Consider n independent binary outcomes where $Y_1, \dots, Y_n$ arise from:
$P(Y_i = 1|x_i) = E(Y_i = 1|x_i) = \beta_0 + \beta_1 x_i$
where $x_i$ is a covariate for subject $i$.
1. Suppose you use least squares to estimate $\beta_0$ and $\beta_1$ (even through the outcomes are binary). Are the
least squares estimators of the regression coefficients unbiased?
2. Write out the likelihood function for an independent sample of n observations arising from the above
model.
3. Considering this same model formulation, suppose that $x_i$ is a binary indicator for the treatment
assignment, coded as 0 for \"untreated\" and 1 for \"treated\". Suppose we observe 5 successes out of 20
outcomes in the untreated group and 25 successes out of 40 in the treated group. Using any method
available, hand calculate the estimators of $\beta_0$ and $\beta_1$ for this case.
4. Finally, does $\beta_1$ have an interpretation related to one of our standard measures of association, namely:
$\bullet$ risk difference $P_{treat} - P_{control}$
$\bullet$ risk ratio $P_{treat}/P_{control}$
$\bullet$ odds ratio $\frac{P_{treat}/(1-P_{treat})}{P_{control}/(1-P_{control})}$
