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add clarifying sentence
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MatthiasSchmidtblaicherQC committed Nov 5, 2024
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"$$\n",
"\\sum_{\\text{event times}}\\log\\left(\\frac{y_{i,t}\\exp(\\eta_{i})}{\\sum_{i \\in \\mathcal{R}_t} \\exp(\\eta_i)} \\right) - 1,\n",
"$$\n",
"which is the same as the partial likelihood in the Cox model, apart from the -1 which drops out when taking derivatives. In short, the Cox partial log likelihood is equivalent to a Poisson log likelihood with the estimate for time period effects fed back in (\"profiled out\"). This means that, to estimate the parameters of the Cox model, one can simply run a Poisson regression with time fixed effects $\\alpha_t$.\n",
"which is the same as the partial likelihood in the Cox model, apart from the -1 which drops out when taking derivatives. In short, the Cox partial log likelihood is equivalent to a Poisson log likelihood with the estimate for time period effects fed back in (\"profiled out\"). This means that, to estimate the parameters of the Cox model, one can simply run a Poisson regression with time fixed effects $\\alpha_t$. The data structures for the two objectives are different: the Cox partial log-likelihood operates on data with one row per observed individual, while the Poisson log-likelihood uses one row per individual and time period.\n",
"\n",
"## 2. Estimating a Cox Model in Glum<a class=\"anchor\"></a>\n",
"\n",
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