Extensions of the absolute standardized hazard ratio and connections with measures of explained variation and variable importance

Abstract

The absolute standardized hazard ratio (ASHR) is a scale-invariant scalar measure of the strength of association of a vector of covariates with the risk of an event. It is derived from proportional hazards regression. The ASHR is useful for making comparisons among different sets of covariates. Extensions of the ASHR concept and practical considerations regarding its computation are discussed. These include a new method to conduct preliminary checks for collinearity among covariates, a partial ASHR to evaluate the association with event risk of some of the covariates conditioning on others, and the ASHR for interactions. To put the ASHR in context, its relationship to measures of explained variation and other measures of separation of risk is discussed. A new measure of the contribution of each covariate to the risk score variance is proposed. This measure, which is derived from the ASHR calculations, is interpretable as variable importance within the context of the multivariable model.

Appendix: Derivation of the standard error of the proportional contributions to the risk score variance

Let $\hat{\delta }_{k}$ and $\hat{\pi }_{k}$ be the kth elements of $\widehat{{\varvec{\updelta}}} = \text{diag} \left( {{\varvec{\Sigma}}_{z}^{1/2} \widehat{{\varvec{\upbeta}}}\widehat{{\varvec{\upbeta}}}^{T} {\varvec{\Sigma}}_{z}^{1/2} } \right)$ and $\widehat{{\varvec{\uppi}}} = {{\widehat{{\varvec{\updelta}}}} \mathord{\left/ {\vphantom {{\widehat{{\varvec{\updelta}}}} {\hat{B}_{\sigma }^{2} }}} \right. \kern-0pt} {\hat{B}_{\sigma }^{2} }},$ where $1 \le k \le p.$ Since $\hat{B}_{\sigma }^{2} = \sum\nolimits_{i = 1}^{p} {\sum\nolimits_{j = 1}^{p} {\hat{\beta }_{i} } } \hat{\beta }_{j} \sigma_{ij} ,$ where $\sigma_{ij}$ denotes the element in row $i$ and column $j$ of ${\varvec{\Sigma}}_{{\mathbf{z}}} ,$ the partial derivative of $\hat{B}_{\sigma }^{2}$ with respect to $\hat{\beta }_{l} ,1 \le l \le p,$ is ${{\partial \left( {\hat{B}_{\sigma }^{2} } \right)} \mathord{\left/ {\vphantom {{\partial \left( {\hat{B}_{\sigma }^{2} } \right)} {\partial \hat{\beta }_{l} }}} \right. \kern-0pt} {\partial \hat{\beta }_{l} }} = 2\sum\nolimits_{i = 1}^{m} {\sigma_{li} \hat{\beta }_{i} } .$ The partial derivative of $\widehat{{\varvec{\upbeta}}}\widehat{{\varvec{\upbeta}}}^{T}$ with respect to $\hat{\beta }_{l}$ is

$$ \frac{\partial }{{\partial \hat{\beta }_{l} }}\left[ {\begin{array}{*{20}c} {\hat{\beta }_{1} \hat{\beta }_{1} } & {\hat{\beta }_{1} \hat{\beta }_{2} } & \cdots & {\hat{\beta }_{1} \hat{\beta }_{p} } \\ {\hat{\beta }_{2} \hat{\beta }_{1} } & {\hat{\beta }_{2} \hat{\beta }_{2} } & \cdots & {\hat{\beta }_{2} \hat{\beta }_{p} } \\ \vdots & \vdots & \vdots & \vdots \\ {\hat{\beta }_{p} \hat{\beta }_{1} } & {\hat{\beta }_{p} \hat{\beta }_{2} } & \cdots & {\hat{\beta }_{p} \hat{\beta }_{p} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} 0 & \cdots & 0 & {\hat{\beta }_{1} } & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & \cdots & 0 & {\hat{\beta }_{l - 1} } & 0 & \cdots & 0 \\ {\hat{\beta }_{1} } & \cdots & {\hat{\beta }_{l - 1} } & {2\hat{\beta }_{l} } & {\hat{\beta }_{l + 1} } & \cdots & {\hat{\beta }_{p} } \\ 0 & \cdots & 0 & {\hat{\beta }_{l + 1} } & 0 & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & \cdots & 0 & {\hat{\beta }_{p} } & 0 & \cdots & 0 \\ \end{array} } \right] $$

The partial derivative of $\hat{\pi }_{k}$ with respect to $\hat{\beta }_{l}$ is thus

$$ \frac{{\partial \hat{\pi }_{k} }}{{\partial \hat{\beta }_{l} }} = \frac{{diag_{k} \left( {{\varvec{\Sigma}}_{z}^{1/2} {\mathbf{D}}_{l} {\varvec{\Sigma}}_{z}^{1/2} } \right)}}{{\hat{B}_{\sigma }^{2} }} - \frac{{2\hat{\pi }_{k} }}{{\hat{B}_{\sigma }^{2} }}\sum\limits_{i = 1}^{p} {\sigma_{li} \hat{\beta }_{i} } . $$

where $diag_{k}$ denotes the kth element of the diagonal, and ${\mathbf{D}}_{l}$ is an $p \times p$ matrix with element $d_{ij}^{(l)} = I_{{\left\{ {i = l} \right\}}} \hat{\beta }_{j} + I_{{\left\{ {j = l} \right\}}} \hat{\beta }_{i}$ in row $i$ and column $j.$ Using the delta method, the standard error of $\hat{\pi }_{k}$ is consistently estimated by

$$ \widehat{SE}\left\{ {\hat{\pi }_{k} } \right\} = \sqrt {\left( {\nabla_{{{\hat{\varvec{\upbeta }}}}} \hat{\pi }_{k} } \right)^{T} {\hat{\mathbf{V}}}\left( {\nabla_{{{\hat{\varvec{\upbeta }}}}} \hat{\pi }_{k} } \right)} ,$$

where $\nabla_{{\hat{{\varvec{\upbeta}}}}} = \left( {{\partial \mathord{\left/ {\vphantom {\partial {\partial \hat{\beta }_{1} ,{\partial \mathord{\left/ {\vphantom {\partial {\partial \hat{\beta }_{2} ,}}} \right. \kern-0pt} {\partial \hat{\beta }_{2} ,}} \ldots ,{\partial \mathord{\left/ {\vphantom {\partial {\partial \hat{\beta }_{p} }}} \right. \kern-0pt} {\partial \hat{\beta }_{p} }}}}} \right. \kern-0pt} {\partial \hat{\beta }_{1} ,{\partial \mathord{\left/ {\vphantom {\partial {\partial \hat{\beta }_{2} ,}}} \right. \kern-0pt} {\partial \hat{\beta }_{2} ,}} \ldots ,{\partial \mathord{\left/ {\vphantom {\partial {\partial \hat{\beta }_{p} }}} \right. \kern-0pt} {\partial \hat{\beta }_{p} }}}}} \right)^{T}$ is the gradient operator.

The partial derivative results can be written in the form of a Jacobian

$$ \nabla_{{\widehat{{\varvec{\upbeta}}}}} \widehat{{\varvec{\uppi}}} = \left( {{{\partial \widehat{{\varvec{\uppi}}}} \mathord{\left/ {\vphantom {{\partial \widehat{{\varvec{\uppi}}}} {\partial \hat{\beta }_{1} ,{{\partial \widehat{{\varvec{\uppi}}}} \mathord{\left/ {\vphantom {{\partial \widehat{{\varvec{\uppi}}}} {\partial \hat{\beta }_{2} ,}}} \right. \kern-0pt} {\partial \hat{\beta }_{2} ,}} \ldots ,{{\partial \widehat{{\varvec{\uppi}}}} \mathord{\left/ {\vphantom {{\partial \widehat{{\varvec{\uppi}}}} {\partial \hat{\beta }_{p} }}} \right. \kern-0pt} {\partial \hat{\beta }_{p} }}}}} \right. \kern-0pt} {\partial \hat{\beta }_{1} ,{{\partial \widehat{{\varvec{\uppi}}}} \mathord{\left/ {\vphantom {{\partial \widehat{{\varvec{\uppi}}}} {\partial \hat{\beta }_{2} ,}}} \right. \kern-0pt} {\partial \hat{\beta }_{2} ,}} \ldots ,{{\partial \widehat{{\varvec{\uppi}}}} \mathord{\left/ {\vphantom {{\partial \widehat{{\varvec{\uppi}}}} {\partial \hat{\beta }_{p} }}} \right. \kern-0pt} {\partial \hat{\beta }_{p} }}}}} \right)^{T} , $$

where

$$ \frac{{\partial \widehat{{\varvec{\uppi}}}}}{{\partial \hat{\beta }_{l} }} = \frac{1}{{\hat{B}_{\sigma }^{2} }}\left\{ {diag\left( {{\varvec{\Sigma}}_{z}^{1/2} {\mathbf{D}}_{l} {\varvec{\Sigma}}_{z}^{1/2} } \right) - 2\widehat{{\varvec{\uppi}}}\left( {{\varvec{\Sigma}}_{z} \widehat{{\varvec{\upbeta}}}} \right)_{l} } \right\}, $$

and ${\mathbf{x}}_{l}$ denotes element $l$ of the vector ${\mathbf{x}}.$ For any subset $S \subset \left\{ {1,2, \ldots ,p} \right\}$ of the covariates, $\nabla_{{{\hat{\varvec{\upbeta }}}}} \left( {\sum\nolimits_{k \in S} {\hat{\pi }_{k} } } \right) = \sum\nolimits_{k \in S} {\nabla_{{{\hat{\varvec{\upbeta }}}}} \hat{\pi }_{k} } ,$ so the standard deviation of the estimator of the proportional contribution of the subset of variables is

$$ \widehat{SE}\left( {\sum\nolimits_{k \in S} {\hat{\pi }_{k} } } \right) = \sqrt {\left( {\sum\nolimits_{k \in S} {\nabla_{{{\hat{\varvec{\upbeta }}}}} \hat{\pi }_{k} } } \right)^{T} {\hat{\mathbf{V}}}\left( {\sum\nolimits_{k \in S} {\hat{\pi }_{k} \nabla_{{{\hat{\varvec{\upbeta }}}}} \hat{\pi }_{k} } } \right)} . $$

Let ${\mathbf{I}}_{S} = \left( {I_{{\{ 1 \in S\} }} ,I_{{\{ 1 \in S\} }} \ldots ,I_{{\{ m \in S\} }} } \right)^{T}$ be a vector of indicator functions for membership of each variable in the subset. Then we can write

$$ \widehat{SE}\left( {{\mathbf{I}}_{S}^{T} \widehat{{\varvec{\uppi}}}} \right) = \sqrt {\left( {\nabla_{{\widehat{{\varvec{\upbeta}}}}} \widehat{{\varvec{\uppi}}}^{T}{\mathbf{I}}_{S} } \right)^{T} {\hat{\mathbf{V}}}\left( {\nabla_{{\widehat{{\varvec{\upbeta}}}}} \widehat{{\varvec{\uppi}}}^{T}{\mathbf{I}}_{S} } \right)} = \sqrt {{\mathbf{I}}_{S}^{T} \left\{ {\left( {\nabla_{{\widehat{{\varvec{\upbeta}}}}} \widehat{{\varvec{\uppi}}}} \right)^{T} {\hat{\mathbf{V}}}\left( {\nabla_{{\widehat{{\varvec{\upbeta}}}}} \widehat{{\varvec{\uppi}}}} \right)} \right\}{\mathbf{I}}_{S} } $$