Sample Size Estimation Wu Gong, MS, MD Wu.gong@Vanderbilt.edu Department of Biostatistics Vanderbilt University Medical Center

Summary Basic Considerations Common Concepts and Terms Two Examples: The t Test And The Chi-Squared Test

Why sample? Often the number of people who meet the selection criteria is too large, and there is a need to select a sample (subset) of the population for study. Generalizing the study findings by designing a study to allow inferences from findings observed in a sample to be applied to a population.

How many subjects to sample? If the sample size is too small, the study may fail to answer its research questions. If the sample size is too large, the study will be too costly.

Research Hypothesis A research hypothesis summarizes the three main element of the study. The target population is the population from which the study sample will be pulled from. It is the population that the study conclusion will be generalized to apply on. The predictor is the intervention, the treatment. It is often a dichotomous variable (YES or NO). The outcome variable is the measurement of the effect of the treatment. It could be a continuous variable, for example, CD+ cell count. It could also be dichotomous variable, for example, died or alive.

Examples of Research Hypothesis A drug treatment will improve FEV1 (forced expiratory volume in 1 second) for asthma patients than the control group. A practice will reduce the risk of back pain for people than the control group. Research hypothesis often states the effect that you want to prove.

The Null Hypothesis The null hypothesis is the opposite of the research hypothesis and is what the investigator hopes to disprove. It is a propose that there is no difference between the groups being compared. By assuming that there is no association between treatment and outcome, the statistical test will estimate the probability that the observed association is due to chance.

The Alternative Hypothesis The alternative hypothesis is the same as the research hypothesis and is what the investigator hopes to prove. The alternative hypothesis is accepted by default if the null hypothesis be rejected.

Type I Error Sometimes by chance alone a sample is not representative of the population and the results in the sample do not reflect reality in the population. A type I error is a false-positive conclusion occurs if the investigator rejects a null hypothesis that is actually true in the population. The maximum probability of committing a type I error is called α (alpha). The alpha is often set as 0.05. It means we design a test which we allow a maximum 5% type I error (false-positive conclusion).

Type II Error Type II error is a false-negative error if an investigator fails to reject a null hypothesis that is actually false in the population. The probability of making a type II error is called β The quantity of 1- β is called power. The power is often set as 0.8, which means we often design a study with at least 80% chance to detect the difference. Although type I and type II errors can never be avoided entirely, the investigator can reduce the errors by increasing the sample size.

Effect Size The effect size the is the actual magnitude of the association between the treatment and outcome. The larger the effect size, the more likelihood that a study will be able to detect the association. On the other side, if the size of the association is small, it will be hard to detect the difference if the study has the same sample size. If the effect size (treatment effect) is big, we will need a smaller number of sample, and if the effect size (treatment effect) is small, we will need a large number of sample.

Variability The greater the variability or spread in the outcome variable among the subjects, the more likely the values in the comparing groups overlap, and the it is more difficult to demonstrate that there exists an overall difference. The variability is often measured with standard deviation.

Sample Size for The t Test: Research Question Aim: Compare the treatment group and the control group regarding a measurement called FEV1 in asthma patients. Existing facts: Previous study shows that the mean of REV1 in control group is 2.0 liters, with a standard deviation of 1.0 liter. Study design: The study is designed to be able to detect a 10% difference in mean FEV1 between the two group.

Sample Size for The t Test: Calculation Null Hypothesis: After treatment, the outcome measurement FEV1 is the same in patients of two groups. Alternative Hypothesis: The means of FEV1 in the two groups are different. Effect Size = 0.2 liters (10%*2.0 liters). Standard Deviation of FEV = 1.0 liter. Two-sided α=0.05, allowing 5% type I error. Power = 0.8, β=1-power=0.2, at least 80% chance to detect the difference. Calculated Sample Size = 393 patients on each arm.

Sample Size for The Chi-Squared Test: Research Question Aim: Compare two practices regarding the risk of back pain. Existing Facts: The literature review suggests that the control group has a risk of back pain 30%. Study design: The study is designed to answer the questions if the treatment practice group has lower risk of developing back pain which means that the treatment practice will reduce at least 10% absolute risk of back pain.

Sample Size for The Chi-Squared Test: Calculation Null Hypothesis: The proportions of people who have back pain are same in the two practice groups. Alternative Hypothesis (two-sided): the proportions of back pain people in the two practice groups are different. The back pain proportion in the control group: 30%. The back pain proportion in the treatment group: 20% (30%-10%). Two-sided α=0.05, allowing 5% type I error. Power = 0.8, β=1-power=0.2, at least 80% chance to detect the difference.

Sample Size for The Chi-Squared Test: Result Calculated Sample Size = 293 patients on each arm.

References Hulley, Stephen B., et al. Designing Clinical Research http://www.bjcancer.org/Sites/Uploaded/File/2016/11/30636160943115667 8937582961.pdf Dupont,William D. & Plummer, Walton D. Jr. PS: Power and Sample Size Calculation. http://biostat.mc.vanderbilt.edu/wiki/Main/PowerSampleSize