# Interpretation of population-level effects in brms -- probit models

I’m new to brms and especially interested in the Probit models for the analysis of Likert items and Likert scales data. I went through previous threads but could not find a clear answer to the following question.

I would like to interpret population-level effects: point estimates of b coefficients and their credible intervals. I understand that the values reported by the analysis are standard-deviation units, but I’m straggling to figure out: (i) what these standard deviations represent and (ii) if there is an easy way to transform these estimates back to the original scale.

For example, consider the analysis of people opinions about funding stem-cell research (in the tutorial of Burkner and Vuorre). For the first analysis presented in the tutorial (rating ~ 1 + belief), the estimates of the two b coefficients are -0.24, 95% CI [-0.43, -0.06] and 0.31, 95% CI [0.13, 0.50], respectively. However, I am not sure which standard deviation these numbers refer to. How could we transform these values into the units of the original scale, where ratings range from 1 to 4?

I would like to communicate such results to an audience of non-experts and provide some clear intuition. I know that the conditional_effects function allows for showing credible intervals in the original scale for each individual group. How can we do the same for contrasts, e.g., difference in ratings between liberals and moderates? Is there something wrong in this logic that I miss?

Hi @fanis,

The effects you mention describe effects on the latent continuous variable assumed to underlie the ordered responses. With a probit link, the latent variable is a standard normal variable, and therefore the effects are on that standard deviation. You can’t go back from that to the 1-4 scale, which is not continuous. It should be sufficient to explain to your audience that the coefficients are on the latent variable, because it directly describes the predicted response probabilities.

An alternative is to describe the implied changes in probabilities at each response category as a function of the predictors. That is, you can obtain the predicted probabilities of each response category at different levels of your predictors, and then take their differences. For example, for a simple model with a single binary predictor and 4 levels of the response variable, this would give you four contrasts. If you have example data and code I might be able to put together an example of that if it sounds reasonable. A downside of it is that then you have four “effects” (and it will lead directly to thinking about category specific effects as described e.g. in the manuscript) to describe to your audience.

Does that help?

Thank you @matti. You are right. I should rather try to explain the role of the latent variable.
(visualizing its distribution over the estimated thresholds might give some good intuition).