Using the Delta Method to Construct Confidence Intervals for Predicted Probabilities, Rates, and Discrete Changes

.Let G(β) be some function, such as predicted probabilities from a logit or ordinal logit model. The Taylor series expansion of G( b β) is G( b β )= G(β )+( b β − β) 0 G 0 (β )+( b β − β) 0 G 00 (β ∗ )( b β − β)/2 (2) ≈ G(β )+( b β − β) 0 G 0 (β) , where G 0 (β) and G 00 (β) are matrices of first and second partial derivatives with respect to β, β ∗ is some value between b and β. Then, √ n h G( b β) − G(β) i ≈ √ n( b β − β) 0 G 0 (β) . (3)