coefficients in vglm(), orm(), polr() and clm() for ologit

Question

I am doing ologit with different packages, they are VGAM, rms, MASS and ordinal, using data set wine from package ordinal.

First is vglm():

library(VGAM)
vglmfit <- vglm(rating ~ temp * contact, data = wine, 
                family=cumulative(parallel=TRUE, reverse=TRUE))

The coefficients are:

  (Intercept):1       (Intercept):2       (Intercept):3       (Intercept):4 
      1.4112568          -1.1435551          -3.3770742          -4.9419773 
       tempwarm          contactyes tempwarm:contactyes 
      2.3212033           1.3474598           0.3595241 

Second is orm():

library(rms)
ormfit <- orm(rating ~ temp * contact, data = wine)

Coef:

                        Coef    S.E.   Wald Z Pr(>|Z|)
y>=2                     1.4113 0.5454  2.59  0.0097  
y>=3                    -1.1436 0.5097 -2.24  0.0248  
y>=4                    -3.3771 0.6382 -5.29  <0.0001 
y>=5                    -4.9420 0.7509 -6.58  <0.0001 
temp=warm                2.3212 0.7009  3.31  0.0009  
contact=yes              1.3475 0.6604  2.04  0.0413  
temp=warm * contact=yes  0.3595 0.9238  0.39  0.6971

Third, polr:

library(MASS)
polrfit <- polr(rating ~ temp * contact, method="logistic", data = wine)

coef:

Coefficients:
           tempwarm          contactyes tempwarm:contactyes 
          2.3211214           1.3474055           0.3596357 

Intercepts:
      1|2       2|3       3|4       4|5 
-1.411278  1.143507  3.377005  4.941901 

Last, clm():

library(ordinal)
clmfit <- clm(rating ~ temp * contact, link="logit", data = wine)

coef:

Coefficients:
           tempwarm          contactyes tempwarm:contactyes 
             2.3212              1.3475              0.3595 

Threshold coefficients:
   1|2    2|3    3|4    4|5 
-1.411  1.144  3.377  4.942 

Besides, when reverse=FALSE in vglm(),

library(VGAM)
vglmfit <- vglm(rating ~ temp * contact, data = wine, 
                family=cumulative(parallel=TRUE, reverse=FALSE))

Coefficients:
      (Intercept):1       (Intercept):2       (Intercept):3       (Intercept):4 
         -1.4112568           1.1435551           3.3770742           4.9419773 
           tempwarm          contactyes tempwarm:contactyes 
         -2.3212033          -1.3474598          -0.3595241 

You may notice that the coefficients in vglm() while reverse=TRUE and those in orm() are the same, and the ones in polr() and clm() are the same. So there are two set of coefficients, the only difference is the sign of intercepts.

And while I set reverse=FALSE, it does reverse the intercepts, but at the same time the parameters of variables, which I don't want.

What's the problem of that? How could I get exactly the same result? or how could I explain it?

Answer 1

This is all just a matter of parametrizations. One classical way to introduce the ordered logistic regression model is to assume that there is a latent continuous response

y* = x'b + e

where e has a standard logistic distribution. Then, it is assumed that not y* itself is observed by only a discretized category y = j if y* falls between cut-offs a_j-1 and a_j. This then leads to the model equation:

logit(P(y <= j)) = a_j - x'b

Other motivations lead to similar equations but with P(y >= j) and/or a_j + x'b. This just leads to switches in the signs of the a and/or b coefficients that you observe in the different implementations. The corresponding models and predictions are equivalent, of course. Which interpretation you find easier is mostly a matter of taste.