Interpreting ordinal GEE coefficients

Question

I have a dataset with an ordinal dependent variable (iws_w) with a range of -3 to +1. I placed it, with two independent variables in an Ordinal Generalized Estimating Equation and got the following output:

                               OrdinalGEE Regression Results                           
===================================================================================
Dep. Variable:                       iws_w   No. Observations:                  352
Model:                          OrdinalGEE   No. clusters:                        2
Method:                        Generalized   Min. cluster size:                 156
                      Estimating Equations   Max. cluster size:                 196
Family:                           Binomial   Mean cluster size:               176.0
Dependence structure:         Independence   Num. iterations:                     2
Date:                     Mon, 21 Nov 2016   Scale:                           1.000
Covariance type:                    robust   Time:                         07:07:29
==============================================================================
                 coef    std err          z      P>|z|      [95.0% Conf. Int.]
------------------------------------------------------------------------------
I(y>-3.0)      0.5991      1.081      0.554      0.580        -1.520     2.718
I(y>-2.0)     -0.6484      0.542     -1.195      0.232        -1.711     0.415
I(y>-1.0)     -1.9218      0.231     -8.334      0.000        -2.374    -1.470
I(y>0.0)      -2.8109      0.090    -31.148      0.000        -2.988    -2.634
HF            -0.2999      0.061     -4.923      0.000        -0.419    -0.181
LF            -0.5081      0.200     -2.541      0.011        -0.900    -0.116
==============================================================================
Skew:                          0.4802   Kurtosis:                      -0.1278
Centered skew:                 0.6044   Centered kurtosis:              0.0444
==============================================================================

It is clear to me that IV's HF and LF decrease with increasing iws_w. However, what I cannot figure out yet are the intercepts I(y>XXX). It seems that with increasing iws_w, the intercept becomes lower, which I don't really understand. I guess that the intercepts are differences compared to iws_w = 1, but then it doesn't make sense that I(y>0.0) has a larger deviation than I(y>-3.0).

How should I interpret the I values?

A similar analysis was performed here, but no explanation was given

Answer 1

I'm not familiar with OrdinalGEE in Python, but I'll assume that the link function is logit, as is perhaps most common in ordinal regression.

If that is the case, the intercepts represent log odds. I(y>-3.0) represent the logged base odds of belonging to categories higher than -3.

We can calculate the probability of belonging to higher categories than -3:

exp(0.5991) / (1+exp(0.5991))
[1] 0.6454504

The probability of belonging to categories higher than -3 is thus 0.645, and the probability of belonging to category -3 is 1 - 0.645.

We can calculate the rest of the probabilities:

# probability of belonging to any category that is  > -2
exp(-0.6484) / (1+exp(-0.6484))
[1] 0.3433502

# probability of belonging to any category that is  > -1
exp(-1.9218) / (1+exp(-1.9218))
[1] 0.127661

# probability of belonging to any category that is  > 0, i.e. 1
exp(-2.8109) / (1+exp(-2.8109))
[1] 0.05673799

To calculate the probability of belonging to category 2, we need to subtract the probability of belonging to categories > -2 (0.3433502) from the probability of belonging to categories > -3 (0.6454504) = 0.3021002.

We can calculate the probabilities for belonging to each category in a similar manner:

-3.0    0.3545496
-2.0    0.3021002
-1.0    0.2156892
 0.0    0.07092301
 1.0    0.05673799

The sum of the probabilities is 1.

So in summary, the intercepts are the logged odds of belonging to the specified categories.