II. Definitions

  1. Sum of probabilities
    1. Sum of probabilities for a given situation = 1
    2. Example: P(infected) + P(not infected) = 1
  2. Conditional probability
    1. Probability of X given Y uses vertical pipe notation (|)
    2. Example: Probability of STD given multiple sexual partners
      1. P(STD | multiple-partners)
      2. Where P (A | B) = Probability of A given B

III. Technique: Decision Tree (Chance Graph)

  1. Definition
    1. Models sequential events with conditional probabilities
  2. Nodes
    1. Decision node (Square)
    2. Chance node (Circle)
      1. Probability assigned to each branch from a chance node
      2. All node branches add to 1
    3. Outcome node (Triangle)
      1. Each outcome node is assigned a value
        1. Values may be relative value, utility, QALY
        2. Values may also be Life (1) or Death (0)
        3. Values may be cost (cost effectiveness analysis)
    4. Rollback Analysis
      1. For a given decision node choice, conditional probabilities are multiplied for each outcome node
    5. Cost Effectiveness Analysis
      1. Outcome nodes are assigned cost unit values
    6. Sensitivity Analysis ("What-if")
      1. Expected values are calculated for a range of chance node probabilities
      2. Example
        1. Treatment success varies between 10 and 30%
        2. Expected values are calculated and plotted for each treatment success probability between 10-30%
        3. Fatal reaction rate is known and plotted
        4. Threshold at which the expected value for treatment success exceeds the risk of fatal reaction

IV. Example

  1. Decision node - Treatment X Given
    1. Chance node - Fatal Reaction: P(rxn) = 0.10
      1. Outcome node: 0 (dies)
    2. Chance node - no fatal reaction: 1-P(rxn) = 0.90
      1. Chance node - Treatment success: P(cure) = 0.20
        1. Outcome node: 1 (survives)
      2. Chance node - Treatment fails: 1 - P(cure) = 0.80
        1. Outcome node: 0 (dies)
    3. Expected Value Calculation if treatment given
      1. Fatal Reaction = (0.1 * 0) = 0
      2. No Fatal Reaction = (tSuccess + tFail) * 0.9
        1. Treatment success (tSuccess) = (0.2 *1) = 0.2
        2. Treatment fails (tFail) = (0.8 * 0) = 0
      3. Expected Value = 0 + (0.2 + 0) * 0.9 = 0.18
  2. Decision node - Treatment X Not Given
    1. Chance node - Improves: P(cure) = 0.15
      1. Outcome node: 1 (survives)
    2. Chance node - Succumbs: P(cure) = 0.85
      1. Outcome node: 0 (dies)
    3. Expected value calculation if treatment not given
      1. Spontaneous cure = (0.15 * 1) = 0.15
      2. Patient succumbs = (0.85 * 0) = 0
      3. Expected Value = (0.15 + 0) = 0.15
  3. Analysis
    1. Expected Value calculated for treatment branch is slightly higher (0.18) compared with non-treatment branch (0.15)

V. Resources

VI. References

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

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