# Core Conversions: Rates, Odds, and Time Rescaling

## Overview

Health economic models require transition probabilities, but clinical literature often reports rates or odds. These are different quantities and cannot be used interchangeably. This vignette explains the conversions and demonstrates them with realistic scenarios.

## Tutorial 1: Rate to Probability

### The Scenario – Anticoagulant Safety (RE-LY Trial)

You are building a Markov model comparing Dabigatran vs Warfarin for atrial fibrillation. The RE-LY trial (Connolly et al., NEJM 2009) reports the incidence rate of major bleeding as:

> 3.36 events per 100 patient-years in the Warfarin arm

### Why Simple Division is Wrong

Dividing 3.36 by 100 gives 0.0336. But this ignores the continuous nature of risk – patients who bleed early in the year are removed from the at-risk pool, meaning the remaining patients face a slightly different risk.

### The Formula

\[p = 1 - e^{-rt}\]

where \(r\) is the instantaneous rate and \(t\) is the time horizon.

### Worked Example

    # RE-LY trial: Warfarin arm major bleeding
    rate_per_100 <- 3.36
    r <- rate_per_100 / 100   # Convert to per-person rate
    t <- 1                     # 1-year model cycle

    p <- 1 - exp(-r * t)
    cat("Rate (per person-year):", r, "\n")
    #> Rate (per person-year): 0.0336
    cat("Annual probability:", round(p, 5), "\n")
    #> Annual probability: 0.03304
    cat("Naive division would give:", r, "(overestimates by", round((r - p)/p * 100, 2), "%)\n")
    #> Naive division would give: 0.0336 (overestimates by 1.69 %)

### In ParCC

1. Navigate to Converters > Rate <-> Probability
2. Input Rate = `3.36`, Multiplier = `Per 100`
3. Time = `1`
4. Result: `0.03304`

## Tutorial 2: Time Rescaling

### The Scenario – UKPDS Risk Engine

You are building a Diabetes model with 1-year cycles. The UKPDS Risk Engine (Clarke et al., Diabetologia 2004) predicts the 10-year probability of coronary heart disease as 20% for a 55-year-old male with HbA1c 8%.

### Why Simple Division is Wrong

Dividing 0.20 by 10 gives 0.02 (2% per year). But risk compounds: if you survive Year 1, you face risk again in Year 2. The correct conversion accounts for this compounding.

### The Formula

\[p_{new} = 1 - (1 - p_{old})^{t_{new}/t_{old}}\]

### Worked Example

    p_10yr <- 0.20
    t_old <- 10
    t_new <- 1

    p_1yr <- 1 - (1 - p_10yr)^(t_new / t_old)
    cat("10-year probability:", p_10yr, "\n")
    #> 10-year probability: 0.2
    cat("Correct 1-year probability:", round(p_1yr, 5), "\n")
    #> Correct 1-year probability: 0.02207
    cat("Naive (divide by 10):", p_10yr / 10, "\n")
    #> Naive (divide by 10): 0.02
    cat("Correct value is", round((p_1yr - 0.02)/0.02 * 100, 1), "% higher\n")
    #> Correct value is 10.3 % higher

### In ParCC

1. Navigate to Converters > Time Rescaling
2. Input Probability = `0.20`, Original Time = `10 Years`
3. New Time = `1 Year`
4. Result: `0.02206`

## Tutorial 3: Odds to Probability

### The Scenario – Logistic Regression Output

A logistic regression predicting post-surgical infection reports an odds ratio of 2.5 for patients with diabetes (vs no diabetes). The baseline infection probability (no diabetes) is 8%.

### The Conversion

\[p = \frac{\text{Odds}}{1 + \text{Odds}}\]

### Worked Example

    p_baseline <- 0.08
    odds_baseline <- p_baseline / (1 - p_baseline)
    odds_diabetes <- odds_baseline * 2.5
    p_diabetes <- odds_diabetes / (1 + odds_diabetes)

    cat("Baseline odds:", round(odds_baseline, 4), "\n")
    #> Baseline odds: 0.087
    cat("Diabetes odds (OR=2.5):", round(odds_diabetes, 4), "\n")
    #> Diabetes odds (OR=2.5): 0.2174
    cat("Diabetes probability:", round(p_diabetes, 4), "\n")
    #> Diabetes probability: 0.1786

## References

- Sonnenberg FA, Beck JR. Markov models in medical decision making: a practical guide. Med Decis Making. 1993;13(4):322-338.
- Fleurence RL, Hollenbeak CS. Rates and probabilities in economic modelling. Pharmacoeconomics. 2007;25(1):3-12.
- Connolly SJ, et al. Dabigatran versus warfarin in patients with atrial fibrillation. N Engl J Med. 2009;361(12):1139-1151.
- Clarke PM, et al. A model to estimate the lifetime health outcomes of patients with Type 2 diabetes. Diabetologia. 2004;47(10):1747-1759.