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

2. 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

3. 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

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

6. 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

7. 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)

8. 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 %
90 %
95 %
99 %

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

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

11. 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

12. SAMPLE SIZE ESTIMATION: Cross-Sectional (One Group)—Proportion
Example:
n = Z21-α/2 P (1-P)/d2
n = * .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).

13. 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

14. SAMPLE SIZE ESTIMATION: Cross-Sectional (One Group)—Mean
Example:
n = Z21-α/2 σ2/d2
n = * (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.

15. 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

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

17. 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

18. SAMPLE SIZE ESTIMATION: Group Comparison (Two Groups)—Mean
Example:
n = Z21-α/2 [2σ2]/d2
n = * [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.

19. 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.

20. 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.

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