Conditional Independence

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Conditional Independence

• Farrokh Alemi Ph.D.Professor of Health
  Administration and PolicyCollege of Health and
  Human Services, George Mason University4400
  University Drive, Fairfax, Virginia 22030703 993
  1929 falemi_at_gmu.edu

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Lecture Outline

• What is probability?
• Assessment of rare probabilities
• Calculus of probability
• Conditional independence
• Definition
• Use
• Methods of verification
• Causal modeling
• Case based learning
• Validation of risk models
• Examples

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Joint Distributions

• Shows probability of co-occurrence

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Joint Distributions
First Event Second Event Second Event Total
First Event Absent Present Total
Absent a b ab
Present c d cd
Total ac bd abcd1

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Example
Medication Error Medication Error Total
No error Error Total
Adequate staffing 50 8 13
Under staffed 7 15 22
Total 12 23 35
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Example
Medication Error Medication Error Total
No error Error Total
Adequate staffing 50 8 13
Under staffed 7 15 22
Total 12 23 35
Medication Error Medication Error Total
No error Error Total
Adequate staffing 0.63 0.1 0.73
Under staffed 0.09 0.19 0.28
Total 0.71 0.29 1
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Reducing Universe of Possibilities
Medication Error Medication Error Total
No error Error Total
Adequate staffing
Under staffed 0.32 0.68 1
Total
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Mathematical Definition of Independence
P(A B) P(A)
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Joint Marginal Distributions
Medication Error Medication Error Total
No error Error Total
Adequate staffing 0.52 0.21 0.73
Under staffed 0.2 0.08 0.28
Total 0.71 0.29 1
P(AB) P(A) P(B)
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CHITEST function
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Comparison of Conditioned Un-conditioned
Probabilities
P( Medication error ) ? P( Medication error
understaffing) 0.29 ? 0.68
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Mathematical Definition of Conditional
Independence
P(A B, C) P(A C)
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Mathematical Definition of Conditional
Independence
P(AB C) P(A C) P(B C) 
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Dependent Events Can Be Conditionally Independent
P( Medication error ) ? P( Medication error Long
shift)
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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)
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Use of Conditional Independence

• Analyze chain of dependent events
• Simplify calculations

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Use of Conditional Independence

• Analyze chain of dependent events
• Simplify calculations

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Use of Conditional Independence

• Analyze chain of dependent events
• Simplify calculations

P(C1,C2,C3, ...,CnH1) P(C1H1)
P(C2H1,C1)
P(C3H1,C1,C2) P(C4H1,C1,C2,C3)
...
P(CnH1,C1,C2,C3,...,Cn-1)    
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Use of Conditional Independence

• Analyze chain of dependent events
• Simplify calculations

P(C1,C2,C3, ...,CnH1) P(C1H1)
P(C2H1,C1)
P(C3H1,C2) P(C4H1,C3)
... P(CnH1,Cn)    
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Verifying Independence

• Reducing sample size
• Correlations
• Direct query from experts
• Separation in causal maps   

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Verifying Independence by Reducing Sample Size

• P(Error Not fatigued) 0.50
• P(Error Not fatigue Long shift) 2/4 0.50

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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

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Example
Case Age BP Weight
1 35 140 200
2 30 130 185
3 19 120 180
4 20 111 175
5 17 105 170
6 16 103 165
7 20 102 155

• 0.91 0.82 0.95 

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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

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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

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Example
Blood pressure does not depend on age given weight
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Take Home Lesson

• Conditional Independence Can Be Verified in
  Numerous Ways

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What Do You Know?

• What is the probability of hospitalization given
  that you are male? 

Case Hospitalized? Gender Age Insured
1 Yes Male gt65 Yes
2 Yes Male lt65 Yes
3 Yes Female gt65 Yes
4 Yes Female lt65 No
5 No Male gt65 No
6 No Male lt65 No
7 No Female gt65 No
8 No Female lt65 No

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What Do You Know?

• Is insurance independent of age?

Case Hospitalized? Gender Age Insured
1 Yes Male gt65 Yes
2 Yes Male lt65 Yes
3 Yes Female gt65 Yes
4 Yes Female lt65 No
5 No Male gt65 No
6 No Male lt65 No
7 No Female gt65 No
8 No Female lt65 No

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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 gt65 Yes
2 Yes Male lt65 Yes
3 Yes Female gt65 Yes
4 Yes Female lt65 No
5 No Male gt65 No
6 No Male lt65 No
7 No Female gt65 No
8 No Female lt65 No

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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 gt65 Yes
2 Yes Male lt65 Yes
3 Yes Female gt65 Yes
4 Yes Female lt65 No
5 No Male gt65 No
6 No Male lt65 No
7 No Female gt65 No
8 No Female lt65 No

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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 gt65 Yes
2 Yes Male lt65 Yes
3 Yes Female gt65 Yes
4 Yes Female lt65 No
5 No Male gt65 No
6 No Male lt65 No
7 No Female gt65 No
8 No Female lt65 No

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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. 

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Minute Evaluations

• Please use the course web site to ask a question
  and rate this lecture