R: LTM: How can I make an odd-behaving hessian matrix converge when standardization fails?

Question

I try to fit a graded response model with the R package ltm. The issue is that the Hessian matrix does not converge, and I don't understand why. Here is the code I use:

  dset %>% 
  select(Apathy5, Apathy6, Apathy7, Apathy8 ) %>%
  grm(IRT.param = TRUE, Hessian = TRUE, start.val = "random") %>%
  summary()

This leads to the error message: "Hessian matrix at convergence contains infinite or missing values; unstable solution."

I have added start.val = "random" as suggested in the help page of grm, and I tried scaling the variables as suggested here using standardize(), but without success.

The culprit is the variable Apathy8, as the code works fine without that variable. The output of the Hessian matrix also shows that the last five columns break down.

            [,1]       [,2]       [,3]       [,4]        [,5]       [,6]       [,7]       [,8]       [,9]       [,10]      [,11]      [,12]      [,13]
 [1,] 112.885423  168.57318  15.121739  29.733845  -43.944168 -30.110538  -48.87855  -9.490120  -5.469535    8.527498 -34.110730  -50.38311 -15.992948
 [2,] 168.573183  668.52572  47.524373  93.447066 -187.886153 -45.249649 -112.95289 -28.018970 -16.976693   40.735345 -55.110927 -122.69238 -48.527770
 [3,]  15.121739   47.52437  33.518296  15.442993  -22.313874  -3.987572  -11.06158  -3.479726  -2.526640    4.539699  -4.739956  -12.18191  -6.179648
 [4,]  29.733845   93.44707  15.442993 145.616224  -53.533768  -7.432701  -20.96739  -7.859794  -7.633892   10.381387  -8.930554  -23.43291 -14.142549
 [5,] -43.944168 -187.88615 -22.313874 -53.533768   89.867650  11.921767   38.91777  11.170036   8.049626  -16.939146  15.689355   43.61965  19.947149
 [6,] -30.110538  -45.24965  -3.987572  -7.432701   11.921767 141.122622  214.78691  42.922011  24.824655  -43.143223 -32.839612  -45.16608 -14.239393
 [7,] -48.878554 -112.95289 -11.061583 -20.967386   38.917771 214.786905  789.77786 122.001160  70.561387 -203.506690 -53.744979 -107.83995 -39.538108
 [8,]  -9.490120  -28.01897  -3.479726  -7.859794   11.170036  42.922011  122.00116 117.428337  28.904687  -60.202784  -9.767986  -24.52154 -12.093539
 [9,]  -5.469535  -16.97669  -2.526640  -7.633892    8.049626  24.824655   70.56139  28.904687  96.874701  -43.127315  -5.590817  -14.56299  -8.576025
[10,]   8.527498   40.73535   4.539699  10.381387  -16.939146 -43.143223 -203.50669 -60.202784 -43.127315  110.411328  10.244042   38.94826  17.207054
[11,] -34.110730  -55.11093  -4.739956  -8.930554   15.689355 -32.839612  -53.74498  -9.767986  -5.590817   10.244042 106.281603  145.39262  48.717527
[12,] -50.383106 -122.69238 -12.181912 -23.432914   43.619649 -45.166085 -107.83995 -24.521540 -14.562993   38.948261 145.392624  500.58555 129.222316
[13,] -15.992948  -48.52777  -6.179648 -14.142549   19.947149 -14.239393  -39.53811 -12.093539  -8.576025   17.207054  48.717527  129.22232 157.761913
[14,]  -7.434605  -23.13389  -3.528695 -10.950332   11.651912  -6.539343  -18.44108  -6.827938  -6.500651    9.723388  22.443312   59.53046  38.698537
[15,]  16.226144   51.88630   6.025721  13.961658  -19.822990  13.267376   42.38738  12.094652   8.965027  -17.567169 -43.954635 -153.44485 -72.100830
[16,]        NaN        NaN        NaN        NaN 2077.264319        NaN        NaN        NaN        NaN 4384.654980        NaN        NaN        NaN
[17,]        NaN        NaN        NaN        NaN 2113.689915        NaN        NaN        NaN        NaN 4419.579473        NaN        NaN        NaN
[18,]        NaN        NaN        NaN        NaN 2070.957253        NaN        NaN        NaN        NaN 4381.196510        NaN        NaN        NaN
[19,]        NaN        NaN        NaN        NaN 2077.112082        NaN        NaN        NaN        NaN 4385.610768        NaN        NaN        NaN
[20,]        NaN        NaN        NaN        NaN 2042.495092        NaN        NaN        NaN        NaN 4355.470331        NaN        NaN        NaN
           [,14]       [,15]    [,16]    [,17]    [,18]    [,19]     [,20]
 [1,]  -7.434605   16.226144      NaN      NaN      NaN      NaN       NaN
 [2,] -23.133895   51.886301      NaN      NaN      NaN      NaN       NaN
 [3,]  -3.528695    6.025721      NaN      NaN      NaN      NaN       NaN
 [4,] -10.950332   13.961658      NaN      NaN      NaN      NaN       NaN
 [5,]  11.651912  -19.822990 2077.264 2113.690 2070.957 2077.112 2042.4951
 [6,]  -6.539343   13.267376      NaN      NaN      NaN      NaN       NaN
 [7,] -18.441078   42.387378      NaN      NaN      NaN      NaN       NaN
 [8,]  -6.827938   12.094652      NaN      NaN      NaN      NaN       NaN
 [9,]  -6.500651    8.965027      NaN      NaN      NaN      NaN       NaN
[10,]   9.723388  -17.567169 4384.655 4419.579 4381.197 4385.611 4355.4703
[11,]  22.443312  -43.954635      NaN      NaN      NaN      NaN       NaN
[12,]  59.530460 -153.444848      NaN      NaN      NaN      NaN       NaN
[13,]  38.698537  -72.100830      NaN      NaN      NaN      NaN       NaN
[14,] 100.203004  -41.326820      NaN      NaN      NaN      NaN       NaN
[15,] -41.326820   80.464068 1274.630 1312.653 1266.878 1274.224 1235.2047
[16,]        NaN 1274.630183      NaN      NaN      NaN      NaN       NaN
[17,]        NaN 1312.652564      NaN      NaN      NaN      NaN       NaN
[18,]        NaN 1266.878092      NaN      NaN      NaN      NaN       NaN
[19,]        NaN 1274.223660      NaN      NaN      NaN      NaN       NaN
[20,]        NaN 1235.204722      NaN      NaN      NaN      NaN -975.8656
> 

It is interesting that the problem occurs for many samples of the data that I have tested. I have 1000 obsdervations, but the problem persists even with 60 random observations, for instance the following (Link to csv):

N   Apathy5 Apathy6 Apathy7 Apathy8
1   1   1   2   1
2   1   1   1   1
3   4   1   1   2
4   3   5   4   5
5   1   2   2   1
6   4   5   4   1
7   4   5   4   5
8   4   3   3   4
9   1   2   2   1
10  2   4   3   3
11  5   5   5   5
12  1   1   1   1
13  3   3   3   2
14  2   2   2   2
15  1   1   1   1
16  2   2   2   3
17  1   1   1   2
18  2   2   2   2
19  2   1   3   2
20  1   1   1   1
21  1   2   2   1
22  2   3   4   2
23  1   1   1   1
24  1   1   1   1
25  1   2   1   1
26  4   3   4   1
27  3   3   3   3
28  3   3   3   2
29  2   2   2   2
30  2   2   2   1
31  2   2   2   2
32  1   1   1   1
33  1   1   1   2
34  2   2   2   2
35  2   2   3   2
36  1   1   1   1
37  1   2   1   1
38  1   1   1   1
39  1   1   1   1
40  1   1   1   1
41  2   2   1   1
42  1   1   1   1
43  2   2   3   4
44  1   1   1   1
45  1   1   2   2
46  1   4   1   1
47  2   2   1   1
48  1   1   1   1
49  2   3   3   2
50  1   1   1   1
51  1   2   1   1
52  1   2   1   1
53  1   1   1   1
54  3   4   5   3
55  5   5   4   5
56  1   2   2   2
57  1   1   1   1
58  2   2   2   1
59  1   1   1   1
60  1   2   1   1

Does anyone know how to solve this?

Answer 1

You could try to use another optimization algorithms using the control parameter of the grm function, for example:

dset %>% 
select(Apathy5, Apathy6, Apathy7, Apathy8) %>%
grm(IRT.param = TRUE, Hessian = TRUE, start.val = "random",
control = list(verbose = T, method = "Nelder-Mead", iter.qN = 2000)) %>% summary()

verbose just outputs what's happening and you can also change the number of iterations the method is doing using iter.qN. Read more about the different optimization algorithms from the optim() documentation. Each of them has its pros and cons. The Nelder-Mead in the example is a heuristic method and seems to run with the example data you provided. However, it is very inconsistent. If you use different starting values, it gives a bit different outcome and might not find the actual global minimum. For the complete data, you perhaps need to do some testing on what kind of method and amount of iterations works. For example, L-BFGS-B with different random starting points.