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

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Point EstimationOdds Ratios, Hazard Ratios,
Risk Differences, Precision
Clinical Trials in 20 Hours

• Elizabeth S. Garrett
• esg_at_jhu.edu
• Oncology Biostatistics
• March 20, 2002

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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

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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, gt 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

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Today

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

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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
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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.

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Odds Ratio

• Odds Ratio for comparing two proportions
• OR gt 1 increased risk of group 1 compared to
  2
• OR 1 no difference in risk of group 1
  compared to 2
• OR lt 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

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Odds Ratio from a 2x2 table
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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

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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 gt 0 increased risk
• log OR 0 no difference in risk
• log OR lt 0 decreased risk

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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

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Related Measures of Risk

• Risk Difference p1 - p2
• Instead of comparing risk via a ratio, we compare
  risks via a difference.
• In many CTs, 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.

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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.

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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, DellAnna, 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

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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
• (1439)/(833) 2.1

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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, DellAnna, Torri, Colombo.
JCO, vol. 20 (5), March 2002, p. 1232.

• 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 doesnt tell the whole
  story..

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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

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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)

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Hazard Ratios

• Hazard Ratio hazard function for Std
• hazard function for New
• Makes the assumption that this ratio is constant
  over time.

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Hazard Ratios

• Hazard Ratio hazard function for Pac
• hazard function for CAP
• Makes the assumption that this ratio is constant
  over time.

HR 2
?
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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
?
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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
?
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Hazard ratio is not always valid .

Hazard Ratio .71
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CAP vs. Paclitaxel

• Hazard Ratio for Progression Free Survival
  0.60 for CAP vs. Paclitaxel

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CAP vs. Paclitaxel

• Hazard Ratio for Overall Survival 0.58 for CAP
  vs. Paclitaxel

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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.

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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 Hodgkins Disease
  Associations with Presenting Features and
  Clinical Outcome, Rassidakis, Mederios, Viviani,
  et al. JCO, March 1, 2002, v. 20(5), p. 1278.

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Standard Deviation

• Mean
• Standard Deviation

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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
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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
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Next time Confidence Intervals

• Measuring precision of statistics
• Central limit theorem
• Confidence intervals for
• means
• proportions
• odds ratios
• etc..