Conditional
Independence
Farrokh Alemi Ph.D.
Professor of Health Administration and Policy
College of Health and Human Services, George Mason University
4400 University Drive, Fairfax, Virginia 22030
703 993 1929 [email protected]
Lecture Outline
What is probability?
Assessment of rare probabilities
Calculus of probability
Conditional independence
1.
2.
3.
4.



5.
6.
7.
8.
Definition
Use
Methods of verification
Causal modeling
Case based learning
Validation of risk models
Examples
Joint Distributions

Shows probability of cooccurrence
Joint Distributions
First
Event
Absent
Present
Total
Second Event
Absent
Present
a
b
c
a+c
d
b+d
Total
a+b
c+d
a+b+c+d=1
Example
Medication Error
No error
Error
Adequate staffing
Under staffed
Total
50
7
12
8
15
23
Total
13
22
35
Example
Medication Error
No error
Error
Adequate staffing
Under staffed
Total
Adequate staffing
Under staffed
Total
50
7
12
8
15
23
Medication Error
No error Error
0.63
0.1
0.09
0.71
0.19
0.29
Total
13
22
35
Total
0.73
0.28
1
Reducing Universe of
Possibilities
Medication Error
No error Error
Adequate staffing
Under staffed
Total
0.32
0.68
Total
1
Mathematical Definition of
Independence
P(A | B) = P(A)
Joint & Marginal Distributions
Medication Error
No error
Adequate staffing
0.52
Under staffed
0.2
Total
0.71
Error
0.21
0.08
0.29
Total
0.73
0.28
1
P(A&B) = P(A) * P(B)
CHITEST function
Comparison of Conditioned &
Un-conditioned Probabilities
P( Medication error ) ≠ P( Medication error| understaffing)
0.29 ≠ 0.68
Mathematical Definition of
Conditional Independence
P(A | B, C) = P(A | C)
Mathematical Definition of
Conditional Independence
P(A&B | C) = P(A | C) * P(B | C)
Dependent Events Can Be
Conditionally Independent
P( Medication error ) ≠ P( Medication error| Long shift)
Dependent Events Can Be
Conditionally Independent
P( Medication error ) ≠ P( Medication error| Long shift)
P( Medication error | Long shift, Not fatigued) =
P( Medication error| Not fatigued)
Use of Conditional
Independence


Analyze chain of dependent
events
Simplify calculations
Use of Conditional
Independence


Analyze chain of dependent
events
Simplify calculations
Use of Conditional
Independence


Analyze chain of dependent
events
Simplify calculations
P(C1,C2,C3, ...,Cn|H1) =
P(C1|H1) *
P(C2|H1,C1) *
P(C3|H1,C1,C2) *
P(C4|H1,C1,C2,C3) *
... *
P(Cn|H1,C1,C2,C3,...,Cn-1)
Use of Conditional
Independence


Analyze chain of dependent
events
Simplify calculations
P(C1,C2,C3, ...,Cn|H1) =
P(C1|H1) *
P(C2|H1,C1) *
P(C3|H1,C2) *
P(C4|H1,C3) *
... *
P(Cn|H1,Cn)
Verifying Independence




Reducing sample size
Correlations
Direct query from experts
Separation in causal maps
Verifying Independence by
Reducing Sample Size
Medication
error
Long shift
No
Yes
No
Yes
No
No
No
No
Yes
Yes
Yes
No
Yes
No
Yes
Yes
No
No
No
No
No
Yes
No
No
No
No
No
No
No
No
No
No
Yes
No
Yes
No
Case
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18


Fatigue
No
No
No
No
No
No
No
No
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
P(Error | Not fatigued) = 0.50
P(Error | Not fatigue & Long shift) = 2/4 = 0.50
Verifying through Correlations




Rab is the correlation between A
and B
Rac is the correlation between
events A and C
Rcb is the correlation between
event C and B
If Rab= Rac Rcb then A is
independent of B given the
condition C
Example
Case
1
Age
35
BP
140
Weight
200
2
3
4
5
30
19
20
17
130
120
111
105
185
180
175
170
6
7
16
20
103
102
165
155
Rage, blood pressure = 0.91
Rage, weight = 0.82
R weight, blood pressure = 0.95
0.91 ~ 0.82 * 0.95
Verifying by Asking Experts







Write each event on a 3 x 5 card
Ask experts to assume a population where
condition has been met
Ask the expert to pair the cards if knowing
the value of one event will make it
considerably easier to estimate the value of
the other
Repeat these steps for other populations
Ask experts to share their clustering
Have experts discuss any areas of
disagreement
Use majority rule to choose the final
clusters
Verifying Independence by Causal
Maps


Ask expert to draw a causal map
Conditional independence: A node
that if removed would sever the flow
from cause to consequence




Any two nodes connected by an arrow
are dependent.
Multiple cause of same effect are
dependent
The consequence is independent of the
cause for a given level of the
intermediary event.
Multiple consequences of a cause are
independent of each other given the
cause
Example
Blood pressure does not depend on age given weight
Take Home Lesson
Conditional Independence Can Be
Verified in Numerous Ways
What Do You Know?

What is the probability of
hospitalization given that you are
male?
Case
Hospitalized?
Gender
Age
Insured
1
Yes
Male
>65
Yes
2
Yes
Male
<65
Yes
3
Yes
Female
>65
Yes
4
Yes
Female
<65
No
5
No
Male
>65
No
6
No
Male
<65
No
7
No
Female
>65
No
8
No
Female
<65
No
What Do You Know?

Is insurance independent of age?
Case
Hospitalized?
Gender
Age
Insured
1
Yes
Male
>65
Yes
2
Yes
Male
<65
Yes
3
Yes
Female
>65
Yes
4
Yes
Female
<65
No
5
No
Male
>65
No
6
No
Male
<65
No
7
No
Female
>65
No
8
No
Female
<65
No
What Do You Know?

What is the likelihood associated of
being more than 65 years old among
hospitalized patients? Please note that
this is not the same as the probability of
being hospitalized given you are 65
years old.
Case
Hospitalized?
Gender
Age
Insured
1
Yes
Male
>65
Yes
2
Yes
Male
<65
Yes
3
Yes
Female
>65
Yes
4
Yes
Female
<65
No
5
No
Male
>65
No
6
No
Male
<65
No
7
No
Female
>65
No
8
No
Female
<65
No
What Do You Know?

In predicting hospitalization, what
is the likelihood ratio associated
with being 65 years old?
Case
Hospitalized?
Gender
Age
Insured
1
Yes
Male
>65
Yes
2
Yes
Male
<65
Yes
3
Yes
Female
>65
Yes
4
Yes
Female
<65
No
5
No
Male
>65
No
6
No
Male
<65
No
7
No
Female
>65
No
8
No
Female
<65
No
What Do You Know?

What is the prior odds for
hospitalization before any other
information is available?
Case
Hospitalized?
Gender
Age
Insured
1
Yes
Male
>65
Yes
2
Yes
Male
<65
Yes
3
Yes
Female
>65
Yes
4
Yes
Female
<65
No
5
No
Male
>65
No
6
No
Male
<65
No
7
No
Female
>65
No
8
No
Female
<65
No
What Do You Know?


Draw what causes medication errors
on a piece of paper, with each cause
in a separate node and arrows
showing the direction of
causality. List all causes, their
immediate effects until it leads to a
medication error.
Analyze the graph you have
produced and list all conditional
dependencies inherent in the graph.
Minute Evaluations

Please use the course web site
to ask a question and rate this
lecture