Available Link Functions

clmstan supports the following link functions for cumulative link models:

Standard links (no additional parameters):

• "logit" - Logistic (proportional odds model)

• "probit" - Normal (latent variable interpretation)

• "cloglog" - Complementary log-log (proportional hazards)

• "loglog" - Log-log (Gumbel minimum)

• "cauchit" - Cauchy (heavy tails)

Flexible links (with additional parameters):

• "tlink" - Student-t (df > 0)

  • df = Inf: equals probit

  • df < 3: increasingly heavy tails; df > 30 is nearly normal

• "aranda_ordaz" - Aranda-Ordaz asymmetric (lambda > 0)

  • lambda = 1: equals logit

  • lambda -> 0: approaches cloglog

• "gev" - Generalized extreme value (shape parameter xi)

  • xi = 0: Gumbel (equals loglog)

  • xi < 0: Weibull (short tail)

  • xi > 0: Frechet (heavy tail)

• "sp" - Symmetric power (r > 0, base distribution)

  • r = 1: equals base distribution

  • 0 < r < 1: positively skewed

  • r > 1: negatively skewed

• "log_gamma" - Log-gamma (lambda)

  • lambda = 0: equals probit

  • lambda > 0 or < 0: asymmetric

• "aep" - Asymmetric exponential power (theta1 > 0, theta2 > 0)

  • alpha = 0.5 fixed for identifiability

  • theta1 = theta2: symmetric distribution

  • theta = 2: Gaussian kernel (but NOT equal to probit due to scaling)

  • theta < 2: heavy tails (leptokurtic)

  • theta > 2: light tails (platykurtic)

Link Parameter Specification

Flexible link parameters can be either fixed or estimated (inferred).

Fixed parameters: Specify a numeric value

clm_stan(y ~ x, link = "tlink", link_param = list(df = 8))
clm_stan(y ~ x, link = "gev", link_param = list(xi = 0))  # equals loglog
clm_stan(y ~ x, link = "aep", link_param = list(theta1 = 2, theta2 = 2))  # symmetric

Estimated parameters: Use "estimate" (with default prior)

clm_stan(y ~ x, link = "tlink", link_param = list(df = "estimate"))
clm_stan(y ~ x, link = "gev", link_param = list(xi = "estimate"))

Custom priors: Combine "estimate" with prior argument

clm_stan(y ~ x, link = "gev",
         link_param = list(xi = "estimate"),
         prior = prior(normal(0, 0.3), class = "xi"))

Default Priors for Link Parameters

When using "estimate", the following default priors are used:

Link	Parameter	Default Prior	Notes
tlink	df	gamma(2, 0.1)	Mode around 10, allows heavy tails
aranda_ordaz	lambda	gamma(0.5, 0.5)	Centered near 1 (logit)
gev	xi	normal(0, 2)	Weakly informative, Wang & Dey (2011)
sp	r	gamma(0.5, 0.5)	Centered near 1 (base distribution)
log_gamma	lambda	normal(0, 1)	Centered at 0 (probit)
aep	theta1	gamma(2, 1)	Mode at 1, symmetric at theta1=theta2
aep	theta2	gamma(2, 1)	Mode at 1, symmetric at theta1=theta2

SP Link Details (Li et al., 2019)

The Symmetric Power link uses a symmetric base distribution F_0, specified via the base argument. Supported bases:

• base = "logit": Logistic base (default)

• base = "probit": Normal base

• base = "cauchit": Cauchy base

• base = "tlink": Student-t base (requires df)

Note: Li et al. (2019) define F_0 as a CDF "whose corresponding PDF is symmetric about 0".