# Extended Weibull-Cox model survival function

I'm reading an article where the authors use an extended Cox-Weibull hazard model with time-varying covariates to model survival time of an individual. This individual is allowed to start at week $$m$$, and covariates are available at weekly granularity.

Assuming the hazard is represented as $$\lambda\left(t ; Z(t)\right)=\lambda_0(t) e^{\beta^\top Z(t)}$$, where the baseline is Weibull, they provide an expression for the probability an individual survives at least $$m^\prime-m$$ weeks, given they started at week $$m$$, through $$\tag{1} S(m^\prime-m|m)=\exp(-\lambda B(m,m^\prime))$$ where $$B(m,m^\prime)=\sum_{i=m+1}^{m^\prime}[(i-m)^c-(i-m-1)^c]\exp(\beta^\top Z(i)).$$ I'm having trouble justifying expression (1), however, and feel it should follow from the relation between survival and hazard functions, namely $$S(m^\prime -m|m)=\exp\left(-\int_0^{m^\prime -m} \exp (\beta^\top Z(u)) \lambda_0(u)~\mathrm d u\right).$$ Using the fact the covariates $$Z(t)$$ are a step function, as they change only at the start of each week, I feel as if I can calculate what I'm after through something like $$\exp\left(-\left[\exp (\beta^\top Z(0)) \int_0^{1} \lambda_0(u) ~\mathrm d u+\ldots+\exp(\beta^\top Z(m^\prime-m-1)) \int_{m^\prime-m-1}^{m^\prime-m} \lambda_0(u) ~\mathrm d u\right]\right),$$ but it doesn't seem like this will correspond to the expression in the paper.

• Can you be explicit with how the Weibull is parameterized please? Sep 29, 2022 at 2:06
• I should have specified that, but for this Weibull parameterization, the hazard is $h(x|k,b)=bkx^{k-1}$, with a CDF of $1-e^{-bx^k}$ Sep 29, 2022 at 14:18

The authors may be expressing that if an individual starts at week $$m$$ then they have survived up until week $$m$$, that is, they have the survival function $$S(t | m) = P(T \ge t | T \ge m)$$. This conditional survival function can be simplified to look like: $$S(t | m) = \frac{S(t)} {S(m)}$$. Their expression looks like this ratio.
• I thought about this as well. I might be naive, but this felt wrong to me because it's not that the individual survived until week $m$, but rather they were born at that time. Are they equivalent? Sep 29, 2022 at 15:31
• Typically the time variable in survival analysis is not an absolute time, but a measure of duration. So it's strange to say they were born at time $m$, normally you would say they were born at time 0, and record from there. However, it sounds like subjects could enter study at week $m$, which would imply survival between time 0 and time m. This is called late entry in the literature. Sep 29, 2022 at 18:10
• I see, this is very helpful. In this case, I worded it as born because these individuals weren't yet added to the population until time $m$, so it feels weird to condition upon survival time from 0 to $m$ in this case Sep 29, 2022 at 19:09