The following exercise is designed to enhance the idea expressed in Figure 8.3 and Figure 8.4 that one way to obtain the proportional odds model is via categorization of a continuous variable. (a)...

The following exercise is designed to enhance the idea expressed in Figure 8.3 and Figure 8.4 that one way to obtain the proportional odds model is via categorization of a continuous variable.

(a) Form the scatterplot of BWT versus LWT.

(b) Fit the linear regression of BWT on LWT and add the estimated regression line to the scatterplot in 2(a). Let λˆ 0 denote the estimate of the intercept, λˆ 1 the estimate of the slope and s the root mean squared error from the linear regression.

(c) It follows from results for the logistic distribution that the relationship between the root mean squared error in the normal errors linear regression and the scale parameter for logistic errors linear regression is approximately σˆ = s √3/π. Use the results from the linear regression in 2(b) and obtain σˆ.

(d) Use the results from 2(b) and 2(c) and show that the estimates presented in Table 8.18 are approximate, and

(e) By hand draw a facsimile of the density function shown in Figure 8.4 with the three vertical lines at the values 2500, 3000, and 3500. Using the results in equation (8.20), equation (8.21) and the estimates in Table 8.18 compute the value of the four areas under the hand-drawn curve. Using these specific areas demonstrate that the relationship shown in equation (8.23) holds at each cut point.