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.
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Acknowledgements
The author wishes to thank Dr. Brian Rini, the Cleveland Clinic Foundation, Dr. Bernard Escudier and Institut Gustave Roussy for the data used in the example and two anonymous peer reviewers for helpful comments and suggestions.
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Appendix: Derivation of the standard error of the proportional contributions to the risk score variance
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
The partial derivative of \( \hat{\pi }_{k} \) with respect to \( \hat{\beta }_{l} \) is thus
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
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
where
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
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
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Crager, M.R. Extensions of the absolute standardized hazard ratio and connections with measures of explained variation and variable importance. Lifetime Data Anal 26, 872–892 (2020). https://doi.org/10.1007/s10985-020-09504-2
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DOI: https://doi.org/10.1007/s10985-020-09504-2