# Dirichlet Distribution & Log-Logistic Survival

## Overview

Standard PSA uses Beta (utilities) and Gamma (costs). Two common modelling situations need specialised distributions: multinomial transition probabilities require the Dirichlet to maintain row-sum constraints, and diseases with hump-shaped hazards need the Log-Logistic survival model.

## Part A: Dirichlet for Transition Matrices

### The Problem

You are building a 3-state Markov model for Chronic Kidney Disease (Stable -> Progressed -> Dead). From a cohort of 200 patients observed for 1 year starting in “Stable”:

- 150 remained Stable
- 35 progressed to CKD Stage 4
- 15 died

These three probabilities (0.75, 0.175, 0.075) must sum to 1.0 in every PSA iteration. If you sample them independently using three Beta distributions, they will almost never sum to 1 – breaking the model.

### The Dirichlet Solution

The Dirichlet distribution is the multivariate generalisation of the Beta. Its parameters are the observed counts:

\[\boldsymbol{\alpha} = (150, 35, 15)\]

Each sample from a Dirichlet is a complete probability vector that sums to exactly 1.0.

### Sampling via Gamma Decomposition

ParCC uses the standard algorithm:

1. Draw \(X_i \sim \text{Gamma}(\alpha_i, 1)\) for each state
2. Compute \(p_i = X_i / \sum_j X_j\)
3. The resulting \((p_1, p_2, p_3)\) is Dirichlet-distributed and sums to 1

### In ParCC

1. Navigate to Uncertainty (PSA) and select Dirichlet (Multinomial).
2. Enter counts: 150, 35, 15.
3. Enter labels: Stable, Progressed, Dead.
4. Click Fit & Sample.

ParCC displays the Dirichlet parameters, mean proportions, a bar chart of sampled proportions, and a ready-to-use R code snippet for your PSA loop.

### When to Use Dirichlet vs Independent Betas

Situation | Use
--- | ---
Single probability (e.g., utility, event rate) | Beta distribution
Two mutually exclusive outcomes | Beta (one parameter determines both)
Three or more mutually exclusive outcomes | Dirichlet – guarantees row-sum = 1
Transition matrix row in a Markov model | Dirichlet for each row

## Part B: Log-Logistic Survival

### The Problem

You are modelling recovery after hip replacement surgery. The hazard of revision is:

- Low immediately after surgery (close monitoring)
- Peaks around year 5-7 (implant loosening)
- Declines after year 10 (survivors have well-fixed implants)

Neither Exponential (constant hazard) nor Weibull (monotonic hazard) can capture this hump-shaped pattern.

### The Log-Logistic Distribution

The survival function is:

\[S(t) = \frac{1}{1 + (t/\alpha)^\beta}\]

The hazard function is:

\[h(t) = \frac{(\beta/\alpha)(t/\alpha)^{\beta-1}}{1 + (t/\alpha)^\beta}\]

When \(\beta > 1\), the hazard rises to a peak then falls – exactly the hump shape needed.

### In ParCC

From a published Kaplan-Meier curve, identify two time-survival points:

- Point 1: At Year 5, implant survival = 92%
- Point 2: At Year 15, implant survival = 78%

1. Navigate to Survival Curves > Fit Survival Curve.
2. Select method: Log-Logistic (From 2 Time Points).
3. Enter the values.
4. ParCC solves for alpha (scale) and beta (shape).
5. Verify beta > 1 in the output to confirm the expected hump-shaped hazard.

### Calibration Method

ParCC uses the log-odds transformation. Since \(S(t) = 1/(1 + (t/\alpha)^\beta)\):

\[\ln\left(\frac{1 - S(t)}{S(t)}\right) = \beta \ln(t) - \beta \ln(\alpha)\]

Two points yield two equations, solved for alpha and beta.

### Choosing the Right Survival Distribution

Distribution | Hazard Shape | Best For
--- | --- | ---
Exponential | Constant | Stable chronic conditions
Weibull | Monotonic (increasing or decreasing) | Cancer mortality, device failure
Log-Logistic | Hump-shaped or decreasing | Post-surgical revision, immune response

### Extrapolation Warning

As with all parametric survival models, extrapolation beyond the observed data requires clinical justification. The Log-Logistic’s long tail means it predicts higher long-term survival than the Weibull – validate this against clinical expectations.

## References

1. Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation. Oxford University Press; 2006. Chapter 4: Probabilistic Sensitivity Analysis.
2. Collett D. Modelling Survival Data in Medical Research. 3rd ed. Chapman & Hall/CRC; 2015. Chapter 5: Log-Logistic Models.
3. NICE Decision Support Unit Technical Support Document 14: Survival Analysis for Economic Evaluations Alongside Clinical Trials. 2013.