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

Lecture Outline 1. What is probability? 2. Assessment of rare probabilities 3. Calculus of probability 4. Conditional independence Definition Definition Use Use Methods of verification Methods of verification 5. Causal modeling 6. Case based learning 7. Validation of risk models 8. Examples

Joint Distributions Shows probability of co- occurrence Shows probability of co- occurrence

Joint Distributions First Event Second Event Total AbsentPresent Absentaba+b Presentcdc+d Totala+cb+da+b+c+d=1

Example Medication Error Total No errorError Adequate staffing50813 Under staffed71522 Total122335

Example Medication Error Total No errorError Adequate staffing50813 Under staffed71522 Total Medication Error Total No errorError Adequate staffing Under staffed Total

Reducing Universe of Possibilities Medication Error Total No errorError Adequate staffing Under staffed Total

Mathematical Definition of Independence P(A | B) = P(A)

Joint & Marginal Distributions Medication Error Total No errorError Adequate staffing Under staffed Total 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 Analyze chain of dependent events Simplify calculations Simplify calculations

Use of Conditional Independence Analyze chain of dependent events Analyze chain of dependent events Simplify calculations Simplify calculations

Use of Conditional Independence Analyze chain of dependent events Analyze chain of dependent events Simplify calculations 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 Analyze chain of dependent events Simplify calculations 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 Reducing sample size Correlations Correlations Direct query from experts Direct query from experts Separation in causal maps Separation in causal maps

Verifying Independence by Reducing Sample Size P(Error | Not fatigued) = 0.50 P(Error | Not fatigued) = 0.50 P(Error | Not fatigue & Long shift) = 2/4 = 0.50 P(Error | Not fatigue & Long shift) = 2/4 = 0.50

Verifying through Correlations R ab is the correlation between A and B R ab is the correlation between A and B R ac is the correlation between events A and C R ac is the correlation between events A and C R cb is the correlation between event C and B R cb is the correlation between event C and B If R ab = R ac R cb then A is independent of B given the condition C If R ab = R ac R cb then A is independent of B given the condition C

Example CaseAgeBPWeight ~ 0.82 * ~ 0.82 * 0.95

Verifying by Asking Experts Write each event on a 3 x 5 card Write each event on a 3 x 5 card Ask experts to assume a population where condition has been met 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 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 Repeat these steps for other populations Ask experts to share their clustering Ask experts to share their clustering Have experts discuss any areas of disagreement Have experts discuss any areas of disagreement Use majority rule to choose the final clusters Use majority rule to choose the final clusters

Verifying Independence by Causal Maps Ask expert to draw a causal map Ask expert to draw a causal map Conditional independence: A node that if removed would sever the flow from cause to consequence Conditional independence: A node that if removed would sever the flow from cause to consequence Any two nodes connected by an arrow are dependent. Any two nodes connected by an arrow are dependent. Multiple cause of same effect are dependent Multiple cause of same effect are dependent The consequence is independent of the cause for a given level of the intermediary event. 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 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? What is the probability of hospitalization given that you are male? CaseHospitalized?GenderAgeInsured 1YesMale>65Yes 2 Male<65Yes 3 Female>65Yes 4 Female<65No 5 Male>65No 6 Male<65No 7 Female>65No 8 Female<65No

What Do You Know? Is insurance independent of age? Is insurance independent of age? CaseHospitalized?GenderAgeInsured 1YesMale>65Yes 2 Male<65Yes 3 Female>65Yes 4 Female<65No 5 Male>65No 6 Male<65No 7 Female>65No 8 Female<65No

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. 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. CaseHospitalized?GenderAgeInsured 1YesMale>65Yes 2 Male<65Yes 3 Female>65Yes 4 Female<65No 5 Male>65No 6 Male<65No 7 Female>65No 8 Female<65No

What Do You Know? In predicting hospitalization, what is the likelihood ratio associated with being 65 years old? In predicting hospitalization, what is the likelihood ratio associated with being 65 years old? CaseHospitalized?GenderAgeInsured 1YesMale>65Yes 2 Male<65Yes 3 Female>65Yes 4 Female<65No 5 Male>65No 6 Male<65No 7 Female>65No 8 Female<65No

What Do You Know? What is the prior odds for hospitalization before any other information is available? What is the prior odds for hospitalization before any other information is available? CaseHospitalized?GenderAgeInsured 1YesMale>65Yes 2 Male<65Yes 3 Female>65Yes 4 Female<65No 5 Male>65No 6 Male<65No 7 Female>65No 8 Female<65No

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. 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. Analyze the graph you have produced and list all conditional dependencies inherent in the graph.

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