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This is a bit longish (I want to be thorough), but my actual question is a short one (in bold below).

BACKGROUND: Suppose I have a conditional logistic discrete time event history model (aka logistic discrete time survival models aka logit hazard model) across $p$ possibly time-varying predictors $\mathbf{X} = \{X_{1t}, \dots, X_{pt}\}$ for $T$ time periods indexed by $t$ from $1, \dots, T$ something like:

$$\text{logit}\left(h_{t,\mathbf{X}}\right) = \alpha_{1}d_{1} + \dots + \alpha_{T}d_{t} + \mathbf{BX_{t}},$$

Where $d_{1}, \dots, d_{T}$ are indicator variables of each time period, $\alpha_{1}, \dots, \alpha_{T}$ are the period-specific intercepts for the hazard function, and $\mathbf{BX}$ are the conditionals and their effects. This model implies that the hazard function is

$$h_{t,\mathbf{X}} = \frac{e^{\alpha_{1}d_{1} + \dots + \alpha_{T}d_{T} + \mathbf{BX_{t}}}}{1 + e^{\alpha_{1}d_{1} + \dots + \alpha_{T}d_{T} + \mathbf{BX_{t}}}}= \frac{e^{\alpha_{t}d_{t} + \mathbf{BX_{t}}}}{1 + e^{\alpha_{t}d_{t} + \mathbf{BX_{t}}}}.$$

A success story:
I would like to employ frequentist inference using confidence intervals around my estimated model quantities. So, using the delta method, I can estimate the variance of $\hat{h}_{t,\mathbf{X}}$ as (see end of question for derivation):

$$\sigma^{2}_{\hat{h}_{t}} = \left[\frac{e^{\hat{\alpha}_{t}+\hat{\mathbf{B}}\mathbf{X}_{t}}}{\left(1+e^{\hat{\alpha}_{t}+\hat{\mathbf{B}}\mathbf{X}_{t}}\right)^{2}}\right]^{2}\mathbf{Z}_{t}\mathbf{\Sigma}_{t}\mathbf{Z}_{t}^{\text{T}},$$

Where the $p+1$ row vector $\mathbf{Z}_{t} = \left[d_{t},\mathbf{X}_{t}\right]$, and $\mathbf{\Sigma}_{t}$ is the variance-covariance matrix of $\mathbf{Z}_{t}$.

When I run simulations of Wald-type confidence intervals ($\hat{h}_{t,\mathbf{X}} \pm t_{\text{CI}}\sigma^{2}_{\hat{h}_{t,\mathbf{X}}}$) using this variance estimator for unconditional models, for models conditioned on two variables, and for models conditioned on those variables plus an interaction (aside: also using different parameterizations of time, more about those at the very end), I get coverage probabilities very close to the nominal confidence level at each time period (i.e. CI of 0.9, 0.95, and 0.99). For example, here are the coverage probabilities across 9 models with a nominal 95% confidence level:

enter image description here

Woot! That looks swell.


A failure story:
The survivor function, $S_{t,\mathbf{X}}$ is also an important quantity in event history models:

$$\hat{S}_{t,\mathbf{X}} = \prod_{i=1}^{t}{\left(1-\hat{h}_i\right)} = \prod_{i=1}^{t}{\frac{1}{1+e^{\hat{\alpha}_{i}+\hat{\mathbf{B}}\mathbf{X}_{i}}}}$$

However, when I go on to derive the variance of the survival function using the delta method to get (see end of question for derivation):

$$\sigma^{2}_{\hat{S}_{t}} = \left[\sum_{i=1}^{t} { \mathbf{Z}_{i}{\mathbf{\Sigma}_{i}}\mathbf{Z}_{i}^{\mathrm{T}} \left(\frac{e^{\hat{\alpha}_{i}+\hat{\mathbf{B}}\mathbf{X}_{i}}}{1+e^{\hat{\alpha}_{i}+\hat{\mathbf{B}}\mathbf{X}_{i}}}\right)^{2}}\right] \left[\prod_{i=1}^{t}{\left(\frac{1}{1+e^{\hat{\alpha}_{i}+\hat{\mathbf{B}}\mathbf{X}_{i}}}\right)}\right]^{2}$$

My coverage probabilities are badly biased. For example, here are the coverage probabilities across 9 models with a nominal 95% confidence level (YUCK!):

enter image description here

I have played around with multiple comparisons adjustments at each succeeding time period... but something fundamental is eluding me. I and a colleague have checked the simulations together, and implemented them in R and Stata so we don't think it is a programming error.

QUESTION: Why am I not getting proper coverage probabilities on $\hat{S}_{t}$ for any but unconditional models with fully discrete parameterizations of time?

Brainstorms are welcome. "You forgot to carry the 2"-type ideas are welcome. Hints are welcome. "Assumption of independence is inappropriate because..." is welcome.


Derivation of $\sigma^{2}_{\hat{h}_{t}}$ using a first-order delta-method approximation: $$\mathbf{V}_{t} = \left.\left[\begin{array}{cccc}\frac{{\partial}h_{t}}{{\partial}\alpha_{t}} & \frac{{\partial}h_{t}}{{\partial}\beta_{1}} & \cdots & \frac{{\partial}h_{t}}{{\partial}\beta_{p}}\end{array}\right]\right|_{\alpha_{t} = \hat{\alpha}_{t}, \mathbf{B} = \hat{\mathbf{B}}}\\ \\ = \left[\begin{array}{ccc}\frac{e^{\left(\hat{\alpha}_{t}+\hat{\mathbf{B}}\mathbf{X}_{t}\right)}}{\left(1+e^{\left(\hat{\alpha}_{t}+\hat{\mathbf{B}}\mathbf{X}_{t}\right)}\right)^{2}} & \frac{X_{1t}e^{\left(\hat{\alpha}_{t}+\hat{\mathbf{B}}\mathbf{X}_{t}\right)}}{\left(1+e^{\left(\hat{\alpha}_{t}+\hat{\mathbf{B}}\mathbf{X}_{t}\right)}\right)^{2}} & \cdots & \frac{X_{pt}e^{\left(\hat{\alpha}_{t}+\hat{\mathbf{B}}\mathbf{X}_{t}\right)}}{\left(1+e^{\left(\hat{\alpha}_{t}+\hat{\mathbf{B}}\mathbf{X}_{t}\right)}\right)^{2}}\end{array}\right]\\ \\ = \frac{e^{\left(\hat{\alpha}_{t}+\hat{\mathbf{B}}\mathbf{X}_{t}\right)}}{\left(1+e^{\left(\hat{\alpha}_{t}+\hat{\mathbf{B}}\mathbf{X}_{t}\right)}\right)^{2}}\left[\begin{array}{cccc}1 & X_{1t} & \cdots & X_{pt}\end{array}\right]$$

$$\sigma^{2}_{\hat{h}_{t,\mathbf{X}}} = \mathbf{V}_{t}\mathbf{\Sigma}_{t}\mathbf{V}_{t}^{\mathrm{T}} \nonumber\\ = \left[\frac{e^{\left(\hat{\alpha}_{t}+\hat{\mathbf{B}}\mathbf{X}_{t}\right)}}{\left(1+e^{\left(\hat{\alpha}_{t}+\hat{\mathbf{B}}\mathbf{X}_{t}\right)}\right)^{2}}\right]^{2}\left[\begin{array}{cccc}1 & X_{1t} & \cdots & X_{pt}\end{array}\right]\mathbf{\Sigma}_{t}\left[\begin{array}{c}1 \\ X_{1t} \\ \vdots \\ X_{pt}\end{array}\right] \\ \\ = \left[\frac{e^{\hat{\alpha}_{t}+\hat{\mathbf{B}}\mathbf{X}_{t}}}{\left(1+e^{\hat{\alpha}_{t}+\hat{\mathbf{B}}\mathbf{X}_{t}}\right)^{2}}\right]^{2}\mathbf{Z}_{t}\mathbf{\Sigma}_{t}\mathbf{Z}_{t}^{\text{T}}$$


Derivation of $\sigma^{2}_{\hat{S}_{t},\mathbf{X}}$ using a first-order delta method approximation:
We start by making $S_{t,\mathbf{X}}$ more tractable by taking the log and assuming (since $h_{t,\mathbf{X}}$ is conditioned on survival to $t$): $$\sigma^{2}_{\ln(\hat{S}_{t,\mathbf{X}})} = \sigma^{2}_{\sum_{i=1}^{t}{\ln\left(1-\hat{h}_{i,\mathbf{X}}\right)}}\\ \\ = \sum_{i=1}^{t}{\sigma^{2}_{\ln\left(1-\hat{h}_{i,\mathbf{X}}\right)}} \quad \mathrm{(by\phantom{0}independence)}$$

Then we get down: $$\sigma^{2}_{\hat{S}_{t,\mathbf{X}}} = \left[\sum_{i=1}^{t}{\sigma^{2}_{\hat{h}_{i,\mathbf{X}}}\left(\frac{1}{\hat{h}_{i,\mathbf{X}}-1}\right)^{2}}\right]\hat{S}_{t,\mathbf{X}}^{2} \\ \\ = \left[\sum_{i=1}^{t}{\mathbf{V}_{i}{\mathbf{\Sigma}_{i}}\mathbf{V}_{i}^{\mathrm{T}}\left(\frac{1}{\hat{h}_{i,\mathbf{X}}-1}\right)^{2}}\right]\hat{S}_{t,\mathbf{X}}^{2}\\ \\ = \left[\sum_{i=1}^{t}{\mathbf{V}_{i}{\mathbf{\Sigma}_{i}}\mathbf{V}_{i}^{\mathrm{T}}\left(1+e^{\hat{\alpha}_{i}+\hat{\mathbf{B}}\mathbf{X}_{i}}\right)^{2}}\right]\hat{S}_{t,\mathbf{X}}^{2}\\ \\ = \left[\sum_{i=1}^{t}{\left[\begin{array}{cccc} 1 & X_{1i} & \cdots & X_{pi}\end{array}\right]\mathbf{\Sigma}_{i}\left[\begin{array}{c}1 \\ X_{1i} \\ \vdots \\ X_{pi}\end{array}\right]\left(\frac{e^{\hat{\alpha}_{i}+\hat{\mathbf{B}}\mathbf{X}_{i}}}{1+e^{\hat{\alpha}_{i}+\hat{\mathbf{B}}\mathbf{X}_{i}}}\right)^{2}}\right] \times \left[\prod_{i=1}^{t}{\left(\frac{1}{1+e^{\hat{\alpha}_{i}+\hat{\mathbf{B}}\mathbf{X}_{i}}}\right)}\right]^{2}\\ \\ \sigma^{2}_{\hat{S}_{t,\mathbf{X}}} = \left[\sum_{i=1}^{t} { \mathbf{Z}_{i}{\mathbf{\Sigma}_{i}}\mathbf{Z}_{i}^{\mathrm{T}} \left(\frac{e^{\hat{\alpha}_{i}+\hat{\mathbf{B}}\mathbf{X}_{i}}}{1+e^{\hat{\alpha}_{i}+\hat{\mathbf{B}}\mathbf{X}_{i}}}\right)^{2}}\right] \left[\prod_{i=1}^{t}{\left(\frac{1}{1+e^{\hat{\alpha}_{i}+\hat{\mathbf{B}}\mathbf{X}_{i}}}\right)}\right]^{2}$$


Alternate parameterizations of time:
The derivations and definitions above are given in terms of a fully discrete paramaterization of time (i.e. the $\alpha_{t}d_{t}$ terms).

In the case of a constant effect of time, the $\alpha_{t}d_{t}$ terms are replaced with a single $\beta_{0}$ term.

In the case of a linear effect of time the $\alpha_{t}d_{t}$ terms are replaced with the terms $\beta_{0} + \beta_{1}t$.

$\mathbf{Z}_{t}$ and $\mathbf{\Sigma}_{t}$ are redefined as appropriate in these cases.

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  • $\begingroup$ Hi there, I'm not sure if this problem is still of interest to you but did you ever try relaxing the assumption of independence when calculating the variance for the log of the survival function to account for the covariance between the hazard at each time? $\endgroup$ – Pseudo_Scientist Mar 31 '16 at 13:16

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