Warning: "Intersecting linear/additive predictors" when fitting non-proportional odds ordinal regression model with VGLM()

Question

I am fitting a partial proportional odds cumulative logit ordinal regression model. Response is an ordinal diagnosis, predictors are two urinary biomarkers. I fit the model using the following command:

fit=vglm(diagnosis ~ creatinine + LYVE1, data=urine.dat, 
    family=cumulative(parallel=F))
summary(fit)

Afterwards, I often get about 20 of the following warnings:

In slot(family, "validparams")(eta, y, extra = extra) :
    It seems that the nonparallelism assumption has resulted in    
    intersecting linear/additive predictors.  
    Try propodds() or fitting a partial nonproportional odds model or  
    choosing some other link function, etc.

Does anyone understand what is meant by "intersecting linear/additive predictors?" From what I have seen, this error is returned very often with non-proportional odds VGLM models. Just trying to understand what is the issue with the model.

Any insight would be helpful.

Answer 1

If you look through the VGAM source code, you'll find the following piece that throws the warning:

    probs <-
      if ( .reverse ) {
        ccump <- cbind(1, eta2theta(eta, .link , earg = .earg ))
        cbind(-tapplymat1(ccump, "diff"), ccump[, ncol(ccump)])
      } else {
        cump <- cbind(eta2theta(eta, .link , earg = .earg ), 1)
        cbind(cump[, 1], tapplymat1(cump, "diff"))
      }
    okay1 <- all(is.finite(probs)) && all(0 < probs & probs < 1)
    if (!okay1)
      warning("It seems that the nonparallelism assumption has ",
              "resulted in intersecting linear/additive ",
              "predictors.  Try propodds() or fitting a partial ",
              "nonproportional odds model or choosing ",
              "some other link function, etc.")

We can boil the guts of this down to a couple of different pieces:

library(VGAM)
#> Loading required package: stats4
#> Loading required package: splines
fit=vglm(cyl ~ wt, data=mtcars, 
         family=cumulative(parallel=F))

eta <- predict(fit, type="link")
cump <- cbind(VGAM:::eta2theta(eta, logitlink ), 1)
probs <- cbind(cump[, 1], tapplymat1(cump, "diff"))

Since cyl has three values, eta is an Nx2 matrix of predicted values on the link scale. cump is the matrix of cumulative probabilities calculated in the usual way for ordered logit. probs is the matrix of category probabilities calculated in the usual way for ordered logit - by subtracting the previous cumulative probability from the current one. Once these are calculated, a flag is generated to identify whether all probabilities are finite and in the theoretical bounds:

okay1 <- all(is.finite(probs)) && all(0 < probs & probs < 1)
#> [1] FALSE

In this case okay1 is FALSE. We can see why below:

all(is.finite(probs))
#> [1] TRUE
all(0 < probs & probs < 1)
#> [1] FALSE

It's because some of the predicted probabilities are negative. We can see which ones below:

ind <- which(probs < 0 | probs > 1, arr.ind=TRUE)[,1]
ind
#> row 
#>  16

probs[ind, ]
#>                    logitlink(P[Y<=2])                    
#>       1.253506e-04      -2.167395e-12       9.998746e-01

Notice here that the predicted probabilities for the second group are negative (though not much different from zero). The takeaway here is that even though you have specified parallel=FALSE, the resulting model is incompatible with the underlying cumulative probability assumption. The warning is encouraging you to use a different model that doesn't calculate probabilities this way, like multinomial logit. For example:

fit2=vglm(cyl ~ wt, data=mtcars, 
         family=multinomial())

which doesn't throw a warning because the probabilities are calculated in a way that won't allow them to be outside [0,1], so long as the exponentiated predicted values on the link scale are finite.

^{Created on 2022-04-26 by the reprex package (v2.0.1)}