On computing standard errors for marginal structural Cox models

Abstract

In recent decades, marginal structural models have gained popularity for proper adjustment of time-dependent confounders in longitudinal studies through time-dependent weighting. When the marginal model is a Cox model, using current standard statistical software packages was thought to be problematic because they were not developed to compute standard errors in the presence of time-dependent weights. We address this practical modelling issue by extending the standard calculations for Cox models with case weights to time-dependent weights and show that the coxph procedure in R can readily compute asymptotic robust standard errors. Through a simulation study, we show that the robust standard errors are rather conservative, though corresponding confidence intervals have good coverage. A second contribution of this paper is to introduce a Cox score bootstrap procedure to compute the standard errors. We show that this method is efficient and tends to outperform the non-parametric bootstrap in small samples.

Appendices

Appendix 1

We now derive expression (10) for $u_i({\hat{\beta }})$ in (9). First, we re-express (5) as follows:

$$\begin{aligned} \tilde{\mathbf{U }}({\hat{\beta }}) =\sum _{i=1}^n w_i(t_i)\delta _i \mathbf{X}_i(t_i) - \int _0^{\infty } \left\{ \frac{{\tilde{S}}^{(1)}({\hat{\beta }},t)}{{\tilde{S}}^{(0)}({\hat{\beta }},t)} \right\} d{\tilde{G}}(t) = 0, \end{aligned}$$

where ${\tilde{G}}(t)$ is as defined in (11). Suppressing the brackets in functions, and taking a first-order Taylor series expansion of (5) around ${\tilde{S}}^{(0)}= S^{(0)}, {\tilde{S}}^{(1)}= S^{(1)}$, and ${\tilde{G}}=G$, we have

$$\begin{aligned} \tilde{\mathbf{U }}({\hat{\beta }})&= \tilde{\mathbf{U }}({\hat{\beta }}) \bigg |_{S^{(0)}, S^{(1)}, G} + \left( {\tilde{S}}^{(0)}-S^{(0)} \right) \frac{{\partial }\tilde{\mathbf{U }}}{{\partial }{\tilde{S}}^{0}} \bigg |_{S^{(0)}, S^{(1)}, G} \nonumber \\&+ \left( {\tilde{S}}^{(1)}-S^{(1)}\right) \frac{{\partial }\tilde{\mathbf{U }}}{{\partial }{\tilde{S}}^{1}} \bigg |_{S^{(0)}, S^{(1)}, G} + \left( {\tilde{G}}-G \right) \frac{{\partial }\tilde{\mathbf{U }}}{{\partial }{\tilde{G}}} \bigg |_{S^{(0)}, S^{(1)}, G} , \end{aligned}$$

(17)

where

$$\begin{aligned} \frac{{\partial }\tilde{\mathbf{U }}}{{\partial }{\tilde{S}}^{(0)}} \bigg |_{S^{(0)}, S^{(1)}, G}&= \int _0^{\infty } \frac{S^{(1)}}{\left( S^{(0)} \right) ^2}dG(t), \\ \frac{{\partial }\tilde{\mathbf{U }}}{{\partial }{\tilde{S}}^{(1)}} \bigg |_{S^{(0)}, S^{(1)}, G}&= - \int _0^{\infty } \frac{1}{S^{(0)}}dG(t), ~\text{ and } \\ \frac{{\partial }\tilde{\mathbf{U }}}{{\partial }{\tilde{G}}} \bigg |_{S^{(0)}, S^{(1)}, G}&= - \frac{S^{(1)}}{S^{(0)}}. \\ \end{aligned}$$

The remainder terms of the Taylor series expansion are negligible because ${\tilde{S}}^{(0)}, {\tilde{S}}^{(1)}$ and ${\tilde{G}}$ are consistent estimates of $S^{(0)}, S^{(1)}$ and $G$ respectively. Plugging in the above partial derivatives, and noting that $\sum _{i=1}^n w_i(t_i)\delta _i (S^{(1)}/S^{(0)}) = \left( S^{(1)}/{S^{(0)}}\right) {\tilde{G}}$, the right-hand side of Eq. (17) can be reduced to

$$\begin{aligned} \sum _{i=1}^n w_i(t_i)\delta _i\mathbf{X}_i(t_i) - \left( \frac{S^{(1)}}{S^{(0)}} \right) {G}(t) + \int _0^{\infty } \frac{{\tilde{S}}^{(0)}S^{(1)}}{\left( S^{(0)}\right) ^2} dG(t) - \int _0^{\infty } \frac{{\tilde{S}}^{(1)}}{S^{(0)}} dG(t). \end{aligned}$$

(18)

After substituting Eqs. (6) for $S^{(0)}$ and $S^{(1)}$ in the last two terms of Eq. (18), approximating the second term with $\int _0^{\infty }(S^{(1)}/S^{(0)})d{\tilde{G}}(t)$, and interchanging the order of integration and summation, we get

$$\begin{aligned} u_i({\hat{\beta }})&= \sum _{i=1}^n w_i(t_i) \delta _i \left\{ \mathbf{X}_i(t_i) - \frac{S^{(1)}({\hat{\beta }},t)}{S^{(0)}({\hat{\beta }},t)} \right\} \nonumber \\&\quad - \sum _{i=1}^n \int _0^{\infty } \left\{ \frac{w_i(t)Y_i(t)\mathbf{X}_i(t)\exp (\beta ^{\prime }\mathbf{X}_i(t))}{{\tilde{S}}^{(0)}({\hat{\beta }},t)} \right\} dG(t) \nonumber \\&\quad + \sum _{i=1}^n \int _0^{\infty } \left\{ \frac{w_i(t)Y_i(t){\tilde{S}}^{(1)}({\hat{\beta }},t) \exp (\beta ^{\prime }\mathbf{X}_i(t))}{\left( {\tilde{S}}^{(0)}({\hat{\beta }},t) \right) ^2} \right\} dG(t). \end{aligned}$$

Hence, (18) is asymptotically equivalent to ${\tilde{U}}({\hat{\beta }})$ in (9), where $u_i({\hat{\beta }})$ is as given in (10), i.e.

$$\begin{aligned} u_i({\hat{\beta }})&= \delta _i \left\{ \mathbf{X}_i(t_i) - \frac{S^{(1)}({\hat{\beta }},t)}{S^{(0)} ({\hat{\beta }},t)} \right\} \nonumber \\&\quad - \frac{1}{w_i(t_i)} \int _0^{\infty } \left\{ \frac{w_i(t)Y_i(t)\mathbf{X}_i(t) \exp (\beta ^{\prime }\mathbf{X}_i(t))}{{\tilde{S}}^{(0)}({\hat{\beta }},t)} \right\} dG(t) \nonumber \\&\quad + \frac{1}{w_i(t_i)} \int _0^{\infty } \left\{ \frac{w_i(t)Y_i(t){\tilde{S}}^{(1)}({\hat{\beta }},t) \exp (\beta ^{\prime }\mathbf{X}_i(t))}{ \left( {\tilde{S}}^{(0)}({\hat{\beta }},t) \right) ^2} \right\} dG(t). \end{aligned}$$

Further, since

$$\begin{aligned} \mathcal{E}[u_i({\hat{\beta }})]&= \sum _{i=1}^n w_i(t_i)u_i({\hat{\beta }}) \\&= \sum _{i=1}^n w_i(t_i) \delta _i \left\{ \mathbf{X}_i(t_i) - \frac{S^{(1)}({\hat{\beta }},t)}{S^{(0)}({\hat{\beta }},t)} \right\} \\&- \int _0^{\infty } \left\{ \frac{{\tilde{S}}^{(1)}({\hat{\beta }},t)}{S^{(0)}({\hat{\beta }},t)} - \frac{S^{(1)}({\hat{\beta }},t)}{S^{(0)}({\hat{\beta }},t)} \left( \frac{{\tilde{S}}^{(0)}({\hat{\beta }},t)}{S^{(0)}({\hat{\beta }},t)} \right) \right\} dG(t), \end{aligned}$$

then we have

$$\begin{aligned} \mathcal{E}[u_i({\hat{\beta }})] \rightarrow \frac{1}{N} \sum _{i=1}^N \delta _i \left\{ \mathbf{X}_i(t_i) - \frac{S^{(1)}({\hat{\beta }},t)}{S^{(0)}({\hat{\beta }},t)} \right\} , \text{ as }\,n \rightarrow N\,\text{ and }\,N \rightarrow \infty , \end{aligned}$$

which by (3) equals zero.

In other words, $\tilde{\mathbf{U }}({\hat{\beta }})$ is a consistent, though not necessarily unbiased estimator of $0$. It can easily be seen that if weights $sw_i^*(t)$ were used, the resulting coefficient estimates would still be consistent and slightly biased.

Appendix 2

In this appendix, we show how one can do parameter estimation using the Breslow approximation, and then provide a toy example that demonstrates that coxph can accommodate time-dependent weights when computing asymptotic standard errors. First, we re-write the partial log-likelihood, score vector and Fisher information matrix such that we can easily compute them from data.

There is a term in the (partial) likelihood function for every event. When there are multiple subjects who have an event at the same time, i.e., event times are tied, the Breslow approximation does not assume that the exact time of any death is unique. Hence the contribution to the likelihood is simply the ratio of each subject’s score to the sum of scores for all subjects at risk just before the event time ( i.e., any subject for which $Y(t_i) = 1$ for event time $t_i$). The log-likelihood is computed as follows (comparing to Eq. (3.1)):

$$\begin{aligned} LL = \sum _i^n W_i A_i, \end{aligned}$$

where $W_i = \delta _i w_i(t_i)$, and $A_i = \left( B_i - \ln \left( \sum _j^n C_j(t_i) \right) \right)$ with

$$\begin{aligned} B_i&= \beta ^{\prime }\mathbf{X}_i(t_i), \text{ and } \\ C_j(t)&= w_j(t)Y_j(t)\exp (\beta ^{\prime }\mathbf{X}_j(t)). \end{aligned}$$

Define

$$\begin{aligned} {\bar{X}}(t)= \frac{\sum _{j=1}^n X_j(t)C_j(t)}{\sum _{j=1}^n C_j(t)}. \end{aligned}$$

It is easy to verify that ${\bar{X}}(t) = {\tilde{S}}^{(1)}({\hat{\beta }}^{\prime },t)/{\tilde{S}}^{(0)}({\hat{\beta }}^{\prime },t)$. Further, we can re-write the respective score vector in Eq. (5) and Fisher information matrix in Eq. (7) as follows:

$$\begin{aligned} U&= \sum _{i=1}^n W_i \{X_i(t_i) - {\bar{X}}_i(t_i) \}, \text{ and }\\ J&= \sum _{i=1}^n W_i \left\{ \frac{ \sum _{j=1}^n (X_j(t_i))^{{\bigotimes }2}C_j(t_i)}{\sum _{j=1}^nC_j(t)} - \left( {\bar{X}}(t_i) \right) ^{{\bigotimes }2} \right\} . \end{aligned}$$

In fact, we can simplify the information matrix even further as follows:

$$\begin{aligned} J_{kl}&= \sum _{i=1}^n W_i \left\{ \frac{ \sum _{j=1}^n X_{kj}(t_i) X_{lj}(t_i) C_j(t_i)}{\sum _{j=1}^n C_j(t)} - {\bar{X}}_{k}(t_i) {\bar{X}}_{l}(t_i) \right\} ,\;\text{ and }\\ J_{kk}&= \sum _{i=1}^n W_i \left\{ \frac{ \sum _{j=1}^n X_{kj}^2(t_i) C_j(t_i) }{ \sum _{j=1}^n C_j(t_i) } - {\bar{X}}_{k}^2(t_i) \right\} , \end{aligned}$$

for $k,l = 1, \ldots , p$ where $X_{kj}(t_i)$ is subject $j$’s value of the $k$th covariate. Similarly, ${\bar{X}}_k(t_i)$ is the $k$th component of ${\bar{X}}(t_i)$. For variance estimation of the model coefficients, we re-write Eq. (9) as:

$$\begin{aligned} {\tilde{U}}({\hat{\beta }})&= \sum _{i=1}^n W_i \left\{ X_i(t_i) - {\bar{X}}(t_i) \right\} \\&- \sum _{i=1}^n \sum _{j=1}^n W_j \frac{X_i(t_j)C_i(t_j)}{\sum _{k=1}^n C_k(t_j)}+ \sum _{i=1}^n \sum _{j=1}^n W_j \frac{{\bar{X}}(t_j) C_i(t_j)}{\sum _{k=1}^n C_k(t_j)} \\&= { \sum _{i=1}^n \left[ W_i \left\{ X_i(t_i) - {\bar{X}}(t_i) \right\} - \sum _{j=1}^n W_j \left\{ X_i(t_i) - {\bar{X}}(t_i) \right\} \frac{C_i(t_j)}{\sum _{k=1}^n C_k(t_j)} \right] }. \end{aligned}$$

Let ${\tilde{U}}$ be a $n \times 1$ vector containing the $i$th contribution to ${\tilde{U}}({\hat{\beta }})$, for $i = 1, \ldots , n$. Then we have,

$$\begin{aligned} {\tilde{U}}_i&= W_i \left\{ X_i(t_i) - {\bar{X}}(t_i) \right\} - \sum _{j=1}^n W_j \left\{ X_i(t_j) - {\bar{X}}(t_j) \right\} \frac{C_i(t_j)}{\sum _{k=1}^n C_k(t_j)}. \end{aligned}$$

The final sandwich estimator for producing robust variance estimates is given by $V = (J^{-1}V_{{\tilde{U}}}J^{-1}) = (J^{-1}{\tilde{U}}^{\prime }) ({\tilde{U}}J^{-1})$. Using the equations detailed in this section, in the examples that follow we will need to compute $W_i, B_i, C_j(t_i), \sum _{j=1}^n C_j(t_i)$ as well as ${\bar{X}}(t_i)$. We will use these quantities in estimating parameters from the data set presented in the next section.

1.1 Worked out implementation of fitting a MSCM to data

The first six columns of Table 7 present a data set that contains eight subjects observed over 1–6 time intervals, comprising 23 observations. There are two covariates: $x_1$ is a binary baseline variable, while $x_2$ is binary but time-dependent. The column ‘wt’ shows the time-dependent weights associated with each subject at each visit. For convenience, subjects are ordered based on their respective failure times. The remaining five columns detail much of the preliminary calculations needed for parameter estimation, and used implicitly in future calculations.

Table 7 Example of survival data with time-dependent weights

Since there are four failures in the data, there are four terms in the log-likelihood. Let $d_1 = 11r_1r_2+5r_1+11, d_3 = 8r_1r_2+2r_1+12, d_5 = 3r_1+12r_2+6$ and $d_6 = 4r_1+8r_2+6$. The corresponding log-likelihood, score vector and Fisher information matrix are as follows:

$$\begin{aligned} LL&= 3(\beta _1 - \ln d_1) + 8(\beta _1 + \beta _2 - \ln d_3) + 4(\beta _2 - \ln d_5) + 8(\beta _2 - \ln d_6) \\&= 11 \beta _1 + 20 \beta _2 - 3 \ln d_1 - 8 \ln d_3 - 4 \ln d_5 - 8 \ln d_6 \\ U_1&= 3 \left( 1-{\bar{X}}_1(1) \right) + 8 \left( 1-{\bar{X}}_1(2) \right) + 4 \left( 0-{\bar{X}}_1(4) \right) + 8 \left( 0-{\bar{X}}_1(5) \right) \\ U_2&= 3 \left( 0-{\bar{X}}_2(1) \right) + 8 \left( 1-{\bar{X}}_2(2) \right) + 4 \left( 1-{\bar{X}}_2(4) \right) + 8 \left( 1-{\bar{X}}_2(5) \right) \\ J_{11}&= 3 \left( {\bar{X}}_1(1) - {\bar{X}}_1^2(1) \right) + 8 \left( {\bar{X}}_1(2) - {\bar{X}}_1^2(2) \right) + 4 \left( {\bar{X}}_1(4) - {\bar{X}}_1^2(4) \right) \\&+8 \left( {\bar{X}}_1(5) - {\bar{X}}_1^2(5) \right) \\ J_{22}&= 3 \left( {\bar{X}}_2(1) - {\bar{X}}_2^2(1) \right) + 8 \left( {\bar{X}}_2(2) - {\bar{X}}_2^2(2) \right) + 4 \left( {\bar{X}}_2(4) - {\bar{X}}_2^2(4) \right) \\&+ 8 \left( {\bar{X}}_2(5) - {\bar{X}}_2^2(5) \right) \\ J_{12}&= 3 {\bar{X}}_2(1)\left( 1 - {\bar{X}}_1(1) \right) + 8 {\bar{X}}_2(2)\left( 1 - {\bar{X}}_1(2) \right) - 4 {\bar{X}}_1(4){\bar{X}}_2(4) \\&-8 {\bar{X}}_1(5){\bar{X}}_2(5) \end{aligned}$$

Setting $U=0$ and solving for $\beta$ we find that ${\hat{\beta }}_1 = 0.5705749$ and ${\hat{\beta }}_2 = 2.1112007$. Before computing $LL, U$ and $J$ we perform preliminary calculations for the four observed failure times in Table 8. The values of these statistics at the initial and final parameter estiamtes are provided in Table 9. Table 10 contains the score residuals for each subject. Finally, we can compute the variance–covariance matrix for the parameters using

$$\begin{aligned} V = J^{-1}{\tilde{U}}^{\prime }{\tilde{U}}J^{-1} = \left( \begin{array}{rr} 1.66533 &{} 0.03275 \\ 0.03275 &{} 2.59323 \end{array} \right) , \end{aligned}$$

giving the standard errors of ${\hat{\beta }}_1$ and ${\hat{\beta }}_2$ as 0.082976 and 1.320177, respectively.

Table 8 Intermediate calculations for updating parameter estimates

Table 9 Update statistics

Table 10 Score residuals

If the data were analyzed in R and stored in a coxph.object called fit, then the quantities evaluated in this section could be compared to the corresponding coxph output as follows:

$$\begin{aligned} {\bar{X}}&= \mathtt{coxph.detail(fit)}\$\mathtt{means} \\ LL&= \mathtt{fit}\$\mathtt{loglik} \\ U&= \mathtt{sum(coxph.detail(fit)}\$\mathtt{score)} \\ J&= \mathtt{fit} \$\mathtt{var} \\ {{\hat{\beta }}}_1, {{\hat{\beta }}}_2&= \mathtt{summary(fit)}\$\mathtt{coeff} \\ {\tilde{U}}&= \mathtt{residuals(fit,collapse=mydata}\$\mathtt{id, weighted=T,}\mathtt{type=\text{`` }score\text{'' })} \\ D = J^{-1}{\tilde{U}}&= \mathtt{residuals(fit,collapse=mydata}\$\mathtt{id, weighted=T,}\mathtt{type=\text{`` }dfbeta\text{'' })} \end{aligned}$$

The MLE calculated here matches that of coxph, as do the standard errors. Hence, this appendix demonstrates that the R code for computing the standard errors in coxph can accommodate time-dependent weights.