What are the chances…
Conditional Probability
&
Introduction to Bayes’
Theorem
Adhir Shroff, MD, MPH
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Agenda





Introduction
Definitions and equations
Odds and probability
Likelihood ratios
Bayes’ Theorem
2
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Examples:



If you flipped a coin 10 times, what is the
probability that the first 5 come up heads?
What is the probability that the 6th toss comes up
heads?
Given a positive dobutamine stress echo, what is
the probability that the patient does NOT have
CAD?
3
Clinical Decision Making: Conditional Probability and Bayes’ Theorem

The probability of an event is the proportion of
times the event is expected to occur in repeated
experiments
–
–
The probability of an event, say event A, is denoted
P(A).
All probabilities are between 0 and 1.
(i.e. 0 < P(A) < 1)
–
The sum of the probabilities of all possible outcomes
must be 1.
4
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Assigning Probabilities




Guess based on prior knowledge alone
Guess based on knowledge of probability
distribution (to be discussed later)
Assume equally likely outcomes
Use relative frequencies
5
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Conditional Probability

The probability of event A occurring, given that
event B has occurred, is called the conditional
probability of event A given event B, denoted
P(A|B)
Example
 Among women with a (+) mammogram, how
often does a patient have breast cancer
–
P(breast CA +|+ mammogram)
6
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Mutually Exclusive Events

Two events are mutually exclusive if their
intersection is empty.

Two events, A and B, are mutually exclusive if
and only if P(AB) = 0
–

a child is a red head and a brunette.
P(A U B) = P(A) + P(B)
“And”
7
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Odds


The concept of "odds" is familiar from gambling
For instance, one might say the odds of a
particular horse winning a race are "3 to 1";
–
–
This means the probability of the horse winning is 3
times the probability of not winning.
Odds of 1 to 1 means a 50% chance of something
happening (as in tossing a coin and getting a head),
and odds of 99 to 1 means it will happen 99 times out
of 100 (as in bad weather on a public holiday).
8
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Odds and Probability


Both are ways to express chance or likelihood of
an event
Example:
–
What is the chance that a coin flip will result in
“heads”?
–
Probability:
–
Odds:
expected number of “heads”
total number of options
expected number of “heads”
expected number of non “heads”
1
2
1
1
9
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Odds and Probability

Example:
–
What is the chance that you will roll a 7 at the craps
table and “crap out”?


Probability:
Odds:
number of ways to roll a 7
6
total number of options
36
number of ways to roll a 7
6
number of ways to not roll a 7
30
16.7%
20%
10
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Odds and Probability

Odds = probability / (1-probability)

Probability = odds / (1+odds)

Use the craps example: if the probability of
rolling a 7 is 16.77777%, what are the odds of
rolling a seven
11
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Likelihood Ratio
Likelihood of a given test result in a patient with
the target disorder compared to the likelihood of
the same result in a patient without that disorder
Gold Standard
= sensitivity / (1-specificity)
= (a/(a+c)) / (b/(b+d))
+
LR-
= (1-sensitivity) / specificity
= (c/(a+c)) / (d/(b+d))
-
+
a
b
-
Test
LR+
c
d
a +c
b+d
12
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Bayes’ Theorem: Definition



Result in probability
theory
Relates the conditional
and marginal probability
distributions of random
variables
In some interpretations of
probability, tells how to
update or revise beliefs in
light of new evidence
Thomas Bayes (1702-1761)
British mathematician and minister
http://en.wikipedia.org/wiki/Bayes'_theorem
13
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Bayes’ Theorem: Definition
P( A | B) P( B)
P( B | A) 
P( A)


Bayes’ Rule underlies reasoning systems in
artificial intelligence, decision analysis, and
everyday medical decision making
we often know the probabilities on the right hand
side of Bayes’ Rule and wish to estimate the
probability on the left.
14
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Example from Wikipedia…
From which bowl is the cookie?
 To illustrate, suppose there are two full bowls of
cookies.
–
–

Fred picks a bowl at random, and then picks a
cookie at random.
–

Bowl #1 has 10 chocolate chip and 30 plain cookies,
Bowl #2 has 20 of each
(Assume there is no reason to believe Fred treats one
bowl differently from another, likewise for the cookies)
The cookie turns out to be a plain one…
15
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Example from Wikipedia…



How probable is it that Fred picked it out of bowl
#1?
Intuitively, it seems clear that the answer should
be more than a half, since there are more plain
cookies in bowl #1.
The precise answer is given by Bayes' theorem.
16
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Example from Wikipedia…



Let B1 correspond to Bowl #1 and B2 to bowl #2
Since the bowls are identical to Fred, P(B1) =
P(B2) and there is a 50:50 shot of picking either
bowl so the P(B1)=P(B2)=0.5
P(C)=probability of a plain cookie
P(B1) * P(C│B1)
P(B1│C) =
P(B1) * P(C│B1) + P(B2) * P(C│B2)
0.5 * 0.75
=
=
0.6
0.5 * 0.75 + 0.5 * 0.5
17
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Bayesian Analysis
Background
Prior
Information
Probability
x
New
Evidence
Information
=
Posterior
Updated
Probability
Information
18
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Bayesian Analysis
Activity
Prior
Borrow money
Buy a stock
Bet a horse
Sentence a criminal
Treat a patient
Interpret a test
Clinical trial analysis
Background
Credit history
Market trends
Past performance
Previous convictions
Past medical history
Pre-test probability
NONE!
19
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Prior Information in Diagnostic Testing
Bayesian Analysis
Women
Prior
Pre-Test Probability
1.0
Typical Angina
0.8
0.6
Atypical Angina
0.4
Nonanginal
0.2
No Pain
0.0
35
55
45
65
Age
N Engl J Med 1979;300:1350
20
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Bayesian Analysis
Women
Pre-Test Probability
1.0
0.2
Prior
0.1
1
10
Prior Odds
0.6
Atypical Angina
0.4
0.17
0.2
0.0
35
0.17
Odds =
0.8
55
45
65
Age
= 0.2
1 – 0.17
N Engl J Med 1979;300:1350
21
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Bayesian Analysis
Men
Pre-Test Probability
1.0
0.8Prior
0.1
1
10
Prior Odds
0.6
Atypical Angina
0.44
0.4
0.2
0.0
35
0.44
Odds =
0.8
55
45
65
Age
= 0.8
1 – 0.44
N Engl J Med 1979;300:1350
22
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Quantifying the Evidence
Bayesian Analysis
x
0.8
0.1
1
Evidence
Test
Disease
+
+
a
b
-
c
d
10
Prior Odds
LR+
= sensitivity / (1-specificity)
= (a/(a+c)) / (b/(b+d))
23
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Quantifying the Evidence
Bayesian Analysis
x
0.8
0.1
1
10
Prior Odds
Test
Disease
+
4.0
0.1
1
+
80
40
-
20
160
100
200
10
Likelihood Ratio
LR+
= sensitivity / (1-specificity)
= (a/(a+c)) / (b/(b+d))
= 80/100 / 40/200
= 4.0
24
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Computing the Post-test Odds
Bayesian Analysis
x
0.8
0.1
1
10
Prior Odds
45 year old man
with atypical angina
CAD probability = 0.8/1.8 = 44%
4.0
0.1
1
Likelihood Ratio
2.0 mm horizontal
ST depression
3.2
=
10
0.1
1
10
Posterior Odds
45 year old man
with atypical angina
and
2.0 mm ST depression
CAD probability = 3.2/4.2 = 76%
25
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Computing the Post-test Odds
Bayesian Analysis
x
0.2
0.1
1
10
Prior Odds
45 year old woman
with atypical angina
CAD probability = 0.2/1.2 = 17%
4.0
0.1
1
Likelihood Ratio
2.0 mm horizontal
ST depression
0.8
=
10
0.1
1
10
Posterior Odds
45 year old woman
with atypical angina
and
2.0 mm ST depression
CAD probability = 0.8/1.8 = 44%
26
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Review
Bayesian Analysis
Prior
Odds Ratio
x
Evidential
Odds Ratio
=
Posterior
Odds Ratio
27
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
A Sample Problem
Bayesian Analysis

Here's a story problem about a situation that doctors
often encounter:
–
–
–


1% of women at age forty who participate in routine screening
have breast cancer.
80% of women with breast cancer will get positive
mammographies.
9.6% of women without breast cancer will also get positive
mammographies.
A woman in this age group had a positive
mammography in a routine screening.
What is the probability that she actually has breast
cancer?
http://www.sysopmind.com/bayes
28
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Bayesian Analysis
Background
Prior
Information
x
New
Evidence
Information
=
Updated
Posterior
Information
29
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Bayesian Analysis

Pre-test probability = .01

Pre-test odds:
–
Odds = probability / (1-probability)
–
= .01/(1-.01)
= 0.01
30
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Bayesian Analysis
Background
Prior Odds
Information
x
0.01
x
New
Evidence
Information
=
Updated
Posterior
Information
31
Clinical Decision Making: Conditional Probability and Bayes’ Theorem

Evidence = Likelihood Ratio
LR+ = sensitivity / (1-specificity)
= (a/(a+c)) / (b/(b+d))
Gold Standard
-
+
a
b
-
Test
+
c
d
a +c
b+d
32
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
A Sample Problem
Bayesian Analysis
Here's a story problem
about a situation that
doctors often encounter:
–
–
+
-
+
80
9.6
-
–
1% of women at age forty
who participate in routine
screening have breast
cancer.
80% of women with breast
cancer will get positive
mammographies.
9.6% of women without
breast cancer will also get
positive mammographies.
Gold Standard
Test

20
90.4
100
100
http://www.sysopmind.com/bayes
33
Clinical Decision Making: Conditional Probability and Bayes’ Theorem

Evidence = Likelihood Ratio
LR+ = sensitivity / (1-specificity)
= (a/(a+c)) / (b/(b+d))
=
8.33
Test
(80/100) / (9.6/100)
+
-
+ 80 (a) 9.6 (b)
-
=
Gold Standard
20 (c)
90.4
(d)
100
100
(a +c)
(b + d)
34
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Bayesian Analysis
Background
Prior Odds
Information
x
0.01
x
New
Evidence
Information
=
Updated
Posterior
Information
Odds
8.33
35
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Bayesian Analysis
Background
Prior Odds
Information
x
0.01
x
New
Evidence
Information
=
8.33
=
Updated
Posterior
Information
Odds
0.0833
36
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Bayesian Analysis
Background
Prior Odds
Information
0.01
x
New
Evidence
Information
=
Updated
Posterior
Information
Odds
x
8.33
=
0.0833
7.7% probability

Given the low pre-test probability, even a + test
did not dramatically effect the post-test
probability
37
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
38
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
7.7%
39
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Conclusions



Probability and odds are different ways to
express chance
Conditional probability allows us to calculate the
probability of an event given another event has
or has not occurred (allows us to incorporate
more information)
Bayes’ theorem incorporates results of
trials/research to update our baseline
assumptions
40
Clinical Decision Making: Conditional Probability and Bayes’ Theorem
Bayesian Analysis
Treatment
Events
+
A
B
a
b
Prior
Risk Ratio
c
d
x
Evidential
Odds Ratio
=
Posterior
Odds Ratio
Odds Ratio = ad/bc
41
Quantifying the Prior
Adhir Shroff, MD, MPH
Quantifying the Prior
Treatment
Events
+
A
B
174 1925
Prior
Risk Ratio
198 1865
x
Evidential
Odds Ratio
=
Posterior
Odds Ratio
PROVE-IT
Odds Ratio = 0.85
Adhir Shroff, MD, MPH
N Engl J Med 2004;350:1495
Quantifying the Prior
x
0.85
0.8
1
Prior
Odds Ratio
Evidential
Odds Ratio
=
Posterior
Odds Ratio
1.25
Adhir Shroff, MD, MPH
Quantifying the Evidence
Treatment
Events
+
0.85
0.8
1
Prior
Odds Ratio
A 309
1956
=
B 343
Posterior
Odds Ratio
1889
1.25
A to Z
Odds RatioAdhir
= 0.87
Shroff, MD, MPH
JAMA 2004;292:1307
Quantifying the Evidence
x
0.85
0.8
1
Prior
Odds Ratio
1.25
0.87
0.8
1
=
Posterior
Odds Ratio
1.25
Evidential
Odds Ratio
Adhir Shroff, MD, MPH
Considering the Uncertainties
x
0.85
0.8
1
Prior
Odds Ratio
1.25
0.87
0.8
1
=
Posterior
Risk Ratio
1.25
Evidential
Odds Ratio
Posterior
Risk Ratio
Adhir Shroff, MD, MPH
Computing the Posterior
x
0.8
1
Prior
Odds Ratio
1.25
=
0.8
1
1.25
Evidential
Odds Ratio
0.8
1
Posterior
Odds Ratio
Adhir Shroff, MD, MPH
1.25
Interpreting the Posterior
Risk Reduction > 10%
Area = 0.8
x
p = 0.10
CI
0.8
1
Prior
Odds Ratio
1.25
Posterior
Risk Ratio
=
0.8
1
1.25
Evidential
Odds Ratio
0.8
1
Posterior
Odds Ratio
Adhir Shroff, MD, MPH
1.25
Interpreting the Posterior
Posterior Probability
1
0.8
1
Prior
Odds Ratio
1.25
0.8
1
1.25
Evidential
Odds Ratio
Area = 0.8
0
0
10
50
Risk Reduction
Threshold
Adhir Shroff, MD, MPH
100
Statins in Acute Coronary Syndromes
PROVE-IT
A to Z
PROVE-IT + A to Z
x
0.8
1
Prior
Odds Ratio
1.25
=
0.8
1
1.25
Evidential
Odds Ratio
0.8
1
1.25
Posterior
Odds Ratio
Adhir Shroff, MD, MPH
JAMA 2004;292:1307
N Engl J Med 2004;350:1495
Statins in Acute Coronary Syndromes
PROVE-IT
A to Z
PROVE-IT + A to Z
Posterior Probability
1.0
0.8
0.6
0.4
0.2
0.0
0.8
1
Prior
Odds Ratio
1.25
0.8
1
1.25
Evidential
Odds Ratio
11
10
10
100
100
Reduction Threshold
RiskRisk
Reduction
Threshold
(%)
(%)
Adhir Shroff, MD, MPH
Tomorrow’s Another Day
TODAY
TODAY
+
TOMORROW
TOMORROW
x
0.8
1
Prior
Odds Ratio
1.25
=
0.8
1
1.25
Evidential
Odds Ratio
0.8
1
Posterior
Odds Ratio
Adhir Shroff, MD, MPH
1.25
Summary
Prior
x
Evidence
=
Posterior
Adhir Shroff, MD, MPH
Conclusions
• Conventional analysis of clinical trials
ignores key background information.
• Bayesian analysis incorporates this
additional information.
• Such analyses help support—but do
not establish—the aggressive use of
statins in ACS.
• The magnitude of benefit is not likely
to be clinically important.
“Excellent sermon.”
Adhir Shroff, MD, MPH