A flexible semiparametric transformation model for survival data

Abstract

I suggest an extension of the semiparametric transformation model that specifies a time-varying regression structure for the transformation, and thus allows time-varying structure in the data. Special cases include a stratified version of the usual semiparametric transformation model. The model can be thought of as specifying a first order Taylor expansion of a completely flexible baseline. Large sample properties are derived and estimators of the asymptotic variances of the regression coefficients are given. The method is illustrated by a worked example and a small simulation study. A goodness of fit procedure for testing if the regression effects lead to a satisfactory fit is also suggested.

Appendix

I here sketch the main arguments of the proof that establishes the asymptotic variances, extending that of Bagdonavicius and Nikulin (1999, 2001) and Chen et al. (2002) to more than one dimension. A detailed consistency proof for the standard transformation model can be found in Dabrowska (2005), and it appears that these arguments can be extended to our setting, but this is no trivial exercise. The key to the multivariate extension is the use of product-limit integration formulae and I here focus on establishing the key formulae that gives expressions for the standard deviations of the estimated quantities.

Assume that subjects are i.i.d. with covariates that are uniformly bounded. Define ${\varvec D}^{**}(\beta,A) = \hbox{diag}(\exp(2 Z_i^{T}\beta)\dot{\lambda}_0(H({t-}|X_i)\exp(Z_i^{T}\beta))), {\varvec S}^{**}(\beta, A) = n^{-1}{\varvec X}^T(t){\varvec D}^{**}(t,\beta,A) {\varvec X}(t), {\varvec S}_j^{**}(\beta, A) = n^{-1}{\varvec X}^{T}(t){\varvec D}(X_{ij}){\varvec D}^{**}(t,\beta,A){\varvec X}(t), {\varvec S}^{Z}(\beta, A) = n^{-1}{\varvec Z}^T(t){\varvec D}(t,\beta,A) {\varvec X}(t)$, and ${\varvec S}_j^{**Z}(\beta, A) = n^{-1}{\varvec Z}^{T}(t) {\varvec D}(X_{ij}){\varvec D}^{**}(t,\beta,A){\varvec X}(t)$.

One of the needed assumptions is that all S matrices converge uniformly to deterministic matrices in both time and the parameter space for β. Let Θ denote an open ball around the true parameter value β₀, then it is assumed that the limit of all large S’s exist and are denoted by small s’s. Such that for example $lim_p {\varvec S}_j^{**Z}(\beta, A) = s_j^{**Z}$ uniformly in $\Theta\times [0,\tau]$. Define the limit of $E_j^{**}(t) = {\varvec S}_0^{-1}(t,\beta){\varvec S}_j^{**}(t,\beta)$ as $e_j^{**}(u,\beta)$. I also assume that the covariates are uniformly bounded.

I start by making the observation that (up to $o_p(n^{-1})$)

$$ \begin{aligned} \tilde{A}(\beta_0,t) - A_0(t) &= \int_{0}^{t}{\varvec X}^{-}(\beta_0,\tilde{A})\,\hbox{d}{\varvec M} \\ &- \int_{0}^{t} \left\{{\varvec X}^{-}(\beta_0,A_0) - {\varvec X}^{-}(\beta_0,\tilde{A})\right\} {\varvec D}(\beta_0,A_0) {\varvec X}\,\hbox{d}A_0(s) \end{aligned} $$

where the time arguments were omitted for notational simplicity. Note that the first term, V _n , by the martingale CLT, can be shown to converge to a Gaussian martingale process V(t) with variance $\sigma^2(t) = \int_{0}^{t}s_{0}^{-1}\,\hbox{d}A_0$. Note also that V _n (t) can be written as

$$ V_n(t) = \sum_i\int_{0}^{t}{\varvec S}_0^{-1}(t,\beta_0) X_i\,\hbox{d}M_{i}. $$

The integrand in the last integral above can be Taylor expanded by the use of matrix derivatives thus leading to (up to $o_p(n^{-1/2})$)

$$ \begin{aligned} ({\varvec S}_0^{-1}(t,\beta_0,\tilde{A}) - {\varvec S}_0^{-1}(t,&\beta_0,A_0)) {\varvec S}_0(\beta_0,A_0) = \\ &- \sum_{j=1}^p {\varvec S}_0^{-1}(t,\beta_0,A_0){\varvec S}_j^{**}(t,\beta_0,A_0)(\tilde{A} _j({t-}) - A_{0j}({t-})). \end{aligned} $$

This implies that $\sqrt{n}(\tilde{A}(\beta_0,t) - A_0(t))$ converges in distribution to a p-dimensional process W that satisfies the integral equation

$$W(t) = \sum_{j=1}^p \int_{0}^{t} - W_j(s-) e_j^{**}(s,\beta_0)\,\hbox{d}A_0(s) + V(t).$$

Denote the elements of $e_{j}^{**}(t)$ as $e_{j,{km}}^{**}(t)$. With the p × p matrices F(t) with elements $F_{jk}(t) = \sum_{m=1}^p e_{j,{km}}^{**}(t)\alpha_{0,m}(t)$ the solution can be written as

$$\hbox{d}W(t) = -W({t-})F\,\hbox{d}t +\hbox{d}V(t). $$

Using the product integration formulae (Gill and Johansen 1990) leads to the solution

$$\begin{aligned} W(t) &= \int_{0}^{t}\,\hbox{d}V(s) {\cal F}(s,t) \\ {\cal F}(s,t) &= \prod_{{]}s,t{]}} (I - F\,\hbox{d}t) \end{aligned}$$

(13)

and where $\mathcal{F}(s,t)$ is the product integral of F. In the one-dimensional (commutative) case this equals $\exp(-\int_{s}^{t} e^{**}(u,\beta_0)\,\hbox{d}A_0(u))$. Note that the matrix $\mathcal{F}(s,t)$ is estimated consistently by the product of (the atoms)

$$ \hat{\mathcal{F}}(s,t) = \prod_{]s,t]}(I - \,\hbox{d}\hat{F}) $$

(14)

where $\hbox{d}\hat{F}_{jk}(t) = \sum_{m=1}^p E_{j,{km}}^{**}(t)\,\hbox{d}\hat{A}_m(t).$ Using the product integral rather than the exponential in the one-dimensional case yields an estimator with the same asymptotic properties.

I now turn to the estimating function for $\beta, \tilde{U}(\beta).$ By the martingale decomposition

$$ \begin{aligned} \tilde{U}(\beta_0) &= \int{\left\{{\varvec Z}^T -{\varvec Z}^{T}{\varvec D}(\beta_0,\tilde{A}){\varvec X}S_0^{-1}(\beta_0,\tilde{A}) {\varvec X}^T \right\}}\,\hbox{d}M \\ &+ \int{\left\{{\varvec Z}^{T} D(\beta_0, A_0){\varvec X} - {\varvec Z}^{T}{\varvec D}(\beta_0, \tilde{A}){\varvec X}\right\}}\,\hbox{d}A_0 \\ &+ \int{{\varvec Z}^{T}{\varvec D}(\beta_0,\tilde{A}){\varvec X}\left\{{\varvec S}_0^{-1}(\beta_0,A_0) - {\varvec S}_0^{-1}(\beta_0,\tilde{A})\right\} {\varvec S}_0(\beta_0,A_0)}\,\hbox{d}A_0. \end{aligned} $$

By a Taylor expansion, as before, it may be shown that the difference (up to $o_p(n^{-1/2})$)

$$ {\varvec Z}^{T} {\varvec D}(\beta_0, A_0) {\varvec X} - {\varvec Z}^{T} {\varvec D}(\beta_0, \tilde{A}) {\varvec X} = -\sum_{j=1}^p {\varvec S}_j^{**Z}(\beta_0, A_0)(\tilde{A}_j({t-}) - A_{0j}({t-})). $$

The last two term therefore equals ( $o_p(n^{-1/2})$)

$$ \sum_{j=1}^p \int{\left\{s_j^{**z} - s^z s_0^{-1} s_j^{**}\right\}}W_j\,\hbox{d}A_0$$

that can be written as

$$ \int{\left\{G_1(t) - G_2(t)\right\}} W(t)\,\hbox{d}t$$

with G ₁(t) and G ₂(t) an q × p matrix with elements $G_{1,kj}(t) = \sum_{m=1}^p s_{j,km}^{**z}\alpha_{0,m}(t)$ and $G_{2,kj}(t) = \sum_{m=1}^p \left\{s^z s_0^{-1}s_j^{**}\right\}_{km} \alpha_{0,m}(t)$. Now changing the order of integration and using the structure of W(t) one gets

$$ \begin{aligned} & \int_{0}^{\tau}\left\{G_1(t)-G_2(t)\right\}W(t)\,\hbox{d}t = \int_{0}^{\tau}\left\{G_1(t)-G_2(t)\right\}\int_{0}^{t}\mathcal{F} (s,t)V(s)\,\hbox{d}s\,\hbox{d}t \\ &= \int_{0}^{\tau} g(s)V(s)\,\hbox{d}s = \sum_{i}\int_{0}^{\tau} g(s) s_0^{-1}(s,\beta) X_i \,\hbox{d}M_i(s), \\ \end{aligned} $$

where

$$ g(s) = \int_{s}^{\tau}\left\{G_1(t)-G_2(t)\right\}\mathcal{F}(s,t)\,\hbox{d}t. $$

Define

$$ \begin{aligned} q_{1i}(t,\beta) &= Z_i - s_z(t,\beta)s_0^{-1}(t,\beta)X_i \\ q_{2i}(t,\beta) &= g(t) s_0^{-1}(t,\beta)X_i \\ q_i(t) &= q_{1i}(t,\beta) + q_{2i}(t,\beta). \end{aligned} $$

(15)

Then one can write the estimating function as follows (up to an o _p (1) term)

$$ n^{-1/2}\tilde{U}(\beta_0) = n^{-1/2}\sum_{i}\int_{0}^{\tau}\left\{q_{1i}(t,\beta_0) +q_{2i}(t,\beta_0)\right\}\,\hbox{d}M_i(t). $$

This is a sum of i.i.d. terms (or a martingale) and therefore converges to a normal distribution with variance that is estimated by the robust estimator given in (11).

Based on a Taylor series expansion (up to an o _p (1) term) we find that

$$ \begin{aligned} \sqrt{n}(\hat{A}(t) - A_0(t)) &= n^{1/2} \int_{0}^{t} \frac{\partial}{\partial\beta}X^{-}(t,\beta_0)\,\hbox{d}{\varvec N}(\hat{\beta} - \beta_0) + n^{-1/2}\sum_{i}\int_{0}^{t}\mathcal{F}(s,t) s_0^{-1}(s,\beta_0) X_i(s)\,\hbox{d}M_i(s) \\ &= n^{-1/2} \sum_{i}L_i(t,\beta_0) \end{aligned} $$

where

$$ \begin{aligned} L_i(t,\beta) &= P(t,\beta_0)I^{-1}(\tau)\int_{0}^{\tau} Y_i(t) q_i(t,\beta_0)\,\hbox{d}M_i(t) \\ &+ \int_{0}^{t} Y_i(s) \mathcal{F}(s,t) s_0^{-1}(s,\beta_0) X_i(s)\,\hbox{d}M_i(s), \end{aligned} $$

(16)

and with P(t,β) the limit of $n^{-1}\tilde{P}(t,\hat{\beta})$ defined in (8), with I(τ) the limit of $n^{-1} \mathcal{I}(\tau,\beta_0)$, and with the product integral $\mathcal{F}(s,t)$ defined in (13).

Note that an estimator of $\mathcal{F}(s,t)$ is given in (14).