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I would like to run a semi-symmetric bi-directional case-crossover method on some generated data using conditional logistic regression.

I generated data from Poisson distribution Poiss(lambda) with

$\textrm{log}(\lambda)=\lambda_0 \textrm{exp}(x\beta), $

where $\lambda_0$ and $\beta$ are known and $x$ is a known time series. Then for each timepoint I split generated value into pairs: case in this timepoint and control in referent period (that is if in generated data there are $5$ cases at day $11$, I get $10$ observations for this timepoint: $5$ $(1$, $x$ at day $11)$ and $5 (0$, $x$ at reference period for day $11)$). Now I run conditional logistic regression on this data, with separate stratum for each pair.

The question is, should the parameter estimates that I get from CLR be $\beta$ from parametrer of Poisson distribution? I found few articles with this procedure applied, but I can't figure out why estimetes from CLR and $\beta$ from Poisson distribution should be the same.

Reference articles:

Thanks for help

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In a case-crossover study, the referent time periods are randomly sampled from the person-time at risk of each individual. Therefore, the odds ratio in a case-crossover estimates the incidence density ratio that would have been obtained in a cohort study - it is just an efficient sampling scheme, similar to an incidence density or risk set sampled case-control study, albeit one that doesn't require any reference people and automatically accounts for between-person confounding and within-person non-time-varying confounding. So, when you use conditional logistic regression your $\beta$ estimates are odds ratios which are in turn estimates of the incidence density ratios, which is what the $\beta$ parameters of your Poisson time-series are.

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