Sample SizeEstimated Power (Percent)Sample Size / Power Considerations Today we will briefly discuss sample size and power calculations for your studies. First we will review the basic sample size and power estimation procedures and then think about how to extend these to regression settings. We will use data provided by Alex Krist to illustrate our calculations. The study is looking at outcomes relating to Prevnar, a vaccine recommended for infants and toddlers which guards against some pneumococcal bacteria that can cause life-threatening meningitis and blood infections. We will consider as the outcome of interest an indicator of whether children receive the recommended 3 vaccinations of Prevnar at any age (prevany: 1 if received all three vaccinations, 0 otherwise). The available predictors are: - Aprvn1: age in months of the child at the first Prevnar vaccination - AlwaysPCV: an indicator of whether the medical office had the vaccine available at the time the child presented for vaccine (1 if Prevnar was present at all office visits, 0 if there was at least one visit with no Prevnar) How you determine your sample size will depend on the goal of your analysis: 1. The purpose of the study is to estimate the prevalence of children receiving the recommended 3 vaccinations of Prevnar to within some specified percentage of the true prevalence with 95% confidence (i.e. to within 5 percent of the true prevalence). 2. The purpose of the study is to compare(test) the difference in the prevalence of children receiving the recommended 3 vaccinations of Prevnar among children who had access to the vaccine at all visits verses those who did not. 3. The purpose of the study is to compare(test) the difference in the prevalence of children receiving the recommended 3 vaccinations of Prevnar among children who had access to the vaccine at all visits verses those who did not after adjusting for the age in months at the first Prevnar vaccination. Sample Size Based on ESTIMATION: We assume that the prevalence estimate is approximately normally distributed and we use the standard formula for a 95% confidence interval to solve for the required sample size: mpnppp ±=−±ˆ)ˆ1(ˆ96.1ˆWe then specify m and solve for n: 22)ˆ1(ˆ96.1mppn−= For instance the following table provides the required sample size to estimate the true prevalence to within m with 95% confidence for a variety of guesses for p. Margin of Error (m) Guess of true prevalence 2 percent 5 percent 10 percent 0.10 865 139 35 0.20 1537 246 62 0.30 2017 323 81 0.40 2305 369 93 0.50 2401 385 96 Sample size based on TESTING THE DIFFERENCE IN TWO PREVALENCES: Now we assume that the two estimates of the prevalence are approximately normally distributed. The null hypothesis is H0: p1 = p1 and the alternative is that H1: p1 ≠ p2. So what do we need to specify to calculate our sample size? 1. The significance level of your test (α) 2. The power that you would like to achieve to detect the difference of interest (1-β) 3. The clinically important difference in p1 – p2 = ∆ that you’d like to be able to detect 4. Initial guesses for p1 and p2. Then the sample size required to detect a difference of ∆ in the two prevalences at the α-level with power 1-β is determined by: 22221112/12∆++=−−qpqpzqpznβα where pqppp −=+= 1,221 The table below presents required sample sizes to detect a difference of ∆ in the two prevalences at the 0.05 level with 80% power, assuming a variety of prevalence values for the children who had access to Prevnar at all vaccination visits.Scientifically Significant Difference (∆) Prevalence among children with access to the vaccine at all visits 0.05 0.10 0.15 0.25 1134 270 113 0.30 1291 313 134 0.35 1417 349 151 0.40 1511 376 165 Sample size based on TESTING THE DIFFERENCE IN TWO PREVALENCES after adjusting for additional covariates: Now, to the more realistic problem where you want to compare the odds of completing all three vaccinations for children with access to the vaccine at all visits vs. those who did not have access to the vaccine at all visits, after adjusting for the age a first vaccination. In this case, our analysis involves building a logistic regression model: AgeaccessIYYLog210)()0Pr()1Pr(βββ++=== where the coefficient of interest is which is the log odds ratio of completing all three vaccinations comparing children with access at all vaccine visits to those who did not have full access to the vaccine, after adjusting for the age a first vaccination. 1β To determine a sample size for this problem is difficult since the sample size will depend on the variability of which is a function of the sample size but also the distribution of the other covariates. 1β I will present a method for determining the sample size (or power) using a simulation study. I will illustrate this method using the Prevnar data provided by Alex. First I will fit the model to the data to determine the actual value of . 1β . logistic prevany aprvn1 alwaysPCV Logistic regression Number of obs = 107 LR chi2(2) = 13.23 Prob > chi2 = 0.0013 Log likelihood = -67.321306 Pseudo R2 = 0.0895 ------------------------------------------------------------------------------ prevany | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- aprvn1 | .7960748 .0599193 -3.03 0.002 .6868869 .9226192 alwaysPCV | 1.145634 .5064119 0.31 0.758 .4817077 2.724632 ------------------------------------------------------------------------------After adjusting for the age of the child at the first vaccination, we estimate that the odds of complete vaccination among children with full access to the vaccine are approximately 15% greater than the odds for children without full access to the vaccine. This difference in the odds is not statistically significant, but may be clinically important. We would like to design a larger study that has power to detect this finding. For this purpose, power is defined to be: Pr(rejecting H0: =0 | = log(1.15)). 1β1β So if we can generate datasets where = log(1.15), then we