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Sample size determination. Nick Barrowman, PhD Senior Statistician Clinical Research Unit, CHEO Research Institute March 29, 2010. Outline. Example: lowering blood pressure Introduction to some statistical issues in sample size determination Two simple approximate formulas
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Sample size determination Nick Barrowman, PhD Senior Statistician Clinical Research Unit, CHEO Research Institute March 29, 2010
Outline • Example: lowering blood pressure • Introduction to some statistical issues in sample size determination • Two simple approximate formulas • Descriptions of sample size calculations from the literature
Example • Physicians design an intervention to reduce blood pressure in patients with high blood pressure • But does it work? Need a study. • How many participants are required? • Too few: may not detect an effect even if there is one. • Too many: may unnecessarily expose patients to risk.
The null hypothesis • For intervention studies, the null hypothesis is usually this: on average there is no effect. • “Innocent until proven guilty” • The physicians who designed the intervention believe the null hypothesis is false. • The study is designed to test the null hypothesis. • Often write H0 for the null hypothesis.
The study • The population is considered to be all people who might be eligible for the intervention (might depend on age, other medical conditions, etc.) • Study participants are viewed as a sample from this population. • Suppose for each study participant we measure blood pressure at baseline, and after 6 weeks of intervention • Outcome is change in blood pressure • H0 is that mean change in BP is 0.
Calculation Sample mean of the change in blood pressure Inference Population vs. sample Random sample Population Population mean of the change in blood pressure
mean ± 1 standard deviation Probability distributions Population distribution of change in blood pressure Recall that variance is the square of the standard deviation, often written as s2
standard error SE = SD N standard deviation Sampling distributionof mean change in blood pressure (N=10) Increasing sample size reduces the variability of the sample mean.
Variance and sample size • As we’ve seen, increasing the sample size is akin to reducing the variance • Equivalently, reducing the variance (e.g. using a more precise measurement device) can reduce the sample size requirements
Hypothesis test Sampling distribution of the mean under the null hypothesis, a.k.a. the null distribution
Hypothesis test Reject the null hypothesis if the observed mean is far in the tails of the null distribution, i.e. we have ruled out chance Observed mean Rejection region
Possible scenarios or or
Four possible scenarios or or
Four possible scenarios or or
Four possible scenarios or or
Four possible scenarios or or
Four possible scenarios or or
Type-I error If the null hypothesis is true, the rejection region of the test representstype-I error. The probability of type-I error is the area of the red region below, and is denoted by .
Type-II error is failing to reject the null hypothesis when it is false. The probability of type-II error is denoted . It depends on how big the true effect is Sample size calculations require specification of an alternative hypothesis, which indicates the size of effect we would like to detect Type-II error
Sample size calculations depend on the tradeoff between type-I and type-II error. We usually fix the probability of type-I error (alpha) at 5% and then try to minimize the probability of type-II error (beta). Define Power = 1 – beta We want to maximize power One way to do this is by increasing the sample size Relationship between type-I and type-II error
An approximate formula for the blood pressure example • Suppose the variance in the change in blood pressure, sigma2, is the same for the null and alternative hypotheses • Suppose alpha is fixed at 0.05 and we use two-sided tests (allowing for the possibility that blood pressure could be either increased or decreased by the intervention) • Then we will have approximately 80% power to detect a mean change in blood pressure delta if we enroll N participants, where N = 8 sigma2 / delta2 (approximately)
Example • Suppose the standard deviation of the change in blood pressure is anticipated to be 7 mmHg (so the variance is 49) • Suppose we fix alpha at 0.05 and we’d like to have approximately 80% power to detect a mean change of 5 mmHg • Then we would need about 16 participants
When there are two groups • So far, the example has used a single group of study participants • Usually we want to compare two groups: a control group that receives “standard of care” or placebo, and an experimental group that receives a new intervention • This is how most randomized controlled trials are set up • In this case, delta is the difference between the means of the two groups. • For simplicity, assume that the variance is the same in the two groups.
An approximate sample size formula for the case of two groups • A similar approximate formula applies, again assuming alpha=0.05 and power=80%: Nper group = 16 sigma2 / delta2 (approximately) • Careful! This is the required sample size per group. • Also, note that the constant is double what is was for the case of a single group. • So the total sample size is 4 times as large.
Example • Suppose we want to compare patients randomized to placebo with patients randomized to a new intervention • Suppose the standard deviation is anticipated to again be 7 mmHg (so the variance is 49) • Suppose we fix alpha at 0.05 and we’d like to have approximately 80% power to detect a change of 5 mmHg • Then we would need about 32 participants per group, for a total of about 64 participants
Summary • increases with variance • decreases with size of effect to detect • decreases with probability of type-I error, alpha • decreases with probability of type-II error, beta Required sample size …
Sample size determination has many other aspects • Different types of outcomes: dichotomous (e.g. mortality), time-to-event (e.g. survival time), etc. • Different designs: observational studies (e.g. case-control), surveys, prevalence studies • Practical considerations: e.g. costs, feasibility of recruitment
α = Probability of type-I error Review: A comedy of errors … Probability of a false conviction (Rejecting the null hypothesis when it is in fact true.) Power = 1 –β Probability of a true conviction (Rejecting the null hypothesis when it is in fact false.)
