This document discusses key concepts in determining sample size and statistical power for clinical trials. It provides examples of calculating sample sizes for dichotomous and continuous outcomes. The importance of choosing an appropriate primary outcome and estimating event rates in control and intervention groups is emphasized. Methods for randomization in clinical trials like simple randomization and blocked randomization are also covered.

1. DESIGN OF CLINICAL TRIALS EPIDEMIOLOGY 2181 “ SAMPLE SIZE DETERMINATION AND STATISTICAL POWER IN CLINICAL TRIALS” S.F.Kelsey/class2181/lecture 4-sample size October 30, 2008 Lecture 4 SHERYL F. KELSEY, PhD Department of Epidemiology

2. S.F.Kelsey/class2181/lecture 4-sample size QUIZ Assume 90% Power, a = 0.05 two-sided (x) more with A (y) more with B (z) the same 1 . Mortality 20% vs 10% 40% vs 30% 2. Mortality 20% vs 10% 20% vs 15% 3. Diastolic 80 vs 85 mmHg 90 vs 95 mmHg BP 4. Diastolic 80 vs 85 mmHg 80 vs 85 mmHg BP A B (x) more with A (y) more with B (z) the same (x) more with A (y) more with B (z) the same (x) more with A (y) more with B (z) the same (St Dev 10) (St Dev 10) (St Dev 10) (St Dev 8) How many subjects?

3. S.F.Kelsey/class2181/lecture 4-sample size 1. More with B 2. More with B 3. The same 4. More with A ANSWERS Variance of the binomial bigger 50% smaller 0% 100% Small difference need more subjects Only standard deviation matters Bigger standard deviation more subjects

4. S.F.Kelsey/class2181/lecture 4-sample size SHALL WE COUNT THE LIVING OR THE DEAD? 40% vs 20% 20% 50% “reduction” in mortality lower mortality 20% vs 10% 10% 50% “reduction” in mortality lower mortality 10% vs 5% 05% 50% “reduction” in mortality lower mortality 60% vs 80% 20% 33% “improvement” in survival higher mortality 80% vs 90% 10% 2.5% “improvement” in survival higher mortality 90% vs 95% 05% 5.6% “improvement” in survival higher mortality Absolute Relative

5. S.F.Kelsey/class2181/lecture 4-sample size Even more confusing with continuous variables Blood pressure (St Dev 10) 5.9% “reduction” 80 vs 85 mmHg 5.3% “reduction” 90 vs 95 mmHg

8. S.F.Kelsey/class2181/lecture 4-sample size Legal null hypothesis: innocent until proven guilty Scientific null hypothesis: no difference in response between treatment groups Innocent Guilty Innocent Guilty Truth Decision of Judge/Jury ok ok guilty goes free type II error (  ) hang the innocent type I error (  ) Treatment Different Treatment Same Truth Same Different Observed Data ok ok miss good treatment type II error (  ) promote worthless Tx type I error (  )

9. FUNDAMENTAL POINT S.F.Kelsey/class2181/lecture 4-sample size Clinical trials should have sufficient statistical power to detect differences between groups considered to be of clinical interest. Therefore, calculation of sample size with provision for adequate levels of significance and power is an essential part of planning.

14. S.F.Kelsey/class2181/lecture 4-sample size To Plan with continuous endpoints:  Clinical difference worth detecting 1–   Power Probability of obtaining a significant result if  is true difference  Significance level, must specify one or two-tailed test (Z   Z  ) 2 Multiplier which depends on level of significance  and Power 1-  n Sample size for each of two groups For continuous measures:  Standard deviation With a little algebra Z  =1.96 for  =.05, two-sided (solve for power) (Solve for difference)

15. S.F.Kelsey/class2181/lecture 4-sample size For two proportions P 1 vs P 2,  = P 1 - P 2 With a little algebra Z  = 1.64 for  .05, one-sided Z  = 1.96 for  .05, two-sided

16. S.F.Kelsey/class2181/lecture 4-sample size TABLE (Z  + Z  ) 2 Needed to determine the size of each sample (Z 2  2.32 1.645 1.28) Desired Two-Tailed Tests One-Tailed Tests Power Level Level Z  P 0.01 0.05 0.10 0.01 0.05 0.10 Two groups of unequal size: Calculate the harmonic mean This n is what is needed for 2 groups of equal size. Note that equal sized groups are the most efficient, that is the harmonic mean is less than the arithmetic mean. References: Snedecor and Cochran, 7th Edition Statistical Methods , 1980, pp 102-1- 5, 120, 130. Fleiss, JL. Statistical Methods for Rates and Proportions, 1981, Chapter 3 & Tables. Schlesselman, JJ. Case Control Studies, 1981, Chapter 6 & Tables. (Z  2.576 1.96 1.645) 0.84 0.80 11.7 7.9 6.2 10.0 6.2 4.5 1.28 0.90 14.9 10.5 8.6 13.0 8.6 6.6 1.645 0.95 17.8 13.0 10.8 15.8 10.8 8.6

17. S.F.Kelsey/class2181/lecture 4-sample size Example: Compare .10 vs .05  = .05 one sided Power 80% arcsin arcsin So total study: 334 x 2 = 668 .10 vs .05 with 200 patients in each group Power = 61% with 100 patients Z  = .28 39% power 50 patients Z  = .68 25% power | | .0963| - 1.64 = .286 | — Z  arcsin

19. ONE-SIDED VERSUS TWO-SIDED TESTS S.F.Kelsey/class2181/lecture 4-sample size I Drug A side effects/expensive Drug B no side effects/cheap A more efficacious A&B the same B more efficacious II X Nutrition Intervention Strategy-Group sessions Y Nutrition Intervention Strategy-Individual program X reduce sodium intake more X&Y the same Y reduce sodium intake more

21. T = Innovative Therapy S = Standard Therapy S.F.Kelsey/class2181/lecture 4-sample size “ Superiority” H 0 : death rate T = death rate S H alt :death rate T < death rate S Equivalence H 0 : death rate T  death rate S +  H alt :death rate T < death rate S +  In general equivalence studies require more patients

22. S.F.Kelsey/class2181/lecture 4-sample size Patients: Acute MI Treatment: Double bolus vs accelerated Alteplace Outcome: 30 day mortality COBALT Equivalence Death rate within 0.4% GUSTO III Superiority Double bolus reduce mortality by 20% WARE AND ANTMAN EDITORIAL

23. MORTALITY RESULTS S.F.Kelsey/class2181/lecture 4-sample size COBALT GUSTO III N 7169 15059 Double bolus 7.98% 7.47% Accelerated 7.53% 7.24% Difference 0.45% 0.23% 95% CI Approx. (-.85%, 1.66%) (-.66%, 1.10%) Action reject equivalence accept null not significantly different from zero

24. DESIGN OF CLINICAL TRIALS EPIDEMIOLOGY 2181 RANDOMIZATION IN CLINICAL TRIALS S.F.Kelsey/class2181/lecture 4-sample size SHERYL F. KELSEY, PH.D

27. STEPS IN THE RANDOMIZATION OF A PATIENT Check eligibility Informed consent Formal identification RANDOMIZE Confirmation of patient entry S.F.Kelsey/class2181/lecture 4-sample size

28. HOW RANDOM TREATMENT ASSIGNMENTS ARE MADE S.F.Kelsey/class2181/lecture 4-sample size Model: Slips in a hat or flipping a coin Masked drugs numbered and given in order: pharmacy, drug manufacturer Envelopes Telephone to central unit Microcomputer at the site Central computer – internet access

29. HOW TO DO THE SCHEME S.F.Kelsey/class2181/lecture 4-sample size Simple randomization Biased coin, urn models Example: Start with 2 balls, one black and one white Draw-replace and add one of opposite color Prevents imbalance with high probability early on Random permuted block Balance at the end of block Could predict with unmasked trial

30. BLOCKS OF SIZE 4 S.F.Kelsey/class2181/lecture 4-sample size 1) 1100 2) 1010 3) 1001 4) 0110 5) 0101 6) 0011

31. HOW TO USE BLOCKS WHEN TREATMENT IS NOT MASKED S.F.Kelsey/class2181/lecture 4-sample size Choose the block sizes at random, too Example: 2 treatment, equal allocation Block sizes 4, 6, and 8 Balance in each block

32. SHOULD YOU STRATIFY? S.F.Kelsey/class2181/lecture 4-sample size Clinical sites - generally yes Prognostic variables - generally no Size Practical considerations Often governed by custom rather than statistical justification Stratified ANALYSIS is generally preferable

33. MINIMIZATION S.F.Kelsey/class2181/lecture 4-sample size Advantages: Balance several prognostic factors Balance marginal treatment totals Good for small trials (<100 patients) Computer makes this fairly easily Disadvantages: Can’t prepare treatment assignment Scheme in advance Need up-to-date record Not really random - could predict but can introduce random element by using say 3/4, 1/4

34. S.F.Kelsey/class2181/lecture 4-sample size TABLE 5.7. - TREATMENT ASSIGNMENTS BY THE FOUR PATIENT FACTORS FOR 80 PATIENTS IN AN ADVANCED BREAST CANCER TRIAL Factor Level No. on each Next treatment patient A B Performance status Ambulatory 30 31 Non-ambulatory 10 9 Age <50 18 17  50 22 23 Disease-free interval <2 years 31 32  2 years 9 8 Dominant metastatic lesion Visceral 19 21 Osseous 8 7 Soft tissue 13 12 Pocock S. Clinical Trials: A Practical Approach. John Wiley & Sons, Chichester, England, 1991, p. 85. Thus, for A this sum = 30 + 18 + 9 + 19 = 76 while for B this sum = 31 + 17 + 8 + 21 = 77

35. S.F.Kelsey/class2181/lecture 4-sample size INTERNAL VALIDITY compare treatments External Validity/ Generalizability extrapolate to other patients Not realistic to find a random sample of patients for recruitment (at the very least they have to consent) More important to establish efficacy of treatment before deciding if it can be broadly applied