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1. GCRC Research-Skills Workshop October 26, 2007 How to do Power & Sample Size Calculations Part 2 **************** William D. Dupont Department of Biostatistics ****************

2. Follow-up time Fate at exit Statistic: Log-rank test Power: Schoenfeld & Richter, Biometrics 1982 Survival Analysis Accrual time A Additional follow-up F Sample Size 0 Time

3. Hemorage-free survival in patients with previous lobular introcerebral hemmorrhage subdivided by apolipoprotein E genotype (O’Donnell et al. 2000).

4. Example: Hemorrhage-free survival and genotype Control group = patients with an e2 or e4 allele Pr[Type I error]  = 0.05 Power = 0.8 Relative Risk (control/experiment) R = 2 Medial survival time of controls m1 = ?

5. Hemorage-free survival in patients with previous lobular intracerebral hemmorrhage subdivided by apolipoprotein E genotype (O’Donnell et al. 2000).

6. Example: Hemorrhage-free survival and genotype Control group = patients with an e2 or e4 allele Pr[Type I error]  = 0.05 Power = 0.8 Relative Risk (control/experiment) R = 2 Medial survival time of controls m1 = 38 Accrual A = 12 months Additional Follow-up F = 24 months

7. What if we wanted to study a treatment of Homozygous e3/e3 patients?

8. = some time = probability of survival at time = If survival has an exponential distribution

10. = some time = probability of survival at time If survival has an exponential distribution = =

11. Relative Risk (Hazard Ratio) for controls relative to experimental subjects Median survival for experimental subjects Median survival for controls = Median Survival 6 3 = = 2 Experimental treatment Control treatment

12. For t tests power calculations for increased or decreased response relative to control response are symmetric. i.e. The power to detect is the same as the power to detect This is not true for Survival Analysis. The power to detect a two-fold increase in hazard does not equal the power to detect a 50% decrease in hazard.

13. Power to detect treatment 1 vs. control greater than treatment 1 vs.control Treatment 1 Control Relative Risk Control vs. Treatment 1 2 Control vs. Treatment 2 0.5 Treatment 2 It is important to read the parameter definitions carefully.

14. Power Calculations for Linear Regression

15. s = 4.75 x 45 · · 40 · · · · · · · 35 · · · · · · · · Birthweight (g/100) · · · · · · 30 · · · · · · · 25 · 20 5 10 15 20 25 30 Estriol (mg/24 hr) Rosner Table 11.1 Am J Obs Gyn 1963;85:1-9 = standard deviation of x variable = standard deviation of y variable

16. = standard deviation of the regression errors Estimated by s= root mean squared error (MSE)

17. r = -1 r = 1 r = 0 0 < r < 1 c) Correlation coefficient is estimated by {1.2}

18. 35 Treatment r A 0.9 B 0.6 30 25 20 Response Variable y Both treatments have identical values of mx = 6, my = 20 sx = 2, sy = 5 15 10 5 1 3 5 6 7 8 9 10 11 12 2 4 Independent Variable x Measures of association between continuous variables Correlation vs. linear regression

19. Treatment l r s A 2.25 0.9 2.18 B 1.5 0.6 4.0 35 30 25 20 Response Variable y Both treatments have identical values of mx = 6, my = 20 sx = 2, sy = 5 15 10 5 1 3 5 6 7 8 9 10 11 12 2 4 Independent Variable x

20. th j Regression Error • y j l g + l x j 1 Unit Patient Response of Dosage g Regression g + l x Line x 0 j Treatment Dose Level

21. c) Slope parameter estimate l (a.k.a. ) is estimated by b = r sy /sx

22. r = -1 r = 1 r = 0 0 < r < 1 l is estimated by b = r sy /sx

23. Studies can be either experimental or observational

24. Normally distributed unless investigator chooses level Chosen by investigator Treatment level

25. Estimating s directly is often difficult If we can estimate or s = = s Warning: If the anticipated value of in your experiment is different from that found in the literature then your value of r will also be different.

26. 35 30 25 Response Variable y 20 15 10 5 1 3 5 6 7 8 9 10 11 12 2 4 Independent Variable x

27. Relationship between BMI and exercise time n = 100 women willing to follow a diet exercise program for six months Interquartile range (IQR) of exercise time = 15 minutes (pilot data) = (IQR / 2) / z0.25 = 7.5 / 0.675 = 11.11 = 4.0 kg/m2 = standard deviation of BMI for women obtained by Kuskowska-Wolk et al. Int J Obes 1992;16:1-9 Would like to detect a true drop in BMI of = -0.0667 kg/m2 per minute of exercise (1/2 hour of exercise per day induces a drop of 2 kg/m2 over 6 months)

28. Interquartile Range In general = (IQR / 2) / z0.25 When s = 1 0.25 0.25 2 0 -2 -0.675 0.675

29. Impaired Antibody Response to Pneumococcal Vaccine after treatment for Hodgkin’s Disease Siber et al. N Engl J Med 1978;299:442-448. n = 17 patients treated with subtotal radiation. vaccinated 8 to 51 months later A linear regression of log antibody response against time from radiation to vaccination gave Suppose we wanted to determine the sample size for a new study with patients randomized to vaccination at 10, 30, or 50 weeks

30. Composing Slopes of Two Linear Regressin Lines Armitage and Berry (1994) gave age and pulmonary vital capacity for 28 cadmium industry workers with > 10 years of exposure 44 workers with no exposure

31. (unexposed) (exposed) pooled error variance How many workers do we need to detect with ratio of unexposed workers m = 44/28 = 1.57? Need 167 exposed workers and 167 x 1.57 = 262 unexposed workers

32. (unexposed) (exposed) pooled error variance How many workers do we need to detect with ratio of unexposed workers m = 44/28 = 1.57? Need 167 exposed workers and 167 x 1.57 = 262 unexposed workers