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

Point Estimation:

Odds Ratios, Hazard Ratios, Risk Differences, Precision

Elizabeth S. Garrett

[email protected]

Oncology Biostatistics

March 20, 2002

Clinical Trials in 20 Hours

3/20/2002 Clinical Trials in 20 Hours 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


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


Today

• Point Estimates commonly seen (and misunderstood) in clinical oncology

– odds ratio

– risk difference

– hazard ratio/risk ratio


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


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.”


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

O Rp p

p p

p p

p p

1 1

2 2

1 2

2 1

1

1

1

1

/ ( )

/ ( )

( )

( )


Odds Ratio from a 2x2 table

SAT No SATWomen a = 298 b = 252 550

Men c = 202 d = 248 450500 500 1000

O Rp p

p p

a d

b c

1 2

2 1

1

1

( )

( )


O Rp p

p p

a d

b c

1 1

2 2

1

1

2 9 8 5 5 0 2 5 2 5 5 0

2 0 2 4 5 0 2 4 8 4 5 0

2 9 8 2 5 2

2 0 2 2 4 82 9 8 2 4 8

2 5 2 2 0 2

/ ( )

/ ( )

( / ) / ( / )

( / ) / ( / )

/

/*

*

SAT No SATWomen a = 298 b = 252 550

Men c = 202 d = 248 450500 500 1000


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”


Odds Ratio0 5 10 15 20

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

Log Odds Ratio-4 -2 0 2 4


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


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.


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.


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


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

CAP Paclitaxel

CR 14 8No CR 33 39

47 47


“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…..


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


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)


Hazard Ratios• Hazard Ratio = hazard function for Std

hazard function for New

• Makes the assumption that this ratio is constant over time.

Time

Ha

zard

Fu

nct

ion

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

Std TherapyNew Drug


Hazard Ratios• Hazard Ratio = hazard function for Pac

hazard function for CAP


HR = 2

Time

Ha

zard

Fu

nct

ion

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

Std TherapyNew Drug


Hazard Ratios• Hazard Ratio = hazard function for Pac

hazard function for CAP


HR = 2

HR = 2

Time

Ha

zard

Fu

nct

ion

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

Std TherapyNew Drug


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

Time

Ha

zard

Fu

nct

ion

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

Std TherapyNew Drug


Hazard ratio is not always valid ….Nelson-Aalen cumulative hazard estimates, by group

analysis time0 10 20 30 40

0.00

1.00

2.00

3.00

4.00

group 0

group 1

Hazard Ratio = .71

Kaplan-Meier survival estimates, by group

analysis time0 10 20 30 40

0.00

0.25

0.50

0.75

1.00

group 0

group 1


CAP vs. Paclitaxel

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


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

for CAP vs. Paclitaxel


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.


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.


Standard Deviation

• Mean:

• Standard Deviation:

x xN ii

N

1

1

s x xN ii

N

1

12

1

( )


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

x


Precision

Standardized Uptake Value in 2-[18F] Fluro-2-Deoxy-D-Glucose in Predicting Outcome in Head and Neck CarcinomasTreated by Radiotherapy With or Without Chemotherapy, Allal, Dulgerov, Allaoua, Haeggeli, Ghazi, Lehmann, Slosman,JCO, March 1, 2002, v. 20(5), p. 1398.

Event =treatmentfailure


Next time: Confidence Intervals

• Measuring precision of statistics

• Central limit theorem

• Confidence intervals for – means– proportions– odds ratios– etc…..

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

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