15.6. Consider the selection model of Example 15.7 when (xi xi1;zi), where zi is a single binary variable predictive of selection but with coefficient

15.6. Consider the selection model of Example 15.7 when (xi ¼ xi1;zi), where zi is a single binary variable predictive of selection but with coefficient zero in the regression on Y1 on X. The following table gives the means of yi1 given xi, classified by zi and by whether yi1 is observed (Mi1 ¼ 0) or missing (Mi1 ¼ 1).

zi 0 1 0 xib1 þ rs1lðgi0Þ xib1 þ rs1lðgi1Þ Mi1 1 xib1 þ rs1lðgi0Þ xib1 þ rs1lðgi1Þ ALL xib1 xib1 In the table, lðÞ is defined as in Example 15.5, and gij is the mean of Y2 for units with values (xi;zi ¼ j) of the covariates.

(a) Derive the expressions in the table.

(b) By considering the difference in means of Y1 between responding and nonresponding cases for several values of gij, show that the model implies that the means in the table have an approximately additive structure. (See Little, 1985a, for more details.)