PROC QLIM to fit Ordered Logit Models and Ordered Probit Models

Suppose that uj, 0..J such that -inf=u0<u1<...<uJ=inf. The observed outcome is a categorization of a latent variable Yi^star=xi.beta+ei s.t. Yi=j iff uj-1<Yi^star<=uj. The probability of response is pj(xi)=Pr(Yi=j|xi)=F(uj-xi.beta)-F((uj-1)-xi.beta).

In the ordered logit model s.t. log(pj(xi)/(1-pj(xi))=uj-xi.beta. The parameter beta describe the effect of a covariate on the log odds of response in the category j, or marginal effect of the covariate E(Yi^star|xi).
In the cumulative probit model s.t. Pi^(-1)=uj-xi.beta.

*Ordered Logit;
proc logistic data=test desc;
class vn0476_M(ref='female') vn0435_34(ref='rent') vn0455_1(ref='unmarried')/param=ref;
model product_pref=&numeric &norminal;
format vn0476_M gender. vn0435_34 rent. vn0455_1 marital.;
run;

Proc qlim fits the ordered logit and ordered probit models. It uses the latent variable formulation. By default, an intercept is included in beta and the first threshold parameter u1=0. The model option limit1=varying overrides the default.

proc qlim data=test covest=qml;
class vn0476_M vn0435_34 vn0455_1;
endogenous product_pref ~ discrete (dist=logistic order=formatted);
*hetero product_pref ~&numeric;
model product_pref=&numeric &norminal/limit1=varying;
format vn0476_M gender. vn0435_34 rent. vn0455_1 marital.;
run;

proc qlim fit the equivalent homoscedastic ordered logit model. However, the signs for the covariates for intercepts are reversed.

Ordered logit and ordered probit models assume the effect of a covariate is identical for all J-1 cumulative logits, which is called the proportional odds property.