Decision Analysis & Decision Trees, Sensitivity Analysis, and Markov Modeling

1. Decision Analysis and Decision Trees

Definition

"Decision analysis is a method developed to provide insight into the processes of diagnosis and management of a complex disease when sufficient data are unavailable... Decision analysis forces the clinician to make quantitative estimates of each of the many factors entering into a clinical decision."

• Bradley and Daroff's Neurology in Clinical Practice

Steps Involved in Decision Analysis

1. Frame the question - Define the clinical problem (e.g., "Should this drug be given?"). Identify the decision to be analyzed and its alternatives.
2. Structure the decision tree - Draw the tree starting from a decision node (usually a square). Each branch represents a possible choice or action.
3. Identify chance nodes - After each decision branch, insert chance nodes (usually circles), which represent probabilistic outcomes (e.g., disease present/absent, treatment success/failure).
4. Assign probabilities - Assign a probability to each branch at every chance node. The probabilities at each node must sum to 1.0. These values are drawn from clinical literature, trials, or expert opinion.
5. Assign utilities (outcomes) - At the terminal end of each branch ("leaf"), assign a utility value reflecting the quality or desirability of that outcome (e.g., quality-adjusted life years, survival, cost).
6. Calculate expected value ("fold back") - Multiply each terminal utility by its probability of occurring. Work backwards (fold back) from the tips to the root, summing expected values at each node.
7. Choose the optimal strategy - The decision with the highest expected utility (or lowest expected cost, depending on the model) is the preferred strategy.
8. Perform sensitivity analysis - Test how robust the conclusions are when key probabilities or utilities are varied (see Section 2).

Applications

• Comparing diagnostic or treatment strategies when a randomized controlled trial is impractical (e.g., public health screening policies)
• Developing cervical cancer screening strategies
• Group B streptococcal screening decisions in obstetrics
• Thromboprophylaxis strategies at cesarean delivery
• Cost-effectiveness analysis (the decision tree is the foundation for formal economic analyses, such as cost-effectiveness and cost-benefit analysis)
• Teaching tool: requires the learner to explicitly identify all assumptions and data gaps
• Evaluating new technologies where prospective trial data are unavailable

"Decision analysis can be applied to decisions in which prospective trials are impractical, such as a public health policy decision, or when a randomized controlled trial would be impractical."

• Goldman-Cecil Medicine

2. Sensitivity Analysis

Definition

"Since probabilities assigned to some of the nodes may not be known and could plausibly be within a range, a technique known as sensitivity analysis can be used to determine if the key conclusions of the decision analysis model would change as the chance node probabilities change within that range."

• Goldman-Cecil Medicine

Types of Sensitivity Analysis

Steps Involved

1. Identify uncertain variables - Select the parameters in the model that are most uncertain or have the widest plausible range (e.g., probability of disease, treatment efficacy, cost estimates).
2. Define the plausible range - Usually derived from the literature (systematic reviews, meta-analyses) or expert opinion when data are sparse.
3. Vary variables one at a time (one-way) - Hold all other variables at their baseline and change the target variable across its range. Record the resulting change in the model outcome.
4. Vary multiple variables (multi-way/Monte Carlo) - For probabilistic sensitivity analysis, assign a statistical distribution to each variable and run thousands of random simulations.
5. Evaluate threshold values - Identify at what value of a variable the decision would switch from one strategy to another (the "threshold" or "break-even" point).
6. Interpret results - If the overall conclusion changes significantly with small changes in an input variable, the model is "sensitive" to that variable - and more precise data on that variable are needed. If the conclusion holds across the full plausible range, it is robust.
7. Present best-case, expected-case, and worst-case scenarios - For financial and clinical planning purposes.

Applications

• Validating the conclusions of decision tree analyses and health economic models
• Cost-effectiveness analyses (testing if the ICER changes with uncertain cost estimates)
• Financial planning in medical practice (e.g., equipment purchase decisions under different revenue scenarios)
• Identifying research priorities: variables to which a model is most sensitive highlight where new clinical research is most needed
• Subgroup analysis to test whether results hold across populations with differing baseline risks
• Policy decisions: identifying which assumptions matter most before allocating health system resources

"If the conclusions of an analysis substantially change with variation in the assumptions of the model, the model is defined to be sensitive to these assumptions. If these assumptions are parameter estimates for variables, then clinical research is needed to more precisely define the value of these variables."

• Yamada's Textbook of Gastroenterology

3. Markov Modeling

Definition

• Health states - discrete conditions a patient can be in (e.g., "disease-free," "mild disease," "severe disease," "dead")
• Transition probabilities - the probability of moving from one state to another in each cycle
• Cycle length - the time period of each transition (e.g., monthly, annually)
• Utilities / costs - values assigned to time spent in each health state

"Markov models allow for movement of hypothetical patients back and forth between various health states in a recursive fashion based on time. Discrete event simulation, conversely, assigns the probability of events occurring independently of time."

• Yamada's Textbook of Gastroenterology

Steps Involved

1. Define health states - Identify all clinically meaningful states a patient can occupy. One state must be absorbing (typically "dead"). States must be mutually exclusive and exhaustive.
2. Define the cycle length - Choose an appropriate time unit (e.g., 1 month, 1 year) consistent with the natural history of the disease.
3. Assign transition probabilities - For each cycle, specify the probability of moving from each health state to every other state. These come from clinical trial data, observational studies, or expert opinion.
4. Assign state values (utilities and costs) - Assign quality-of-life weights (utilities, 0-1 scale, used to calculate QALYs) and costs to each health state.
5. Define the starting distribution - Specify what proportion of the hypothetical cohort starts in each health state.
6. Run the model (cycle through time) - At each cycle, multiply the current state distribution by the transition probability matrix to generate the new distribution. Repeat for all cycles.
7. Accumulate outcomes - Sum QALYs, life-years, or costs accrued across all cycles and all states.
8. Calculate the ICER - Compare the cumulative cost and health outcomes of competing strategies:
   • ICER = (Cost A - Cost B) / (Effect A - Effect B)
9. Apply discounting - Future costs and health outcomes are discounted back to present value (typically 3-5% per year).
10. Perform sensitivity analysis - Test how much the transition probabilities and utilities affect the results (see Section 2).

Key Features That Distinguish Markov from Decision Trees

Applications

• Modeling chronic diseases with recurring events (cancer recurrence, heart failure exacerbations, HIV progression)
• Comparing long-term costs and effectiveness of treatments for chronic conditions
• Screening program evaluation (e.g., colorectal cancer screening, cervical cancer screening)
• Cost-utility analyses generating QALYs for health technology assessment
• Informing national health policy and reimbursement decisions (e.g., NICE in the UK)
• Oncology - modeling treatment pathways and disease-free survival
• Infectious disease modeling (e.g., HIV, hepatitis C treatment strategies)
• Epidemiological modeling of disease progression at the population level

Summary Table

1. Decision Analysis & Decision Tree

• Square node = Decision node (clinician chooses: Treat vs. Do Not Treat)
• Circle node = Chance node (nature decides: Disease Present vs. Absent, with probabilities summing to 1.0)
• Rectangle (terminal) = Outcome node with assigned utility value
• Fold-back: multiply each terminal utility by its probability, sum at each chance node, then compare the two decision branches - choose the branch with the highest expected value

2. Sensitivity Analysis

• Tornado diagram (left): Variables ranked by their influence on the result. The widest bar = most sensitive variable = most important target for further research.
• Threshold analysis (right): Shows where the optimal strategy switches. At the crossover point (threshold), both strategies yield equal expected value. Left of the line, one strategy dominates; right of the line, the other does.

3. Markov Model State-Transition Diagram

• Each circle is a health state patients can occupy
• Arrows show transitions between states each cycle, each labelled with a transition probability
• Self-loops show patients who remain in the same state
• Dead is the absorbing state (no exit)
• The model runs for N cycles; at each cycle the cohort redistributes across states, accumulating QALYs and costs that are summed to produce the final cost-effectiveness result

The Markov Model - Full Explanation with Example

What Is a Markov Model?

"Markov models assume that a patient is always in one of a finite number of discrete health states, each with a specified value and/or cost, and can transition from one health state to another over time based on known or estimated probabilities of events."

• Sabiston Textbook of Surgery

Step-by-Step Process

Worked Example: Colorectal Cancer Screening

Step 1 - Define Health States

Step 2 - Set Cycle Length

Step 3 - Assign Transition Probabilities (Annual)

Step 4 - Define Starting Cohort

Step 5 - Assign Utilities and Costs

Step 6 - Run the Cycles (Cohort Redistribution)

• QALYs = (Healthy × 1.0) + (Polyp × 0.85) + (Cancer × 0.50) + (Dead × 0)
• Cost = (Healthy × $200) + (Polyp × $500) + (Cancer × $40,000) + (Dead × $0)

Step 7 - Accumulate Over 30 Years (with 3% annual discount)

Step 8 - Calculate the ICER

State-Transition Diagram

Key Markov Model Rules

Real-World Clinical Applications

"By modeling the time dependence of probabilities for entering several different health states and outcomes, Markov models allow for many different scenarios to be estimated for any given clinical decision and are thus a valuable and versatile method for modeling the trade-offs between competing strategies."

• Sabiston Textbook of Surgery