Background

Cutpoints in ordered logistic regression are not terrifically substantively informative, but they do contain statistical information.

This handout draws heavily on the Stata documentation for ologit.

Ordered Logistic Regression

Setup

. clear all // clear the workspace

Get The Data

. use http://www.stata-press.com/data/r15/fullauto // use auto data set from Stata documenta
> tion
(Automobile Models)

Codebook

. codebook rep77 foreign // codebook

────────────────────────────────────────────────────────────────────────────────────────────
rep77                                                                     Repair Record 1977
────────────────────────────────────────────────────────────────────────────────────────────

                  type:  numeric (int)
                 label:  repair

                 range:  [1,5]                        units:  1
         unique values:  5                        missing .:  8/74

            tabulation:  Freq.   Numeric  Label
                             3         1  Poor
                            11         2  Fair
                            27         3  Average
                            20         4  Good
                             5         5  Excellent
                             8         .  

────────────────────────────────────────────────────────────────────────────────────────────
foreign                                                                              Foreign
────────────────────────────────────────────────────────────────────────────────────────────

                  type:  numeric (int)
                 label:  foreign

                 range:  [0,1]                        units:  1
         unique values:  2                        missing .:  0/74

            tabulation:  Freq.   Numeric  Label
                            52         0  Domestic
                            22         1  Foreign

Run The Model

. ologit rep77 foreign // estimate ordered logistic regression

Iteration 0:   log likelihood = -89.895098  
Iteration 1:   log likelihood = -85.951765  
Iteration 2:   log likelihood = -85.908227  
Iteration 3:   log likelihood = -85.908161  
Iteration 4:   log likelihood = -85.908161  

Ordered logistic regression                     Number of obs     =         66
                                                LR chi2(1)        =       7.97
                                                Prob > chi2       =     0.0047
Log likelihood = -85.908161                     Pseudo R2         =     0.0444

─────────────┬────────────────────────────────────────────────────────────────
       rep77 │      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
─────────────┼────────────────────────────────────────────────────────────────
     foreign │   1.455878   .5308951     2.74   0.006     .4153425    2.496413
─────────────┼────────────────────────────────────────────────────────────────
       /cut1 │  -2.765562   .5988208                     -3.939229   -1.591895
       /cut2 │  -.9963603   .3217706                     -1.627019   -.3657016
       /cut3 │   .9426153   .3136398                      .3278925    1.557338
       /cut4 │   3.123351   .5423257                      2.060412     4.18629
─────────────┴────────────────────────────────────────────────────────────────

. predict yhat* // predicted probabilities for different levels of dv
(option pr assumed; predicted probabilities)

. tabstat yhat1 yhat2 yhat3 yhat4 yhat5, by(foreign) // table of predicted probabilities

Summary statistics: mean
  by categories of: foreign (Foreign)

 foreign │     yhat1     yhat2     yhat3     yhat4     yhat5
─────────┼──────────────────────────────────────────────────
Domestic │  .0592137  .2104439    .44997  .2382181  .0421543
 Foreign │  .0144652  .0648099   .295154  .4668096  .1587614
─────────┼──────────────────────────────────────────────────
   Total │  .0459101  .1671473  .4039436  .3061777  .0768213
─────────┴──────────────────────────────────────────────────

Calculations

We can use the cutpoints as another way of calculating these probabilities, with the logistic formula \(1/(1 + e^{u_j})\)

For example, the Stata documentation notes that

“For a foreign car, the probability of a poor record is the probability that \(1.46 + u_j <= -2.77\), or equivalently, \(u_j <= -4.23\). Making this calculation requires familiarity with the logistic distribution: the probability is \(1/(1+e^{4.23}) = 0.014\). On the other hand, for domestic cars, the probability of a poor record is the probability \(u_j <= -2.77\), which is 0.059 [\(1/(1 + e^{2.77})\)].”