HR-Based Probability Converter

The Problem

One of the most common tasks in health economic modelling is converting a control group probability to an intervention group probability using a Hazard Ratio (HR) from a clinical trial. This arises whenever:

A common mistake is to multiply the probability directly by the HR (i.e., \(p_{int} = p_{ctrl} \times HR\)). This is mathematically incorrect because the HR operates on the instantaneous rate, not on the cumulative probability.

The Three-Step Method

Under the proportional hazards assumption (the HR is constant over time):

Step 1: Convert control probability to rate

\[r_{control} = -\frac{\ln(1 - p_{control})}{t}\]

Step 2: Apply the Hazard Ratio

\[r_{intervention} = r_{control} \times HR\]

Step 3: Convert back to probability

\[p_{intervention} = 1 - e^{-r_{intervention} \times t_{cycle}}\]

Worked Example 1: PLATO Trial (Ticagrelor vs Aspirin)

The PLATO trial (Wallentin et al., NEJM 2009) compared Ticagrelor to Aspirin in Acute Coronary Syndrome. Key results at 12 months:

Endpoint Aspirin (Control) HR (95% CI)
CV Death 5.25% 0.79 (0.69 - 0.91)
MI 6.43% 0.84 (0.75 - 0.95)
Stroke 1.38% 1.01 (0.79 - 1.28)
Major Bleeding 11.43% 1.04 (0.95 - 1.13)

CV Death Conversion

# PLATO trial - CV Death
p_control <- 0.0525   # 5.25% at 12 months
hr <- 0.79             # Ticagrelor vs Aspirin
t <- 1                 # 1 year

# Step 1: Probability -> Rate
r_control <- -log(1 - p_control) / t
cat("Step 1 - Control rate:", round(r_control, 5), "per year\n")
#> Step 1 - Control rate: 0.05393 per year

# Step 2: Apply HR
r_intervention <- r_control * hr
cat("Step 2 - Intervention rate:", round(r_intervention, 5), "per year\n")
#> Step 2 - Intervention rate: 0.0426 per year

# Step 3: Rate -> Probability
p_intervention <- 1 - exp(-r_intervention * t)
cat("Step 3 - Intervention probability:", round(p_intervention, 5), "\n\n")
#> Step 3 - Intervention probability: 0.04171

# Compare with naive method
p_naive <- p_control * hr
cat("Correct method:", round(p_intervention, 5), "\n")
#> Correct method: 0.04171
cat("Naive (p x HR):", round(p_naive, 5), "\n")
#> Naive (p x HR): 0.04147
cat("Difference:", round((p_intervention - p_naive) * 10000, 2), "per 10,000 patients\n")
#> Difference: 2.34 per 10,000 patients

With 95% Confidence Interval

# Apply CI bounds
hr_low <- 0.69
hr_high <- 0.91

p_low <- 1 - exp(-r_control * hr_low * t)
p_high <- 1 - exp(-r_control * hr_high * t)

cat("Intervention probability: ", round(p_intervention, 5),
    " (95% CI:", round(p_low, 5), "to", round(p_high, 5), ")\n")
#> Intervention probability:  0.04171  (95% CI: 0.03653 to 0.04789 )

# Clinical impact
arr <- p_control - p_intervention
nnt <- ceiling(1 / arr)
cat("ARR:", round(arr * 100, 2), "%\n")
#> ARR: 1.08 %
cat("NNT:", nnt, "\n")
#> NNT: 93

Worked Example 2: High Event Rate (Oncology)

The error from naive multiplication grows with higher event rates. Consider an oncology model where the control arm 2-year mortality is 40%:

p_control_onc <- 0.40
hr_onc <- 0.75
t_onc <- 2

# Correct method
r_ctrl <- -log(1 - p_control_onc) / t_onc
r_int <- r_ctrl * hr_onc
p_int_correct <- 1 - exp(-r_int * t_onc)

# Naive method
p_int_naive <- p_control_onc * hr_onc

cat("Control 2-year mortality:", p_control_onc, "\n")
#> Control 2-year mortality: 0.4
cat("HR:", hr_onc, "\n\n")
#> HR: 0.75
cat("Correct intervention probability:", round(p_int_correct, 4), "\n")
#> Correct intervention probability: 0.3183
cat("Naive (p x HR):", round(p_int_naive, 4), "\n")
#> Naive (p x HR): 0.3
cat("Error:", round(abs(p_int_correct - p_int_naive) * 100, 2), "percentage points\n")
#> Error: 1.83 percentage points
cat("This affects", round(abs(p_int_correct - p_int_naive) * 10000), "patients per 10,000\n")
#> This affects 183 patients per 10,000

Worked Example 3: Rescaling to Monthly Cycles

Often you need monthly transition probabilities for your Markov model. You can combine HR conversion with time rescaling:

# PLATO CV Death, but for a monthly model
p_annual_ctrl <- 0.0525
hr <- 0.79

# Convert annual probability to rate
r_ctrl <- -log(1 - p_annual_ctrl) / 1
r_int <- r_ctrl * hr

# Convert to monthly probabilities
t_month <- 1/12
p_monthly_ctrl <- 1 - exp(-r_ctrl * t_month)
p_monthly_int <- 1 - exp(-r_int * t_month)

cat("Monthly control probability:", round(p_monthly_ctrl, 6), "\n")
#> Monthly control probability: 0.004484
cat("Monthly intervention probability:", round(p_monthly_int, 6), "\n")
#> Monthly intervention probability: 0.003544

Batch Conversion

For models with multiple health states and transitions, you can upload a CSV with control probabilities and HRs. Navigate to Bulk Conversion in ParCC and select “HR -> Intervention Probability”.

Example CSV format:

Endpoint,Control_Prob,Hazard_Ratio,Time_Horizon
CV Death,0.0525,0.79,1
MI,0.0643,0.84,1
Stroke,0.0138,1.01,1
Major Bleeding,0.1143,1.04,1

Multi-HR Comparison

Use the Multi-HR Comparison tab to evaluate how different HRs from subgroup analyses or network meta-analyses translate into absolute probabilities. This is useful for:

Key Assumptions and Limitations

  1. Proportional Hazards: The HR is assumed constant over time. If the Kaplan-Meier curves cross, this assumption is violated and alternative methods (e.g., piecewise HRs) should be considered.

  2. Exponential within cycle: The conversion assumes exponential survival within each model cycle. This is standard practice for short cycles (1 month, 1 year) but may be less appropriate for very long cycles.

  3. Independent events: The conversion treats each event independently. For competing risks models, additional adjustments may be needed.

References