The cumulative-logit analogue of ordinal_probit(): the same
Wright/Falconer/Dempster-Lerner threshold model for K-category ordinal
data with K >= 3, fitted on the latent (logistic) scale instead of the
probit scale. The latent variable representation is
\(y^* = \eta + \varepsilon\), with \(\varepsilon \sim
\text{Logistic}(0, 1)\) and the observed category \(y = k\) iff
\(\tau_{k-1} < y^* \le \tau_k\), using cutpoints \(\tau_0 =
-\infty\), \(\tau_1 = 0\) (fixed for identifiability), \(\tau_2,
\ldots, \tau_{K-1}\), \(\tau_K = +\infty\). A K-category trait
therefore estimates K - 2 free cutpoints, reconstructed and reported
via the SAME extract_cutpoints() convention as ordinal_probit().
Value
A family object with class c("ordinal_logit", "family").
The cutpoints are estimated as part of the model fit; recover them
with extract_cutpoints().
Details
Because the standard logistic distribution has variance
\(\pi^2 / 3\), the link-residual variance
\(\sigma^2_d = \pi^2 / 3\) exactly – the logit analogue of
ordinal_probit()'s \(\sigma^2_d = 1\). As with ordinal_probit(),
this fixes the ordinal liability scale; it does not by itself make
estimates directly comparable with an observed continuous response.
The K = 2 case reduces exactly to binomial(link = "logit"), the
logit analogue of the probit family's Hadfield (2015) eqn 10 reduction
(ordinal_probit() with K = 2 reduces to binomial(link = "probit")).
Use binomial() for binary outcomes and ordinal_logit() for K >= 3.
Not to be confused with cumulative_logit().
cumulative_logit() is an unrelated predictor-model family tag
consumed by impute_model(family = cumulative_logit()) for an ORDERED
CATEGORICAL missing covariate inside mi(); it declares no response
family and never appears in the top-level family = argument of
gllvmTMB(). ordinal_logit(), declared here, is a RESPONSE family
for gllvmTMB()'s family = argument, exactly parallel to
ordinal_probit(). Both happen to use a cumulative-logit cell
probability, but one models a missing predictor's marginal distribution
and the other models the observed multivariate response; they are not
interchangeable and share no code path.
Cross-engine note: GLLVModels.jl's ordinal family is a cumulative-logit
model, so ordinal_logit() is the gllvmTMB family that is directly
comparable to it on the link scale (unlike ordinal_probit(), which
differs by a factor of roughly \(\pi / \sqrt{3}\); see the
cross-engine note on ordinal_probit()).
References
Hadfield, J. D. (2015). Increasing the efficiency of MCMC for hierarchical phylogenetic models of categorical traits using reduced mixed models. Methods Ecol. Evol. 6:706-714. doi:10.1111/2041-210X.12354
See also
ordinal_probit() for the probit threshold family;
extract_cutpoints() to recover \(\tau_2, \ldots, \tau_{K-1}\)
after fitting; cumulative_logit() for the UNRELATED
missing-predictor family of the same name shape.