---
title: "Dirichlet Distribution & Log-Logistic Survival"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Dirichlet Distribution & Log-Logistic Survival}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
```

## 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.