SAMPLE DESIGN: HOW MANY WILL BE IN THE SAMPLE—DESCRIPTIVE STUDIES ? Lu Ann Aday, Ph.D. The University of Texas School of Public Health

TYPES OF OBJECTIVES DESCRIPTIVE ANALYTICAL Describes Explains Is more exploratory Profiles characteristics of group(s) Focuses on what? Assumes no hypothesis Does not require comparisons (between groups or over time) Try to maximize precision of estimates ANALYTICAL Explains Is more explanatory Analyzes why group(s) have characteristics Focuses on why? Assumes an hypothesis Requires comparisons (between groups or over time) Try to maximize power to detect differences, if they exist

RELATING SAMPLE SIZE ESTIMATION TO STUDY OBJECTIVES Select the sample size estimation procedure that best matches the study design underlying the respective study objectives Compute the sample size required to address each objective Based on the sample sizes required to address each of the objectives, appropriate sample size adjustments, as well as time and resource constraints, recommend an overall sample size Discuss possible limitations in terms of statistical precision or power in addressing any specific study objective(s), given the recommended sample size

CRITERIA: Descriptive Studies Objective: to estimate a parameter, i.e., provide a precise estimate for selected variable(s) Framework: normal sampling distribution

NORMAL SAMPLING DISTRIBUTION Sampling Distribution: distribution of estimates, e.g., mean, for all possible simple random samples of a certain size that could be hypothetically drawn from the target population Population Mean: grand mean of all possible simple random samples of a certain size that could be hypothetically drawn from the target population

STANDARD ERROR Definition: average variation of all possible simple random samples of a certain size that could be hypothetically drawn from the target population Formula: SE = s/n, where, SE = standard error s = sample standard deviation n = sample size  = square root (sqrt)

CONFIDENCE INTERVAL Definition: range of values in which the population mean is likely to be contained, with a given level of probability, defined by the standard errors of the sampling distribution Confidence Interval Standard Errors (Z) 68 % 1.00 90 % 1.645 95 % 1.96 99 % 2.58

EXAMPLE: Mean Estimate Population Mean: 5 visits Standard Error: .50 visits Confidence Interval: 68 % = 1.00 * .50 = +/- .50 visits 90 % = 1.645 * .50 = +/- .82 visits 95 % = 1.96 * .50 = +/- .98 visits 99 % = 2.58 * .50 = +/- 1.29 visits

EXAMPLE: Proportion Estimate Population Proportion: .50, i.e., 50% Standard Error: .025, i.e., 2.5% Confidence Interval: 68 % = 1.00 * 2.5 = +/- 2.5% 90 % = 1.645 * 2.5 = +/- 4.1% 95 % = 1.96 * 2.5 = +/- 4.9% 99 % = 2.58 * 2.5 = +/- 6.4%

SAMPLE SIZE ESTIMATION: Cross-Sectional (One Group)—Proportion Formula: n = Z21-α/2 P(1-P)/d2, where, n = sample size Z21-α/2 = confidence interval P = estimated proportion d = desired precision

SAMPLE SIZE ESTIMATION: Cross-Sectional (One Group)—Proportion Example: n = Z21-α/2 P (1-P)/d2 n = 1.962 * .50(1-.50)/.052 n = 384 Note: See Table 7.1B, Aday & Cornelius, 2006, for sample size estimates based on different estimated proportions (P) and levels of desired precision (d).

SAMPLE SIZE ESTIMATION: Cross-Sectional (One Group)—Mean Formula: n = Z21-α/2 σ2/d2, where, n = sample size Z21-α/2 = confidence interval σ = estimated standard deviation d = desired precision

SAMPLE SIZE ESTIMATION: Cross-Sectional (One Group)—Mean Example: n = Z21-α/2 σ2/d2 n = 1.962 * (2.5 2) /1 2 n = 24 Note: To estimate σ when not known, estimate the inter-quartile range by dividing the possible range of values by 4, e.g., if range is 0-10, then 10/4 = 2.5.

SAMPLE SIZE ESTIMATION: Group-Comparison (Two Groups)—Proportion Formula: n = Z21-α/2[P1(1-P1) + P2(1-P2)]/d2, where, n = sample size Z21-α/2 = confidence interval P1 = estimated proportion (larger) P2 = estimated proportion (smaller) d = desired precision

SAMPLE SIZE ESTIMATION: Group-Comparison (Two Groups)—Proportion Example: n = Z21-α/2 [P1(1-P1) + P2(1-P2)]/d2 n = 1.962 [(.70)(.30) + (.50)(.50)]/.052 n = 707 (in each group)

SAMPLE SIZE ESTIMATION: Group Comparison (Two Groups)—Mean Formula: n = Z21-α/2 [2σ2]/d2, where, n = sample size Z21-α/2 = confidence interval σ = estimated standard deviation d = desired precision

SAMPLE SIZE ESTIMATION: Group Comparison (Two Groups)—Mean Example: n = Z21-α/2 [2σ2]/d2 n = 1.962 * [2 * (2.5 2)] /1 2 n = 48 (in each group) Note: To estimate σ when not known, estimate the inter-quartile range by dividing the possible range of values by 4, e.g., if range is 0-10, then 10/4 = 2.5.

SUMMARY: Steps in Estimating Sample Size – Descriptive Studies 1. Identify the major study variables. 2. Determine the types of estimates of study variables, such as means or proportions. 3. Select the population or subgroups of interest (based on study objectives and design). 4a. Indicate what you expect the population value to be. 4b. Estimate the standard deviation of the estimate.

SUMMARY: Steps in Estimating Sample Size – Descriptive Studies 5. Decide on a desired level of confidence in the estimate (confidence interval). 6. Decide on a tolerable range of error in the estimate (desired precision). 7. Compute sample size, based on study assumptions.

SAMPLE SIZE ESTIMATION: EXCEL SPREADSHEET See EXCEL file with spreadsheet for computing sample sizes.