Review of Day 1 Concepts

Decision Trees

Strengths

  • They are easy to describe and understand
  • Works well with limited time horizon
  • Decision trees are a powerful framework for analyzing decisions and can provide rapid/useful insights, but they have limitations.

Limitations

  • No explicit accounting for the elapse of time.
    • Recurrent events must be separately built into model.
    • Fine for short time cycles (e.g., 12 months) but we often want to model over a lifetime.
  • Difficult to incorporate real clinical detail - Tree structure can quickly become complex.

At what probability (p_B) of rain for the beach are you indifferent between the two options?

At what probability (p_B) of rain for the beach are you indifferent between the two options?

  • Earlier, we solved for the expected payoff of remaining at home: 0.74 (which was a lower expected value than going to the beach when the chance of rain at both was 30%)

  • What would p_B need to be to yield an expected payoff at the beach of 0.74?

    • In other words, at what probability of rain at the beach would you be indifferent between staying at home & going to the beach?

At what probability (p_B) of rain for the beach are you indifferent between the two options?

Set 0.74 (expected value of remaining at home) equal to the beach payoffs and solve for p_B


pB * 0.4 + (1 - pB) * 1.0 = 0.74

At what probability (p_B) of rain for the beach are you indifferent between the two options?

pB * 0.4 + (1 - pB) * 1.0 = 0.74

pB * 0.4 + 1 - pB = 0.74

At what probability (p_B) of rain for the beach are you indifferent between the two options?

pB * 0.4 + 1 - pB = 0.74

pB * -0.6 = -0.26

At what probability (p_B) of rain for the beach are you indifferent between the two options?

pB * -0.6 = -0.26

pB = -0.26 / -0.6 = 0.43

At what probability (p_B) of rain for the beach are you indifferent between the two options?

When the probability of rain at the beach is 43% (probability of rain at home remains at 30%), we would be indifferent between staying at home & going to the beach.

If the probability of rain at the beach in > 43%, then we would stay home

Threshold Example: Screening Program

Decision Tree: Pulmonary Embolism

Decision Tree: Pulmonary Embolism

Parameter Uncertainty

  • Suppose we are not certain about the probability of fatal hemorrhage.

  • At what value of p_fatal_hem would expected survival be equal?

Threshold Analysis: Idea

  • Allow the value of p_fatal_hem to vary over a range.

  • Find the value of p_fatal_hem along this range where expected survival is equal for the “Anticoagulant” and “No Anticoagulant” strategies.

Threshold Analysis

Threshold Analysis

Threshold Analysis

Threshold Analysis

Threshold Analysis

Example 2: Screening Program

Screening Program

  • The Ministry of Health is considering implementation of a population-wide treatment program for a costly disease that affects a subgroup of the population.
  • The prevalence of the disease is not well-established.

Screening Program

  • An inexpensive screening test is available, but it is not perfect at detecting individuals with the disease.
  • A more expensive (perfect) diagnostic test is available.

Decision Problem

  1. Do nothing.
  2. Population screening with the inexpensive test.
  3. Expensive diagnostic test for everyone.

Decision Problem

  • Given that we do not know the underlying probability of disease (p_disease), can we make a policy decision?
  • Perhaps! We can use a threshold analysis.

Threshold Analysis: Steps

  1. Allow p_disease to vary over a plausible range.
  2. Find the threshold at which we would be indifferent between:
  • Do nothing vs. population screening with inexpensive test.
  • Population screening vs. diagnostic test for everyone.

Threshold Analysis

Decision

  • We can now solicit expert opinion on a likely range of the disease prevalance.
  • If this range falls within the thresholds, we can make a decision despite uncertainty in the underlying disease prevalence.

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