Likelihood of a multiple-spell survival model from Tutz & Schmid (2016)

Question

I am reading Tutz & Schmid "Modeling Discrete Time-to-Event Data" (2016) chapter 10 Multiple Spell Analysis section 10.1.1 Estimation. On p. 215 the closed form of the total likelihood is provided: $$ L = \Pi_{i=1}^n \Pi_{k=1}^{k_i} \Pi_{r=1}^m \left[ \lambda_{y_k}^{(k)}(t_{ik}\vert H_{k-1}, x_{ik}) \right]^{\delta_{ikr}} \Pi_{s=t_{i,k-1}+1}^{t_k-1} (1-\lambda^{(k)}(s\vert H_{k-1}, x_{ik})) P(x_k \vert H_{k-1}) ^{\epsilon_{ik}} $$ I think it has some problems with indices/subscripts and parentheses. A version I would find more logical is the following: $$ L = \Pi_{i=1}^n \Pi_{k=1}^{k_i} \Pi_{r=1}^m \left[ \lambda_\color{red}{r}^{(k)}(t_{ik}\vert H_{\color{red}{i}k-1}, x_{ik}) \right]^{\delta_{ikr}} \Pi_{s=t_{i,k-1}+1}^{t_{\color{red}{i}k-1}} \left[ (1-\lambda^{(k)}(s\vert H_{\color{red}{i}k-1}, x_{ik})) P(x_{\color{red}{i}k} \vert H_{\color{red}{i}k-1}) \right]^{\epsilon_{ik}} $$ Do my corrections make sense? Did I miss anything?

_{P.S. How do I color \left[ and \right] that I added in the second part?
And how do I color the word color in the last sentence?}

Answer 1

Your proposed replacement of $y_k$ with $r$ in $\left[ \lambda_{y_k}^{(k)}(t_{ik}\vert H_{k-1}, x_{ik}) \right]^{\delta_{ikr}}$ is equivalent to what Tutz and Schmid write. $\delta_{ikr}=0$ unless "the $i$th individual ends at $t_{ik}$ in state $r$," in which case $y_k=r$.

Confusion does arise from inconsistent subscripting of the covariate vector $\pmb{x}_k$ "that determines duration in spell $k$" (page 214). At the top of page 215, Tutz and Schmid say that "the subscript $i$ [indexing observations] is suppressed on the right-hand side" of the likelihood equation presented in Section 10.1.1, but they then go on to re-introduce that subscript in the equation you show, it seems inconsistently.

There might be an argument in favor of omitting some subscripting. For example, the history $H_{k-1}$ is defined as "the history of the process up till $t_{k-1}$ [the time of the transition to the current state]," including prior covariate vectors $\pmb{x}_1$ through $\pmb{x}_{k-1}$. If those covariate vectors are taken to include the values for all individuals at risk, I suppose there is no need to subscript the $H_{k-1}$ further by individual.

The re-introduction of subscripting via ${x}_{ik}$ in the equation you show is presumably to emphasize the product over all observations in the likelihood. I don't know why they didn't carry that over to the final factor $P(\pmb{x}_k \vert H_{k-1})$; I might be missing something subtle.

For the superscript of the product $\Pi_{s=t_{i,k-1}+1}^{t_k-1}$ as written by Tutz and Schmid, I think that you intended to write $\Pi_{s=t_{i,k-1}+1}^{t_{\color{red}i,k}-1}$. That seems to be the correct way to write the contribution to the likelihood from the probability of surviving up to time $t_{ik}$ during the spell in question.