Hello,

I have been using Probabilistic Index models introduced by Thas et al (2012) as an extension of the Mann-Whitney that accommodates covariates in several analyses. More recently, I have learned about the Cumulative Probability models introduced Liu et al (2016) that is also an extension of the Mann-Whiney test that accommodates covariates. They became interesting to me because CPM can also accommodate random effects and I also learned that there is a relationship between Probabilistic Index (PI) and the common Odds Ratio (cOR) presented in De Neve et al (2019) as follow:

I tried to validate this relationship in R:

```
library(ordinal)
library(pim)
find_pi <- function(cOR){
aux <- function(cOR){
if (cOR != 1)
out <- cOR*(cOR - log(cOR) - 1)/((cOR - 1)^2)
if (cOR == 1)
out <- 0.5
return(out)
}
aux <- Vectorize(aux)
out <- aux(cOR)
return(out)
}
set.seed(1234)
z_clm <- rep(NA, 100)
z_pim <- rep(NA, 100)
pi_clm <- rep(NA, 100)
pi_pim <- rep(NA, 100)
for (i in 1:100){
y0 <- rnorm(100, 4, 1)
y1 <- rnorm(100, 5, 1)
y <- c(y0, y1)
x <- c(rep(0, length(y0)), rep(1, length(y1)))
dt <- data.frame(y = as.factor(y), x = as.factor(x))
fit <- clm(y ~ x, data = dt)
sm_clm <- summary(fit)
cor_clm <- exp(sm_clm$beta)
pi_clm[i] <- find_pi(cor_clm)
z_clm[i] <- sm_clm$coefficients[nrow(sm_clm$coefficients), 3]
dt <- data.frame(y, x = as.factor(x))
fit <- pim(y ~ x, data = dt)
sm_pim <- summary(fit)
pi_pim[i] <- plogis(sm_pim@coef)
z_pim[i] <- sm_pim@zval
}
plot(pi_clm, pi_pim)
abline (coef = c (0, 1))
plot(z_clm, z_pim)
abline (coef = c (0, 1))
```

There are small differences when estimating the probabilistic index. I assumed that the observed differences are numerical. Does that make sense?

However, the differences in Z-score make me a bit more concerned when p-values are borderline.

Then, I tried to re-run a few previous analyses and I found some discrepancies in the conclusions. For example,

```
library(ordinal)
library(pim)
data('FEVData')
fit_pim <- pim(FEV~ Smoke + Sex + Age + Height, data=FEVData)
sm_pim <- summary(fit_pim)
pi_pim <- plogis(sm_pim@coef)
fit_clm <- clm(as.factor(FEV)~ Smoke + Sex + Age + Height, data=FEVData)
sm_clm <- summary(fit_clm)
cor_clm <- exp(sm_clm$beta)
pi_clm <- find_pi(cor_clm)
pi_pim
# Smoke Sex Age Height
# 0.3928110 0.5361332 0.5512311 0.5915537
sm_pim
# Estimate Std. Error z value Pr(>|z|)
# Smoke -0.43551 0.23093 -1.886 0.0593 .
# Sex 0.14479 0.11098 1.305 0.1920
# Age 0.20565 0.02940 6.994 2.68e-12 ***
# Height 0.37039 0.02312 16.018 < 2e-16 ***
pi_clm
# Smoke Sex Age Height
# 0.4193120 0.5513324 0.5432701 0.5895691
sm_clm
# Estimate Std. Error z value Pr(>|z|)
# Smoke -0.48797 0.26103 -1.869 0.0616 .
# Sex 0.30897 0.14530 2.126 0.0335 *
# Age 0.26021 0.04000 6.505 7.75e-11 ***
# Height 0.54269 0.02735 19.843 < 2e-16 ***
```

As you can see, there are differences on the conclusion for the variable Sex even though the estimates of PI are very close. What am I missing?

References:

Thas O, Neve JD, Clement L, Ottoy JP. Probabilistic index models. Journal of the Royal Statistical Society Series B: Statistical Methodology. 2012 Sep;74(4):623-71.

Liu Q, Shepherd BE, Li C, Harrell Jr FE. Modeling continuous response variables using ordinal regression. Statistics in medicine. 2017 Nov 30;36(27):4316-35.

De Neve J, Thas O, Gerds TA. Semiparametric linear transformation models: Effect measures, estimators, and applications. Statistics in medicine. 2019 Apr 15;38(8):1484-501.