What are the chances� by xakJDHF

VIEWS: 6 PAGES: 55

• pg 1
```									What are the chances…

Conditional Probability
&
Introduction to Bayes’
Theorem

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
   Given a positive dobutamine stress echo, what is
the probability that the patient does NOT have

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
–   Probability:          expected number of “heads”       1
total number of options          2
–   Odds:                 expected number of “heads”       1
expected number of non “heads”   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:     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

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
LR+        = sensitivity / (1-specificity)                                  +       -
= (a/(a+c)) / (b/(b+d))                                         a       b
+

Test
c       d

-
LR-        = (1-sensitivity) / specificity
= (c/(a+c)) / (d/(b+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
–   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
–   (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
   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                              New                          Posterior
Updated
x         Evidence               =
Information
Probability                       Information                      Probability
Information

18
Clinical Decision Making: Conditional Probability and Bayes’ Theorem

Bayesian Analysis

Activity                     Background
Borrow money                    Credit history
Bet a horse                     Past performance
Prior
Sentence a criminal             Previous convictions
Treat a patient                 Past medical history
Interpret a test                Pre-test probability
Clinical trial analysis                NONE!

19
Clinical Decision Making: Conditional Probability and Bayes’ Theorem

Prior Information in Diagnostic Testing
Bayesian Analysis

1.0
Women

Pre-Test Probability
Typical Angina
0.8

0.6                          Atypical Angina
Prior
0.4

Nonanginal
0.2
No Pain
0.0
35   45         55         65
Age

N Engl J Med 1979;300:1350
20
Clinical Decision Making: Conditional Probability and Bayes’ Theorem

Bayesian Analysis

1.0
Women

Pre-Test Probability
0.8

0.6                                Atypical Angina
Prior
0.2
0.4

0.2        0.17

0.1        1          10
0.0
Prior Odds
35          45         55       65
0.17                                                                 Age
Odds =                 = 0.2
1 – 0.17                                                                           N Engl J Med 1979;300:1350

21
Clinical Decision Making: Conditional Probability and Bayes’ Theorem

Bayesian Analysis

1.0
Men

Pre-Test Probability
0.8

0.6                               Atypical Angina
Prior
0.8                                                         0.44
0.4

0.2

0.1        1          10
0.0
Prior Odds
35          45         55       65
0.44                                                                 Age
Odds =                   = 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
Disease
+     -
+   a     b

Test
0.8                   x         Evidence
-   c    d

0.1         1         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
Disease
+     -
+   80    40

Test
0.8                   x                  4.0
-   20    160
0.1         1         10          0.1          1           10
Prior Odds                       Likelihood Ratio                   100   200
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

0.8                         x                4.0               =                   3.2

0.1            1             10       0.1          1            10         0.1            1              10
Prior Odds                         Likelihood Ratio                      Posterior Odds
45 year old man
45 year old man                      2.0 mm horizontal                     with atypical angina
with atypical angina                    ST depression                               and
2.0 mm ST depression
CAD probability = 0.8/1.8 = 44%                                            CAD probability = 3.2/4.2 = 76%

25
Clinical Decision Making: Conditional Probability and Bayes’ Theorem

Computing the Post-test Odds
Bayesian Analysis

0.2                      x                4.0               =            0.8

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

26
Clinical Decision Making: Conditional Probability and Bayes’ Theorem

Review
Bayesian Analysis

Prior                          Evidential                        Posterior
x                                =
Odds Ratio                       Odds Ratio                       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                             New                           Updated
Prior                 x         Evidence               =        Posterior
Information                        Information                     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                             New                           Updated
Prior Odds                x         Evidence               =        Posterior
Information                        Information                     Information

0.01                x

31
Clinical Decision Making: Conditional Probability and Bayes’ Theorem

   Evidence = Likelihood Ratio

LR+ = sensitivity / (1-specificity)
Gold Standard
= (a/(a+c)) / (b/(b+d))
+       -
+    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
doctors often encounter:
Gold Standard
–   1% of women at age forty
who participate in routine                                     +             -
screening have breast
80           9.6
cancer.                                                +

Test
–   80% of women with breast
20          90.4

-
cancer will get positive
mammographies.                                              100          100
–   9.6% of women without
breast cancer will also get
positive mammographies.
http://www.sysopmind.com/bayes

33
Clinical Decision Making: Conditional Probability and Bayes’ Theorem

    Evidence = Likelihood Ratio

LR+ = sensitivity / (1-specificity)
Gold Standard
= (a/(a+c)) / (b/(b+d))
+         -
=          (80/100) / (9.6/100)                                      + 80 (a) 9.6 (b)

Test
=          8.33                                                         20 (c)    90.4    (d)

-
100       100
(a +c)    (b + d)

34
Clinical Decision Making: Conditional Probability and Bayes’ Theorem

Bayesian Analysis

Background                             New                           Updated
Prior Odds                x         Evidence               =        Posterior
Information                        Information                     Information
Odds

0.01                x             8.33

35
Clinical Decision Making: Conditional Probability and Bayes’ Theorem

Bayesian Analysis

Background                             New                           Updated
Prior Odds                x         Evidence               =        Posterior
Information                        Information                     Information
Odds

0.01                x             8.33               =         0.0833

36
Clinical Decision Making: Conditional Probability and Bayes’ Theorem

Bayesian Analysis

Background                            New                           Updated
Prior Odds               x         Evidence               =        Posterior
Information                       Information                     Information
Odds

0.01                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
   Bayes’ theorem incorporates results of
trials/research to update our baseline
assumptions

40
Clinical Decision Making: Conditional Probability and Bayes’ Theorem

Bayesian Analysis

Events
+    -
Treatment

A       a         b
Prior                        Evidential                        Posterior
x                                =
Risk Ratio                      Odds Ratio                       Odds Ratio
B      c       d

41
Quantifying the Prior

Quantifying the Prior

Events
+    -
Treatment

A    174 1925
Prior                 Evidential          Posterior
x                =
Risk Ratio               Odds Ratio         Odds Ratio
B    198 1865

PROVE-IT

Odds Ratio = 0.85              Adhir Shroff, MD, MPH

N Engl J Med 2004;350:1495
Quantifying the Prior

Evidential        Posterior
0.85             x                =
Odds Ratio       Odds Ratio

0.8       1        1.25
Prior
Odds Ratio
Quantifying the Evidence

Events
Treatment
+    -
A 309     1956
Posterior
0.85                                                =
Odds Ratio
B 343     1889
0.8       1         1.25
Prior                               A to Z
Odds Ratio
Odds Ratio = 0.87   Shroff, MD, MPH

JAMA 2004;292:1307
Quantifying the Evidence

Posterior
0.85              x            0.87             =
Odds Ratio

0.8       1         1.25       0.8       1        1.25
Prior                        Evidential
Odds Ratio                     Odds Ratio
Considering the Uncertainties

Posterior
0.85             x            0.87             =
Risk Ratio

0.8       1        1.25       0.8       1        1.25
Prior                       Evidential                    Posterior
Odds Ratio                    Odds Ratio                    Risk Ratio
I figure 40% chance of
and a 10% chance we know
Computing the Posterior

x                             =

0.8       1         1.25       0.8       1        1.25       0.8       1        1.25
Prior                        Evidential                     Posterior
Odds Ratio                     Odds Ratio                    Odds Ratio
Interpreting the Posterior

Risk Reduction > 10%

Area = 0.8

Posterior
x                                     =
Risk Ratio
p = 0.10
CI
0.8       1         1.25       0.8         1              1.25       0.8            1             1.25
Prior                        Evidential                                Posterior
Odds Ratio                     Odds Ratio                               Odds Ratio
Interpreting the Posterior

1

Posterior Probability
Area = 0.8

0
0.8       1         1.25   0.8       1        1.25                            0   10          50        100
Prior                    Evidential                                            Risk Reduction
Odds Ratio                 Odds Ratio                                               Threshold
Statins in Acute Coronary Syndromes

PROVE-IT                        A to Z                 PROVE-IT + A to Z

x                             =

0.8        1        1.25       0.8       1        1.25        0.8        1         1.25
Prior                       Evidential                      Posterior
Odds Ratio                    Odds Ratio                     Odds Ratio
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
1.0

Posterior Probability
0.8

0.6

0.4

0.2

0.0
0.8        1        1.25   0.8       1        1.25                                 1
1           10
10           100
100
Prior                   Evidential                                           Risk Reduction Threshold
Risk Reduction Threshold
Odds Ratio                Odds Ratio                                                     (%)
(%)
Tomorrow’s Another Day

TODAY
+
TODAY                          TOMORROW
TOMORROW

x                              =

0.8       1         1.25       0.8        1        1.25       0.8        1        1.25
Prior                         Evidential                      Posterior
Odds Ratio                      Odds Ratio                     Odds Ratio
Summary

Prior   x    Evidence   =    Posterior

Conclusions

• Conventional analysis of clinical trials
ignores key background information.
• Bayesian analysis incorporates this