Binary Outcome Models for Experimental Research
When $Y\in{0,1}$, its conditional mean is a probability:
\[E[Y\mid X]=P(Y=1\mid X).\]Linear probability model
The linear probability model specifies
\[P(Y=1\mid X)=X\beta.\]Each coefficient is a change in probability for a one-unit change in the corresponding predictor, holding the others fixed. Its simplicity is useful, but fitted values may fall outside $[0,1]$ and errors are inherently heteroskedastic, so heteroskedasticity-robust standard errors are generally used.
Logistic model
Logistic regression keeps fitted probabilities between zero and one:
\[P(Y=1\mid X)=\frac{1}{1+e^{-X\beta}}.\]Equivalently, it models the log-odds linearly:
\[\log\frac{p}{1-p}=X\beta.\]For a one-unit increase in $X_j$, holding other variables fixed, the odds are multiplied by
\[OR=e^{\beta_j}.\]An odds ratio is not a risk ratio and is not a probability difference. Marginal effects are needed to express results on the probability scale.
Treatment effects with binary outcomes
In a randomized two-arm experiment, the unadjusted difference in sample proportions is also the difference in mean outcomes:
\[\hat\tau=\bar Y_1-\bar Y_0=\hat p_1-\hat p_0.\]This is the estimated risk difference. Logistic regression instead naturally reports conditional odds ratios, so the estimand and reporting scale should be stated explicitly.
Propensity-score connection
Logistic regression may model treatment assignment rather than the outcome:
\[\hat e(X)=P(W=1\mid X).\]That use belongs to Propensity Scores and should not be confused with an outcome model.