Background
I am working on a project where my outcome variables consist of sum scores from responses on a series of likert questionnaire items.
One of my outcomes only has 8 possible outcomes, and my cumulative-probit model fits perfectly fine there.
But for another sum scored outcome, which has 18 possible outcomes, the cumulative-probit model has issues with initialization in brms (which has been reported here, here and here already).
The standard solutions of setting init, init_r and strong priors doesn’t help.
Simulations
To investigate further, I’ve created a little simulation below which seems to affirm my suspicion that the MCMC struggles with the cumulative probit model when there are more than about ~12 ordered categories in the outcome variable, even in large sample sizes. I’ve tried settings inits = 0, changing inits_r, and setting up my own initialization parameters.
The cumulative logit model seems to work even when I set n_thresholds to something quite high like 25, though it is a bit slow to fit. Furthermore the cumulative probit model works if i set threshold=“equidistant”.
Question
I am seeking any advice or recommendations for modelling ordinal outcomes with many (10-30) categories. I appreciate that the number of thresholds we need to estimate is K - 1 (K = number of outcome categories), so I would need a large sample size for these models. Perhaps there are other response distributions I should consider that i’ve missed?
For now i’m not considering the approach described in Burkner and Vuorre (2019) where instead of sumscoring responses on likert questions, responses on each question are modelled, and item and participant random intercepts are added to the model. This is just to keep things simple and the sumscores have been benchmarked to clinical and general populations - though its a cool approach i want to try out sometime.
rm(list = ls())
library(brms)
# Paremeters -------------------------------------------------------------------
set.seed(20)
N = 10000 # Sample Size
x1 = rnorm(N) # Predictor Variable
eta = .5*x1 # Linear Predictor Term
n_thresholds = 10 # Number of cut points
threshold = c(-Inf,sort(qnorm(runif(n_thresholds))),Inf)
error = rnorm(N)
y_latent = eta + error
y_observed = cut(y_latent, breaks = threshold, include.lowest = TRUE, ordered_result = TRUE)
if(length(which(is.na(y_observed)))!=0) warning("STOP")
table(y_observed)
dat = data.frame(y_observed, x1)
# Try altering initialisation of thresholds ------------------------------------
init_func <- function(n_intercepts = 3, chain_id = 1) {
uni_intercepts = 1:n_intercepts/(n_intercepts + 1)
# Create a list with initial values for each Intercept based on means
intercept_inits <- setNames(qnorm(uni_intercepts), paste0("Intercept[", 1:length(uni_intercepts), "]"))
# Combine with other initial values if necessary
c(intercept_inits, list(beta = 0, sigma = 1))
}
init_list <- list(
init_func(n_intercepts = n_thresholds, chain_id = 1)
)
# Probit Model -----------------------------------------------------------------
test =
brm(
data = dat,
formula = "y_observed ~ x1",
family = cumulative("probit", threshold = "flexible"),
# family = cumulative("probit", threshold = "equidistant"),
chains = 1,
cores = 2,
iter = 4000,
init = init_list,
# init = 0,
init_r = .5, # Also tried setting this to 0
backend = "rstan",
control = list(adapt_delta = 0.95)
)
# Logit Model ------------------------------------------------------------------
test_logit =
brm(
data = dat,
formula = "y_observed ~ x1",
family = cumulative("logit", threshold = "flexible"),
# family = cumulative("probit", threshold = "equidistant"),
chains = 1,
cores = 2,
iter = 4000,
# init = init_list,
init = 0,
init_r = 1,
backend = "rstan",
control = list(adapt_delta = 0.95)
)
Error message:
> test =
+ brm(
+ data = dat,
+ formula = "y_observed ~ x1",
+ family = cumulative("probit", threshold = "flexible"),
+ # family = cumulative("probit", threshold = "equidistant"),
+ chains = 1,
+ cores = 2,
+ iter = 4000,
+ init = init_list,
+ # init = 0,
+ init_r = .5, # Also tried setting this to 0
+ backend = "rstan",
+ control = list(adapt_delta = 0.95)
+ )
Compiling Stan program...
Start sampling
SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
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Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
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Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
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Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
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Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
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Chain 1: Rejecting initial value:
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Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
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Chain 1: Rejecting initial value:
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Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
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Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
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Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
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Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
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Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
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Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
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Chain 1: Rejecting initial value:
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Chain 1: Rejecting initial value:
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Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
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Chain 1: Rejecting initial value:
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Chain 1: Rejecting initial value:
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Chain 1: Rejecting initial value:
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Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
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Chain 1: Rejecting initial value:
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Chain 1: Rejecting initial value:
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Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
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Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
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Chain 1: Rejecting initial value:
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Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
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Chain 1: Rejecting initial value:
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Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
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Chain 1: Rejecting initial value:
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Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
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Chain 1: Stan can't start sampling from this initial value.
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Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
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Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
Chain 1: Log probability evaluates to log(0), i.e. negative infinity.
Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
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Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
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Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
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Chain 1: Stan can't start sampling from this initial value.
Chain 1: Rejecting initial value:
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Chain 1: Rejecting initial value:
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Chain 1:
Chain 1: Initialization between (-0.5, 0.5) failed after 100 attempts.
Chain 1: Try specifying initial values, reducing ranges of constrained values, or reparameterizing the model.
[1] "Error : Initialization failed."
[1] "error occurred during calling the sampler; sampling not done"