1. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.1 Sample Size and Power

2. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.2 Sample Size Considerations A pharmaceutical company calls and says, “We believe we have found a cure for the common cold. How many patients do I need to study to get our product approved by the FDA?”

3. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.3 Where to begin? N = (Total Budget / Cost per patient)? Hopefully not!

4. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.4 Where to begin?  Understand the research question  Learn about the application and the problem.  Learn about the disease and the medicine.  Crystal Ball  Visualize the final analysis and the statistical methods to be used.

5. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.5 Where to begin?  Analysis determines sample size. Sample size calculations are based upon the planned method of analysis.  If you don’t know how the data will be analyzed (e.g., 2-sample t-test), then you cannot accurately estimate the sample size.

6. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.6 Sample Size Calculation  Formulate a PRIMARY research question.  Identify: 1.A hypothesis to test (write down H 0 and H A ), or 2.A quantity to estimate (e.g., using confidence intervals)

7. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.7 Sample Size Calculation  Determine the endpoint or outcome measure associated with the hypothesis test or quantity to be estimated.  How do we “measure” or “quantify” the responses?  Is the measure continuous, binary, or a time- to-event?

8. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.8 Sample Size Calculation  Based upon the PRIMARY outcome  Other analyses (i.e., secondary outcomes) may be planned, but the study may not be powered to detect effects for these outcomes.

9. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.9 Sample Size Calculation  Two strategies  Hypothesis Testing  Estimation with Precision

10. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.10 Sample Size Calculation Using Hypothesis Testing  By far, the most common approach.  The idea is to choose a sample size such that both of the following conditions simultaneously hold:  If the null hypothesis is true, then the probability of incorrectly rejecting is (no more than) α  If the alternative hypothesis is true, then the probability of correctly rejecting is (at least) 1- β = power.

11. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.11 Reality H o TrueH o False Test Result Reject H o Type I error ( α ) Power (1- β) Do not reject H o 1- α Type II error ( β )

12. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.12 Determinants of Sample Size: Hypothesis Testing Approach  α  β  An “effect size” to detect  Estimates of variability

13. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.13 What is Needed to Determine the Sample-Size?  α  Up to the investigator or FDA regulation (often = 0.05)  How much type I (false positive) error can you afford?

14. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.14 What is Needed to Determine the Sample-Size?  1- β (power)  Up to the investigator (often 80%-90%)  How much type II (false negative) error can you afford?  Not regulated by FDA

15. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.15 Choosing α and β  Weigh the cost of a Type I error versus a Type II error.  In early phase clinical trials, we often do not want to “miss” a significant result and thus often consider designing a study for higher power (perhaps 90%) and may consider relaxing the α error (perhaps 0.10).  In order to approve a new drug, the FDA requires significance in two Phase III trials strictly designed with α error no greater than 0.05 (Power = 1- β is often set to 80%).

16. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.16 Effect Size  The “minimum difference (between groups) that is clinically relevant or meaningful”.  Not readily apparent  Requires clinical input  Often difficult to agree upon  Note for noninferiority studies, we identify the “maximum irrelevant or non-meaningful difference”.

17. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.17 Estimates of Variability  Often obtained from prior studies  Explore the literature and data from ongoing studies for estimates needed in calculations  Consider conducting a pilot study to estimate this  May need to validate this estimate later

18. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.18 Other Considerations  1-sample vs. 2-sample  Independent samples or paired  1-sided vs. 2-sided

19. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.19 Example: Cluster Headaches  A experimental drug is being compared with placebo for the treatment of cluster headaches.  The design of the study is to randomize an equal number of participants to the new drug and placebo.  The participants will be administered the drug or matching placebo. One hour later, the participants will score their pain using the visual analog scale (VAS) for pain.  A continuous measure ranging from 0 (no pain) to 10 (severe pain).

20. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.20 Example: Cluster Headaches  The planned analysis is a 2-sample t- test (independent groups) comparing the mean VAS score between groups, one hour after drug (or placebo) initiation  H 0 : μ 1 =μ 2 vs. H A : μ 1 ≠ μ 2

21. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.21 Example: Cluster Headaches  It is desirable to detect differences as small as 2 units (on the VAS scale).  Using α=0.05 and β=0.80, and an assumed standard deviation (SD) of responses of 4 (in both groups), 63 participants per group (126 total) are required.  STATA Command: sampsi 0 2, sd(4) a(0.05) p(.80)  Note: you just need a difference of 2 in the first two numbers  http://newton.stat.ubc.ca/~rollin/stats/ssize/n2.html

22. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.22 Example: Part 2  Let’s say that instead of measuring pain on a continuous scale using the VAS, we simply measured “response” (i.e., the headache is gone) vs. non-response.

23. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.23 Example: Part 2  The planned analysis is a 2-sample test (independent groups) comparing the proportion of responders, one hour after drug (or placebo) initiation  H 0 : p 1 =p 2 vs. H A : p 1 ≠ p 2

24. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.24 Example: Part 2  It is desirable to detect a difference in response rates of 25% and 50%.  Using α=0.05 and β=0.80,  STATA Command: sampsi 0.25 0.50, a(0.05) p(.80)  66 per group (132 total) w/ continuity correction  http://newton.stat.ubc.ca/~rollin/stats/ssize/b2.html http://newton.stat.ubc.ca/~rollin/stats/ssize/b2.html  58 per group (116 total) without continuity correction

25. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.25 Notes for Testing Proportions  One does not need to specify a variability since it is determined from the proportion.  The required sample size for detecting a difference between 0.25 and 0.50 is different from the required sample size for detecting a difference between 0.70 and 0.95 (even though both are 0.25 differences) because the variability is different.  This is not the case for means.

26. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.26 Caution for Testing Proportions  Some software computes the sample size for testing the null hypothesis of the equality of two proportions using a “continuity correction” while others calculate sample size without this correction.  Answers will differ slightly, although either method is acceptable.  STATA uses a continuity correction  The website does not

27. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.27 Sample Size Calculation Using Estimation with Precision  Not nearly as common, but equally as valid.  The idea is to estimate a parameter with enough “precision” to be meaningful.  E.g., the width of a confidence interval is narrow enough

28. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.28 Determinants of Sample Size: Estimation Approach  α  Estimates of variability  Precision  E.g., The (maximum) desired width of a confidence interval

29. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.29 Example: Evaluating a Diagnostic Examination  It is desirable to estimate the sensitivity of an examination by trained site nurses relative to an oral medicine specialist for the diagnosis of Oral Candidiasis (OC) in HIV-infected people.  Precision: It is desirable to estimate the sensitivity such that the width of a 95% confidence interval is 15%.

30. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.30 Example: Evaluating a Diagnostic Examination  Note: sensitivity is a proportion  The (large sample) CI for a proportion is:

31. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.31 Example: Evaluating a Diagnostic Examination  We wish the width of the CI to be <0.15  Using an estimated proportion of 0.25 and α =0.05, we can calculate n=129.  Since sensitivity is a conditional probability, we need 129 that are OC+ as diagnosed by the oral health specialist. If the prevalence of OC is ~20%, then we would need to enroll or screen ~129/(0.20)=645.

32. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.32 Sensitivity Analyses  Sample size calculations require assumptions and estimates.  It is prudent to investigate how sensitive the sample size estimates are to changes in these assumptions (as they may be inaccurate).  Thus, provide numbers for a range of scenarios and various combinations of parameters (e.g., for various values combinations of α, β, estimates of variance, effect sizes, etc.)

33. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.33 Example: Sample Size Sensitivity Analyses for the Study of Cluster Headaches μ1μ1 μ2μ2 SDPower=80%Power=90% 023.54965 024.06385 024.580107 033.52229 034.02838 034.53648

34. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.34 Effects of Determinants  In general, the following increases the required sample size (with all else being equal):  Lower α  Lower β  Higher variability  Smaller effect size to detect  More precision required

35. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.35 Caution  In general, higher sample size implies higher power.  Does this mean that a higher sample size is always better?  Not necessarily. Studies can be very costly. It is wasteful to power studies to detect between-group differences that are clinically irrelevant.

36. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.36 Sample Size Adjustments  Complications (e.g., loss-to-follow-up, poor adherence, etc.) during clinical trials can impact study power.  This may be less of a factor in lab experiments.  Expect these complications and plan for them BEFORE the study begins.  Adjust the sample size estimates to account for these complications.

37. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.37 Complications that Decrease Power  Missing data  Poor Adherence  Multiple tests  Unequal group sizes  Use of nonparametric testing (vs. parametric)  Noninferiority or equivalence trials (vs. superiority trials)  Inadvertent enrollment of ineligible subjects or subjects that cannot respond

38. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.38 Adjustment for Lost-to-Follow-up  Loss-to-Follow-Up (LFU) refers to when a participants endpoint status is not available (missing data).  If one assumes that the LFU is non-informative or ignorable (i.e., random and not related to treatment), then a simple sample size adjustment can be made.  This is a very strong assumption as LFU is often associated with treatment. The assumption is further difficult to validate.  Researchers need to consider the potential bias of examining only subjects with non-missing data.

39. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.39 Adjustment for Lost-to-Follow-up  Calculate the sample size N.  Let x=proportion expected to be lost-to-follow- up.  N adj =N/(1-x)  Note: no LFU adjustment is necessary if you plan to impute missing values. However, if you use imputation, an adjustment for a “dilution effect” may be warranted.

40. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.40 Adjustment for Poor Adherence  Adjustment for the “dilution effect” due to poor adherence or the inclusion (perhaps inadvertently) of subjects that cannot respond:  Calculate the sample size N.  Let x=proportion expected to be non-adherent.  N adj =N/(1-x) 2

41. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.41 Inflation Factor for Non-adherence Proportion non- Adherent 0.050.100.200.300.50 Inflation Factor1.111.231.562.044.00

42. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.42 Adjustment for Unequal Allocation  When comparing groups, power is maximized when groups sizes are equal (with all else being equal)  There may be other reasons however, to have some group sizes larger than others  E.g., having more people on an experimental therapy (rather than placebo) to obtain more safety information of the product

43. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.43 Adjustment for Unequal Allocation  Adjustment for unequal allocation in two groups:  Let Q E and Q C be the sample fractions such that Q E +Q C =1.  Note power is optimized when Q E =Q C =0.5  Calculate sample size N bal for equal sample sizes (i.e., Q E =Q C =0.5)  N unbal =N bal ((Q E -1 +Q C -1 )/4)

44. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.44 Adjustment for Nonparametric Testing  Most sample-size calculations are performed expecting use of parametric methods (e.g., t- test).  This is often done because formulas (and software) for these methods are readily available  However, parametric assumptions (e.g., normality) do not always hold.  Thus nonparametric methods may be required.

45. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.45 Adjustment for Nonparametric Testing  Pitman Efficiency  Applicable for 1 and 2 sample t-tests  Method  Calculate sample size N par.  N nonpar = N par /(0.864)

46. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.46 Example: Cluster Headaches  Recall the cluster headache example in which the required sample size was 126 (total) for detecting a 2 unit (VAS scale) difference in means.  If we expect 10% of the participants to be non-adherent then an appropriate inflation is needed  126/(1-0.1) 2 =156  If we further expect that we will have to perform a nonparametric test (instead of a t-test) due to non- normality, then further inflation is required:  156/(0.864)=181  Round to 182 to have an equal number (81) in each group

47. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.47 Adjustment: Noninferiority/Equivalence Studies  Calculate sample size for standard superiority trial but reverse the roles of α and β.  Works for large sample binary and continuous data.  Does not work for time-to-event data.

48. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.48 More Adjustments?  Adjustments are needed if:  You plan interim analyses  Group sequential designs  You have more than one primary test to be conducted  Multiple comparison adjustments  E.g., Bonferroni (if 2 tests or comparisons are to be made, then power each at α/2.  Additional adjustments may be needed for stratification, blocking, or matching.

49. Introduction to Biostatistics, Harvard Extension School © Scott Evans, Ph.D.49 Sample Size Re-estimation  Hot Topic in clinical trials  Re-estimating sample size based on interim data  Complicated  Must be done carefully to maintain scientific integrity and blinding.