BINF702 SPRING 2014 - George Mason...

BINF702 SPRING 2014- Chapter 3 Probability

BINF702 – SPRING 2014

Chapter 3 - Probability


3.1 Introduction - An Example to Hang Our Hat On

Example 3.1

Cancer One theory concerning the etiology of breast cancer states that women in a given age group who give birth to their first child relatively late in life (after 30) are at greater risk for eventually developing breast cancer over some time period t than are women who give birth to their first child early in life (before 20). Because women in the upper social classes tend to have children later, this theory has been used to explain why these women have a higher risk of developing breast cancer than women in the lower social classes. To test this hypothesis, we might identify 2000 women from a particular census tract who are currently ages 45-54 and have never had breast cancer, of whom 1000 had their first child before the age of 20 (call this group A) and 1000 after the age of 30 (group B). These 2000 women might be followed for 5 years to assess if they developed breast cancer during this period. Suppose there are 4 new cases of breast cancer in group A and 5 new cases in group B.


3.2 Definition of Probability

Def. 3.1 - The sample space is the set of all possible outcomes. In referring to probabilities of events, and event is any set of outcomes of interest. The probability of an event is the relative frequency of this set of outcomes over a an indefinitely large (or infinite) number of trials.

Ex - Toss a fair coin and observe the uppermost side. Since we expect that heads is as likely to come up as tails, we conclude that the empirical probability distribution is

P(H) = 1/2, P(T) = 1/2.

Can you provide another example?


Section 3.2 -Definition of Probability

Eq. 3.1

1. The probability of an event E, denoted by Pr(E), always satisfies

2. If outcomes A and B are two events that cannot both happen at the same time, then Pr(A or B occurs) = Pr(A) + Pr(B)

0 Pr( ) 1E

Can you provide an example of two events that can’t happen at the same time?


Section 3.2 -Definition of Probability

Def. 3.2 - Two events A and B are mutually exclusive if they cannot both happen at the same time.

A B mutually exclusive A B not mutually exclusive


Section 3.3 - Some Useful Probabilistic Notation

Def. 3.3 - The symbol {} is used as shorthand for the event.

Def. 3.4 - A U B is the event that either A or B occurs, or they both occur.

Def. 3.5 - A B is the event that both A and B occurs simultaneously.

A B

Can you provide an example of two events and their intersection?


Section 3.3 - Some Useful Probabilistic Notation

Def. 3.6 - The complement of A is denoted

We note

A

Pr 1 PrA A

A

Can you provide an example of an event and its complement?


Section 3.4 - The Multiplicative Law of Probability Hypertension, Genetics Suppose we are conducting a hypertension-screening program

in the home. Consider all possible pairs of DBF measurements of the mother and father

within a given family, assuming that the mother and father are not genetically related.

This sample space consists of all pairs of numbers of the form (X, Y) where X > 0, Y >

0. Certain specific events might be of interest in this context. In particular, we might be

interested in whether the mother or father is hypertensive, which is described,

respectively, by events A = {mother's DBF > 95), B = {father's DBF > 95). These events

are diagrammed in Figure 3.4.Suppose we know that Pr(A) - .1, Pr(B) = .2. What can

we say about Pr(A n B) = Pr(mother's DBF > 95 and father's DBF > 95) = Pr(both

mother and father are hypertensive)? We can say nothing unless we are willing to make

certain assumptions.


Section 3.4 - The Multiplicative Law of Probability

Def. 3.7 - Two events A and B are called independent events if

Def. 3.8 - Two events A, B are dependent if

Eq. 3.2 Multiplication Law of Probability

If A1, …, Ak are mutually exclusive events, then

Pr Pr PrA B A B

Pr Pr PrA B A B

1 2 1 2Pr Pr Pr Prk kA A A A A A

Can you provide an example of two independent events?

Can you provide an example of two dependent events?


Section 3.5 - The Addition Law of Probability

Eq. 3.3 - Addition Law of Probability

If A and B are any events, then

Pr Pr Pr PrA B A B A B

What happens if A and B are mutually exclusive?


Section 3.5 - The Addition Law of Probability

Eq. 3.4 - Addition Law of Probability for Independent Events

If two events A and B are independent, then

Pr Pr Pr( ) 1 PrA B A B A

How does this come about?


Section 3.6 - Conditional Probability

Def. 3.9 - The conditional probability of B given A written Pr(B|A) is defined as

Eq. 3.5

1. If A and B are independent events, then

2. If two events A, B are dependent, then

PrPr |

Pr

A BB A

A

Pr | Pr Pr |B A B B A

Pr | Pr Pr |

Pr Pr Pr

B A B B A and

A B A B

?=A B B A



Def. 3.10 - The relative risk (RR) of B given A is given by

N. B. - A and B independent implies RR is 1. The larger the dependence of the two events the further the relative risk is different from 1.

Pr |

|Pr |

B ARR B A

B A



Eq. 3.6 - Total-Probability rule

For any events A and B.

N. B. - This is a very useful rule.

Def. 3.11 - A set of events A1, …, Ak is exhaustive if at least one of the events must occur.

Eq. 3.7 - Total-Probability Rule

Let A1, …, Ak be mutually exclusive and exhaustive events. The unconditional probability of B Pr(B) can be written as a weighted average of the conditional probabilities of B given Ai (Pr(B|Ai)) as follows

Pr Pr | Pr Pr | PrB B A A B A A

1

Pr Pr | Prk

i i

i

B B A A



Eq. 3.8 - Generalized Multiplicative Law of Probability

If A1, …,Ak are an arbitrary set of events then

1 2 1 2 1 3 2 1 1 2 1Pr Pr Pr | Pr | Prk k kA A A A A A A A A A A A A


Section 3.7 - Bayes’ Rule and Screening Tests

Def. 3.12 - The predictive value positive (PV+) of a screening test is the probability that a person has a disease given that the test is positive.

Pr(disease|test+)

The predictive value negative (PV-) of a screening test is the probability that a person does not have a disease given that the test is negative.

Pr(no disease|test-)


Section 3.7 - Bayes Rule and Screening Tests

Def. 3.13 - The sensitivity of a symptom (or set of symptoms or screening test) is the probability that the symptom is present given that the person has a disease.

Def. 3.14 - The specificity of a symptom (or set of symptoms or screening test) is the probability that the symptom is not present given that the person does not have a disease.

Def. 3.15 - A false negative is defined as a person who tests out as negative but who is actually positive. A false positive is defined as a person who tests out as positive but who is actually negative.

Section 3.7 - Bayes Rule and Screening Tests

Eq. 3.9 Bayes’ Rule

Let A = symptom and B = disease,

Pr | Pr

Pr |Pr | Pr Pr | Pr

A B BPV B A

A B B A B B

In words, we can write this as

sensitivity *

sensitivity * 1 specificity * 1-x

xPV

x

where x = Pr(B) = prevalence of disease in the reference

population. Similarly,

specificity * 1

specificity * 1- 1-sensitivity *

xPV

x x



Section 3.7 - Bayes’ Rule and Screening Tests

Eq. 3.10 - Generalized Bayes’ Rule

Let B1, B2, …, Bk be a set of mutually exclusive and exhaustive

disease states; that is, at lease one disease state must occur and

no two disease states can occur at the same time. Let A represent

the presence of a symptom or set of symptoms. Then

1

Pr | PrPr |

Pr | Pr

i i

i k

j j

j

A B BB A

A B B


Section 3.8 - Bayesian Inference

Def. 3.16 - The prior probability of an event is the best guess by the observer of an event’s probability in the absence of data. This prior probability may be a single number, or it may be a range of likely values for the probability, perhaps with weights attached to each possible value.

Def. 3.17 - The posterior probability of an event is the probability of an event after collecting some empirical data. It is obtained by integrating information from the prior probability with additional data related to the event in question.


Section 3.9 - ROC Curves

Def. - A receiver operating characteristic (ROC) curve is a plot of the sensitivity versus (1 - specificity) of a screening test, where the different points on the curve correspond to different cutoff points used to designate test positive.


Section 3.10 - Prevalence and Incidence

De. 3.19 - The prevalence of a disease is the probability of currently having the disease regardless of the duration of time one has had the disease.It is obtained by dividing the number of people who currently have the disease by the number of people in the study population.

Def. 3.20 - The cumulative incidence of a disease is the probability that a person with no prior disease will develop a new case of the disease over some specified period.

Occupational Health

Ex. 3.29 pg. 69



Occupational Health

Ex. 3.30


Genetics

Ex 3.31


Pulmonary Disease

Ex. 3.52


Pulmonary Disease

3.53


Pulmonary Disease

Ex. 3.54


Pulmonary Disease

Ex. 3.55


Pulmonary Disease

Ex. 3.57


Pulmonary Disease

Ex. 3.58


Pulmonary Disease

Ex. 3.59


Pulmonary Disease

3.60


Pulmonary Disease

Ex. 3.61


Pulmonary Disease

Ex. 3.62


Pulmonary Disease

Ex. 3.70


Pulmonary Disease

Ex. 3.71


Pulmonary Disease

Ex. 3.73

Pulmonary Disease

Ex. 3.74



Hypertension

Ex. 3.120


Hypertension

3.121


Hypertension

Ex. 3.122


Hypertension

Ex. 3.122 Ex. 3.122


Hypertension

Ex. 3.123


Orthopedics

Ex. 3.134


Orthopedics

Ex. 3.135


Orthopedics

Ex. 3.136


Orthopedics

Ex. 3.136

Can you produce R code to produce a ROC curve from this data?


Orthopedics

Ex. 3.137


Homework Problems Chapter 3

3.13, 3.16, 3.20, 3.62, 3.63, 3.79, 3.80, 3.81, 3.82, 3.100, 3.101, 3.102, 3.103, 3.104, 3.105, 3.106

BINF702 SPRING 2014 - George Mason...

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