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**II. Definitions**

- Sum of probabilities
- Sum of probabilities for a given situation = 1
- Example: P(infected) + P(not infected) = 1

- Conditional probability
- Probability of X given Y uses vertical pipe notation (|)
- Example: Probability of STD given multiple sexual partners
- P(STD | multiple-partners)
- Where P (A | B) = Probability of A given B

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**III. Technique: Decision Tree (Chance Graph)**

- Definition
- Models sequential events with conditional probabilities

- Nodes
- Decision node (Square)
- Chance node (Circle)
- Probability assigned to each branch from a chance node
- All node branches add to 1

- Outcome node (Triangle)
- Each outcome node is assigned a value
- Values may be relative value, utility, QALY
- Values may also be Life (1) or Death (0)
- Values may be cost (cost effectiveness analysis)

- Each outcome node is assigned a value
- Rollback Analysis
- For a given decision node choice, conditional probabilities are multiplied for each outcome node

- Cost Effectiveness Analysis
- Outcome nodes are assigned cost unit values

- Sensitivity Analysis ("What-if")
- Expected values are calculated for a range of chance node probabilities
- Example
- Treatment success varies between 10 and 30%
- Expected values are calculated and plotted for each treatment success probability between 10-30%
- Fatal reaction rate is known and plotted
- Threshold at which the expected value for treatment success exceeds the risk of fatal reaction

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**IV. Example**

- Decision node - Treatment X Given
- Chance node - Fatal Reaction: P(rxn) = 0.10
- Outcome node: 0 (dies)

- Chance node - no fatal reaction: 1-P(rxn) = 0.90
- Chance node - Treatment success: P(cure) = 0.20
- Outcome node: 1 (survives)

- Chance node - Treatment fails: 1 - P(cure) = 0.80
- Outcome node: 0 (dies)

- Chance node - Treatment success: P(cure) = 0.20
- Expected Value Calculation if treatment given
- Fatal Reaction = (0.1 * 0) = 0
- No Fatal Reaction = (tSuccess + tFail) * 0.9
- Treatment success (tSuccess) = (0.2 *1) = 0.2
- Treatment fails (tFail) = (0.8 * 0) = 0

- Expected Value = 0 + (0.2 + 0) * 0.9 = 0.18

- Chance node - Fatal Reaction: P(rxn) = 0.10
- Decision node - Treatment X Not Given
- Chance node - Improves: P(cure) = 0.15
- Outcome node: 1 (survives)

- Chance node - Succumbs: P(cure) = 0.85
- Outcome node: 0 (dies)

- Expected value calculation if treatment not given
- Spontaneous cure = (0.15 * 1) = 0.15
- Patient succumbs = (0.85 * 0) = 0
- Expected Value = (0.15 + 0) = 0.15

- Chance node - Improves: P(cure) = 0.15
- Analysis
- Expected Value calculated for treatment branch is slightly higher (0.18) compared with non-treatment branch (0.15)

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**V. Resources**

- Decision Tree (Wikipedia)

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**VI. References**

- Desai (2014) Clinical Decision Making, AMIA’s CIBRC Online Course