# Robust variance/covariance matrix in Poisson regression

Suppose I have survival data with more than one row per subject, because I have splitted the follow-up time of each subject into pieces (maybe because I have one or more time-varying variables or maybe just because I want to fit a Poisson model with a non-constant hazard over time).

Do I have to use the robust variance / covariance estimator (that is implemented for example in Stata with the option vce(cluster clustvar)) to take into account that I have more than one observation per subject (i.e. they're not independent)?

Edit (15 March 2012):

Lambert and Royston in their book carry out this analysis: they split each subject's follow-up on one time-scale (let's say attained age)[*] and fit a Poisson regression including attained age as the dependent variable (modelled for example using splines) plus the offset, so that it's possible to model the incidence of some disease according to attained age.

They do not use the robust variance/covariance estimator, but I haven't found in the text any explanation of why the single rows (or episodes) can be considered independent.

The question: Can someone explain to me why the single rows (or episodes) can be considered as independent?

[*]To clarify what's been done, let's take for example subject number 1001. He/she enters the study at 80.00219 years of age and develop the disease at 85.037236 years (_d==1). This is what happens to the record of this subject after splitting it up. (The offset variable is defined as ln(_t-_t0))

  id   _d          _t        _t0
1001    0          81   80.00219
1001    0          82         81
1001    0          83         82
1001    0          84         83
1001    0          85         84
1001    1   85.037236         85

• If you're happy with proportional hazards assumptions, you don't need to use clustering in this way; contributions to the partial likelihood at each time point only reflect people in the risk set at that time point, and (in your case) what covariates they had at that time point. Robust variance estimators (i.e. , robust in Stata) can still be useful though; they give you standard errors that appropriately summarize variability in your point estimates, regardless of whether the hazard ratio is truly log-linear in the covariates. – guest Mar 15 '12 at 23:59

It is important to note that we do not assume that the $d_{ij}$ have independent Poisson distributions, because they clearly do not. If individual $i$ died in interval $j(i)$, then it must have been alive in all prior intervals $j < j(i)$, so the indicators couldn't possibly be independent. Moreover, each indicator can only take the values one and zero, so it couldn't possibly have a Poisson distribution, which assigns some probability to values greater than one. The result is more subtle. It is the likelihood functions that coincide. Given a realization of a piece-wise exponential survival process, we can find a realization of a set of independent Poisson observations that happens to have the same likelihood, and therefore would lead to the same estimates and tests of hypotheses.
We [Royston and Lambert] are not actually assuming that the $d_{ij}$ have independent Poisson distributions. However, when we express the model in this way, it is the likelihood function that is equivalent to that from a piecewise exponential model.