Power and Sample Size Bios 662

Transcription

1 Power and Sample Size Bios 662 Michael G. Hudgens, Ph.D. mhudgens :06 BIOS Power and Sample Size

2 Outline Introduction One sample: continuous outcome, Z test continuous outcome, t test binary outcome, Z test binary outcome, exact test BIOS Power and Sample Size

3 Introduction Read Chs 5.8 and of text In a test of a hypothesis, we are testing whether some population parameter has a particular value H 0 : θ = θ 0, where θ 0 is a known constant Generally, H A : θ θ 0 BIOS Power and Sample Size

4 Introduction Once the data are collected, we will compute a statistic related to θ, say S(ˆθ) S(ˆθ) is a random variable, since it is computed from a sample (and hence it has a probability distribution) BIOS Power and Sample Size

5 Power Pr[Type I error] = α = Pr[S(ˆθ) C α H 0 ] Pr[Type II error] = β = Pr[S(ˆθ) / C α H A ] P ower = 1 β = Pr[S(ˆθ) C α H A ] BIOS Power and Sample Size

6 One sample Z test Example: One sample test Study: collect data on continuous outcome (Y ) on N individuals E(Y ) = µ, V (Y ) = σ 2 H 0 : µ = µ 0 vs H A : µ > µ 0 S(ˆθ) = Z = Ȳ µ 0 σ/ N BIOS Power and Sample Size

7 One sample Z test Pr[S(ˆθ) C α H 0 ] = Pr[Z > z 1 α H 0 ] = α Pr[S(ˆθ) C α H A ] = Pr[Z > z 1 α H A ] = 1 β Choose a value µ A H A Q: what sample size do we need to detect this alternative with 1 β power? BIOS Power and Sample Size

8 One sample Z test Under H A : µ = µ A Note Thus Z = Y µ A σ/ N N(0, 1) Z + µ A µ 0 σ/ N = Y µ 0 σ/ N = Z Z N ( ) µ A µ 0 σ/ N, 1 BIOS Power and Sample Size

9 Distribution of Z-statistic under H 0 and H A Density H 0 :µ=µ 0 H A :µ=µ 0 +3σ N Pr(Z>1.65 Ha) Pr(Z>1.65 Ho) Z BIOS Power and Sample Size

10 Power is given by One sample Z test 1 β = Pr[Z > z 1 α µ = µ A ] Therefore = Pr[Z > z 1 α + z 1 α + N(µ0 µ A ) σ N(µ0 µ A ) σ µ = µ A ] = z β = z 1 β BIOS Power and Sample Size

11 Equivalently For a two sided test One sample Z test N = σ2 (z 1 α + z 1 β ) 2 (µ 0 µ A ) 2 N = σ2 (z 1 α/2 + z 1 β ) 2 (µ 0 µ A ) 2 BIOS Power and Sample Size

12 Equivalent form One sample Z test N = ( z1 α/2 + z 1 β ) 2 where = µ 0 µ A σ is called the standardized distance or difference BIOS Power and Sample Size

13 One sample Z test To apply this formula, we need to know α, β, µ 0 µ A, and σ 2 Example: a study to determine the effect of a drug on blood pressure. BP measured, drug administered, and BP measured 2 hours later Y = (BP after BP before ) H 0 : µ = 0 vs H A : µ 0 BIOS Power and Sample Size

14 One sample Z test Choose α = 0.05, 1 β = 0.9 Estimate of σ 2 : need data from literature or pilot study; note we need variance of difference after-before The choice of µ A is a subject matter decision. In example: a drug that changes BP only 1 mmhg would not be of practical importance, but a drug that changes BP 5 mmhg might be of interest BIOS Power and Sample Size

15 One sample Z test Suppose α = 0.05, 1 β =.9, σ 2 = 225, and µ A = 5 225( )2 N = 5 2 = Often compute N for various different values of α, β, σ 2, and µ A Note ( ) , so that N BIOS Power and Sample Size

16 One sample Z test α 1 β σ 2 µ A N α 1 β σ 2 µ A N BIOS Power and Sample Size

17 One sample Z test α N 1 β N σ 2 N µ 0 µ A N BIOS Power and Sample Size

18 One sample Z test Sometimes N is fixed and we estimate the power of the test N = σ2 (z 1 α/2 + z 1 β ) 2 (µ 0 µ A ) 2 z 1 β = µ 0 µ A N z σ 1 α/2 { µ 1 β = Φ 0 µ A } N z σ 1 α/2 BIOS Power and Sample Size

19 One sample Z test Investigator says there are only 50 patients, α =.05, σ 2 = 225, µ 0 µ A = 5 z 1 β = = β = BIOS Power and Sample Size

20 One sample t test In practice, σ is not know, so we use a t test instead of a Z test Results above should be viewed as approximations in this case Power of t test is as follows BIOS Power and Sample Size

21 One sample t test Need the following result: if U N(λ, 1) and V χ 2 ν where U V, then U V/ν t ν,λ i.e., a non-central t distribution with df ν and noncentrality parameter λ BIOS Power and Sample Size

22 One sample t test Consider H 0 : µ = 0 versus H A : µ = µ A for µ A 0 Under H A, Thus Recall Y N(µ A, σ 2 /N) Y N σ (N 1)s 2 σ 2 N(µ A N σ, 1) χ 2 N 1 BIOS Power and Sample Size

23 Since Y s 2, One sample t test T = where λ = µ A N/σ Y s/ N t N 1,λ So power of two-sided t-test for µ A > 0 where T t N 1,µA N/σ Pr[T t N 1,0;1 α/2 ] BIOS Power and Sample Size

24 One sample t test: R # by hand > 1-pt(qt(.975,49), 49, 5/15*sqrt(50)) [1] > power.t.test(n=50, sd=15, delta=5, type="one.sample") One-sample t test power calculation n = 50 delta = 5 sd = 15 sig.level = 0.05 power = alternative = two.sided BIOS Power and Sample Size

25 proc power; onesamplemeans mean = 5 ntotal = 50 stddev = 15 power =.; run; One sample t test: SAS The POWER Procedure One-sample t Test for Mean Fixed Scenario Elements Distribution Normal Method Exact Mean 5 Standard Deviation 15 Total Sample Size 50 Number of Sides 2 Null Mean 0 Alpha 0.05 Computed Power Power BIOS Power and Sample Size

26 Null hypothesis One sample Z test: Binary Outcome H 0 : π = π 0 Test statistic Z = ˆp π 0 π0 (1 π 0 )/N Sample size formula ( z 1 α/2 π0 (1 π 0 ) + z 1 β πa (1 π A ) N = (π A π 0 ) 2 ) 2 BIOS Power and Sample Size

27 One sample Z test: Binary Outcome Example: Study of risk of breast cancer in sibs. Prevalence in general population of women years old: 2% Sample sisters of women with breast cancer H 0 : π = 0.02 vs H A : π.02 BIOS Power and Sample Size

28 One sample Z test: Binary Outcome Suppose α =.05, 1 β =.9, π A =.05 Then N = ( (.98) ) 2.05(.95) (.05.02) 2 = BIOS Power and Sample Size

29 One sample Z test with binary outcome: SAS proc power; onesamplefreq test = z method = normal nullp =.02 p =.05 power =.9 ntotal =.; run; The POWER Procedure Z Test for Binomial Proportion Fixed Scenario Elements Method Normal approximation Null Proportion 0.02 Binomial Proportion 0.05 Nominal Power 0.9 Number of Sides 2 Alpha 0.05 Actual N Power Total BIOS Power and Sample Size

30 One sample exact test: binary outcome What is power of exact test? Let Y, the number of successes, be test statistic. Under H 0, Y Binomial(N, π 0 ) Under H A, Y Binomial(N, π A ) Power Pr[Y y 1 α/2 π = π A ] + Pr[Y y α/2 π = π A ] where y α/2 and y 1 α/2 determined as in Counting Data slides BIOS Power and Sample Size

31 One sample exact test: binary outcome Exact example. Suppose N = 20, π 0 =.2, π A =.5, α = What is exact power of two-sided test? Based on CDF of Binomial(20,.2), choose y α/2 = 0 and y 1 α/2 = 9 such that power equals Pr[Y 9 π =.5] + Pr[Y 0 π =.5] = BIOS Power and Sample Size

32 One sample exact test, binary outcome: SAS proc power; onesamplefreq test = exact method = exact nullp =.2 p =.5 power =. ntotal = 20; run; Computed Power Lower Upper Crit Crit Actual Val Val Alpha Power BIOS Power and Sample Size