91Ó°ÊÓ

since the logit transformation (14.18) linearizes the logistic response function, why can't this transformation be used on the individual responses \(Y_{l}\) and a linear response function then fitted? Explain.

a. Plot the logistic mean response function (14.16) when \(\beta_{0}=20\) and \(\beta_{1}=-2\) b. For what value of \(X\) is the mean response equal to \(.5 ?\) c. Find the odds when \(X=125,\) when \(X=126,\) and the ratio of the odds when \(X=126\) to the odds when \(X=125 .\) Is the odds ratio equal to \(\exp \left(\beta_{1}\right)\) as it should be?

Consider the multiple logistic regression model with \(X^{\prime} \beta=\beta_{0}+\beta_{1} X_{1}+\beta_{2} X_{2}+\beta_{3} X_{1} X_{2}\) Derive an expression for the odds ratio for \(X_{1} .\) Does \(\exp \left(\beta_{1}\right)\) have the same meaning there as for a regression model containing no interaction term?