The Wright/Falconer/Dempster-Lerner threshold model for K-category ordinal data with K >= 3, fitted on the latent (probit) scale. The latent variable representation is \(y^* = \eta + \varepsilon\), with \(\varepsilon \sim N(0, 1)\) 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.
Value
A family object with class c("ordinal_probit", "family").
The cutpoints are estimated as part of the model fit; recover them
with extract_cutpoints().
Details
Because \(\varepsilon\) has unit variance by construction, the
link-residual variance \(\sigma^2_d = 1\) exactly (no trigamma
correction needed). This is the central selling point of
ordinal_probit() for phylogenetic / threshold-trait analyses:
variance components estimated on the latent scale are directly
comparable to those of a continuous trait, giving the
Dempster-Lerner heritability formula
\(H^2 = \sigma^2_{\text{phy}} / (\sigma^2_{\text{phy}} + 1)\)
without approximation.
Hadfield (2015) eqn 10 shows that the K = 2 case reduces exactly to
binomial(link = "probit"), so use binomial() for binary outcomes
and ordinal_probit() for K >= 3.
Convention: the engine follows Hadfield's notation (\(\tau_1 = 0\)
fixed, K - 2 free cutpoints reported as cutpoint_2, ...,
cutpoint_{K-1}). This differs from brms::cumulative(), which
reports K - 1 cutpoints as Intercept[1..K-1].
References
Dempster, E. R. and Lerner, I. M. (1950). Heritability of threshold characters. Genetics 35:212-236.
Falconer, D. S. and Mackay, T. F. C. (1996). Introduction to Quantitative Genetics, 4th ed. Longman.
Felsenstein, J. (2005). Using the quantitative genetic threshold model for inferences between and within species. Phil. Trans. R. Soc. B 360:1427-1434.
Felsenstein, J. (2012). A comparative method for both discrete and continuous characters using the threshold model. Am. Nat. 179:145-156.
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
Mizuno, A. et al. (2025). Phylogenetic comparative methods for threshold traits. J. Evol. Biol. 38(12):1699-1712.
See also
extract_cutpoints() to recover \(\tau_2, \ldots,
\tau_{K-1}\) after fitting.