Democratic Confidence Intervals for comparing two Proportions

We offer methods to create an accurate graph for comparing confidence intervals for two proportions. The firstmethod involves using weighted sum contrasts when fitting logistic regression with a binary explanatory variable. Since it iscomplicated to estimate standard errors of the logit function, the second method is thus based on applying a delta method tothe logit function of the proportions to get their individual estimated standard errors and shrink them to get their comparisonstandard errors. The two methods give an accurate approximation of confidence intervals for comparing proportions that isconsistent with the p-values near 0.05 provided by logistic regression.