Firth’s penalized likelihood for proportional hazards regressions for complex surveys
Section 2. Weight scaling

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Let $w_i$ be the weight for unit $i.$ We propose to use ${\tilde{w}}_i\mathrm{<mo>=</mo>}\left({\displaystyle {\sum }_{A_N}}1/{\displaystyle {\sum }_{A_N}}{w}_i\right)w_i\mathrm{<mo>=</mo>}\left(n/{\displaystyle {\sum }_{A_N}}{w}_i\right)w_i$ as the scaled weight. By construction, the scaled weights are invariant to the scale of the weight. That is, ${\tilde{w}}_i^{*}\mathrm{<mo>=</mo>}\left(n/{\displaystyle {\sum }_{A_N}}\gamma w_i\right)\gamma w_i\mathrm{<mo>=</mo>}\left(n/{\displaystyle {\sum }_{A_N}}{w}_i\right)w_i\mathrm{<mo>=</mo>}{\tilde{w}}_i$ for all $\gamma \ne 0.$

Firth’s penalized likelihood is given by $L_p\left(\beta \right)\mathrm{<mo>=</mo>}L\left(\beta \right){\left|I\left(\beta \right)\right|}^{0.5},$ where $L\left(\beta \right)$ and $I\left(\beta \right)$ are the unpenalized likelihood and information matrix, respectively. The penalized log likelihood is

$$l_p\left(\beta \right)\mathrm{<mo>=</mo>}l\left(\beta \right)+0.5\log \left(\left|I\left(\beta \right)\right|\right).$$

In particular, when the scaled weights are used, the Breslow unpenalized log partial likelihood (Breslow, 1974) is

$$l\left(\beta \right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\left\{{\beta }^{\prime }{\displaystyle \sum _{i\in D_k}}{\tilde{w}}_i{Z}_i\left(t_k\right)-\left({\displaystyle \sum _{i\in D_k}}{\tilde{w}}_i\right)\log {\displaystyle \sum _{i\in R_k}}{\tilde{w}}_i\exp \left({\beta }^{\prime }{Z}_i\left(t_k\right)\right)\right\}$$

where $w_i$ is the unscaled weight for unit $i.$

Denote

$$S_k^{\left(a\right)}\left(\beta \right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in R_k}}{\tilde{w}}_i\exp \left({\beta }^{\prime }{Z}_i\left(t_k\right)\right){\left[Z_i\left(t_k\right)\right]}^{\otimes a}$$

where $k$ is the $k^{\text{th}}$ -ordered event time, $a\mathrm{<mo>=</mo>}\mathrm{0,}\mathrm{1,}2,$ ${\left[Z_i\left(t_k\right)\right]}^{\otimes 0}$ is 1, ${\left[Z_i\left(t_k\right)\right]}^{\otimes 1}$ is the vector $Z_i\left(t_k\right),$ and ${\left[Z_i\left(t_k\right)\right]}^{\otimes 2}$ is the matrix $\left[Z_i\left(t_k\right)\right]{\left[Z_i\left(t_k\right)\right]}^{\prime }.$

Then the score function is given by

$$\begin{array}{ll}U\left(\beta \right)\hfill & \equiv {\left(U\left({\beta }_1\right)\mathrm{,}\dots \mathrm{,}U\left({\beta }_p\right)\right)}^{\prime }\hfill \\ \hfill & \mathrm{<mo>=</mo>}\frac{\partial l\left(\beta \right)}{\partial \beta }\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\left\{{\displaystyle \sum _{i\in D_k}}{\tilde{w}}_i{Z}_i\left(t_k\right)-{\displaystyle \sum _{i\in D_k}}{\tilde{w}}_i\frac{S_k^{\left(1\right)}\left(\beta \right)}{S_k^0\left(\beta \right)}\right\}\hfill \end{array}$$

and the Fisher information matrix is given by

$$\begin{array}{ll}I\left(\beta \right)\hfill & =-\frac{{\partial }^{2}l\left(\beta \right)}{\partial \beta {}^2}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}{\displaystyle \sum _{i\in D_k}}{\tilde{w}}_i\left\{\frac{S{}_k^{\left(2\right)}\left(\beta \right)}{S_k^{\left(0\right)}\left(\beta \right)}-\left[\frac{S{}_k^{\left(1\right)}\left(\beta \right)}{S_k^{\left(0\right)}\left(\beta \right)}\right]{\left[\frac{S{}_k^{\left(1\right)}\left(\beta \right)}{S_k^{\left(0\right)}\left(\beta \right)}\right]}^{\prime }\right\}.\hfill \end{array}$$

Denote

$$Q_{kp}^{\left(a\right)}\left(\beta \right)\mathrm{<mo>=</mo>}{\displaystyle \sum _{i\in R_k}}{\tilde{w}}_i\exp \left({\beta }^{\prime }{Z}_i\left(t_k\right)\right)Z_{i\mathrm{,}p}\left(t_k\right){\mathrm{<mo\; stretchy="false">[</mo>}{Z}_i\left(t_k\right)\mathrm{<mo\; stretchy="false">]</mo>}}^{\otimes a}$$

where $a\mathrm{<mo>=</mo>}\mathrm{0,}\mathrm{1,}2;$ $p\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}P;$ and $Z_i\left(t\right)\mathrm{<mo>=</mo>}\left(Z_{i\mathrm{,}1}\left(t\right)\mathrm{,}\dots \mathrm{,}{Z}_{i\mathrm{,}p}\left(t\right)\right).$ Then

$$\begin{array}{ll}\frac{\partial I\left(\beta \right)}{\partial {\beta }_p}\hfill & ={\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}{\displaystyle \sum _{i\in D_k}}{\tilde{w}}_i\{{\left[\frac{Q_{kp}^{\left(2\right)}\left(\beta \right)}{S_k^{\left(0\right)}\left(\beta \right)}-\frac{Q{}_{kp}^{\left(0\right)}\left(\beta \right)}{S_k^{\left(0\right)}\left(\beta \right)}\frac{S_k^{\left(2\right)}\left(\beta \right)}{S_k^{\left(0\right)}\left(\beta \right)}\right]}_{}^{{}^{}}\hfill \\ \hfill & -\left[\frac{Q{}_{kp}^{\left(1\right)}\left(\beta \right)}{S_k^{\left(0\right)}\left(\beta \right)}-\frac{Q_{kp}^{\left(0\right)}\left(\beta \right)}{S_k^{\left(0\right)}\left(\beta \right)}\frac{S{}_k^{\left(1\right)}\left(\beta \right)}{S_k^{\left(0\right)}\left(\beta \right)}\right]{\left[\frac{S_k^{\left(1\right)}\left(\beta \right)}{S_k^{\left(0\right)}\left(\beta \right)}\right]}^{\prime }\hfill \\ \hfill & -\left[\frac{S_k^{\left(1\right)}\left(\beta \right)}{S_k^{\left(0\right)}\left(\beta \right)}\right]{\left[\frac{Q_{kp}^{\left(1\right)}\left(\beta \right)}{S_k^{\left(0\right)}\left(\beta \right)}-\frac{Q_{kp}^{\left(0\right)}\left(\beta \right)}{S_k^{\left(0\right)}\left(\beta \right)}\frac{S_k^{\left(1\right)}\left(\beta \right)}{S_k^{\left(0\right)}\left(\beta \right)}\right]}^{\prime }\}\hfill \end{array}$$

where $p\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}P.$

Point estimates and Taylor linearized standard errors for the penalized likelihood are obtained from the score functions and the Hessian as described in Section 1.2. The jackknife standard errors are obtained by maximizing the penalized likelihood in every replicate sample.

Appendix 1 shows that under certain regularity conditions, the point estimators obtained by maximizing Firth’s penalized likelihood are design-consistent.

2.1  Penalized likelihoods and the scale of weights

In this section, we derive a relationship between the penalized log likelihood that uses scaled weights and the penalized log likelihood that uses unscaled weights, and we demonstrate that Firth’s penalized likelihood using unscaled weights does not have the invariance property.

Let $l\left({\beta }_{\tilde{w}}\mathrm{<mo>;</mo>}\tilde{w}\right)$ be the log likelihood using weights $\tilde{w},$ and let $l\left({\beta }_w\mathrm{<mo>;</mo>}w\right)$ be the log likelihood using weights $w,$ where ${\tilde{w}}_i\mathrm{<mo>=</mo>}\alpha w_i$ for all $i$ and $\alpha \ne 0.$ The Breslow log likelihood can be written as

$$\begin{array}{ll}l\left({\beta }_{\tilde{w}}\mathrm{<mo>;</mo>}\tilde{w}\right)\hfill & ={\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\left\{{\beta }_{\tilde{w}}^{\prime }{\displaystyle \sum _{i\in D_k}}{\tilde{w}}_i{Z}_i\left(t_k\right)-\left({\displaystyle \sum _{i\in D_k}}{\tilde{w}}_i\right)\log {\displaystyle \sum _{i\in R_k}}{\tilde{w}}_i\exp \left({\beta }_{\tilde{w}}^{\prime }{Z}_i\left(t_k\right)\right)\right\}\hfill \\ \hfill & \mathrm{<mo>=</mo>}{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\left\{{\beta }_{\tilde{w}}^{\prime }\alpha {\displaystyle \sum _{i\in D_k}}{w}_i{Z}_i\left(t_k\right)-\left(\alpha {\displaystyle \sum _{i\in D_k}}{w}_i\right)\log {\displaystyle \sum _{i\in R_k}}\alpha w_i\exp \left({\beta }_{\tilde{w}}^{\prime }{Z}_i\left(t_k\right)\right)\right\}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\alpha {\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\left\{{\beta }_{\tilde{w}}^{\prime }{\displaystyle \sum _{i\in D_k}}{w}_i{Z}_i\left(t_k\right)-\left({\displaystyle \sum _{i\in D_k}}{w}_i\right)\log {\displaystyle \sum _{i\in R_k}}{w}_i\exp \left({\beta }_{\tilde{w}}^{\prime }{Z}_i\left(t_k\right)\right)\right\}\hfill \\ \hfill & -{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\left(\alpha {\displaystyle \sum _{i\in D_k}}{w}_i\right)\log \alpha \hfill \\ \hfill & \mathrm{<mo>=</mo>}\alpha l\left({\beta }_{\tilde{w}}\mathrm{<mo>;</mo>}w\right)-{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\left(\alpha {\displaystyle \sum _{i\in D_k}}{w}_i\right)\log \alpha .\hfill \end{array}$$

Because the second term on the right-hand side does not contain $\beta ,$ the derivative and the Hessian of the log likelihood are only a multiplier of $\alpha$ and the parameter estimates and standard errors are invariant to the scale of the weights.

However, the following relation shows that the point estimates that are obtained by maximizing the penalized log likelihood are not invariant to the scale of the weights:

$$\begin{array}{ll}{l}_p\left({\beta }_{\tilde{w}}\mathrm{<mo>;</mo>}\tilde{w}\right)\hfill & \mathrm{<mo>=</mo>}l\left({\beta }_{\tilde{w}}\mathrm{<mo>;</mo>}\tilde{w}\right)+0.5\log \left|I\left({\beta }_{\tilde{w}}\mathrm{<mo>;</mo>}\tilde{w}\right)\right|\hfill \\ \hfill & \mathrm{<mo>=</mo>}\alpha l\left({\beta }_{\tilde{w}}\mathrm{<mo>;</mo>}w\right)+0.5\log \left|\alpha I\left({\beta }_{\tilde{w}}\mathrm{<mo>;</mo>}w\right)\right|-{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\left(\alpha {\displaystyle \sum _{i\in D_k}}{w}_i\right)\log \alpha \hfill \\ \hfill & \mathrm{<mo>=</mo>}\alpha l\left({\beta }_{\tilde{w}}\mathrm{<mo>;</mo>}w\right)+0.5\left\{\log \left|I\left({\beta }_{\tilde{w}}\mathrm{<mo>;</mo>}w\right)\right|+p\log \alpha \right\}-{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\left(\alpha {\displaystyle \sum _{i\in D_k}}{w}_i\right)\log \alpha \hfill \\ \hfill & \mathrm{<mo>=</mo>}\alpha \left\{l\left({\beta }_{\tilde{w}}\mathrm{<mo>;</mo>}w\right)+0.5\log \left|I\left({\beta }_{\tilde{w}}\mathrm{<mo>;</mo>}w\right)\right|\right\}-{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\left(\alpha {\displaystyle \sum _{i\in D_k}}{w}_i\right)\log \alpha \hfill \\ \hfill & +0.5\left\{p\log \alpha +\left(1-\alpha \right)\log \left|I\left({\beta }_{\tilde{w}}\mathrm{<mo>;</mo>}w\right)\right|\right\}\hfill \\ \hfill & \mathrm{<mo>=</mo>}\alpha l_p\left({\beta }_{\tilde{w}}\mathrm{<mo>;</mo>}w\right)-{\displaystyle \sum _{k\mathrm{<mo>=</mo>\; 1}}^K}\left(\alpha {\displaystyle \sum _{i\in D_k}}{w}_i\right)\log \alpha \hfill \\ \hfill & +0.5\left\{p\log \alpha +\left(1-\alpha \right)\log \left|I\left({\beta }_{\tilde{w}}\mathrm{<mo>;</mo>}w\right)\right|\right\}.\hfill \end{array}$$

The additional term in the right hand side of the preceding equation involves the regression parameters. Thus the point estimates and the standard errors are not invariant to the scale of the weights.

By construction, point estimates that use the penalized log likelihood and the scaled weights are invariant to the scale of the weights.

2.2  Example that uses scaled weights

Consider the myeloma study described in Section 1.1. We refit the same proportional hazards regression model using LogBUN, HGB, and Contrived as explanatory variables, but now we use scaled weights in constructing Firth’s penalized likelihood.

Table 2.1 displays point estimates and standard errors from Firth’s penalized likelihood using scaled weights and the Taylor linearized variance estimator. These statistics are invariant to the scale of the weights.

Standard errors using jackknife replicates are also invariant to the scale of the weights. For replicate variance estimation methods, every set of replicate weights must be scaled using the same scaling factor that is used to scale the full sample weights. Table 2.2 displays point estimates and standard errors from Firth’s penalized likelihood using scaled weights and the jackknife replicate variance estimator.

Estimates from the penalized log likelihood using the scaled weights also have the closeness property. The ratios of jackknife standard errors to Taylor linearized standard errors are 1.16, 1.17, and 1.40 for all three sets of weights for the variables LogBUN, HGB, and Contrived, respectively (Tables 2.1 and 2.2).

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