Health Care Law and Legislation, Statistics Policies

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Chapter 8

Power and Sample Size Determination

Learning Objectives

Provide examples demonstrating how the margin of error, effect size, and variability of the outcome affect sample size computations
Compute the sample size required to estimate population parameters with precision
Interpret statistical power in tests of hypothesis
Compute the sample size required to ensure high power in tests of hypothesis

Sample Size Determination

Adequate sample size is needed to ensure precision in analysis.
Sample size determined based on type of planned analysis.
Confidence interval estimate
Test of hypothesis

Determining Sample Size for
Confidence Interval Estimates (1 of 2)

Goal is to estimate an unknown parameter using a confidence interval estimate.
Plan a study to sample individuals, collect appropriate data and generate CI estimate.
How many individuals should we sample?

Confidence intervals

point estimate ± margin of error

Determine n to ensure small margin of error (precision).
Must specify desired margin of error, confidence level, and variability of parameter.

Determining Sample Size for
Confidence Interval Estimates (2 of 2)

Example 8.1. Find n for One Sample, Continuous Outcome (1 of 2)

Planning study to estimate mean systolic blood pressure in children with congenital heart disease.
Want estimate within 5 units of true mean
Will use 95% confidence level; estimate of standard deviation is 20.

Example 8.1. Find n for One Sample, Continuous Outcome (2 of 2)

Need sample size of 62 children with congenital heart disease

Example 8.3. Find n for One Sample, Dichotomous Outcome (1 of 2)

Planning study to estimate proportion of freshmen who currently smoke
Want estimate within 5% of the true proportion; will use 95% confidence level

Example 8.3. Find n for One Sample, Dichotomous Outcome (2 of 2)

Formula requires estimate of proportion, p. If unknown, use p = 0.5 to produce largest n (most conservative).
Need sample size of 385 freshmen

Example 8.5. Find n for Two Independent Samples, Continuous Outcome (1 of 5)

Planning a study to assess the efficacy of a new drug to raise HDL cholesterol
Participants will be randomized to receive either the new drug or placebo and followed for 12 weeks.
Goal is to estimate the difference in mean HDL between groups (m1 – m2).

Want estimate of the difference to be no more than 3 units
We will use a 95% confidence interval.
The estimate of the (common) standard deviation in HDL is 17.1.
We expect 10% attrition over 12 weeks.

Example 8.5. Find n for Two Independent Samples, Continuous Outcome (2 of 5)

Need n1 = 250 and n2 = 250 with complete outcome data

Example 8.5. Find n for Two Independent Samples, Continuous Outcome (3 of 5)

Need n1 = 250 and n2 = 250 with complete outcome data (at end of study)
Need to account for 10% attrition
How many subjects must be enrolled?

Example 8.5. Find n for Two Independent Samples, Continuous Outcome (4 of 5)

Need n1 = 250 and n2 = 250 with complete outcome data
Account for 10% attrition:

N (to enroll)*(% retained) = 500

Need to enroll 500/0.90 = 556

Participants enrolled N = ?

Complete study (500)

Lost to follow-up

10%

90%

Example 8.5. Find n for Two Independent Samples, Continuous Outcome (5 of 5)

Example 8.7. Find n for Two Matched Samples, Continuous Outcome (1 of 5)

Planning study to estimate the mean difference in weight lost between two diets (low-fat versus low-carb) over 8 weeks
A crossover trial is planned where each participant follows each diet for 8 weeks and weight loss is measured.
Goal is to estimate the mean difference in weight lost (md).

Need to specify the margin of error (E), decide on the confidence level, and estimate the variability in the difference in weight lost between diets

Example 8.7. Find n for Two Matched Samples, Continuous Outcome (2 of 5)

Want estimate of the difference in weight lost to be within 3 pounds of the true difference
We will use a 95% confidence interval.
The standard deviation of the difference in weight lost is estimated at 9.1.
Expect 30% attrition over 16 weeks.

Example 8.7. Find n for Two Matched Samples, Continuous Outcome (3 of 5)

Need n = 36 with complete outcome data

Example 8.7. Find n for Two Matched Samples, Continuous Outcome (4 of 5)

Need n = 36 with complete outcome data
Account for 30% attrition:

N (to enroll)*(% retained) = 36

Need to enroll 36/0.70 = 52

Participants enrolled N = ?

Complete study (36)

Lost to follow-up

30%

70%

Example 8.7. Find n for Two Matched Samples, Continuous Outcome (5 of 5)

Example 8.8. Find n for Two Independent Samples, Dichotomous Outcome (1 of 2)

Planning study to estimate the difference in proportions of premature deliveries in mothers who smoke compared to those who do not
Want estimate within 4% of the true difference, will use 95% confidence level and assume that 12% of infants are born prematurely

Need n1 = 508 women who smoke during pregnancy and n2 = 508 who do not, with complete outcome data

Example 8.8. Find n for Two Independent Samples, Dichotomous Outcome (1 of 2)

Determining Sample Size for
Hypothesis Testing (1 of 2)

a = P(Type I error) = P(Reject H0|H0 true)
b = P(Type II error)

= P(Do not reject H0|H0 false)

Power = 1 – b = P(Reject H0|H0 false)

Determining Sample Size for
Hypothesis Testing (2 of 2)

b and power are related to the sample size, level of significance (a), and the effect size (difference in parameter of interest under H0 versus H1).

a, b, and Power

Determining Sample Size for
Hypothesis Testing

b and power are related to the sample size, level of significance (a), and the effect size (difference in parameter of interest under H0 versus H1).
Power is higher with larger a.
Power is higher with larger effect size.
Power is higher with larger sample size.

Example 8.11.
Find n to Test H0: m = m0 (1 of 2)

Planning study to test

H0: m = $3302 vs.

H1: m ≠ $3302 at a = 0.05

Determine n to ensure 80% power to detect a difference of $150 in mean expenditures on health care and prescription drugs (assume standard deviation is $890).

Example 8.11.
Find n to Test H0: m = m0 (2 of 2)

Need sample size of 272

Example 8.12.
Find n to Test H0: p = p0 (1 of 2)

Planning study to test

H0: p = 0.26 vs.

H1: p ≠ 0.26 at a = 0.05

Determine n to ensure 90% power to detect a difference of 5% in the proportion of patients with elevated LDL cholesterol.

Example 8.12.
Find n to Test H0: p = p0 (2 of 2)

Need sample size of 869

Example 8.14.
Find n1, n2 to Test H0: m1 = m2 (1 of 2)

Planning study to test

H0: m1 = m2 vs.

H1: m1 ≠ m2 at a = 0.05

Determine n1 and n2 to ensure 80% power to detect a difference of 5 units in means (assume standard deviation is 19.0).
Expect 10% attrition.

Example 8.14.
Find n1, n2 to Test H0: m1 = m2 (2 of 2)

Need samples of size n1 = 232 and n2 = 232
Account for 10% attrition:

N (to enroll)*(% retained) = 464

Need to enroll 464/0.90 = 516

Example 8.16.
Find n to Test H0: md = 0 (1 of 2)

Planning study to test

H0: md = 0 vs.

H1: md ≠ 0 at a = 0.05

Determine n to ensure 80% power to detect a 3-pound difference between diets (assume standard deviation of differences is 9.1).

Example 8.16.
Find n to Test H0: md = 0 (2 of 2)

Need sample of size n = 72

Example 8.18.
Find n1, n2 to Test H0: p1 = p2 (1 of 2)

Planning study to test

H0: p1 = p2 vs.

H1: p1 ≠ p2 at a = 0.05

Determine n1 and n2 to ensure 80% power to detect a difference in proportions of hypertensives on the order of 24% versus 30% in the new drug and placebo treatments.

Need samples of size n1 = 861 and n2 = 861

Example 8.18.
Find n1, n2 to Test H0: p1 = p2 (2 of 2)

)

(

384

)

(

p(1

249

)

(

)

(

507

)]

(

)

(

[

)]

)

[

271.

0.17

0.84

1.96

0.17

890

150

α/2

868

0.11

282

1.96

)

0.26

0.26(1

0.05

)

α/2

232.0

0.26

0.84

1.96

0.26

19.0

α/2

72.0

0.33

0.84

1.96

0.33

9.1

α/2

0.4

0.135

0.84

1.96

0.135

0.27)

0.27(1

0.06

p(1

α/2