= Solution
For user $j$ and offer $r$, both fits use
$$
\operatorname{logit}(p_j)=\beta_0+\beta_1\operatorname{age}_j+\beta_2\operatorname{fitness}_j
+\gamma_{\operatorname{region}_j}+\delta_{\operatorname{sex}_j}.
$$
The first model treats each click as Bernoulli$(p_j)$. The grouped model treats the click count $C_j=m_j\operatorname{prop}_j$ as $\operatorname{Bin}(m_j,p_j)$. Their likelihoods differ only by binomial coefficients independent of the parameters, so their fitted coefficients agree.
Back to article page
