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I estimate a proportional odds model in R with the polr model. The regression is basically the categorical educational achievements of parents on the categorical educational achievements of children:

polrmodel <- polr(eisced ~ eiscedmax, data= finalordinal, 
method="logistic")

For both variables, I have 6 categories. When I thus estimate the polr Model in R I get 5 coefficients and 5 intercepts:

Coefficients:
          Value Std. Error t value
eiscedmax.L  2.8847    0.11205  25.744
eiscedmax.Q -0.1661    0.09529  -1.744
eiscedmax.C -0.1184    0.08832  -1.340
eiscedmax^4  0.1598    0.07549   2.118
eiscedmax^5  0.1279    0.04906   2.608

Intercepts: 
     Value    Std. Error t value 
1|2  -5.3166   0.1352   -39.3290
2|3  -2.8792   0.0505   -56.9896
3|4   0.0454   0.0374     1.2136
4|5   1.1844   0.0387    30.6408
5|6   1.8200   0.0412    44.1326

I do now predict the probabilities for this model and the first eiscedmax category (i.e. the lowest education category for parents) with predict:

predict(polrmodel, newdata = data.frame(eiscedmax="1"), type="p")

which yields

         1          2          3          4          5          6 
0.02742981 0.21657597 0.61337588 0.09205802 0.02313072 0.02742959 

Now, I want to verify this probabilities manually. I tried the logit formula as it is done in many applications such as here for instance: Predicting ordered logit in R

However, I cannot get the correct result for this categorical case. I seem to be blocked. Does anyone know how to get the probabilities shown above?

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It's hard to tell exactly what's happening since the example here isn't reproducible, but it looks like the variable eiscedmax is set-up using the ordered contrasts for an ordinal variable. Thus, when you put in "1" as your predictor, it takes that and turns it into another variable in the design matrix, which it is then multiplying by some coefficient. Details on how these contrasts are constructed can be found here. Basically, R constructs orthogonal polynomials (L = linear, Q=quadratic, etc.) from your ordered factor.

If you look at the design matrix going into polr, you may find out what you should multiply by the coefficient and then you may get the correct probabilities.

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