How to deal with clmm warning: "hessian is numerically singular"?

Question

I am using R's ordinal package to run a mixed regression model with an ordinal dependent variable. The data I am working with looks like this:

      x y z
1  S153 A 2
2   S11 A 2
3   S40 A 2
4  S112 A 1
5  S150 A 2
6   S40 A 2
7   S40 A 2
8  S150 A 2
9   S40 A 2
10  S39 A 2
11 S150 A 2
12  S53 A 2
13 S150 A 2
14 S150 A 2
15  S23 A 2
16  S36 A 1
17  S79 A 2
18 S150 A 2
19  S70 A 2
20 S133 A 1
21  S40 A 2
22 S150 A 2
23  S48 A 2
24  S53 A 2
25 S150 A 2
26  S12 A 2
27 S150 A 1
28  S80 B 2
29 S147 B 3
30  S92 C 2
31   S2 D 2
32  S37 D 2
33  S14 D 2
34  S56 D 2
35  S14 D 2

structure(list(x = structure(c(8L, 1L, 14L, 2L, 7L, 14L, 14L,  7L, 14L, 
13L, 7L, 16L, 7L, 7L, 10L, 11L, 19L, 7L, 18L, 4L, 14L,  7L, 15L, 16L, 7L, 3L, 
7L, 20L, 6L, 21L, 9L, 12L, 5L, 17L, 5L), .Label = c("S11",  "S112", "S12", 
"S133", "S14", "S147", "S150", "S153", "S2", "S23",  "S36", "S37", "S39", 
"S40", "S48", "S53", "S56", "S70", "S79", "S80", "S92"), class = "factor"), y = 
structure(c(1L, 1L, 1L,  1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L,  1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 3L, 4L, 4L, 4L, 4L, 4L ), 
.Label = c("A", "B", "C", "D"), class = "factor"), z = c(2L,  2L, 2L, 1L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 2L,  2L, 2L, 1L, 2L, 2L, 2L, 2L, 
2L, 2L, 1L, 2L, 3L, 2L, 2L, 2L, 2L,  2L, 2L)), .Names = c("x", "y", "z"), 
class = "data.frame", row.names = c(NA, -35L))

Variable 'z' is my response variable (ordinal factor). Variable 'y' is my predictor and I want to include 'x' as random effects. To do this, I am using clmm as follows:

m1 <- clmm(factor(z, ordered=T) ~ y + (1|x) , data=df)

However, this results in the following warning message:

Warning message:
(1) Hessian is numerically singular: parameters are not uniquely determined 
In addition: Absolute convergence criterion was met, but relative criterion was not met

I have tried running this with clm excluding the random effects and I keep getting the same warning.

Here is the table of the predictor and response variabes:

table(df$z,df$y)

     A  B  C  D
  1  4  0  0  0
  2 23  1  1  5
  3  0  1  0  0

I am not sure if this is a problem of complete separation or not. Why am I getting this warning and how can I deal with it?

Answer 1

Is that your complete data set? If so, I think I can see the problem. Factor z almost always takes value 2, and value 3 is achieved only once. Your response isn't changing very much, so it's not easy to estimate the parameters. The random effect also seems to load heavily on factor A. Your data set is not really informative about factor A and the response y. The optimizer knows this.

The Hessian is singular because the optimizer can't distinguish some of your parameters. This makes sense, given that your data set does not really distinguish them either. So you need a more balanced data set. It would also be nice to have a larger sample size.