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
p1 / (1 − p1 )
OR =
p2 / (1 − p2 )
p (1 − p2 )
= 1
p2 (1 − p1 )
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
Odds Ratio from a 2x2 table
SAT No SAT
Women a = 298 b = 252 550
Men c = 202 d = 248 450
500 500 1000
p1 (1 − p2 ) ad
OR = =
p2 (1 − p1 ) bc
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”
More on the Odds Ratio
Sometimes, we see the log odds ratio instead of the odds
ratio.
0 5 10 15 20 -4 -2 0 2 4
Odds Ratio
Log 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
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.
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.
“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…..
Example: Assume hazard ratio is 0.7.
Patients in temsirolimus group are at 0.7 times the risk
of death as those in the interferon-alpha arm, at any
given point in time.
Hazard function= probably of dying at time
t given you survived to time t
Survival (S(t)) vs. Hazard (h(t))
Not the same thing.
Hard to ‘envision’ the difference
technically, it is the the negative of the slope of the log
of the survival curve
h(t ) = d
dt (− log(S (t ))
Yikes….don’t worry though
the statisticians deal with this stuff
just remember:
the hazard ratio is not the ratio of the survival curves
it is a ratio of some function of the survival curves
Hazard Ratios
Hazard Ratio = hazard function for T
hazard function for IA
Makes the assumption that this ratio is constant over
time.
0.30
0.25
0.20
Hazard function
0.15
0.10
0.05
0.00
0 5 10 15 20 25 30
Time (months)
Hazard Ratios
Hazard Ratio = hazard function for T
hazard function for IA
Makes the assumption that this ratio is constant over
time.
0.30
0.25
0.20
Hazard function
0.15
HR=0.7
0.10
0.05
0.00
0 5 10 15 20 25 30
Time (months)
Hazard Ratios
Hazard Ratio = hazard function for T
hazard function for IA
Makes the assumption that this ratio is constant over
time.
0.30
0.25
0.20
Hazard function
HR=0.7
0.15
HR=0.7
0.10
0.05
HR=0.7
0.00
0 5 10 15 20 25 30
Time (months)
Interpretation Again
For any fixed point in time, individuals in the T therapy
group are at 0.7 times the risk of death as the IA group.
0.30
0.25
0.20
Hazard function
HR=0.7
0.15
HR=0.7
0.10
0.05
HR=0.7
0.00
0 5 10 15 20 25 30
Time (months)
Hazard ratio is not always valid ….
Kaplan-Meier survival estimates, by group Nelson-Aalen cumulative hazard estimates, by group
1.00 4.00
0.75 3.00
group 0
0.50 2.00
group 1
0.25 1.00
group 1
0.00 group 0 0.00
0 10 20 30 40 0 10 20 30 40
analysis time analysis time
Hazard Ratio = .71
