pmatrix.msm error ("Infinite matrix entry") and sojourn.msm gives identical results for both sexes

[tomato-git-hub]

Hi Prof. Jackson,
I hope all is well.
I am using msm to fit a four-state disease model (states 1,2,3 are transient, state 4 absorbing). The covariate sex is essential to compare male vs. female transition patterns in a specific region.

The model runs fine on the full dataset, but after subsetting to the region of interest (sample size decreases), I encounter:

1. pmatrix.msm fails with: Error in balance(baP$z, "s") : R_dgebal(*, type="S"): Infinite matrix entry.
2. sojourn.msm returns identical estimates for covariates = list(sex="male") and sex="female", indicating the covariate effect is not captured.

sojourn.msm(data.msm_region, covariates=list(sex="male"))
Warning message:
In msm.parse.covariates(x, covariates, x$qmodel) :
Ignoring covariates - no covariates in this part of the model

Diagnostic evidence:

1. Fitted Q-matrix for the regional data (extremely wide CIs for some transitions):
   State 1 -> State 3: 9.745e-07 (6.921e-17, 1.372e+164)
   State 3 -> State 1: 1.164e-09 (0.000e+00, Inf)
   These suggest highly unstable parameter estimates.
2. Sex-specific transition frequencies (regional subset):
   Males: 1→3 (162), 3→1 (125)
   Females: 1→3 (369), 3→1 (216)
   Although non-zero, the per-sex sample size for these transitions may be insufficient to stably estimate covariate effects.

Attempted fix:
I switched to L-BFGS-B with maxit = 100000,but the problem persists.

control = list(method = "L-BFGS-B", control = list(maxit = 100000, trace = 1))

My questions:

1. What are the likely reasons for this behavior? Is it due to sparse data per sex for certain transitions, leading to flat likelihoods or boundary estimates for covariate parameters?
2. What should be the next steps?

Thank you for your help!

[chjackson]

The large CIs around the intensities tell me that your model fit hasn't converged to the maximum likelihood, and the likelihood is flat, as you say, so there's no point in trying to get results out of it.

A common reason for this is choosing a transition structure based on the transitions that are allowed in discrete time (over an interval). This is inappropriate for msm, which works with allowed transitions in continuous time (real time). For example, if you have a progressive disease like mild-moderate-severe, you shouldn't allow transitions from mild to severe. The transition frequencies you show above are for the transitions over an interval, so are a poor guide to how much data you have to inform the intensities in continuous time.

What is your transition structure (qmatrix) and what do your state observations represent? Do you know when the transitions happened in continuous time, or do you just know the state at particular times?

[tomato-git-hub]

Dear Professor Jackson,

I hope this message finds you well.
Thank you for your invaluable advice! My states are defined as: 1=normal (<10), 2=mild risk (10-20), 3=severe risk (20-30), 4=disease (>30), with state 4 being absorbing. Since my study focuses on disease progression without drug intervention, I have removed transitions from the disease state back to earlier states (i.e., 4→3, 4→2, 4→1). Transitions among states 1, 2, and 3 represent the natural course, while transitions from state 4 to other states would require drug intervention.
Observations are recorded at discrete times; exact transition times are unknown.

# Original matrix (which caused previous errors)
Q_1 <- rbind(
  c(0, 1, 1, 1),
  c(1, 0, 1, 1),
  c(1, 1, 0, 1),
  c(0, 0, 0, 0)
)

Following your guidance, I modified the transition matrix (disallowing direct transitions between normal and severe risk), and the model now runs without errors.

I have a new question regarding the transition structure and would greatly appreciate your insight.
I am considering two candidate structures, with Q matrices defined in R as follows:

# Candidate structure 1: allows 1, 2, and 3 to enter the absorbing state, i.e., 1→4, 2→4, 3→4
Q_2 <- rbind(
  c(0, 1, 0, 1),
  c(1, 0, 1, 1),
  c(0, 1, 0, 1),
  c(0, 0, 0, 0)
)

# Candidate structure 2: allows only state 3 to enter the absorbing state (3→4)
Q_3 <- rbind(
  c(0, 1, 0, 0),
  c(1, 0, 1, 0),
  c(0, 1, 0, 1),
  c(0, 0, 0, 0)
)

I have seen studies in the diabetes literature that define three states (normal, borderline elevated, diabetes) and allow a direct transition from "normal" to "diabetes" (1→3). In my disease context, should I allow direct transitions from state 1 and state 2 to the absorbing state 4? From both clinical and statistical perspectives, how should I decide whether to retain or remove these direct pathways?

I look forward to your guidance. Thank you again for your time and help!

[chjackson]

I think the issue is confusion between transitions over an interval and transitions in continuous time (I have edited my previous comment a bit to clarify). I don't know much about diabetes development, but I suspect that transition structure 2 (Q_3) is the correct one for your situation. That is, transitions in continuous time should only be allowed between clinically-adjacent states.

Note that under this structure, a person in state "normal" at the current time is still allowed to be in the diabetes state at any interval of time in the future (one day, one week, even one second!). We just assume they went through the mild and severe states quickly in the middle of this interval.

See these course notes for another example.

[tomato-git-hub]

Thank you for your detailed explanation and the course notes. I have carefully read your documentation on the psoriatic arthritis (PsA) example, which models four stateswith a unidirectional adjacent-only transition structure (only 1→2→3→4 allowed).

I have a specific question regarding the interpretation of the results:

Background:
In your example, since the model specification does not allow direct 1→3 transitions (i.e., transition intensity q₁₃=0), the 1→3 transition probability calculated by pmatrix.msm() is nonetheless non-zero.

My questions:
1.Terminology: When reporting this result, should I interpret it as "cumulative transition probability" or "indirect transition probability"—representing the probability of reaching state 3 from state 1 via intermediate state 2, rather than by direct jump?
2.Visualization: If I want to draw a state transition probability diagram, should I keep the 1→3 arrow but explicitly label it as multi-step cumulative probability?

I look forward to your guidance. Thank you!

[chjackson]

I don't think either of those terms are very clear by themselves. I would just call it the transition probability. But then, make sure your readers can see clearly how it is defined - the probability that someone is in state s at time t+u given that they are in state r at time t. This depends on the time interval u.

Visualisation is entirely your choice, and depends on what you want to communicate.

[tomato-git-hub]

Thank you for your guidance and assistance, it has been very helpful for my research.