Notes on Power and Sample Size

D. Keith Williams PhD

Department of Biostatistics

Area = 0.16

1.00

Area = 0.47

2.00

Area = 0.81

3.00

3.87

Area = 0.955

Goals Remove the ‘mystery’ of power

and sample size Introduce the main ideas See how a ‘statistician’ views the

topic Provide information on how to do it

yourself, or at least get started (Its no big deal)

Buzzwords Alpha (α) = P(Type I error) = P(Conclude

experimental groups are different when they really are the same)

Beta () = P(Type II error) = P(Conclude the experimental groups are the same when they really are different)

Power = 1 - = P(Conclude experimental groups are different when they really are!)

Thoughts When planning an experiment, one

should determine a sample size that results in a statistical test powerful enough to declare significance for a reasonable difference in the means…if that difference truly exists in the population

Thoughts Generally speaking...in order to

calculate power/sample size, one needs a ‘guess’ about the pattern of the population means and an estimate of their variance

Otherwise the statistician feels that they have the role of dreaming up what the population means and variances are…YIKES!

Thoughts α 1- Variance Population means N:

Represent the five items involved in power and sample size.

One needs to recognize that that you must input 4 of these items to get the fifth.

One or the other…. Input ⇒ (α, 1- , variance,population means) ⇒

gives N Input ⇒ (α, variance,population means, N) ⇒

gives 1-

One usually ends up iterating between the above to arrive at a sample size that has a desirable level of power.

How to Help Your Statistician Help You!

Usually a study has several questions to be answered…and a statistical test that goes with each.

Prioritize which of these are most important and arguably the ones power should be based on.

Organize your best bet on the population means and their variances…or some scenarios that are clinically important that you wish to detect (if they truly exist in the population).

How to Help Your Statistician Help You! Determine what the resources of the study are…

how many subjects can you afford. Communicate this up front.

Try to do some preliminary power calculations on your own.

The Non Centrality ParameterTwo Group t-test

21

21

11nn

221 nndf

Scenario 1

Alpha =0.05, sigma=2 |mu1 – mu2| = 2, that is, a two unit diff in means

for a population Propose n1 = 10 and n2 = 10

236.2

101

101

2

2

11

21

21

nn

Rejection region for two tailed t-test alpha=0.05, df = 18

Noncentrality value =2.236, Critical value = |2.101|Table B.5, Values between 2.0 and 3.0, alpha = 0.05, df = 18Power between 0.47 and 0.81, SAS calculation 0.56195

Now one has a couple of choices

Decide that a 2 unit difference in the means is reasonable and you can afford 30 subjects in each group

87.3

301

301

2

2

11

21

21

nn

Rejection region for two tailed t-test alpha=0.05, df = 58

Noncentrality value =3.87, Critical value = |2.00|

Table B.5, Values between 3.0 and 4.0, alpha = 0.05, df = 58 (60)Power between 0.84 and 0.98, SAS calculation 0.97044

Now one has a couple of choices

Decide that a 3 unit difference in the means is reasonable and you can only afford 10 subjects in each group

35.3

101

101

2

3

11

21

21

nn

Noncentrality value =3.35 , Critical value = |2.101|Table B.5, Values between 3.0 and 4.0, alpha = 0.05, df = 18Power between 0.81 and 0.97, SAS calculation 0.88621

Now Lets Turn it AroundSample Sizes for Given Power Values

Δ = max(mu) – min(mu) Determine k = Δ/s.d. (‘effect size’) Use table B.12 Different levels of alpha = 0.2, 0.1. 0.05, and 0.01 1 – β = 0.7, 0.8, 0.9, and 0.95 r : number of treatments, 2, 3, …., 10

From Our Earlier Anova Setting..

mu1 = 20, mu2 = 15, mu3 = 15, mu4 = 12 Δ = 20 – 12 = 8, sigma = 4 K = Δ/sigma = 8/4 = 2 , r = 4

Power n per group Total N

70 6 24

80 7 28

90 9 36

95 10 40