Theoretical Background

Overview

This vignette provides the theoretical foundation for the CorOncoEndpoints package, covering:

• The Fleischer model for OS and PFS
• Copula-based modeling of dependence
• Correlation bounds (Fréchet-Hoeffding)
• Mathematical derivations

The Fleischer Model

Model Specification

The Fleischer model (2009) provides a framework for modeling the dependence between overall survival (OS) and progression-free survival (PFS).

Key Components:

1. Overall Survival (OS): $OS \sim \text{Exp}(\lambda_{OS})$
2. Time to Progression (TTP): $TTP \sim \text{Exp}(\lambda_{TTP})$
3. Progression-Free Survival (PFS): $PFS = \min(OS, TTP)$

where OS and TTP are independent.

Important Properties

Property 1: PFS follows an exponential distribution $$PFS \sim \text{Exp}(\lambda_{PFS})$$ where $\lambda_{PFS} = \lambda_{OS} + \lambda_{TTP}$

Proof: Since $PFS = \min(OS, TTP)$ with independent exponentials, $$P(PFS > t) = P(OS > t) \cdot P(TTP > t) = e^{-\lambda_{OS}t} \cdot e^{-\lambda_{TTP}t} = e^{-(\lambda_{OS} + \lambda_{TTP})t}$$

Property 2: Correlation between OS and PFS $$\text{Corr}(OS, PFS) = \frac{\lambda_{OS}}{\lambda_{PFS}} = \frac{\lambda_{OS}}{\lambda_{OS} + \lambda_{TTP}}$$

This ensures $0 < \text{Corr}(OS, PFS) < 1$ and automatically satisfies $PFS \leq OS$.

Median Survival Times

For exponentially distributed time-to-event variables: $$\text{Median} = \frac{\log(2)}{\lambda}$$

Therefore: - Median OS = $\frac{\log(2)}{\lambda_{OS}}$ - Median PFS = $\frac{\log(2)}{\lambda_{PFS}}$

Copula-Based Dependence Modeling

What is a Copula?

A copula is a function that links marginal distributions to their joint distribution. For two random variables $X$ and $Y$ with marginal distributions $F_X$ and $F_Y$, the copula $C$ satisfies:

$$F_{X,Y}(x,y) = C(F_X(x), F_Y(y))$$

Why Copulas?

Copulas allow us to:

1. Separate marginal behavior from dependence structure
2. Model non-linear dependencies
3. Handle different types of tail dependence

Clayton Copula

Definition: $$C(u, v; \theta) = (u^{-\theta} + v^{-\theta} - 1)^{-1/\theta}, \quad \theta > 0$$

Properties:

• Lower tail dependence: $\lambda_L = 2^{-1/\theta}$
• Upper tail independence: $\lambda_U = 0$
• Kendall’s tau: $\tau = \theta / (\theta + 2)$
• Cannot model negative dependence (requires $\theta > 0$)

Conditional Distribution: $$C_{2|1}(v|u; \theta) = \frac{\partial C(u,v;\theta)}{\partial u} = u^{-\theta-1}(u^{-\theta} + v^{-\theta} - 1)^{-1/\theta - 1}$$

This is used in the generation algorithm.

Frank Copula

Definition: $$C(u, v; \theta) = -\frac{1}{\theta} \log\left(1 + \frac{(e^{-\theta u} - 1)(e^{-\theta v} - 1)}{e^{-\theta} - 1}\right)$$

Properties:

• Symmetric: No tail dependence ($\lambda_L = \lambda_U = 0$)
• Kendall’s tau: $\tau = 1 - \frac{4}{\theta}\left[1 - D_1(\theta)\right]$ where $D_1$ is the Debye function
• Can model negative dependence ($\theta$ can be negative)

Conditional Distribution: $$C_{2|1}(v|u; \theta) = \frac{(e^{-\theta v} - 1)(e^{-\theta} - 1)}{(e^{-\theta v} - 1) + (e^{-\theta u} - 1)(e^{-\theta} - 1)}$$

Correlation Bounds

Fréchet-Hoeffding Bounds

For any copula $C$: $$\max(u + v - 1, 0) \leq C(u,v) \leq \min(u, v)$$

These bounds correspond to:

• Lower bound: Perfect negative dependence (countermonotonic copula)
• Upper bound: Perfect positive dependence (comonotonic copula)

Correlation Bounds for TTE and Binary Response

For a time-to-event variable $T \sim \text{Exp}(\lambda)$ and binary response $R \sim \text{Bernoulli}(p)$:

Lower Bound: $$\rho_{lower} = -\sqrt{\frac{p}{1-p}} \int_0^{q_p} e^{-\lambda t} dt$$

where $q_p$ is the $p$-th quantile of the exponential distribution.

Upper Bound: $$\rho_{upper} = \sqrt{\frac{1-p}{p}} \int_{q_p}^\infty e^{-\lambda t} dt$$

These simplify to: $$\rho_{lower} = -\sqrt{\frac{p}{1-p}}(1 - p)$$$$\rho_{upper} = \sqrt{\frac{1-p}{p}} \cdot p$$

Important: These bounds depend only on $p$, not on $\lambda$ (for the TTE-Response case).

Correlation Bounds for PFS and Response

In the three-endpoint framework (OS + PFS + Response), the bounds for PFS-Response correlation are more complex and depend on both $p$ and the hazard rates:

$$\rho_{lower}^{PFS} = \sqrt{\frac{1-p}{p}} \frac{\lambda_{OS}}{\lambda_{TTP}} \left[(1-p)^{\lambda_{TTP}/\lambda_{OS}} - 1\right]$$

$$\rho_{upper}^{PFS} = \sqrt{\frac{p}{1-p}} \frac{\lambda_{OS}}{\lambda_{TTP}} \left[1 - p^{\lambda_{TTP}/\lambda_{OS}}\right]$$

where $\lambda_{TTP} = \lambda_{PFS} - \lambda_{OS}$.

Data Generation Algorithm

Step 1: Generate Uniform Random Variables

For copula-based generation:

1. Generate $U_1 \sim \text{Uniform}(0,1)$
2. Generate $U_2 \sim \text{Uniform}(0,1)$

Step 2: Apply Copula Transform

Use the conditional copula to transform $U_2$: $$V = C_{2|1}(U_2|U_1; \theta)$$

Now $(U_1, V)$ have the desired dependence structure.

Step 3: Apply Inverse CDF Transform

Transform uniforms to target distributions:

• For TTE: $T = -\frac{1}{\lambda}\log(1 - U_1)$
• For Response: $R = \mathbb{1}\{V > 1 - p\}$

where $\mathbb{1}\{\cdot\}$ is the indicator function.

Step 4: Enforce PFS ≤ OS Constraint

When generating all three endpoints:

1. Generate $OS \sim \text{Exp}(\lambda_{OS})$
2. Generate $TTP \sim \text{Exp}(\lambda_{TTP})$ independently
3. Set $PFS = \min(OS, TTP)$

This automatically ensures $PFS \leq OS$.

Calculating Copula Parameters

From Correlation to Theta

Given desired correlation $\rho$ between TTE and Response, we need to find copula parameter $\theta$.

For Clayton Copula:

Solve numerically: $$\rho = \text{Corr}(T, R) = f_{Clayton}(\theta, p, \lambda)$$

using Hoeffding’s formula for covariance.

For Frank Copula:

Solve numerically: $$\rho = \text{Corr}(T, R) = f_{Frank}(\theta, p, \lambda)$$

The package implements these using numerical optimization (bisection method).

Validation Metrics

Bias

$$\text{Bias}(\hat{\theta}) = E[\hat{\theta}] - \theta$$

Measures systematic error. Should be close to 0 for unbiased estimators.

Mean Squared Error (MSE)

$$\text{MSE}(\hat{\theta}) = E[(\hat{\theta} - \theta)^2] = \text{Var}(\hat{\theta}) + \text{Bias}^2(\hat{\theta})$$

Combines variance and bias into a single measure.

Root Mean Squared Error (RMSE)

$$\text{RMSE}(\hat{\theta}) = \sqrt{\text{MSE}(\hat{\theta})}$$

Error measure in the original scale.

Relative Bias

$$\text{Relative Bias}(\hat{\theta}) = \frac{\text{Bias}(\hat{\theta})}{\theta} \times 100\%$$

Expresses bias as a percentage of the true value.

Mathematical Derivations

Derivation of Corr(OS, PFS)

Given $PFS = \min(OS, TTP)$ with $OS \sim \text{Exp}(\lambda_1)$ and $TTP \sim \text{Exp}(\lambda_2)$ independent:

$$E[PFS \cdot OS] = E[PFS] \cdot E[OS|PFS = OS] \cdot P(PFS = OS) + E[PFS] \cdot E[OS|PFS < OS] \cdot P(PFS < OS)$$

After integration: $$\text{Cov}(OS, PFS) = \frac{1}{(\lambda_1 + \lambda_2)^2}$$

And since: $$\text{Var}(OS) = \frac{1}{\lambda_1^2}, \quad \text{Var}(PFS) = \frac{1}{(\lambda_1 + \lambda_2)^2}$$

We get: $$\text{Corr}(OS, PFS) = \frac{\text{Cov}(OS, PFS)}{\sqrt{\text{Var}(OS) \cdot \text{Var}(PFS)}} = \frac{\lambda_1}{\lambda_1 + \lambda_2}$$

Derivation of PFS-Response Correlation

In the three-endpoint framework, the correlation between PFS and Response is derived from:

1. The specified correlation between OS and Response
2. The Fleischer model relationship $PFS = \min(OS, TTP)$
3. The copula linking OS and Response

The derivation involves: $$\text{Cov}(PFS, R) = E[PFS \cdot R] - E[PFS] \cdot E[R]$$

This requires computing integrals over the copula-linked distributions, which is done numerically in the package.

References

1. Fleischer, F., Gaschler-Markefski, B., & Bluhmki, E. (2009). A statistical model for the dependence between progression-free survival and overall survival. Statistics in Medicine, 28(21), 2669-2686.

2. Trivedi, P. K., & Zimmer, D. M. (2005). Copula modeling: an introduction for practitioners. Foundations and Trends in Econometrics, 1(1), 1-111.

3. Nelsen, R. B. (2006). An introduction to copulas (2nd ed.). Springer.

4. Hofert, M., Kojadinovic, I., Maechler, M., & Yan, J. (2018). Elements of copula modeling with R. Springer.

5. Joe, H. (2014). Dependence modeling with copulas. CRC Press.

Summary

This vignette provided the mathematical foundation for:

• The Fleischer model for OS and PFS
• Copula-based dependence modeling (Clayton and Frank)
• Correlation bounds based on Fréchet-Hoeffding bounds
• Data generation algorithms
• Validation metrics

Understanding these concepts helps users: - Choose appropriate parameter values - Interpret simulation results - Validate model assumptions - Extend the methodology