# R ordered logit regression polr

I have result from R: polars

polr(formula = cat ~ log_pop21 + birth + ngrw, data = new_df,
na.action = na.omit, Hess = T, method = "logistic")

Coefficients:
log_pop21      birth       ngrw
3.1346164  1.0315305 -0.2486664

Intercepts:
0|1      1|2      2|3      3|4      4|5
10.04250 12.66218 15.00460 18.26514 20.93464


How to interpret intercepts and coefficients specifically for my data? I've read so much documentation but can't understand.

Here's how I understand it, which is a little hand-waivy but, I think it's broadly correct. I'd be happy to be corrected, or somebody to explain it a little more technically but I hope this helps:

##### Intercepts

Ordered logistic regression works by modelling an unobserved continuous variable, that is put into its observed categories by the various cutpoints (which are called Intercepts here). Any value below 10.04 on the unobserved variable will be assigned 0, between 10.04 and 12.66 will be assigned 1, and so on.

##### Coefficients

It uses a logistic distribution as the underlying distribution, the nice thing about that its units are log odds of increasing from category C or less to cateogry C + 1 or more. It is an assumption of the model that the coefficients are the same to move from 0|1-5 as it is to move from 0-1|2-5, (The 'proportional odds assumption'). Otherwise, you would need one coefficient for each variable and for each cutpoint. This is possible, but is a different model and is very hard to interpret.

So this leaves you with this interpretation (taking log_pop1 as an example):

For a 1 unit increase in log_pop1 there is a 3.13 increase in the log odds of the outcome being categories C or less compared to being category C or more. (whatever C is, this is the same, as I explained above)

But because it's difficult to interpret log odds, it is often best to take the exponent of the coefficients.

exp(3.134) = 22.97, so for a one unit increase in log_pop21, the odds of being category 0 compared to categories 1-5, or from being categories 0 or 1 compared to being categoies 2, 3, 4 or 5, increases by a factor of 22.97 (because we have exponentiated the log odds, we now multiply the odds by exp(coefficient) for every one unit increase in it).