Binary Choice

Start with the probability contrast

Let $Y=1$ mean degree completion. The policy question may be

an average risk difference. A logit coefficient is a change in log-odds, not this quantity.

Three models, three conveniences

For causal work, all three still require a credible treatment-assignment argument. Robust standard errors address heteroskedasticity in an LPM; they do not repair confounding.

Odds are not risks

Suppose completion rises from 0.20 to 0.30:

• risk difference: $0.30-0.20=0.10$;
• risk ratio: $0.30/0.20=1.50$;
• odds ratio: $(0.30/0.70)/(0.20/0.80)=1.71$.

“A 71% increase” would misdescribe the risk. State the scale.

Convert nonlinear coefficients into quantities people can read

For a logit model $p_i=\Lambda(X_i'\beta)$ and continuous $x_k$,

If $\beta_k=0.8$, the derivative is 0.20 at $p=0.5$ but only 0.072 at $p=0.10$. A single coefficient does not imply a constant probability change.

For a binary scholarship offer, prefer a discrete change:

This averages the same treatment contrast over the observed covariate distribution.

The odds ratio $e^{0.8}=2.23$ is constant, while the probability change is not.

Interactions are contrasts of predictions

In a nonlinear model, the coefficient on $D\times G$ is generally not the interaction effect on probability. Compute four predictions:

Report uncertainty for this contrast, preferably through the model’s delta method or resampling scheme aligned with the design.

Fit is not only discrimination

AUC can be high while probabilities are poorly calibrated. For a causal effect, neither calibration nor AUC proves exchangeability.

Quick check

The estimated odds ratio is 1.5 in both a low-risk and high-risk group. Are the risk differences equal?

Next: Multinomial Choice