what are the chances…

Adhir Shroff, MD, MPH

What are the chances…

Conditional Probability&

Introduction to Bayes’ Theorem

Clinical Decision Making: Conditional Probability and Bayes’ Theorem

2

Agenda

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


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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 < P(A) < 1)– 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


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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)


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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”


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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).


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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 options2

– Odds: expected number of “heads” 1

expected number of non “heads” 1


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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 = 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


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Likelihood Ratio

LR+ = sensitivity / (1-specificity)

= (a/(a+c)) / (b/(b+d))

LR- = (1-sensitivity) / specificity

= (c/(a+c)) / (d/(b+d))

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

a b

c d

+ -

Gold Standard

Tes

t +

-

a +c b + d


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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

http://en.wikipedia.org/wiki/Bayes'_theorem

Thomas Bayes (1702-1761)

British mathematician and minister


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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.

)(

)()|()|(

AP

BPBAPABP


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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…


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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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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│C) =P(B1) * P(C│B1)

P(B1) * P(C│B1) + P(B2) * P(C│B2)

=0.5 * 0.75

0.5 * 0.75 + 0.5 * 0.5 = 0.6


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Bayesian AnalysisBayesian Analysis

x =BackgroundInformation

NewInformation

UpdatedInformation

PriorProbability

EvidencePosterior

Probability


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Borrow money Credit historyBuy a stock Market trendsBet a horse Past performanceSentence a criminal Previous convictionsTreat a patient Past medical historyInterpret a test Pre-test probability

Activity BackgroundActivity Background

PriorPrior

Clinical trial analysis NONE!

Borrow money Credit historyBuy a stock Market trendsBet a horse Past performanceSentence a criminal Previous convictionsTreat a patient Past medical historyInterpret a test Pre-test probability

Activity BackgroundActivity Background


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Prior Information in Diagnostic TestingPrior Information in Diagnostic TestingBayesian AnalysisBayesian Analysis

0.0

0.2

0.4

0.6

0.8

1.0

35 45 55 65

Age

Pre

-Tes

t P

rob

abili

ty

WomenWomen

No Pain

Nonanginal

Typical Angina

Atypical Angina

PriorPrior

N Engl J Med 1979;300:1350


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PriorPrior

Prior Odds0.1 1 10

N Engl J Med 1979;300:1350

0.0

0.2

0.4

0.6

0.8

1.0

35 45 55 65

Age

Pre

-Tes

t P

rob

abili

ty

0.17Odds = = 0.2 1 – 0.17

0.2

WomenWomen

0.17

Atypical Angina


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N Engl J Med 1979;300:1350

PriorPrior

Prior Odds0.1 1 10

0.8

0.44Odds = = 0.8 1 – 0.44

0.0

0.2

0.4

0.6

0.8

1.0

35 45 55 65

Age

Pre

-Tes

t P

rob

abili

ty

MenMen

0.44 Atypical Angina


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Quantifying the EvidenceQuantifying the EvidenceBayesian AnalysisBayesian Analysis

EvidenceEvidence0.8

Prior Odds0.1 1 10

a b

c d

Disease + -

T

est +

-x

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


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Quantifying the EvidenceQuantifying the EvidenceBayesian AnalysisBayesian Analysis

4.00.8

80 40

20 160

Disease + -

T

est +

-x

100 200Likelihood Ratio0.1 1 10

Prior Odds0.1 1 10

LR+ = sensitivity / (1-specificity) = (a/(a+c)) / (b/(b+d))= 80/100 / 40/200= 4.0


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Computing the Post-test OddsComputing the Post-test OddsBayesian AnalysisBayesian Analysis

4.00.8

Prior Odds0.1 1 10

x =

Posterior Odds0.1 1 10

Likelihood Ratio0.1 1 10

3.2

45 year old manwith atypical angina

CAD probability = 0.8/1.8 = 44%

45 year old manwith atypical angina

and 2.0 mm ST depression


2.0 mm horizontalST depression


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Computing the Post-test OddsComputing the Post-test OddsBayesian AnalysisBayesian Analysis

0.84.00.2 x =

Likelihood Ratio Posterior Odds0.1 1 100.1 1 10

45 year old womanwith atypical angina


2.0 mm horizontalST depression

45 year old womanwith atypical angina

and 2.0 mm ST depression


Prior Odds0.1 1 10


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ReviewBayesian AnalysisBayesian Analysis

PosteriorPosteriorOdds RatioOdds Ratio

EvidentialEvidentialOdds RatioOdds Ratio

PriorPriorOdds RatioOdds Ratio

x =


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A Sample Problem

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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NewInformation

UpdatedInformation

Prior Evidence Posterior


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Pre-test probability = .01

Pre-test odds:– Odds = probability / (1-probability)

– = .01/(1-.01) = 0.01



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NewInformation

UpdatedInformation

Prior Odds Evidence Posterior

0.01 x


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Evidence = Likelihood Ratio


= (a/(a+c)) / (b/(b+d))

a b

c d

+ -

Gold Standard

Tes

t +

-

a +c b + d


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A Sample Problem

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.


http://www.sysopmind.com/bayes

+ -

Gold Standard

Tes

t +

-

80

100

20

9.6

90.4

100


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Evidence = Likelihood Ratio


= (a/(a+c)) / (b/(b+d))

80 (a) 9.6 (b)

20 (c) 90.4 (d)

+ -

Gold Standard

Tes

t +

-

100(a +c)

100(b + d)

= (80/100) / (9.6/100)

= 8.33


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NewInformation

UpdatedInformation

Prior Odds Evidence Posterior Odds

0.01 x 8.33


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NewInformation

UpdatedInformation

Prior Odds Evidence Posterior Odds

0.01 x 8.33 = 0.0833


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



NewInformation

UpdatedInformation

Prior Odds Evidence PosteriorOdds

0.01 x 8.33 = 0.08337.7% probability


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39

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


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PriorPriorRisk RatioRisk Ratio



a b

c d

Events + -

Tre

atm

ent

A

B

x =

Odds Ratio = ad/bc


Quantifying the PriorQuantifying the Prior


PriorPriorRisk RatioRisk Ratio




174 1925

198 1865

Events + -

Tre

atm

ent

A

B

x =

N Engl J Med 2004;350:1495

PROVE-IT

Odds Ratio = 0.85





x =

PriorOdds Ratio

0.85

0.8 1 1.25



Quantifying the EvidenceQuantifying the Evidence

Events + -

Tre

atm

entA

B

=

309 1956

343 1889

PriorOdds Ratio

0.85

A to Z

JAMA 2004;292:1307

Odds Ratio = 0.87

0.8 1 1.25



x =

PriorOdds Ratio

EvidentialOdds Ratio

Quantifying the EvidenceQuantifying the Evidence

0.85 0.87

0.8 1 1.25 0.8 1 1.25


PosteriorPosteriorRisk RatioRisk Ratio

x =

PriorOdds Ratio


PosteriorRisk Ratio

Considering the UncertaintiesConsidering the Uncertainties

0.870.85

0.8 1 1.25 0.8 1 1.25


x =

PriorOdds Ratio


Computing the PosteriorComputing the Posterior

PosteriorOdds Ratio

0.8 1 1.25 0.8 1 1.25 0.8 1 1.25


PosteriorPosteriorRisk RatioRisk Ratio

x =

PriorOdds Ratio


Interpreting the PosteriorInterpreting the Posterior

PosteriorOdds Ratio

Area = 0.8

Risk Reduction > 10%

p = 0.10CI

0.8 1 1.25 0.8 1 1.25 0.8 1 1.25

Adhir Shroff, MD, MPHP

oste

rior

Pro

babi

lity

1

0

Area = 0.8

Risk Reduction Threshold

0 50 100

PriorOdds Ratio


Interpreting the Posterior Interpreting the Posterior

100.8 1 1.25 0.8 1 1.25


Statins in Acute Coronary SyndromesStatins in Acute Coronary Syndromes

x =

0.8 1 1.25 0.8 1 1.25

PriorOdds Ratio


PosteriorOdds Ratio

0.8 1 1.25

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

JAMA 2004;292:1307N Engl J Med 2004;350:1495


1 10 100

Risk Reduction Threshold(%)

PROVE-IT + A to Z

Risk Reduction Threshold(%)

Pos

terio

r P

roba

bilit

y

1 10 100

1.0

0.8

0.6

0.4

0.2

0.0

0.8 1 1.25 0.8 1 1.25

PriorOdds Ratio


A to ZPROVE-IT

Statins in Acute Coronary SyndromesStatins in Acute Coronary Syndromes


TomorrowTomorrow’’s Another Days Another Day

0.8 1 1.25

PriorOdds Ratio


x =

TOMORROW

TODAY+

TOMORROW

PosteriorOdds Ratio

0.8 1 1.250.8 1 1.25

TODAY


SummarySummary

x =PriorPrior EvidenceEvidence PosteriorPosterior


• 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.

ConclusionsConclusions

“Excellent sermon.”

what are the chances…

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