What are the chances

1
What are the chances

• Conditional ProbabilityIntroduction to Bayes
  Theorem

2
Agenda

• Introduction
• Definitions and equations
• Odds and probability
• Likelihood ratios
• Bayes Theorem

3
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?

4

• 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
• The sum of the probabilities of all possible
  outcomes must be 1.

5
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

6
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(AB)

• Example
• Among women with a () mammogram, how often does
  a patient have breast cancer
• P(breast CA mammogram)

7
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
8
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).

9
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 expected number of heads 1
• total number of options 2
• Odds expected number of heads 1
• expected number of non heads 1

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Odds and Probability

• Example
• What is the chance that you will roll a 7 at the
  craps table and crap out?
• Probability number of ways to roll a 7 6 16.7
• total number of options 36
• Odds number of ways to roll a 7 6 20
• number of ways to not roll a 7 30

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Odds and Probability

• Odds probability / (1-probability)
• Probability odds / (1odds)
• Use the craps example if the probability of
  rolling a 7 is 16.77777, what are the odds of
  rolling a seven

12
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

• LR sensitivity / (1-specificity)
• (a/(ac)) / (b/(bd))
• LR- (1-sensitivity) / specificity
• (c/(ac)) / (d/(bd))

Gold Standard
Test
13
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
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Bayes Theorem Definition

• 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.

15
Example from Wikipedia

• From which bowl is the cookie?
• To illustrate, suppose there are two full bowls
  of cookies.
• Bowl 1 has 10 chocolate chip and 30 plain
  cookies,
• Bowl 2 has 20 of each
• Fred picks a bowl at random, and then picks a
  cookie at random.
• (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

16
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.

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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 5050 shot of picking either
  bowl so the P(B1)P(B2)0.5
• P(C)probability of a plain cookie

P(B1) P(CB1)
P(B1C)

P(B1) P(CB1) P(B2) P(CB2)
0.5 0.75

0.6

0.5 0.75 0.5 0.5
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Bayesian Analysis
New Information
Background Information
Updated Information
x

19
Bayesian Analysis
Prior
Clinical trial analysis NONE!
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Prior Information in Diagnostic Testing
Bayesian Analysis
Prior
N Engl J Med 19793001350
21
Bayesian Analysis
0.1 1
10
Prior Odds
N Engl J Med 19793001350
22
Bayesian Analysis
Prior
0.1 1
10
Prior Odds
N Engl J Med 19793001350
23
Quantifying the Evidence
Bayesian Analysis
Evidence
0.8
x
0.1 1
10
Prior Odds
LR sensitivity / (1-specificity)
(a/(ac)) / (b/(bd))
24
Quantifying the Evidence
Bayesian Analysis
Disease -
-
80
40
4.0
0.8
x
Test
20
160
100 200
0.1 1
10
0.1 1
10
Likelihood Ratio
Prior Odds
LR sensitivity / (1-specificity)
(a/(ac)) / (b/(bd)) 80/100 / 40/200 4.0
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Computing the Post-test Odds
Bayesian Analysis
x
4.0
0.8

3.2
0.1 1
10
0.1 1
10
0.1 1
10
Prior Odds
Posterior Odds
Likelihood Ratio
45 year old man with atypical angina and 2.0 mm
ST depression CAD probability 3.2/4.2 76
45 year old man with atypical angina CAD
probability 0.8/1.8 44
2.0 mm horizontal ST depression
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Computing the Post-test Odds
Bayesian Analysis
x
0.8
4.0
0.2

0.1 1
10
0.1 1
10
0.1 1
10
Posterior Odds
Prior Odds
Likelihood Ratio
45 year old woman with atypical angina and 2.0
mm ST depression CAD probability 0.8/1.8 44
45 year old woman with atypical angina CAD
probability 0.2/1.2 17
2.0 mm horizontal ST depression
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Review
Bayesian Analysis
Posterior Odds Ratio
Evidential Odds Ratio
Prior Odds Ratio
x

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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
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Bayesian Analysis
New Information
Background Information
Updated Information
x

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Bayesian Analysis

• Pre-test probability .01
• Pre-test odds
• Odds probability / (1-probability)
• .01/(1-.01) 0.01

31
Bayesian Analysis
New Information
Background Information
Updated Information
x

x
0.01
32

• Evidence Likelihood Ratio
• LR sensitivity / (1-specificity)
• (a/(ac)) / (b/(bd))

Gold Standard
Test
33
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. 

Gold Standard
80
9.6
Test
20
90.4
100
100
http//www.sysopmind.com/bayes
34

• Evidence Likelihood Ratio
• LR sensitivity / (1-specificity)
• (a/(ac)) / (b/(bd))

Gold Standard
(80/100) / (9.6/100) 8.33
Test
35
Bayesian Analysis
New Information
Background Information
Updated Information
x

x
0.01
8.33
36
Bayesian Analysis
New Information
Background Information
Updated Information
x

x
0.01
8.33

0.0833
37
Bayesian Analysis
New Information
Background Information
Updated Information
x

x

0.01
8.33
0.0833
7.7 probability

• Given the low pre-test probability, even a test
  did not dramatically effect the post-test
  probability

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7.7
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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

41
Bayesian Analysis
Events -
a
b
A B
Posterior Odds Ratio
Prior Risk Ratio
Evidential Odds Ratio
x

Treatment
c
d
Odds Ratio ad/bc
42
Bayesian Analysis of Clinical Trials
Quantifying the Prior
43
Bayesian Analysis of Clinical Trials
Quantifying the Prior
Events -
A B
174
1925
Posterior Odds Ratio
Prior Risk Ratio
Evidential Odds Ratio
x

Treatment
198
1865
PROVE-IT
Odds Ratio 0.85
N Engl J Med 20043501495
44
Bayesian Analysis of Clinical Trials
Quantifying the Prior
Posterior Odds Ratio
Evidential Odds Ratio
x

0.85
0.8 1
1.25
Prior Odds Ratio
45
Bayesian Analysis of Clinical Trials
Quantifying the Evidence
Events -
A B
309
1956
Posterior Odds Ratio

0.85
Treatment
343
1889
0.8 1
1.25
Prior Odds Ratio
A to Z
Odds Ratio 0.87
JAMA 20042921307
46
Bayesian Analysis of Clinical Trials
Quantifying the Evidence
x

0.85
0.87
0.8 1
1.25
0.8 1
1.25
Prior Odds Ratio
Evidential Odds Ratio
47
Bayesian Analysis of Clinical Trials
Considering the Uncertainties
x

0.87
0.85
0.8 1
1.25
0.8 1
1.25
Posterior Risk Ratio
Prior Odds Ratio
Evidential Odds Ratio
48
Bayesian Analysis of Clinical Trials
Computing the Posterior
x

0.8 1
1.25
0.8 1
1.25
0.8 1
1.25
Posterior Odds Ratio
Prior Odds Ratio
Evidential Odds Ratio
49
Bayesian Analysis of Clinical Trials
Interpreting the Posterior
Risk Reduction 10
Area 0.8
x

0.8 1
1.25
0.8 1
1.25
0.8 1
1.25
Posterior Odds Ratio
Prior Odds Ratio
Evidential Odds Ratio
50
Bayesian Analysis of Clinical Trials
Interpreting the Posterior
1 0
Area 0.8
Posterior Probability
10
0 50
100
0.8 1
1.25
0.8 1
1.25
Risk Reduction Threshold
Prior Odds Ratio
Evidential Odds Ratio
51
Bayesian Analysis of Clinical Trials
Statins in Acute Coronary Syndromes
A to Z
PROVE-IT
PROVE-IT A to Z
x

0.8 1
1.25
0.8 1
1.25
0.8 1
1.25
Posterior Odds Ratio
Prior Odds Ratio
Evidential Odds Ratio
JAMA 20042921307 N Engl J Med 20043501495
52
Bayesian Analysis of Clinical Trials
Statins in Acute Coronary Syndromes
PROVE-IT A to Z
A to Z
PROVE-IT
1.0 0.8 0.6 0.4 0.2 0.0
Posterior Probability
0.8 1
1.25
0.8 1
1.25
1 10
100
Prior Odds Ratio
Evidential Odds Ratio
Risk Reduction Threshold ()
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Bayesian Analysis of Clinical Trials
Tomorrows Another Day
TODAY
x

0.8 1
1.25
0.8 1
1.25
0.8 1
1.25
Posterior Odds Ratio
Prior Odds Ratio
Evidential Odds Ratio
54
Bayesian Analysis of Clinical Trials
Summary
x

Prior
Evidence
Posterior
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Bayesian Analysis of Clinical Trials
Conclusions

• Conventional analysis of clinical trials
  ignores key background information.
• Bayesian analysis incorporates this
  additional information.
• Such analyses help supportbut do not
  establishthe aggressive use of statins in
  ACS.
• The magnitude of benefit is not likely to be
  clinically important.

Excellent sermon.