1. Point Estimation: Odds Ratios, Hazard Ratios, Risk Differences, Precision Elizabeth S. Garrett esg@jhu.edu Oncology Biostatistics March 20, 2002 Clinical Trials in 20 Hours

2. 3/20/2002Clinical Trials in 20 Hours2 Point Estimation Definition: A “point estimate” is a one- number summary of data. If you had just one number to summarize the inference from your study….. Examples: –Dose finding trials: MTD (maximum tolerable dose) –Safety and Efficacy Trials: response rate, median survival –Comparative Trials: Odds ratio, hazard ratio

3. 3/20/2002Clinical Trials in 20 Hours3 Types of Variables The point estimate you choose depends on the “nature” of the outcome of interest Continuous Variables –Examples: change in tumor volume or tumor diameter –Commonly used point estimates: mean, median Binary Variables –Examples: response, progression, > 50% reduction in tumor size –Commonly used point estimate: proportion, relative risk, odds ratio Time-to-Event (Survival) Variables –Examples: time to progression, time to death, time to relapse –Commonly used point estimates: median survival, k-year survival, hazard ratio Other types of variables: nominal categorical, ordinal categorical

4. 3/20/2002Clinical Trials in 20 Hours4 Today Point Estimates commonly seen (and misunderstood) in clinical oncology –odds ratio –risk difference –hazard ratio/risk ratio

5. 3/20/2002Clinical Trials in 20 Hours5 Point Estimates: Odds Ratios “Age, Sex, and Racial Differences in the Use of Standard Adjuvant Therapy for Colorectal Cancer”, Potosky, Harlan, Kaplan, Johnson, Lynch. JCO, vol. 20 (5), March 2002, p. 1192. Example: Is gender associated with use of standard adjuvant therapy (SAT) for patients with newly diagnosed stage III colon or stage II/III rectal cancer? –53% of men received SAT* –62% of women received SAT* How do we quantify the difference? * adjusted for other variables

6. 3/20/2002Clinical Trials in 20 Hours6 Odds and Odds Ratios Odds = p/(1-p) The odds of a man receiving SAT is 0.53/(1 - 0.53) = 1.13. The odds of a woman receiving SAT is 0.62/(1 - 0.62) = 1.63. Odds Ratio = 1.63/1.13 = 1.44 Interpretation: “A woman is 1.44 times more likely to receive SAT than a man.”

7. 3/20/2002Clinical Trials in 20 Hours7 Odds Ratio Odds Ratio for comparing two proportions OR > 1: increased risk of group 1 compared to 2 OR = 1: no difference in risk of group 1 compared to 2 OR < 1: lower risk (“protective”) in risk of group 1 compared to 2 In our example, –p 1 = proportion of women receiving SAT –p 2 = proportion of men receiving SAT

8. 3/20/2002Clinical Trials in 20 Hours8 Odds Ratio from a 2x2 table

9. 3/20/2002Clinical Trials in 20 Hours9

10. 3/20/2002Clinical Trials in 20 Hours10 More on the Odds Ratio Ranges from 0 to infinity Tends to be skewed (i.e. not symmetric) –“protective” odds ratios range from 0 to 1 –“increased risk” odds ratios range from 1 to  Example: –“Women are at 1.44 times the risk/chance of men” –“Men are at 0.69 times the risk/chance of women”

11. 3/20/2002Clinical Trials in 20 Hours11 More on the Odds Ratio Sometimes, we see the log odds ratio instead of the odds ratio. The log OR comparing women to men is log(1.44) = 0.36 The log OR comparing men to women is log(0.69) = -0.36 log OR > 0: increased risk log OR = 0: no difference in risk log OR < 0: decreased risk

12. 3/20/2002Clinical Trials in 20 Hours12 Related Measures of Risk Relative Risk: RR = p 1 /p 2 –RR = 0.62/0.53 = 1.17. –Different way of describing a similar idea of risk. –Generally, interpretation “in words” is the similar: “Women are at 1.17 times as likely as men to receive SAT” –RR is appropriate in trials often. –But, RR is not appropriate in many settings (e.g. case-control studies) –Need to be clear about RR versus OR: p 1 = 0.50, p 2 = 0.25. RR = 0.5/0.25 = 2 OR = (0.5/0.5)/(0.25/0.75) = 3 Same results, but OR and RR give quite different magnitude

13. 3/20/2002Clinical Trials in 20 Hours13 Related Measures of Risk Risk Difference: p 1 - p 2 –Instead of comparing risk via a ratio, we compare risks via a difference. –In many CT’s, the goal is to increase response rate by a fixed percentage. –Example: the current success/response rate to a particular treatment is 0.20. The goal for new therapy is a response rate of 0.40. –If this goal is reached, then the “risk difference” will be 0.20.

14. 3/20/2002Clinical Trials in 20 Hours14 Why do we so often see OR and not others? (1) Logistic regression: –Allows us to look at association between two variables, adjusted for other variables. –“Output” is a log odds ratio. –Example: In the gender ~ SAT example, the odds ratios were evaluated using logistic regression. In reality, the gender ~ SAT odds ratio is adjusted for age, race, year of dx, region, marital status,….. (2) Can be more globally applied. Design of study does not restrict usage.

15. 3/20/2002Clinical Trials in 20 Hours15 Another Example “Randomized Controlled Trial of Single- Agent Paclitaxel Versus Cyclophosphamide, Doxorubicin, and Cisplatin in Patients with Recurrent Ovarian Cancer Who Responded to First-line Platinum-Based Regimens”, Cantu, Parma, Rossi, Floriani, Bonazzi, Dell’Anna, Torri, Colombo. JCO, vol. 20 (5), March 2002, p. 1232. Groups: paclitaxel (n = 47) versus CAP (n = 47) “14 patients in the CAP group and 8 patients in the paclitaxel group had complete responses p1 = 14/47 = 0.30; p2 = 8/47 = 0.17 OR = (0.30/0.70)/(0.17/0.83) = 2.1

16. 3/20/2002Clinical Trials in 20 Hours16 Odds Ratio via 2x2 table “14 patients in the CAP group and 8 patients in the paclitaxel group had complete responses “Patients in the CAP group are twice as likely to have a CR as those in the paclitaxel group.” 2x2 Table approach: OR = ad/bc = (14*39)/(8*33) = 2.1

17. 3/20/2002Clinical Trials in 20 Hours17 “Randomized Controlled Trial of Single-Agent Paclitaxel Versus Cyclophosphamide, Doxorubicin, and Cisplatin in Patients with Recurrent Ovarian Cancer Who Responded to First-line Platinum-Based Regimens”, Cantu, Parma, Rossi, Floriani, Bonazzi, Dell’Anna, Torri, Colombo. JCO, vol. 20 (5), March 2002, p. 1232. Point Estimates: Hazard Ratios “What is the effect of CAP on overall survival as compared to paclitaxel?” –Median survival in CAP group was 34.7 months. –Median survival in paclitaxel group was 25.8 months. But, median survival doesn’t tell the whole story…..

18. 3/20/2002Clinical Trials in 20 Hours18 Hazard Ratio Compares risk of event in two populations or samples Ratio of risk in group 1 to risk in group 2 First things first….. –Kaplan-Meier Curves (product- limit estimate) –Makes a “picture” of survival

19. 3/20/2002Clinical Trials in 20 Hours19 Hazard Ratios Assumption: “Proportional hazards” The risk does not depend on time. That is, “risk is constant over time” But that is still vague….. Hypothetical Example: Assume hazard ratio is 2. –Patients in standard therapy group are at twice the risk of death as those in new drug, at any given point in time. Hazard function= P(die at time t | survived to time t)

20. 3/20/2002Clinical Trials in 20 Hours20 Hazard Ratios Hazard Ratio = hazard function for Std hazard function for New Makes the assumption that this ratio is constant over time.

21. 3/20/2002Clinical Trials in 20 Hours21 Hazard Ratios Hazard Ratio = hazard function for Pac hazard function for CAP Makes the assumption that this ratio is constant over time.  HR = 2

22. 3/20/2002Clinical Trials in 20 Hours22 Hazard Ratios Hazard Ratio = hazard function for Pac hazard function for CAP Makes the assumption that this ratio is constant over time. HR = 2  

23. 3/20/2002Clinical Trials in 20 Hours23 Interpretation Again For any fixed point in time, individuals in the standard therapy group are at twice the risk of death as the new drug group. HR = 2  

24. 3/20/2002Clinical Trials in 20 Hours24 Hazard ratio is not always valid …. Hazard Ratio =.71

25. 3/20/2002Clinical Trials in 20 Hours25 CAP vs. Paclitaxel Hazard Ratio for Progression Free Survival: 0.60 for CAP vs. Paclitaxel

26. 3/20/2002Clinical Trials in 20 Hours26 CAP vs. Paclitaxel Hazard Ratio for Overall Survival: 0.58 for CAP vs. Paclitaxel

27. 3/20/2002Clinical Trials in 20 Hours27 Introduction to Precision Issues PrecisionVariability Two kinds of variability we tend to deal with –variation in the population: how much do individuals tend to differ from one another? –variance of statistics: how certain are we of our estimate of the odds ratio? There might be great variability in the population, but with a large sample size, we can have very good precision for a sample statistic.

28. 3/20/2002Clinical Trials in 20 Hours28 Standard Deviation Standard deviation measures how much variability there is in a variable across individuals in the population CD20 Expression in Hodgkin and Reed-Sternberg Cells of Classical Hodgkin’s Disease: Associations with Presenting Features and Clinical Outcome, Rassidakis, Mederios, Viviani, et al. JCO, March 1, 2002, v. 20(5), p. 1278.

29. 3/20/2002Clinical Trials in 20 Hours29 Standard Deviation Mean: Standard Deviation:

30. 3/20/2002Clinical Trials in 20 Hours30 Other measures of precision for continuous variables Range: the smallest and largest values of x IQR (interquartile range): 25% percentile and 75% percentile of the data 25%-tile 75%-tile

31. 3/20/2002Clinical Trials in 20 Hours31 Precision Standardized Uptake Value in 2-[18F] Fluro-2-Deoxy-D- Glucose in Predicting Outcome in Head and Neck Carcinomas Treated by Radiotherapy With or Without Chemotherapy, Allal, Dulgerov, Allaoua, Haeggeli, Ghazi, Lehmann, Slosman, JCO, March 1, 2002, v. 20(5), p. 1398. Event = treatment failure

32. 3/20/2002Clinical Trials in 20 Hours32 Next time: Confidence Intervals Measuring precision of statistics Central limit theorem Confidence intervals for –means –proportions –odds ratios –etc…..