About the Models

Description of models fitted to the FXa inhibition data.

Model description

In this Expo, we will fit models to the percent FXa inhibition as a direct effect of drug concentration. The models will differ with respect to the random effects distribution (uncorrelated and correlated), parameterization (centered and non-centered), and whether additional covariates are included to explain interindividual variability in the model parameters.

The basic statistical model will be:

\[\begin{align*} Y_{ij} &\sim N(\mu_{ij}, \sigma) \\ \mu_{ij} &= \text{e0}_i + \frac{(\text{Emax}-e0_i) \cdot C_{ij}^\gamma}{\text{EC50}_i^\gamma + C_{ij}^\gamma} \end{align*}\]

where \(i=1, \dots, n\) indexes individuals and \(j\) indexes observation within an individual.

The parameters are subject specific and (possibly conditional on covariates) are assumed to follow a normal distribution in the population:

\[\begin{align*} \{\text{e0}_i, \log \text{EC50}_i \}^T & \sim N(\{ \text{tv\_e0},\text{tv\_log\_ec50} \}^T, \boldsymbol{\Omega}) \end{align*}\]

where \(\boldsymbol{\Omega}\) is the variance-covariance matrix of the parameters.

The primary way the models differ will be in the structure of \(\boldsymbol{\Omega}\) and how the models for the inter-subject parameters are coded.

Non-centered, uncorrelated parameterization (mod0)

The initial model will assume the subject-level parameters are uncorrelated and use a non-centered parameterization. Specifically,

\[\begin{align*} \text{e0}_i &= \text{tv\_e0} + \omega_\text{e0} \cdot \eta_{1i} \\ \log \text{EC50}_i &= \text{tv\_log\_ec50} + \omega_\text{log\_ec50} \cdot \eta_{2i} \\ \eta_{1i} &\sim N(0,1)\\ \eta_{2i} &\sim N(0,1) \end{align*}\]

For this model, we assign the following prior distributions:

\[\begin{align*} \text{tv\_e0} &\sim N(0, 10) \\ \text{tv\_log\_ec50} &\sim N(4,2) \\ \text{Emax} &\sim \text{Uniform}(0,100) \\ \log \gamma &\sim N(0,1) \\ \sigma &\sim N_+(0,10) \\ \omega_\text{e0} &\sim N_+(0,2) \\ \omega_\text{log\_ec50} &\sim N_+(0,1) \end{align*}\]

where \(N(a,b)\) denotes a normal distribution with mean \(a\) and standard deviation \(b\) and \(N_+(a,b)\) denotes a half-normal distribution with mean \(a\) and standard deviation \(b\).

Centered, uncorrelated parameterization (mod0_centered)

To explore the effect of model parameterization, we will fit a statistical model identical to the model above but use a centered parameterization. Specifically,

\[\begin{align*} \text{e0}_i &\sim N(\text{tv\_e0} , \omega_\text{e0}) \\ \log \text{EC50}_i &\sim N(\text{tv\_log\_ec50} , \omega_\text{log\_ec50}) \end{align*}\]

Non-centered, correlated parameterization

The next two models assume the subject-level parameters are correlated and use a non-centered parameterization. Specifically,

\[\begin{align*} \text{e0}_i &= \text{tv\_e0} + \eta_{1i} \\ \log \text{EC50}_i &= \text{tv\_log\_ec50} + \eta_{2i} \end{align*}\]

where \(\boldsymbol{\eta}_i = (\eta_{1i},\eta_{2i})^T \sim N(\mathbf{0}, \boldsymbol{\Omega})\). Furthermore, we factor the variance-covariance matrix as \(\boldsymbol{\Omega} = \text{diag}{(\omega_\text{e0},\omega_\text{log\_ec50})} \times \boldsymbol{\Delta} \times \text{diag}{(\omega_\text{e0},\omega_\text{log\_ec50})}\) where \(\boldsymbol{\Delta}\) is the correlation matrix for \(\boldsymbol{\eta}\).

For this model, we assign the following prior distributions:

\[\begin{align*} \omega_\text{e0} &\sim N_+(0,2) \\ \omega_\text{log\_ec50} &\sim N_+(0,1) \\ \boldsymbol{\Delta} &\sim \text{LKJ}(2) \end{align*}\]

where \(\text{LKJ}(2)\) is the LKJ prior with 2 degrees of freedom. More information about the LKJ prior can be found in Section 27 of the Stan Functions Reference .

Version 1 (mod1)

In this model, we code the prior distribution for \(\boldsymbol{\eta}_i\) directly:

\[ \boldsymbol{\eta}_i = (\eta_{1i},\eta_{2i})^T \sim N(\mathbf{0}, \boldsymbol{\Omega}) \]

Version 2 (mod2)

In this model, we code the prior distribution for \(\boldsymbol{\eta}_i\) as a standard bivariate normal and scale the values accordingly:

\[ \boldsymbol{\eta}_i^* = (\eta_{1i},\eta_{2i})^T \sim N(\mathbf{0}, \mathbf{I}_2) \] and derive the individual parameters as

\[ \boldsymbol{\eta}_i = \text{Chol}(\boldsymbol{\Omega}) \boldsymbol{\eta}_i^* \] where \(\text{Chol}(\mathbf{A})\) is the Cholesky factorization of the matrix \(\mathbf{A}\).

Non-centered, correlated parameterization with covariates

Lastly, we fit a model which incorporates covariate effects on the individual parameters. This model takes the form:

\[\begin{align*} \text{e0}_i &= \text{tv\_e0} + \mathbf{X}_{\text{e0},i} \beta_{\text{e0}} + \eta_{1i} \\ \log \text{EC50}_i &= \text{tv\_log\_ec50} + \mathbf{X}_{\text{EC50},i} \beta_{\text{EC50}} + \eta_{2i} \end{align*}\]

where \(\mathbf{X}_{\text{e0}}\) and \(\mathbf{X}_{\text{EC50}}\) are design matrices (one row per individual) for covariate effects on \(\text{e0}\) and \(\text{EC50}\), respectively, and \(\beta_{\text{e0}}\) and \(\beta_{\text{EC50}}\) are the corresponding effects. For this example, both matrices will include a single covariate for sex (a two-level categorical variable).