Core Conversions: Rates, Odds, and Time Rescaling

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

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

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