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Point Estimation: Odds Ratios, Hazard Ratios, Risk Differences, Precision

Clinical Trials in 20 Hours. Point Estimation: Odds Ratios, Hazard Ratios, Risk Differences, Precision. Elizabeth S. Garrett esg@jhu.edu Oncology Biostatistics March 20, 2002. Point Estimation. Definition : A “point estimate” is a one-number summary of data.

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Point Estimation: Odds Ratios, Hazard Ratios, Risk Differences, Precision

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

  2. 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 Clinical Trials in 20 Hours

  3. 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 Clinical Trials in 20 Hours

  4. Today • Point Estimates commonly seen (and misunderstood) in clinical oncology • odds ratio • risk difference • hazard ratio/risk ratio Clinical Trials in 20 Hours

  5. 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 Clinical Trials in 20 Hours

  6. 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.” Clinical Trials in 20 Hours

  7. 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, • p1 = proportion of women receiving SAT • p2 = proportion of men receiving SAT Clinical Trials in 20 Hours

  8. Odds Ratio from a 2x2 table Clinical Trials in 20 Hours

  9. Clinical Trials in 20 Hours

  10. 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” Clinical Trials in 20 Hours

  11. 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 Clinical Trials in 20 Hours

  12. Related Measures of Risk • Relative Risk: RR = p1/p2 • 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: • p1 = 0.50, p2 = 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 Clinical Trials in 20 Hours

  13. Related Measures of Risk • Risk Difference: p1 - p2 • 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. Clinical Trials in 20 Hours

  14. 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. Clinical Trials in 20 Hours

  15. 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 Clinical Trials in 20 Hours

  16. “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 Odds Ratio via 2x2 table Clinical Trials in 20 Hours

  17. “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….. Point Estimates: Hazard Ratios “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. Clinical Trials in 20 Hours

  18. 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 Hazard Ratio Clinical Trials in 20 Hours

  19. 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) Clinical Trials in 20 Hours

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

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

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

  23. 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  HR = 2  Clinical Trials in 20 Hours

  24. Hazard ratio is not always valid …. Hazard Ratio = .71 Clinical Trials in 20 Hours

  25. CAP vs. Paclitaxel • Hazard Ratio for Progression Free Survival: 0.60 for CAP vs. Paclitaxel Clinical Trials in 20 Hours

  26. CAP vs. Paclitaxel • Hazard Ratio for Overall Survival: 0.58 for CAP vs. Paclitaxel Clinical Trials in 20 Hours

  27. Introduction to Precision Issues • Precision Variability • 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. Clinical Trials in 20 Hours

  28. 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. Clinical Trials in 20 Hours

  29. Standard Deviation • Mean: • Standard Deviation: Clinical Trials in 20 Hours

  30. 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 75%-tile 25%-tile Clinical Trials in 20 Hours

  31. 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 Clinical Trials in 20 Hours

  32. Next time: Confidence Intervals • Measuring precision of statistics • Central limit theorem • Confidence intervals for • means • proportions • odds ratios • etc….. Clinical Trials in 20 Hours

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