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

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

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### 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.
- 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

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

Today

- Point Estimates commonly seen (and misunderstood) in clinical oncology
- odds ratio
- risk difference
- hazard ratio/risk ratio

Clinical Trials in 20 Hours

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

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

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

Odds Ratio from a 2x2 table

Clinical Trials in 20 Hours

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

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

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

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

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

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

“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 tableClinical Trials in 20 Hours

“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

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 RatioClinical Trials in 20 Hours

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

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

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

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

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

CAP vs. Paclitaxel

- Hazard Ratio for Progression Free Survival: 0.60 for CAP vs. Paclitaxel

Clinical Trials in 20 Hours

CAP vs. Paclitaxel

- Hazard Ratio for Overall Survival: 0.58 for CAP vs. Paclitaxel

Clinical Trials in 20 Hours

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

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

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

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

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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