Materials: Protocol unavailable Results unavailable – grant not funded

1. Case Study 13Designing a simple sequential trial on frequentist principles: a two-stage design with proper a for assessing hyperthermia toxicities in cancer radiotherapyPrincipal Investigator:Richard Steeves, MD, University of Wisconsin

2. Materials: Protocol unavailable Results unavailable – grant not funded Information on hyperthermia in cancer treatment available at https://www.cancer.gov/about-cancer/treatment/types/surgery/hyperthermia-fact-sheet Key Words: Power, Size (Type I Error), two-stage designs, single-arm trials, phase II trials. Rick Chappell, Ph.D. Professor, Department of Biostatistics and Medical Informatics University of Wisconsin Medical School Stat 542 – Spring 2018

3. A single-dose non-escalating toxicity study on concurrent hyperthermia (microwave) and high-dose-rate irradiation for near-surface tumors (UWCCC protocol RO 9471, R. Steeves, M.D., P.I.). We want to show that the toxicity rate isn’t too high using historical controls – so it is a single-arm trial.

4. To test that the toxicity rate isn’t too high: Let p = Pr(Toxicity). The null hypothesis in this one-arm trial is H0: p = .3 vs. the alternative hypothesis of HA: p = .1 . We hope to reject H0 and conclude HA.

5. H0: p = .3 vs. HA: p = .1 . What mistakes can we make? Reject H0 when it is true: Type I Error, or False positive Pr(Type I Error) = a = size. Don’t reject H0 when HA is true: Type II Error, or False negative. Pr(Type II Error) = b = 1 - power. (Why don’t I ever write “Accept H0”?)

6. Suppose we want a simple 1-stage single-arm trial with one-sided a <= .05 and power >= .8 (These are common choices). Use 29 patients and a rejection rule of: Reject H0 if 0 - 4 toxicities; Fail to reject H0 if 5 or more toxicities. Under the binomial formula, a = Pr(Reject H0 | H0) = Pr(0 - 4 toxicities | p = .3) = .038. This isn’t exactly .05, but moving up to 5 toxicities would raise it to .093 which is too high.

7. Suppose we want a simple 1-stage single-arm trial with one-sided a <= .05 and power >= .8. Under the binomial formula, a = Pr(False positive) = Pr(Reject H0 | H0) = Pr(0 - 4 toxicities | p = .3) = .038. 1 - b = Power = Pr(True positive) = Pr(Reject H0 | HA) = Pr(0 – 4 toxicities | p = .1) = .84.

8. Now suppose we want a 2-stage single-arm trial with one-sided a = .05 and power = .8, so we can stop early for success or futility. This can be done using 28 patients, with two stages of 14 patients each using the following rules: 0 toxicities in Stage 1: stop and reject H0; 5 or more toxicities in Stage 1: stop and don’t reject H0; 1 - 4 toxicities in Stage 1: continue to Stage 2; 5 or more total toxicities after Stage 2: don’t reject H0; Less than 5 total toxicities after Stage 2: reject H0.

9. How to Figure Out? Under H0: π = .3 # of events in 2nd cohort of 14 0 1 2 3 4 5 6 7 …. 14 0 1 2 3 4 5 6 . . . 14 # of events in 1st cohort of 14

10. How to Figure Out? Under H0: π = .3 # of events in 2nd cohort of 14 0 1 2 3 4 5 6 7 …. 14 0 1 2 3 4 5 6 . . . 14 Pr (Reject at Stage 1) = .68% Pr (Reject at Stage 2) Pr (Go to Stage 2, don’t reject) = 4.3% = 53.3% # of events in 1st cohort of 14 Pr (Stop, don’t reject at Stage 1) = 41.7% a = Size = Total False Rejection Rate = 4.3% + .68% = 5.0%

11. How to Figure Out? Under HA: π = .1 # of events in 2nd cohort of 14 0 1 2 3 4 5 6 7 …. 14 0 1 2 3 4 5 6 . . . 14 Pr (Stop, reject at Stage 1) = 23% Pr (Reject at Stage 2) Pr (Go to Stage 2, don’t reject) = 63% = 13% # of events in 1st cohort of 14 Pr (Stop, don’t reject at Stage 1) = .9% Power to detect (π= .1) = 23% + 63% = 86%