C-Index for Proportional Odds models

[Alice Toll]

I'm having trouble understanding the C-Index for proportional odds models.  I've made an example where I increase the number of categories Y can take on.  As the number of response categories increases, the C-index decreases.  Since we know categorization reduces power I would have expected the C-index to increase.  Can anyone explain why we see this behavior?

x <- rnorm(1000, 100, 10)

y <- x + rnorm(1000, 0, 10)

plot(x, y)

for(i in 3:10){

  y_temp <- cut2(y, g=i)

  assign(paste0('y', i), y_temp)

  assign(paste0('f', i), lrm(y_temp ~ x))

}

getStats <- function(stat){

  c(f3$stats[stat], f4$stats[stat], f5$stats[stat], f6$stats[stat],

    f7$stats[stat], f8$stats[stat], f9$stats[stat], f10$stats[stat])

}

nquant <- 3:10

plot(nquant, getStats('C'))

[Frank Harrell]

Excellent question.  Increasing categories generally means more power/information but it doesn't necessarily increase every index of association.  Different indexes penalized differently for ties.  But more at the heart of your question is what it means to discriminate two values of Y.  Discriminating diseased vs. not diseased is easier than discriminating level 3 disease from level 4 disease on a 10-point scale.  c goes down when Y is more continuous because discriminating from among more levels of Y is more difficult.