Editing Logistic regression (section)

===In terms of binomial data===
A closely related model assumes that each ''i'' is associated not with a single Bernoulli trial but with ''n''<sub>''i''</sub> [[independent identically distributed]] trials, where the observation ''Y''<sub>''i''</sub> is the number of successes observed (the sum of the individual Bernoulli-distributed random variables), and hence follows a [[binomial distribution]]:

:<math>Y_i \,\sim  \operatorname{Bin}(n_i,p_i),\text{ for }i = 1, \dots , n</math>

An example of this distribution is the fraction of seeds (''p''<sub>''i''</sub>) that germinate after ''n''<sub>''i''</sub> are planted.

In terms of [[expected value]]s, this model is expressed as follows:

:<math>p_i = \operatorname{\mathbb E}\left[\left.\frac{Y_i}{n_{i}}\,\right|\,\mathbf{X}_i \right]\,, </math>

so that

:<math>\operatorname{logit}\left(\operatorname{\mathbb E}\left[\left.\frac{Y_i}{n_i}\,\right|\,\mathbf{X}_i \right]\right) = \operatorname{logit}(p_i) = \ln \left(\frac{p_i}{1-p_i}\right) = \boldsymbol\beta \cdot \mathbf{X}_i\,,</math>

Or equivalently:

:<math>\Pr(Y_i=y\mid \mathbf{X}_i) = {n_i \choose y} p_i^y(1-p_i)^{n_i-y} ={n_i \choose y} \left(\frac{1}{1+e^{-\boldsymbol\beta \cdot \mathbf{X}_i}}\right)^y \left(1-\frac{1}{1+e^{-\boldsymbol\beta \cdot \mathbf{X}_i}}\right)^{n_i-y}\,.</math>

This model can be fit using the same sorts of methods as the above more basic model.