Statistics in Health Care Management
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chapter8.pdf
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
• Need adequate sample size 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
• 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?
Determining Sample Size for Confidence
Interval Estimates
• 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
Example 8.1.
Find n for One Sample, Continuous Outcome
• 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 and estimate of
standard deviation is 20.
Example 8.1.
Find n for One Sample, Continuous Outcome
5.61 5
)20(96.1
E
Zσ n
22
Need sample size of 62 children with congenital heart disease
Example 8.3.
Find n for One Sample, Dichotomous Outcome
• Planning study to estimate proportion of
freshmen who currently smoke.
• Want estimate within 5% of the true
proportion and will use 95% confidence level.
Example 8.3.
Find n for One Sample, Dichotomous Outcome
2.384 05.0
96.1 )5.01(5.0
E
Z p)p(1n
22
Need sample size of 385 freshmen.
Formula requires estimate of proportion, p. If unknown, use p=0.5 to produce largest n (most conservative).
Example 8.5.
Find n for Two Independent Samples,
Continuous Outcome
• 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)
Example 8.5.
Find n for Two Independent Samples,
Continuous Outcome
• 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 also expect 10% attrition over 12 weeks.
Example 8.5.
Find n for Two Independent Samples,
Continuous Outcome
6.249 3
)1.17(96.1 2
E
Zσ 2n
22
i
Need n1=250 and n2=250 with complete outcome data
Example 8.5.
Find n for Two Independent Samples,
Continuous Outcome
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 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 8.7.
Find n for Two Matched Samples, Continuous
Outcome
• 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)
2
d
E
Zσ n
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
• 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 also 30% attrition over 16 weeks.
Example 8.7.
Find n for Two Matched Samples, Continuous
Outcome
3.35 3
)1.9(96.1
E
Zσ n
22
d
Need n=36 with complete outcome data
Example 8.7.
Find n for Two Matched Samples, Continuous
Outcome
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
Example 8.8.
Find n for Two Independent Samples,
Dichotomous Outcome
• Planning study to estimate the difference in
proportions of premature deliveries in mothers
who smoke as 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.
Example 8.8.
Find n for Two Independent Samples,
Dichotomous Outcome
1.507 04.0
96.1 )]12.01(12.0)12.01(12.0[
E
Z )]p-(1p)p(1p[n
2
2
2211i
Need n1=508 women who smoke during pregnancy and n2=508 who do not with
complete outcome data
Determining Sample Size for Hypothesis
Testing
a=P(Type I error)=P(Reject H0|H0 true)
b=P(Type II error)
=P(Don’t reject H0|H0 false)
• Power=1-b=P(Reject H0|H0 false)
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)
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: mm0
• 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: mm0
3271. 0.17
0.841.96
ES
ZZ n
0.17 890
150
σ
μ-μ ES
22
β-1α/2-1
01
Need sample size of 272.
Example 8.12.
Find n to Test H0: pp0
• 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: pp0
6.868 0.11
282.11.96
ES
ZZ n
11.0 )0.26-0.26(1
0.05
)p-(1p
p-p ES
22
β-1α/2-1
00
01
Need sample size of 869.
Example 8.14.
Find n1, n2 to Test H0: m1m2
• Planning study to test
H0: m1m2 vs.
H1: m1 ≠ m2 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: m1m2
232.0 0.26
0.841.96 2
ES
ZZ 2n
0.26 19.0
5
σ
μ-μ ES
22
β-1α/2-1
21
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: md0
• Planning study to test
H0: md0 vs.
H1: md ≠ 0 a=0.05
• Determine n to ensure 80% power to detect a
difference of 3 pounds difference between
diets (assume standard deviation of differences
is 9.1).
Example 8.16.
Find n to Test H0: md0
72.0 0.33
0.841.96
ES
ZZ n
0.33 9.1
3
σ
μ ES
22
β-1α/2-1
d
d
Need sample of size n=72.
Example 8.18.
Find n1, n2 to Test H0: p1p2
• Planning study to test
H0: p1p2 vs.
H1: p1 ≠ p2 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.
Example 8.18.
Find n1, n2 to Test H0: p1p2
0.486 0.135
0.841.96 2
ES
ZZ 2n
0.135 0.27)-0.27(1
0.06
p)-p(1
p-p ES
22
β-1α/2-1
21
Need samples of size n1=861 and n2=861.
