Equation (13) is estimated subject to the constraint that the
relative ordering of the estimated rankings (Rj^) agrees with
those revealed in Step 1. That is, for all i and j between 1
and ALT, the sign of Ri^ RJ^ must be the same as for R1-
Rj.
Most monotonic regression procedures choose W to
minimize a loss function of the following type:
(14) L = p dij (Ri^ Rj^)k
ij
where di,j = 1 if (Ri Rj)*(Ri^ Rj^) < 0
= 0 otherwise.
When k = 0, this can be accomplished with integer programm-
ing (Pekelman and Sen) or nonlinear programming (Garrod,
Garrod and Miklius). When k = 1, linear programming may be
employed (Srinivasan, Srinivasan and Shocker, and Pekelman
and Sen). If k = 2, iterative procedures (R. Johnson) or the
minimization of stress (Kruskal) are feasible estimation
approaches.
An alternative to the above is to convert the reported
rankings to binary data and then use a probability model such
as logit or probit, to estimate W. This involves the following
steps:
a. Form the following ALT*((ALT-1)/2) equations.
(15) Ri Rj = Wk(gk(qk,1)) Wk(gk(qk,1)) for all i > j.
b. Next, transform the righthand side into a Bernoulli
variable that has the value of 1 if R. R. > 0 and 0 otherwise.
1 j