Understanding the disc parameter in ordered logit models

Interfaces brms
9

As you say, it is no longer the case that β1 is in a standardized-mean difference metric because we no longer have a common standard deviation for the two levels of group .
Correct?

I guess you can still call it a standardized mean difference metric. (What does that mean anyway?) But more precisely, it is the difference in means of the latent variable in terms of the SD of the reference group’s latent variable (1). When in doubt; conditional_effects(model), and take a look at how the response proportions differ across groups–that usually makes your effects really clear to interpret.

Also, given we have two scales for σ , how does one interpret the τ parameters? Are they still based on the Φ(z) for both levels of group or are they interpreted conditional on the scale for each level of group somehow?

The thresholds are also on the scale of the reference group’s latent variable. There is now an option in brms to model different thresholds for groups, but I am not familiar with how it works, and if and how it would impact this issue.

In general, you can take a look at

make_stancode(
  data = data.frame(group = c(rep(0, 4), rep(1, 4)), Y = c(1,2,3,4,3,4,4,4)),
  family = cumulative(probit),
  formula = bf(Y ~ 1 + group) +
    lf(disc ~ 0 + group, cmc = FALSE),
  prior = c(prior(normal(0, 4), class = Intercept),
            prior(normal(0, 2), class = b),
            prior(normal(0, 1), class = b, dpar = disc))
)

together with

make_standata(
    data = data.frame(group = c(rep(0, 4), rep(1, 4)), Y = c(1,2,3,4,3,4,4,4)),
    family = cumulative(probit),
    formula = bf(Y ~ 1 + group) +
      lf(disc ~ 0 + group, cmc = FALSE),
    prior = c(prior(normal(0, 4), class = Intercept),
              prior(normal(0, 2), class = b),
              prior(normal(0, 1), class = b, dpar = disc))
)

to verify that your model equations are correct (and generally to see how things are working.) But of course this assumes you are familiar with Stan code.

I hope that helps… Ordinal models can be very tricky to interpret if you’re not used to them (I am not sure I still am).

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