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Biostatistics
Probability
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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.
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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
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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
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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
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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.
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Probability Rules
Let P(A) = the probability that event A
occurs.1. P(S) =1
2. 0 < = P(A)
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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
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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
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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
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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
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)(
)(
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) =
=
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Multiplicative rule
P(AB) =P(A)*P(B) if A and B are independent
P(A/B) =)(
)(
BP
BAP =
)()(
)(*)(AP
BP
BPAP
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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
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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.
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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
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Elementary Properties of Probabilities - VII
For two events A and B, where P(A) + P(B) =
1, then )(1)( APAP
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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
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Male Female Total
Medical; 35 25 60
Dental 16 24 40
Total 51 49 100
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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)
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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%
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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%
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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?
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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)
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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-)
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Bayes Theorem
Predictive value positive
Predictive value negative
)()|()()|(
)()|()|(
DPDTPDPDTP
DPDTPTDP
)()|()()|(
)()|()|(
DPDTPDPDTP
DPDTPTDP
))(1()1()(
)()|(
DPxyspecificitDPxysensitivit
DPxysensitivitTDP
)()1())(1(
))(1()|(
DPxysensitivitDPxyspecificit
DPxyspecificitTDP
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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
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