Understanding p values in ordinal logistic regression in R

Question

I'm trying to wrap my head around ordinal logistic regression outputs in R. I've seen some similar posts before and read many tutorials, but I feel like some things are missing. What I'm looking for is a complete non-math heavy breakdown of the output so any explanations with formulae please explain what the math means in laymans terms.

Here is the code I use to get the output

#install if not already installed and load package 
if (!require('ordinal')) install.packages('ordinal'); require('ordinal')
if (!require('carData')) install.packages('carData'); require('carData')

#load in the WVS dataset from carData
data(WVS) 

#run the analysis 
propodds_coves_nwu=clm(poverty ~ age, data=WVS)
summary(propodds_coves_nwu)

The output is like this

> propodds_coves_nwu=clm(poverty ~ age, data=WVS)
> summary(propodds_coves_nwu)
formula: poverty ~ age
data:    WVS

 link  threshold nobs logLik   AIC      niter max.grad cond.H 
 logit flexible  5381 -5332.58 10671.16 5(0)  2.52e-10 4.0e+04

Coefficients:
    Estimate Std. Error z value Pr(>|z|)    
age 0.013172   0.001522   8.656   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Threshold coefficients:
                       Estimate Std. Error z value
Too Little|About Right  0.60346    0.07377    8.18
About Right|Too Much    2.33749    0.08094   28.88

My main question is how is the p value even calculated here and what does a significant p value mean in terms of hypotheses here?

This data is taken from the World Values Survey. The outcome variable poverty is asking "Do you think that what the government is doing for people in poverty in this country is about the right amount, too much, or too little?"

The estimate is positive but does that mean as age goes up every .013 units people are more likely to think the government is doing too much to reduce poverty? In addition what are these "threshold coefficients" referring to?

Threshold coefficients:
                       Estimate Std. Error z value
Too Little|About Right  0.60346    0.07377    8.18
About Right|Too Much    2.33749    0.08094   28.88

Answer 1

One way to think of the proportional odds form of ordinal logistic regression is that you are studying a continuous latent variable which we might call Attitude to government policy on poverty. What you have observed though is not the latent variable itself but a coarsened form of it with three categories. The threshold coefficients account for where the cut-offs between the categories are and the coefficient for your covariate (age) tells you (when exponentiated) the odds ratio of being in the higher category versus the lower at every cut-off point. The model assumes this is the same for each cut-off hence the name proportional odds.

Answer 2

The threshold coefficients are similar to what the coefficient for the intercept would be in binary logistic regression but, since you have three levels of the dependent variable, there is a coefficient for each threshold.

The 0.013 is the coefficient for age. The odds ratio for age would be $e^{0.013} = 1.013$ per year (don't rely on just adding 1 to the coefficient, that is only correct when the coefficient is close to 0). So, the odds increase by about 1.3\% for each year of age.