# Sample Size Planning, Calculation, and Justification

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1 Sample Size Planning, Calculation, and Justification Theresa A Scott, MS Vanderbilt University Department of Biostatistics Theresa A Scott, MS (Vandy Biostatistics) Sample Size 1 / 24 Introduction After you ve decided what and whom you re going to study and the design to be used, you must decide how many subjects to sample. Even the most rigorously executed study may fail to answer its research question if the sample size is too small. If the sample size is too large, the study will be more difficult and costly than necessary while unnecessarily exposing a number of subjects to possible harm. Goal: to estimate an appropriate number of subjects for a given study design. ie, the number needed to find the results you re looking for. IMPORTANT: Although a useful guide, sample size calculations give a deceptive impression of statistical objectivity. Really only making a ballpark estimate. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 2 / 24

2 Introduction, cont d Only as accurate as the data and estimates on which they are based, which are often just informed guesses. Often reveals that the research design is not feasible or that different predictor or outcome variables are needed. TAKE HOME MESSAGE: Sample size should be estimated early in the design phase of the study, when major changes are still possible. In addition to the statistical analysis plan, the sample size section is critical to an IRB proposal and any kind of grant. 42% of R01s examined in one review paper were criticized for the sample size justifications or analysis plans. 1 Much more involved than a cut-and-paste paragraph. 1 Inouye & Fiellin, An Evidence-Based Guide to Writing Grant Proposals for Clinical Research, Annals of Internal Medicine, (2005): Theresa A Scott, MS (Vandy Biostatistics) Sample Size 3 / 24 Underlying principles Research hypothesis: Specific version of the research question that summarizes the main elements of the study the sample, and the predictor and outcome variables in a form that establishes the basis for the statistical hypothesis tests. 2 Should be simple (ie, contain one predictor and one outcome variable); specific (ie, leave no ambiguity about the subjects and variables or about how the statistical hypothesis will be applied); and stated in advance. Example: Use of tricyclic antidepressant medications, assessed with pharmacy records, is more common in patients hospitalized with an admission of myocardial infarction at Longview Hospital in the past year than in controls hospitalized for pneumonia. 2 NOTE: Hypotheses are not needed for descriptive studies more to come. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 4 / 24

3 Underlying principles, cont d Null hypothesis: Formal basis for testing statistical significance; 3 states that there is no association, difference, or effect. eg, Alcohol consumption (in mg/day) is not associated with a risk of proteinuria (>300 mg/day) in patients with diabetes. Alternative hypothesis: Proposition of an association, difference, effect. Can be one-sided (ie, specifies a direction). eg, Alcohol consumption is associated with an increased risk of proteinuria in patients with diabetes. However, most often two-sided no direction mentioned. Expected by most reviewers; very critical of a one-sided. 3 Hypothesis testing discussed in more detail in the Biostatistics: Types of Data Analysis lecture. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 5 / 24 Underlying principles, cont d General process used in hypothesis testing: Presume the null hypothesis (eg, no association between the predictor and outcome variables in the population). Based on the data collected in the sample, use statistical tests to determine whether there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis (eg, there is an association in the population). Reaching a wrong conclusion: Type I error: false-positive; rejecting the null hypothesis that is actually true in the population. Type II error: false-negative; failing to reject the null hypothesis that is actually not true in the population. Neither can be avoided entirely. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 6 / 24

4 Underlying principles, cont d Effect size: Size of the association/difference/effect you expect/wish to be present in the sample. Selecting an appropriate size is most difficult aspect of sample size planning. REMEMBER: Sample size calculation only as accurate as the data/estimates on which they are based. Find data from prior studies to make an informed guess needs to be as similar as possible to what you expect to see in your study. Pilot study/studies sometimes needed first. Good rule of thumb: choose the smallest effect size that would be clinically meaningful (and you would hate to miss). Will be okay if true effect size ends up being larger. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 7 / 24 Underlying principles, cont d Establish the maximum chance that you will tolerate of making wrong conclusions: α: probability of committing a type I error; aka level of statistical significance. β: probability of making a type II error. Power: 1 β; probability of correctly rejecting the null hypothesis in the sample if the actual effect in the population is equal to (or greater than) the effect size. Aim: choose a sufficient number of subjects to keep α and β at an acceptably low level without making the study unnecessarily large (ie, expensive or difficult). α and β decrease as sample size increases. Often α = 0.05, 0.10; β = 0.20, 0.10 (power = 80, 90%). Theresa A Scott, MS (Vandy Biostatistics) Sample Size 8 / 24

5 Additional considerations Variability of the effect size: Statistical tests depend on being able to show a difference between the groups being compared. The greater the variability (spread) in the outcome variable among the subjects, the more likely it is that the values in the groups will overlap, and the more difficult it will be to demonstrate an overall difference between them. Use the most precise measurements/variables possible. Often have >1 hypothesis, but should specify a single primary hypothesis for sample size planning. Helps to focus the study on its main objective and provides a clear basis for the main sample size calculation. Useful to rank other research questions/specific aims as secondary, etc. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 9 / 24 Calculating sample size Specific method used depends on The specific aim(s)/objective(s). The study design, including the planned number of measurements per subject. The outcome(s) and predictor(s). The proposed statistical analysis plan. Will also need to consider: Accrual/Enrollment (response rate for questionnaires). Drop-outs (ie, lost to follow-up) and missing data. Budgetary constraints. Requires you to make assumptions. Assume specific effect size (variability), α, power, etc. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 10 / 24

6 Calculating sample size for analytic studies Often want to show a significant difference/association between 2 groups. Most traditional recipe in this case: 1 State the null and 1- or 2-sided alternative hypothesis. 2 Select the appropriate statistical test based on the type of predictor and outcome variables in the hypotheses. 3 Choose a reasonable effect size (and variability, if necessary). 4 Specify α and power. 5 Use an appropriate table, formula, or software program to estimate the sample size. Most common used statistical tests for comparing 2 groups: The t-test. The Chi-squared test. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 11 / 24 Calculating sample size for analytic studies, cont d The t-test: Commonly used to determine whether the mean value of a continuous outcome variable in one group differs significantly from that in another group. Assumes that the distribution of the variables in each of the 2 groups is approximately normal (bell-shaped). Assumptions: Whether the 2 groups are paired or independent. Example paired : Comparing the mean BMI of subjects before and after a weight loss program. Example independent : Comparing the mean depression score in subjects treated with 2 different antidepressants. The mean value of the variable in each group. Calculation actually uses the effect size (the difference in the mean values between the 2 groups). Theresa A Scott, MS (Vandy Biostatistics) Sample Size 12 / 24

7 Calculating sample size for analytic studies, cont d The t-test, cont d: Assumptions, cont d: The standard deviation (SD) of the variable. If 2 groups are paired: SD of the difference in the variable between matched pairs. If 2 groups are independent: SD of the variable itself. Rules of thumb: Smaller sample size needed for paired groups SD of the difference in a variable usually smaller than the SD of a variable. Sample size decreases as the difference in the mean values increases (holding SD constant). Sample size increases as SD increases (holding the difference in the mean values constant). Also have standardized effect size = effectsize SD. Sample size decreases as standardized effect size increases. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 13 / 24 Calculating sample size for analytic studies, cont d Example using the t-test: Research question: Is there a difference in the efficacy of salbutamol and ipratropium bromide for the treatment of asthma? Planned study: randomized trial of the effect of these drugs on FEV 1 (forced expiratory volume in 1 second) after 2 weeks of treatment. Previous data: mean FEV 1 in persons with asthma treated with ipratropium was 2.0 liters, with a SD of 1.0 liter. Wish: to be able to detect a difference of 10% in mean FEV 1 between the 2 treatment groups. Assumptions: α (two-sided) = 0.05; power = 0.80; effect size = 0.2 liters (10% X 2.0 liters); SD = 1.0 liter. Calculation: A sample size of 394 patients per group is needed to detect a difference of 10% in mean FEV 1 between the 2 (independent) treatment groups with 80% power, using a two-sample t-test and assuming a (two-sided) α of 0.05, a mean FEV 1 of 2.0 liters in the ipratropium group, and a SD of 1.0 liter. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 14 / 24

8 Calculating sample size for analytic studies, cont d The Chi-square-test: Commonly used to determine whether the proportion of subjects who have a binary outcome variable in one group differs significantly from that in another group. Assumptions: Whether the 2 groups are matched/paired or independent. The proportion with the outcome in each group. Rule of thumb: both the value of the proportions and the difference between them affect sample size sample size much larger for small proportions. Can also calculate assuming a relative risk (instead of 2 proportions). IMPORTANT: a relative risk (ie, risk ratio) equals an odds ratio in only certain cases. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 15 / 24 Calculating sample size for analytic studies, cont d Example using the (uncorrected) Chi-square-test: Research question: Is there a difference in the incidence of skin cancer between elderly smokers and non-smokers? Previous data: the 5-year incidence of skin cancer is about 0.20 in elderly non-smokers. Wish: to determine that the 5-year skin cancer incidence is 0.30 in elderly smokers. Assumptions: α (two-sided) = 0.05; power = 0.80; P 1 (incidence in smokers) = 0.30; P 2 (incidence in non-smokers) = Calculation: A sample size of 293 elderly smokers and 293 elderly non-smokers is needed to determine that the 5-year skin cancer incidence is 0.30 in elderly smokers with 80% power, using an (uncorrected) Chi-square test and assuming a (two-sided) α of 0.05 and a 5-year skin cancer incidence of 0.20 in elderly smokers. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 16 / 24

9 Calculating sample size for analytic studies, cont d Alternative recipes useful when sample size is fixed or the number of subjects who are available or affordable is limited: Calculate the power for a given effect and sample size. Calculate the effect size for a given power and sample size. Good general rule: always assume 80% power. Can also calculate the sample size required to determine whether the correlation coefficient between 2 continuous variables is significant (ie, significantly differs from 0) see Designing Clinical Research. Adjusting for covariates : when designing a study, you may decide that 1 variable will confound the association between the predictor and outcome, and plan to use regression analysis to adjust for these confounders. Calculated sample size will need to be increased ( 10:1 rule ) see a statistician. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 17 / 24 Calculating sample size for descriptive studies Usually do not involve hypotheses; goal is to calculate descriptive statistics (eg, means and proportions). Approach: calculate sample size required to estimate a confidence interval (CI) of a specified confidence level (eg, 95%) and total width (ie, precision). For a continuous variable: interested in the CI around the mean value of that variable. For a binary variable: interested in the CI around the estimated proportion of subjects with one of the values. Rules of thumb: A larger sample size is needed for a tighter (ie, smaller total width; more precise) CI of any confidence level. For a given total width, a larger sample size is needed for a CI with a higher confidence level. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 18 / 24

10 Sample size for descriptive studies, cont d Assumptions made for calculating the sample size: For a continuous variable: the standard deviation of the variable. For a binary variable: the expected proportion with the variable of interest in the population. If more than half of the population is expected to have the characteristic, then plan the sample size based on the proportion of expected not to have the characteristic. For both: (1) the desired precision (total width) of the CI. (2) the confidence level for the interval (eg, 95 or 99%). Example: A sample size of 166 subjects is needed to estimate the mean IQ among 3rd graders in an urban area with a 99% CI of ±3 points (ie, a total with of 6 points), assuming a standard deviation of 15 points. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 19 / 24 Sample size for descriptive studies, cont d IMPORTANT: descriptive studies of binary variables include studies of the sensitivity and specificity of a diagnostic test. Goal of a diagnostic test (or procedure): to correctly classify subjects as having or not having a disease (or symptom). Sensitivity: proportion of true positives; probability of testing positive when you truly have the disease. Specificity: proportion of true negatives; probability of testing negative when you truly do not have the disease. Example: A sample size of 246 subjects is needed to estimate a 95% CI for the sensitivity of a new diagnostic test for pancreatic cancer, assuming 80±5% of the patients with pancreatic cancer will have positive tests. When studying the specificity of a test, must estimate the required number of subjects who do not have the disease. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 20 / 24

11 Sample Size Software Programs PS: Power and Sample Size Calculation Free from Available for Windows and can run on Linux using Wine. Handles several common analyses including t-test, Chi-square test ( Dichotomous ), and Survival. Can generate graphs (eg, power vs sample size) and keeps log. nquery Advisor Not free; handles complex calculations including >2 groups (ie, ANOVA), non-parametric tests, and cross-over designs. Has a statement button, can generate graphs, and keeps log. Simulations can also be done for any statistical technique. Most valuable for complex analyses, such as mixed effects or GEE models statistician (most likely) needed. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 21 / 24 Additional thoughts/considerations Consider strategies for minimizing sample size and maximizing power, which include using Continuous variables, Paired measurements, Unequal group sizes, and A more common (ie, prevalent) binary outcome. Useful to calculate (and report) a range of sample sizes by assuming different combinations of parameter values take the largest sample size to cover all bases. Always justify the feasibility of the calculated sample size. How long would it take to accrue/enroll the subjects? Need to consider the source of subjects, the inclusion/exclusion criteria, the prevalence of the outcome, etc. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 22 / 24

12 What do I include in my sample size write-up? Key: state all the information assumed such that anyone reading your Sample Size section would be able to re-calculate the sample size given. This includes The (primary) specific aim. The outcome and predictor variable(s). Primary comparison of interest (if applicable). Parameter estimates (ie, α, power, effect size, variability, etc). Data on which you based your assumptions. Statistical test used (if applicable). Including graphs can be very helpful. Show the reviewers that you have solid reasoning behind your calculations (as well as your statistical analysis plan). Acknowledge whether study will be a pilot or feasibility study. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 23 / 24 Take home message Sample size justification is an essential part of every research study, and thus any IRB proposal or grant application. Calculating the appropriate sample size is rarely as simple as plugging numbers into a formula. Sample size should be estimated early in the design phase of the study, when major changes are still possible. Involve a statistician from the beginning you re more likely to be funded/approved if you do so. References & acknowledgments: Designing Clinical Research (3rd edition) by Hulley, et al. Ayumi Shintani & Jennifer Thompson. Theresa A Scott, MS (Vandy Biostatistics) Sample Size 24 / 24

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