# Ordinal model cumulative logit parameter estimates

I noticed a difference in the parameter estimates for a centred versus non-centred version of an ordered logistic model. This became apparent when I implemented a model per the stan users guide (1.8 Ordered logistic and probit regression | Stan User’s Guide and 12.6 Ordered Logistic Distribution | Stan Functions Reference) and then compared it to an equivalent brms version.

Start with polr and a simple dataset as a reference point.

library(data.table)
library(MASS)
library(foreign)

dat <- data.table(
)

# mean centre per brms approach to prop odds
dat[, pared2 := pared - mean(pared)]
dat[, gpa2 := gpa - mean(gpa)]
dat[, apply2 := as.numeric(apply)]

m1 <- polr(apply ~ pared + gpa, data = dat, Hess=TRUE)
m2 <- polr(apply ~ pared2 + gpa2, data = dat, Hess=TRUE)


The two models have the same parameter estimates but different intercepts

# same ors, different intercepts
summary(m1)
Coefficients:
Value Std. Error t value
pared 1.0457     0.2656   3.937
gpa   0.6042     0.2539   2.379

Intercepts:
Value  Std. Error t value
unlikely|somewhat likely    2.1763 0.7671     2.8370
somewhat likely|very likely 4.2716 0.7922     5.3924

summary(m2)
Coefficients:
Value Std. Error t value
pared2 1.0457     0.2656   3.936
gpa2   0.6043     0.2539   2.379

Intercepts:
Value   Std. Error t value
unlikely|somewhat likely     0.1995  0.1030     1.9372
somewhat likely|very likely  2.2948  0.1719    13.3494



but both models produce the same probabilities when you run predict.

cbind(
predict(
m1,
newdata = data.table(pared = 0, gpa = mean(dat$gpa)), type = "p"), predict( m2, newdata = data.table(pared2 = min(dat$pared2), gpa2 = 0),
type = "p")
)
[,1]       [,2]
unlikely        0.59005138 0.59005227
somewhat likely 0.33120156 0.33120059
very likely     0.07874706 0.07874714


Initially, I implemented the model using stan. I have aligned the variable names and formatting to be as similar as possible to the brms model.

data {
int<lower=1> N;
int Y[N];
int<lower=2> nthres;
int<lower=1> K;
matrix[N, K] X;
int prior_only;
}
parameters {
vector[K] b;
ordered[nthres] Intercept;
}
model {
if (!prior_only) {
vector[N] mu = X * b;
for (n in 1:N) {
target += ordered_logistic_lpmf(Y[n] | mu[n], Intercept);
}
}
target += normal_lpdf(b | 0, 10);
target += student_t_lpdf(Intercept | 3, 0, 2.5);
}
generated quantities {}


next, using brms::make_stancode(apply2 ~ pared + gpa, data = dat, family=brms::cumulative("logit")) I produced the following (I removed the custom functions as they are unnecessary here).

data {
int<lower=1> N;  // total number of observations
int Y[N];  // response variable
int<lower=2> nthres;  // number of thresholds
int<lower=1> K;  // number of population-level effects
matrix[N, K] X;  // population-level design matrix
int prior_only;  // should the likelihood be ignored?
}
transformed data {
int Kc = K;
matrix[N, Kc] Xc;  // centered version of X
vector[Kc] means_X;  // column means of X before centering
for (i in 1:K) {
means_X[i] = mean(X[, i]);
Xc[, i] = X[, i] - means_X[i];
}
}
parameters {
vector[Kc] b;  // population-level effects
ordered[nthres] Intercept;  // temporary thresholds for centered predictors
}
transformed parameters {
real<lower=0> disc = 1;  // discrimination parameters
}
model {
// likelihood including constants
if (!prior_only) {
// initialize linear predictor term
vector[N] mu = Xc * b;
for (n in 1:N) {
target += ordered_logistic_lpmf(Y[n] | mu[n], Intercept);
}
}
// priors including constants
target += student_t_lpdf(Intercept | 3, 0, 2.5);
}
generated quantities {
// compute actual thresholds
vector[nthres] b_Intercept = Intercept + dot_product(means_X, b);
}


The main difference between the two is that the latter centres the covariates.

ld <- brms::make_standata(apply2 ~ pared + gpa,
data = dat,
family=brms::cumulative("logit"))
f1 <- rstan::sampling(object  = mod1,
data    = ld,
chains  = 1,
iter    = 5000,
warmup  = 1000,
refresh = 1000)
f2 <- rstan::sampling(object  = mod2,
data    = ld,
chains  = 1,
iter    = 5000,
warmup  = 1000,
refresh = 1000)


After sampling from the two models I get:

standard rstan version:

                mean se_mean   sd    2.5%     25%     50%     75%   97.5% n_eff Rhat
b[1]            1.06    0.01 0.28    0.53    0.87    1.06    1.24    1.62  2724    1
b[2]            0.47    0.01 0.24   -0.01    0.30    0.46    0.63    0.95  1315    1
Intercept[1]    1.76    0.02 0.73    0.36    1.26    1.75    2.24    3.20  1353    1
Intercept[2]    3.85    0.02 0.75    2.40    3.34    3.85    4.35    5.32  1339    1


brms (centred covariate):

                  mean se_mean   sd    2.5%     25%     50%     75%   97.5% n_eff Rhat
b[1]              1.05    0.00 0.27    0.53    0.87    1.05    1.23    1.58  3474    1
b[2]              0.60    0.00 0.26    0.11    0.43    0.60    0.77    1.10  3914    1
b_Intercept[1]    2.17    0.01 0.77    0.66    1.66    2.17    2.69    3.65  4010    1
b_Intercept[2]    4.27    0.01 0.80    2.73    3.74    4.27    4.81    5.84  4031    1


the brms (centred covariate) version appears to align with the m1 output from polr, whereas the other version does not. However, as with polr when I predict the probability of being in each group for a set of covariates I get the same values using both models and the results are similar to the frequentist version.

post1b <- as.matrix(f1, pars = "b")
post2b <- as.matrix(f2, pars = "b")

post1c <- as.matrix(f1, pars = "Intercept")
post2c <- as.matrix(f2, pars = "b_Intercept")

q1 <- exp(
post1c - mean(dat$gpa)*post1b[, 2])/ (1 + exp(post1c - mean(dat$gpa)*post1b[, 2]))

q2 <- exp(
post2c - mean(dat$gpa)*post2b[, 2])/ (1 + exp(post2c - mean(dat$gpa)*post2b[, 2]))

p1 <- cbind(q1[,1],q1[,2]-q1[,1])
p1 <- cbind(p1,1-rowSums(p1))

p2 <- cbind(q2[,1],q2[,2]-q2[,1])
p2 <- cbind(p2,1-rowSums(p2))

cbind(
colMeans(p1),
colMeans(p2)
)
[,1]       [,2]
[1,] 0.58882504 0.59000145
[2,] 0.33111789 0.33097647
[3,] 0.08005707 0.07902209


It seems like I would be unable to compare the inference on the basis of the parameter estimates across these two parmaterisations, even though the predictions from each model are the same. For example, in the first model, the parameter associated with the gpa is smaller than what is reported in the second model. Is this true or have I simply made a mistake in the implementation/interpretation?

I think you are not actually getting the same predicted probabilities (although they are close at the mean gpa), and the difference is driven by the prior on the Intercept.

e.g. If you calculate the predicted probabilities at the minimum gpa:

q1 <- exp(
post1c - min(dat$gpa)*post1b[, 2])/ (1 + exp(post1c - min(dat$gpa)*post1b[, 2]))

q2 <- exp(
post2c - min(dat$gpa)*post2b[, 2])/ (1 + exp(post2c - min(dat$gpa)*post2b[, 2]))

p1 <- cbind(q1[,1],q1[,2]-q1[,1])
p1 <- cbind(p1,1-rowSums(p1))

p2 <- cbind(q2[,1],q2[,2]-q2[,1])
p2 <- cbind(p2,1-rowSums(p2))

cbind(
colMeans(p1),
colMeans(p2)
)
[,1]       [,2]
[1,] 0.70270343 0.73319564
[2,] 0.24658863 0.22312593
[3,] 0.05070794 0.04367843


You are right that a shift in x (e.g. x_{ni}^* = x_{ni} + s_i) shouldn’t affect the slope parameters, but — as you can see in the brms intercept correction — the intercepts are shifted by c_k^* = c_k + \sum_i s_i b_i.

From the polr output, the Intercept MLEs when using the uncentred covariates are 2.17 and 4.27, which are towards the tails of your t_{3,0,2.5} prior. So I think that when you run the uncentred Stan code (f1), the likelihood doesn’t completely overwhelm the prior and you end up with the posterior means of 1.76 and 3.85; shrunken towards zero. This has a flow-on effect for the slope parameter estimation. You can see even more dramatic differences if you shift your x values by a larger amount, such that the Intercept MLEs are even further from zero.

If you remove the prior from Intercept (and also the one from b, which doesn’t appear in the brms-generated code), i.e. so that Stan uses a flat prior, you should get something much closer to the polr and brms results:

                mean se_mean   sd    2.5%     25%     50%     75%   97.5% n_eff Rhat
b[1]            1.05    0.01 0.27    0.53    0.88    1.05    1.24    1.57  2145    1
b[2]            0.60    0.01 0.25    0.11    0.43    0.59    0.76    1.10  1967    1
Intercept[1]    2.15    0.02 0.76    0.68    1.64    2.14    2.66    3.67  1944    1
Intercept[2]    4.26    0.02 0.78    2.75    3.74    4.25    4.80    5.86  1917    1


Hope this helps