Multi-state robust design and unequal time intervals

[mhollanders]

Hello,

I am working on a frog project with a robust design. Our primary sessions are between spring and fall and missing from winter, and furthermore, time intervals in the active seasons are also very uneven. Here's an example from one site since 2017, with time intervals expressed in months: c(8.22, 0.69, 1.78, 2.53, 0.82, 6.31, 1.51, 3.06)

We are interested in identifying state transitions between frogs getting infected with and recovering from chytrid fungus, and survival probabilities of infected and uninfected frogs. I am trying to wrap my head around analysing this data correctly before commencing the next two years of sampling.

I understand that unequal time intervals are applied to survival probabilities but can't be applied to state transitions and emigration parameters. The options I've found to deal with unequal time intervals are 1) adding dummy variables for missing occasions and 2) using the log-link function. I think option 1 won't work because of just how jagged the time intervals are, so I think that leaves me with using the log-link function.

Unfortunately, I don't fully understand how it works. Do you use the time intervals when setting up the model, then include the time intervals as a covariate, and relate them using the log-link function to the state transitions and emigration parameters? I'm assuming this is totally wrong and I'd really appreciate a resource or some (dumbed down) explanation of how to go about this.

Kind regards,
Matt

[cooch]

Its not that you can't build unequal intervals for parameters other than S (although you have to be clever in implementing it - the log link, which is described in chapter 4, is one approach), its that with unequal intervals, the state transition parameters aren't interpretable without (fairly) heroic assumptions. I suspect you have already, but if not, (re)-read section 15.9 in the robust design chapter, and section 10.6 in chapter 10 (multi-state models).

So, MARK 'turns off' the usual mechanism for handling unequal intervals for state transitions for MS and RD models (and permutations thereof -- this includes the dynamic occupancy models) as a way of 'protecting' you from 'telling stories' about transitions, which may make little to no biological sense with unequal intervals.

[mhollanders]

Okay, I was under the impression that you could actually make the state transitions interpretable by implementing something like the loglink function on it.

Yes, I've read and re-read those sections, and I do understand the principle, and I was wondering if there's a way around it.

Does this imply that one should really steer clear of multistate models that have unequal time intervals? Or is there still a way to analyse the data to get some meaningful results about the state transitions?

[cooch]

Okay, I was under the impression that you could actually make the state transitions interpretable by implementing something like the loglink function on it.

You can use the log-link as a 'way around' the hard-coded restrictions in MARK on specifying the interval lengths explicitly (which is the normal way forward for unequal intervals), but, just because you can do something mechanically doesn't mean you necessarily will come up with an estimate that you can make sense of.

Yes, I've read and re-read those sections, and I do understand the principle, and I was wondering if there's a way around it.

In my opinion, a qualified 'no'. If you knew a priori something about transition probabilities (which might be possible for disease models) there are some technical things you can do that would allow you to do some of what you're after, but again, only given said informatioon, and perhaps some strong assumptions. If the variation in intervals among primary samples is relatively small, perhaps you might feel comfortable holding your nose. But if the difference(s) are large, relative to the process you're trying to model, you might be setting yourself up.

Not everyone is so pessimistic (Jim Nichols, for example, has worked on a problem of this sort, and for the system in question, he felt the assumptions were reasonable), but I tend to err on the side of caution.

Does this imply that one should really steer clear of multistate models that have unequal time intervals? Or is there still a way to analyse the data to get some meaningful results about the state transitions?

In my opinion, a qualified 'yes' to steering clear, and a qualified 'maybe' to the 'is there anything I can do?' query. To the former, if its me, I either 'get the design right in the first place', or I deecide how long I can go holding my nose (if I absolutely have to proceed). Knowing the challenges of meaningfully applying MS models (and permutations thereto), you should strive to conduct a sampling protocol where the intervals between primary samples is the same over the course of the study (or, at least nearly so). Unfortunately, as might be the situation you're faced with, the data weren't collected this way, and you're left wondering what to do. Low hanging fruit is to model unequal intervals, present estimates, and then do a *lot* of arm-waving about uncertianty in what comes out the other end. Not particularly satisfying, but might be what you have.

The other brick you can pile on (piling up bricks not being the best way to 'build a scientific house') is to try simulating under various plausibility scenarios, and see how much trouble you might get into by trying to explicitly model unequal intervals. Again, in the disease context, where transitions might be modelled as a function of various things, this might be possible (although, in my personal experience with this sort of stuff -- in an earlier incarantion I did some work with disease models -- the transition structure for most disease systems is semi-Markov, making most of the standard MS approaches pretty bogus -- including my own early attempts -- since they typically assume simple first-order Markov. For alternate and arguably better approaches, see work by Langrock and King).

Quoting Darryl MacKenzie: 'these methods are statistical, not magical'. Meaning, in context, that 'statistical cleverness' can't always 'magically' pull good stuff from data collected under a less-than-optimal design.

[mhollanders]

Thanks so much for the information, this has been very insightful. At the moment, it still isn't entirely clear to me whether the survival estimates of each state can be somewhat correctly interpreted when you have unequal time intervals. If the state transitions can't be interpreted very well, does this extend to the respective S probabilities?

[cooch]

The answer is generally, yes, survival estimates are likely to be more robust, but with a caveat. The issue being dicussed is that if there is time-dependence to movments among states, then with unequal intervals, you have more uncertainty about whether or not an individual may have moved 'back and forth' among states, over the interval. If survival is state-dependent, this could, in theory, add noise (uncertainty) to your survival estimates as well.