Slide 4 - Probability

7/29/2019 Slide 4 - Probability

1/31

Biostatistics

Probability


2/31

Probability

Probability provides a mathematical description

of randomness. A phenomenon is calledrandom

if the outcome of an experiment is uncertain.However, random phenomena often followrecognizable patterns. This long-run regularity of

random phenomena can be describedmathematically. The mathematical study ofrandomness is called probability theory.


3/31

Basic Probability Concepts

Foundation of statistics because of the

concept of sampling and the concept ofvariation or dispersion and how likely anobserved difference is due to chance

Probability statements used frequently in

statistics e.g., we say that we are 90% sure that an

observed treatment effect in a study is real


4/31

Elementary Properties of Probabilities - I

Probability of an event is a non-negative number

Given some process (or experiment) with n mutuallyexclusive outcomes (events), E1, E2, , En, theprobability of any event Ei is assigned a nonnegativenumber

P(Ei) 0

key concept is mutually exclusive outcomes - cannotoccur simultaneously

Given previous definition, not clear how to construct anegative probability


5/31

Experiments and Events

A well-defined procedure resulting in an outcome, e.g., rolling a die,tossing a coin, dealing cards.

Experiment: An experiment with the following characteristics: The set of all possible outcomes is known before the experiment. The outcome of the experiment is not known beforehand.

Space. The set of all possible outcomes of the experiment. We use Sto denote the sample space.

Event. Any subset of the sample space. A and B are events, then : A B, called the union of A and B is the event consisting of all

outcomes that are in A or in B or in both A and B. AB, called the intersection of A and B. It consists of all outcomes

that are in both A and B

For any even A, Ac is called the complement of A, consists of alloutcomes in S that are not in A


6/31

Relative Frequency Interpretation of

Probability

If I flip a fair coin hundreds and hundreds of times,the fraction of heads will be very close to 0.5. Themore I repeat the experiment, the closer to 0.5 therelative frequency will be. This is the same result theclassical definition gives us. The relative frequencyinterpretation of probability works especially well for

repeatable events, e.g., flipping a coin, rolling dice,drawing cards, etc.


7/31

Probability Rules

Let P(A) = the probability that event A

occurs.1. P(S) =1

2. 0 < = P(A)


8/31

Characteristics of Probabilities

Probabilities are expressed as fractions between 0.0and 1.0 e.g., 0.01, 0.05, 0.10, 0.50, 0.80

Probability of a certain event = 1.0

Probability of an impossible event = 0.0

Application to biomedical research e.g., ask if results of study or experiment could be due to

chance alone

e.g., significance level and power

e.g., sensitivity, specificity, predictive values


9/31

Elementary Properties of Probabilities - II

Sum of the probabilities of mutually exclusiveoutcomes is equal to 1

Property of exhaustiveness refers to the fact that the observer of the process must allow for

all possible outcomes

P(E1) + P(E2) + + P(En) = 1

key concept is still mutually exclusive outcomes


10/31

Elementary Properties of Probabilities - III

Probability of occurrence of either of twomutually exclusive events is equal to the

sum of their individual probabilities Given two mutually exclusive events A and

B

P(A or B) = P(A) + P(B) If not mutually exclusive, then problem

becomes more complex


11/31

Elementary Properties of Probabilities - IV

For two independent events, A and B, occurrence ofevent A has no effect on probability of event B

P(A B) = P(B) + P(A)

P(A | B) = P(A)

P(B | A) = P(B)

P(A

B) = P(A) x P(B)* * Key concept in contingency table analysis


12/31

)(

)(

BP

BAP

)(

)(

AP

BAP )(

)(

)(*)(BP

AP

BPAP

Conditional Probability and IndependenceThe conditional probability of A given B is P(A/B) =

if A and B are independent

P(B/A) =

=


13/31

Multiplicative rule

P(AB) =P(A)*P(B) if A and B are independent

P(A/B) =)(

)(

BP

BAP =

)()(

)(*)(AP

BP

BPAP


14/31

Elementary Properties of Probabilities - V

Conditional probability

Conditional probability of B given A is givenby:

P(B | A) = P(A B) / P(A)

Probability of the occurrence of event Bgiven that event A has already occurred.

Ex. given that a test for bladder cancer ispositive, what is the probability that the

patient has bladder cancer


15/31


16/31

Relative Frequency Interpretation of Probability

If I flip a fair coin hundreds and hundreds of times,the fraction of heads will be very close to 0.5. The

more I repeat the experiment, the closer to 0.5 therelative frequency will be. This is the same result theclassical definition gives us. The relative frequencyinterpretation of probability works especially well for

repeatable events, e.g., flipping a coin, rolling dice,drawing cards, etc.


17/31

Elementary Properties of Probabilities - VI

Given some variable that can be broken down into m

categories designated A1, A2, , Am and anotherjointly occurring variable that is broken down into ncategories designated by B1, B2, , Bn, the marginalprobability of Ai, P(Ai), is equal to the sum of the joint

probabilities of Ai with all the categories of B. That is,

jBiAPiAP


18/31

Elementary Properties of Probabilities - VII

For two events A and B, where P(A) + P(B) =

1, then )(1)( APAP


19/31

Elementary Properties of Probabilities - VIII

Multiplicative Law For any two events A and B,

P(A B) = P(A) P(B | A) Joint probability of A and B = Probability of B times Probability

of A given B

Addition Law

For any two events A and B P(A B) = P(A) + P(B) - P(A B)

Probability of A or B = Probability of A plus Probability of Bminus the joint Probability of A and B


20/31

Male Female Total

Medical; 35 25 60

Dental 16 24 40

Total 51 49 100


21/31


22/31


23/31


24/31

Disease

+ -

Test

+

9990True Positive(TP)

990False Positive(FP)

All withPositive TestTP+FP

PositivePredictive Value=TP/(TP+FP)9990/(9990+990)=91%

-

10False Negative(FN)

989,010True Negative(TN)

All withNegative TestFN+TN

NegativePredictive Value=TN/(FN+TN)989,010/(10+989,010)=99.999%

All with Disease10,000

All withoutDisease999,000

Everyone=TP+FP+FN+TN

Sensitivity=

TP/(TP+FN)9990/(9990+10)

Specificity=

TN/(FP+TN)989,010/

Pre-Test Probability=

(TP+FN)/(TP+FP+FN+TN)(in this case = prevalence)


25/31

Disease

+ -

Test+ 999(TP) 999(FP)

All withPositive TestTP+FP1998Positive PredictiveValue=TP/(TP+FP)=50%

-1(FN) 998,001(TN)

All withNegativeTestFN+TN

NegativePredictive Value=TN/(FN+TN)=99.999%All withDisease1000

All withoutDisease999,000EveryoneTP+FP+FN+TN

Sensitivity99.9% Specificity99.9% Pre-Test Probability0.1%


26/31

Disease+ -

Test

+ 99,900(TP) 900(FP)All withPositive Test100,800

Positive PredictiveValue=TP/(TP+FP)99,900/100,800=99%

- 100(FN) 899,100(TN)All withNegativeTest899,200

Negative PredictiveValue=TN/(FN+TN)899,100/899,200=99.99%All with Disease100,000

All withoutDisease900,000

EveryoneTP+FP+FN+TNSensitivity99.9% Specificity99.9% Pre-Test Probability10%


27/31

Questions about Screening Tests

Given that a patient has the disease, what is theprobability of a positive test results?

Given that a patient does not have the disease, whatis the probability of a negative test result?

Given a positive screening test, what is theprobability that the patient has the disease?

Given a negative screening test, what is theprobability that the patient does not have thedisease?


28/31

Sensitivity and Specificity

Sensitivity of a test is the probability of a positive testresult given the presence of the disease a / (a + c)

Specificity of a test is the probability of a negative testresult given the absence of the disease d / (b + d)


29/31

Predictive Values

Predictive value positive of a test is the probabilitythat the subject has the disease given that thesubject has a positive screening test P(D | T)

Predictive value negative of a test is the probabilitythat a subject does not have the disease, given that

the subject has a negative screening test P(D- | T-)


30/31

Bayes Theorem

Predictive value positive

Predictive value negative

)()|()()|(

)()|()|(

DPDTPDPDTP

DPDTPTDP

)()|()()|(

)()|()|(

DPDTPDPDTP

DPDTPTDP

))(1()1()(

)()|(

DPxyspecificitDPxysensitivit

DPxysensitivitTDP

)()1())(1(

))(1()|(

DPxysensitivitDPxyspecificit

DPxyspecificitTDP


31/31

Prevalence and Incidence

Prevalence is the probability of having the

disease or condition at a given point in timeregardless of the duration

Incidence is the probability that someone without

the disease or condition will contract it during aspecified period of time

Documents

Slide 4 - Probability