Monotonic effects with ordered predictors and ordinal response variables

Modeling
1

I have been reading about the estimation of monotonic effects with the brms library by @paul.buerkner

I found it a very nice and interesting paper. The case study in the vignette
shows a linear model with an ordered predictor (income).

However, the paper also makes it clear that the method can be used with other link functions and GLMs.
I have doubts about how the coefficients would be interpreted in a binary logistic regression or an ordinal model. (i.e., with a binary or ordered response variable but also an ordered predictor). For example, I have fit the following model assessing life satisfaction vs income (4 level ordered variable). How could I interpret the coefficients for the slope and simplex parameters in terms of odds ratios??

library(brms)
fit1 <- brm(ordered(ls) ~ mo(income), data = dat, chains=1, iter=1000,family = cumulative(link = "logit"))

summary(fit1)
 Family: cumulative 
  Links: mu = logit; disc = identity 
Formula: ordered(ls) ~ mo(income) 
   Data: dat (Number of observations: 100) 
  Draws: 1 chains, each with iter = 1000; warmup = 500; thin = 1;
         total post-warmup draws = 500

Population-Level Effects: 
             Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept[1]    -2.87      0.45    -3.91    -2.07 1.00      453      369
Intercept[2]    -1.30      0.35    -2.05    -0.67 1.00      584      398
Intercept[3]    -0.53      0.32    -1.20     0.07 1.00      645      468
Intercept[4]     0.33      0.31    -0.30     0.91 1.00      601      437
Intercept[5]     1.92      0.41     1.11     2.75 1.00      667      392
moincome        -0.25      0.15    -0.58     0.03 1.00      613      344

Simplex Parameters: 
             Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
moincome1[1]     0.31      0.23     0.01     0.81 1.01      521      226
moincome1[2]     0.34      0.22     0.02     0.81 1.02      680      222
moincome1[3]     0.34      0.23     0.03     0.82 1.00      722      203

Family Specific Parameters: 
     Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
disc     1.00      0.00     1.00     1.00   NA       NA       NA
2

since nobody is pitching in, I’ll give you a quick answer. Generally as soon as you go generalized linear models you cannot really interpret in any easy way the coefficients (as they have non-linear effects on the outcome scale). What you need to do is to simulate data from the model with different values of interest for the predictors and then compare the predicted outcomes. the diff in predicted outcomes is the effect you are interested in.

3 Likes
3

thank you very much for your answer
I have a very basic doubt related to this. Sorry if it is a silly question. When you get derived measures from the main estimates, for example in this case I will obtain odds ratios from the predictions of a logistic model or an ordinal model, are the posterior probability distributions of that derived measures valid and allow to obtain credibility intervals if the priors were set for the primary estimates ?

4

Not sure I understand your question, so forgive me for being too pedagogical.
When you set priors on e.g. beta 1 and beta 2, you are implicitly setting a prior also for secondary/derived parameters (beta1 - beta2) and in a sense for the outcome (usually called prior predictive distributions). When you then fit the model, the model learns from the data and adjust those estimates, both for parameters, derived parameters, and expected outcomes. In this sense you will have posterior distributions for parameters and for outcomes (just called posterior predictive distributions). All of these distributions are valid and can be described in terms of credible intervals (given the data and model assumptions). So I think the answer is yes :-)

5

Thanks for your comments. I think everything is clear now.