GPQ Algorithm for Incremental Cost-Effectiveness Ratio

Research updated on August 6, 2025
Author: Santhosh Ramaraj

Imagine you are comparing two migraine treatments, Treatment A and Treatment B. Suppose Treatment A costs $12,000 on average, while Treatment B costs $10,000. The average effectiveness is measured in Quality-Adjusted Life Years (QALYs), with Treatment A providing 2.2 QALYs and Treatment B providing 2.0 QALYs. The Incremental Cost-Effectiveness Ratio (ICER) measures the additional cost for each additional QALY gained by choosing Treatment A over Treatment B.

The ICER formula is:

$ICER = \frac{\mu_{C1} - \mu_{C2}}{\mu_{E1} - \mu_{E2}}$

Substituting the values:

$ICER = \frac{12,000 - 10,000}{2.2 - 2.0} = \frac{2,000}{0.2} = 10,000$

This means Treatment A costs $10,000 per extra QALY compared to Treatment B. But to make informed decisions, you also need confidence intervals around this ICER estimate, which is where the GPQ algorithm comes in.

What is GPQ?

The Generalized Pivotal Quantity (GPQ) method, introduced by Weerahandi (1993), is a modern approach to calculate confidence intervals without relying heavily on asymptotic assumptions. A GPQ for a parameter is constructed such that it has a distribution independent of unknown parameters when conditioned on the observed data. It is particularly useful in health economics where cost and effectiveness data may follow different distributions, like log-normal costs and normally distributed effectiveness.

Step 1: Modeling Cost and Effectiveness

In our example, we consider the random variables:

$C_j \sim \text{Cost in group } j \quad (j = 1,2)$
$E_j \sim \text{Effectiveness in group } j \quad (j = 1,2)$

Let the population means be:

$\mu_{Cj} = \mathbb{E}[C_j], \quad \mu_{Ej} = \mathbb{E}[E_j]$

To stabilize cost data, which is often skewed, we apply a log transformation:

$\ln(C_j) \sim N(\mu_{lCj}, \sigma_{lCj}^2)$

Here, $\mu_{lCj}$ is the mean of the log-transformed cost.

Step 2: GPQ Construction

Suppose we have samples of cost and effectiveness from each treatment group:

Sample mean vector and covariance matrix for group

$X_{ji} = \begin{pmatrix} \ln C_{ji} \ E_{ji} \end{pmatrix}, \quad i = 1,2,\dots,n_j$

$\bar{X}_j = \frac{1}{n_j} \sum_{i=1}^{n_j} X_{ji}, \quad \hat{\Sigma}_j = \frac{1}{n_j - 1} \sum_{i=1}^{n_j} (X_{ji} - \bar{X}_j)(X_{ji} - \bar{X}_j)^{T}$

GPQs

$T_{\Sigma j} = (n_j - 1) \hat{\Sigma}_j W_j^{-1}, \quad T_{\mu j} = \bar{X}_j + \frac{1}{\sqrt{n_j}} \, T_{\Sigma j}^{1/2} Z_j$

Where:

$W_j \sim \text{Wishart}(I_2, n_j - 1), \quad Z_j \sim N(0, I_2)$

Step 3: GPQ for ICER

The ICER is a function of the mean costs and effectiveness. Using GPQs, we compute:

$T_{ICER} = \frac{ \exp\left( T_{\mu 1, lC} + \frac{1}{2} T_{\Sigma 1, 11} \right) - \exp\left( T_{\mu 2, lC} + \frac{1}{2} T_{\Sigma 2, 11} \right) }{ T_{\mu 1, E} - T_{\mu 2, E} }$

Here:

$T_{\mu j, lC}$ is the GPQ for log-transformed cost,
$T_{\mu j, E}$ is the GPQ for effectiveness.

By repeating this computation many times (e.g., 10,000 Monte Carlo simulations), we obtain a distribution for $T_{ICER}$ . The 2.5th and 97.5th percentiles of this distribution give the 95% confidence interval for the ICER.

Step 4: Applying GPQ to Our Example

Let us assume:

Treatment A (group 1): cost samples $C_1$ and effectiveness $E_1$ ,
Treatment B (group 2): cost samples $C_2$ and effectiveness $E_2$ .

We compute $\bar{X}_1, \bar{X}_2$ and $\hat{\Sigma}_1, \hat{\Sigma}_2$ .
We generate $W_1, W_2$ and $Z_1, Z_2$ .
We build $T_{\mu 1}, T_{\mu 2}$ and $T_{\Sigma 1}, T_{\Sigma 2}$ .
We evaluate $T_{ICER}$ for each simulation.

If after simulation we find that 95% of $T_{ICER}$ values fall between $8,500 and $11,500, we interpret this as the confidence range of the ICER for Treatment A over Treatment B.

Step 5: Why Use GPQ?

Traditional methods like Fieller’s theorem may fail when the denominator (effectiveness difference) is small, leading to undefined intervals. GPQ avoids this by directly simulating the distribution of the parameter, ensuring that the confidence interval is always finite. This makes it more robust for health economic evaluations, especially when cost and effectiveness data are skewed.

Step 6: Monte Carlo Simulation

Monte Carlo simulation is at the core of GPQ-based inference. We repeatedly sample from the distributions of $W_j$ and $Z_j$ , reconstruct the GPQs, and compute the ICER. The final confidence interval is simply the range where most of these simulated values lie.

For instance, if we simulate 10,000 values of $T_{ICER}$ , the lower bound is the 250th smallest value, and the upper bound is the 9,750th value.

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