Sample Size I: 1 Sample Size Determination In the Context of Estimation

Sample Size I: 2 For a confidence interval on a population mean, , the width of the interval depends upon: The confidence level (1 –  )  confidence coefficient, z 1-  /2 or t 1-  /2;n-1 The population standard deviation,  or it’s estimator, s The sample size, n or Width of interval is 2 times this !

Sample Size I: 3 In fact, the width of a confidence interval, w, is w = 2 [ z 1-  /2 (  /  n) ] ( ) x x + z 1-  /2 (  /  n) x – z 1-  /2 (  /  n) w

Sample Size I: 4 Sample Size in the context of Estimation When planning a research study, a common first question is: “What sample size is necessary for a good estimate?” Three pieces of information are needed: 1.What precision is required – that is, what is the desired width of the confidence interval? (This is what is meant by “good”) 2.What is our desired confidence level, 1 –  ? 3.What is the underlying variability in the population, e.g., what is the standard deviation,  ?

Sample Size I: 5 Since we know the width of the interval is we can use algebra to solve for the sample size, n:

Sample Size I: 6 Let’s illustrate with an example: Previous studies have shown the standard deviation of test scores to be 25 points. How large a sample is needed to find a 95% confidence interval for the mean test score, with a width of 3 points? ( ) x w = 3

Sample Size I: 7 We have a desired width, w = 3 points the standard deviation,  = 25 points, a confidence level, 1 –  =.95  z 1-  /2 = z.975 = 1.96 We can solve for n: n = { 4(z 1-a/2 ) 2 (  2 ) / w 2 } = { 4 (1.96) 2 (25) 2 / 3 2 } = Always round up for sample size  n = 1068 We need 1068 subjects to estimate the mean test score to within  1.5 points (a width of 3 points) with 95% confidence.

Sample Size I: 8 Notes on sample size estimation: It is highly dependent upon the information you determine is important If you don’t have a good estimate of the population standard deviation use previous studies, published articles you may want to calculate n for a range of possible values of  You may need to conduct a pilot study to get a good estimate of the standard deviation before starting a larger study

Sample Size I: 9 You may need to adjust your confidence level, or your desired precision to get a more realistic sample size – or decide you can’t do the study! Continuing the example: I decide I can’t possibly recruit over 1000 subjects for my study – I may decide that I can be content with a confidence width of 5 points rather than 3: n = { 4(z 1-  /2 ) 2 (s 2 ) /w 2 } = { 4 (1.96) 2 (25) 2 /5 2 } which gives me n=385 subjects, a more do-able study.

Sample Size I: 10 I may also decide than I can accept a lower confidence level, say 90% confidence, so that z 1-  /2 = z.95 = 1.645, my sample size estimate is now: n = { 4(z 1-  /2 ) 2 (  2 ) /w 2 } = { 4 (1.645) 2 (25) 2 /5 2 } or an estimate of n=271 subjects.

Sample Size I: 11 The choice of a confidence level is arbitrary. By custom, the standard is usually 95% confidence However, if your study is more exploratory in nature, or the consequences of an error are not great,  you may choose a lower confidence level, such as 90% or even less. If the consequences of an error are great, e.g., very costly in terms of risk or money,  you may wish to choose a higher confidence level, 99%, or even higher.

Sample Size I: 12 We will revisit sample size estimation later for studies where we wish to estimate a population proportion in the context of hypothesis testing, where our goal is not only estimation, but testing a particular hypothesis. In this context, issues of power of a study will be defined.