s, A ad B. The case of kow populatio SDs Suppose A ad B are kow. The X Ȳ oral µ A µ B, Test statistic: Z = X Ȳ If H 0 is true µ A = µ B, Z oral0,1 = C = z α/2 so that Pr Z > C µ A = µ B = α For exaple, for α = 0.05, C = qor0.975 = 1.96.

3 Power whe the populatio SDs are kow If µ A µ B =, the Z = X Ȳ A 2 oral0,1 Pr X Ȳ > 1.96 = Pr X Ȳ > Pr X Ȳ < 1.96 = Pr X Ȳ > Pr X Ȳ < 1.96 = Pr Z > Pr Z < 1.96 Calculatios i R Power = Pr Z > Pr Z < 1.96 C <- qor0.975 se <- sqrt sigaaˆ2/ + sigabˆ2/ power <- 1 - porc - delta/se + por-c - delta/se Power curves Power =20 =10 =5

4 Power depeds o... Power = Pr Z > C + Pr Z < C Choice of α which affects C Larger α less striget greater power. = µ A µ B = the true effect. Larger greater power. Populatio SDs, A ad B Saller s greater power. Saple sizes, ad Larger, greater power. Choice of saple size We ostly ifluece power via ad. Power is greatest whe 2 A is as sall as possible. Suppose the total saple size N = + is fixed. is iiized whe = A A + B N ad = B A + B N For exaple: If A = B, we should choose =. If A = 2 B, we should choose = 2. e.g., if A = 4 ad B = 2, we ight use =20 ad =10

5 Calculatig the saple size Suppose we seek 80% power to detect a particular value of µ A µ B =, i the case that A ad B are kow. For coveiece here, let s preted that A = B ad that we pla to have equal saple sizes for the two groups. Power Pr Z > C Fid such that Pr Z > Thus = Pr Z > = qor0.2 = = 80%. = = [ ] 2 = = Equal but ukow populatio SDs X 1,...,X iid oralµ A, Y 1,...,Y iid oralµ B, Test H 0 : µ A = µ B vs H a : µ A µ B at α = ˆ p = s 2 A 1+s2 B ŜD X Ȳ = ˆ p Test statistic: T = X Ȳ ŜD X Ȳ. I the case µ A = µ B, T follows a t distributio with + 2 d.f. Critical value: C = qt0.975, +-2

6 Power: equal but ukow pop SDs Power = Pr X Ȳ ˆ p > C I the case µ A µ B =, the statistic X Ȳ ˆ p This distributio has two paraeters: degrees of freedo as before the o-cetrality paraeter, follows a o-cetral t distributio C <- qt0.975, se <- siga * sqrt 1/ + 1/ power <- 1 - ptc, +-2, cp=delta/se + pt-c, +-2, cp=delta/se Power curves Power = 20 = 10 = 5 kow SDs ukow SDs

7 A built-i fuctio: power.t.test Calculate power or deterie the saple size for the t-test whe: Saple sizes equal Populatio SDs equal Arguets: = saple size delta = = µ 2 µ 1 sd = = populatio SD sig.level = α = sigificace level power = the power type = type of data two-saple, oe-saple, paired alterative = two-sided or oe-sided test Exaples A. = 10 for each group; effect = = 5; pop SD = = 10 power.t.test=10, delta=5, sd=10 = 18% B. power = 80%; effect = = 5; pop SD = = 10 power.t.testdelta=5, sd=10, power=0.8 = = 63.8 = 64 for each group C. power = 80%; effect = = 5; pop SD = = 10; oe-sided power.t.testdelta=5, sd=10, power=0.8, alterative="oe.sided" = = 50.2 = 51 for each group

8 Ukow ad differet pop SDs X 1,...,X iid oralµ A, A Y 1,...,Y iid oralµ B, B Test H 0 : µ A = µ B vs H a : µ A µ B at α = Test statistic: T = X Ȳ s 2 A + s2 B To calculate the critical value for the test, we eed the ull distributio of T that is, the distributio of T if µ A = µ B. To calculate the power, we eed the distributio of T give the value of = µ A µ B. We do t really kow either of these. Power by coputer siulatio Specify,, A, B, ad = µ A µ B, ad the sigificace level, α. Siulate data uder the odel. Perfor the proposed test ad calculate the P-value. Repeat ay ties. Exaple: = 5, = 10, A = 1, B = 2, = 0.0, 0.5, 1.0, 1.5, 2.0 or 2.5.

9 = 0 = 0.5 = 1.0 = 1.5 = 2.0 = Power

10 Deteriig saple size The thigs you eed to kow: Structure of the experiet Method for aalysis Chose sigificace level, α usually 5% Desired power usually 80% Variability i the easureets If ecessary, perfor a pilot study, or use data fro prior experiets or publicatios The sallest eaigful effect Reducig saple size Reduce the uber of treatet groups beig copared. Fid a ore precise easureet e.g., average survival tie rather tha proportio dead. Decrease the variability i the easureets. Make subjects ore hoogeous. Use stratificatio. Cotrol for other variables e.g., weight. Average ultiple easureets o each subject.