What are the chances� by xakJDHF

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									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:          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
   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…

                                                                       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                              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
                                Buy a stock                     Market trends
                                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
     about a situation that
     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
     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

                    Events
                    +    -
Treatment




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



                Odds Ratio = ad/bc


                                                                                           41
Quantifying the Prior




           Adhir Shroff, MD, MPH
                                    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
                                   Adhir Shroff, MD, MPH
                   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
                                                 Adhir
                                        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
                                             Adhir Shroff, MD, MPH
              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
                                            Adhir Shroff,aMD, MPHrain,
                                                   I figure 40% chance of
                                                        and a 10% chance we know
                                                         what we’re talking about.
                   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
                                             Adhir Shroff, MD, MPH
                   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
                                               Adhir Shroff, MD, MPH
                   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
                                         Adhir Shroff, MD, MPH
      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
                                             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
                                                                             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                                                     (%)
                                                                                               (%)
                                         Adhir Shroff, MD, MPH
                   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
                                              Adhir Shroff, MD, MPH
            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.
                       Adhir Shroff, MD, MPH
“Excellent sermon.”

								
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