Arithmetic mean, lambda(.), or geometric mean of lambda?

[Eric Janney]

If one is interested in assessing cumulative gain or loss to a population over time using Pradel’s temporal symmetry model it seems that using the arithmetic mean of lambda or the constrained value from the lambda(.) model could potentially be misleading? Isn’t it possible to have an arithmetic mean, Beta-hat (from a random effects model), or lambda(.) value greater than one when in fact a net population loss occurred over the period in question? Because the order in which the various lambda values fall is important in determining cumulative change in N over time, wouldn’t the geometric mean be a better way to assess the % population gain or loss over time? I poked around a bit but couldn’t find any discussion of this issue.

[cooch]

If one is interested in assessing cumulative gain or loss to a population over time using Pradel’s temporal symmetry model it seems that using the arithmetic mean of lambda or the constrained value from the lambda(.) model could potentially be misleading? Isn’t it possible to have an arithmetic mean, Beta-hat (from a random effects model), or lambda(.) value greater than one when in fact a net population loss occurred over the period in question? Because the order in which the various lambda values fall is important in determining cumulative change in N over time, wouldn’t the geometric mean be a better way to assess the % population gain or loss over time? I poked around a bit but couldn’t find any discussion of this issue.

Some quick thoughts (I'm already late):

1. Pradel's lambda gives the realized growth rate of the age class used in the analysis. It isn't hard to concoct a scenario where the realized trajectory of one age class differs from the population as a whole (especially if the time-series is short)

2. you have to distinguish between the retrospective use of the time series (what *has* happened) versus the prospective (what *will happen* if the underlying stochastic process is stationary). This is an important distinction to remember

3. Eric is correct - the appropriate average for what has happened is the geometric mean of individual realized lambda values. And, for the usual reasons, the geometric mean is always less than the arithmetic mean. All you need to do is for (t) time steps is take 1/t times the sum of log(lambda). SE's can be derived in several ways.

But, despite the technical issue Eric raises, I would like to emphasize (1) and (2). Pradel's lambda is a very slick 'result', but you have to really think about what it is, and how useful it might be for your purposes. For some populations I've looked at, it can lead to quite misleading 'results' unless you're careful.

[dhewitt]

[Note: I'm assuming that Evan's points 1 and 2 are understood.]

I hope (sort of) that I'm missing something obvious here, but I'm surprised that more has not been said about this issue in the literature related to the Pradel models. Perhaps it's because most applications have realized lambdas close to 1.0 in each year (so there are only small differences among ways of getting a measure of central tendency for lambda), or the focus is on annual estimates (and covariates) and the data support this higher model structure.

For fish populations, contributions of single year classes can be big and generate lambdas that are quite large (say, > 3.0 in some cases). And, model selection might often indicate less than time-specific structure on parameters, perhaps due to low recaptures, potentially leading to a constant lambda model.

I'm still not clear on whether the estimate of lambda from a dot model would be closer to the geometric mean, the arithmetic mean, or perhaps an estimate from a random effects model. The differences can be big in terms of characterizing what the status has been over a reasonable length of time, say 15 years.

Given the definition of realized lambda (successive changes in population size), we clearly need a geometric mean to generate a time series of population sizes (in the background) that matches the overall lambda across that period. But is this what a dot model gets you (ignoring the issues with the earlier and later estimates being trash)? My gut says it is not, but I'm not an expert on likelihood. If it isn't equivalent to a geometric mean, what is a dot model estimate good for?

[cooch]

[Note: I'm assuming that Evan's points 1 and 2 are understood.]

I hope (sort of) that I'm missing something obvious here, but I'm surprised that more has not been said about this issue in the literature related to the Pradel models. Perhaps it's because most applications have realized lambdas close to 1.0 in each year (so there are only small differences among ways of getting a measure of central tendency for lambda), or the focus is on annual estimates (and covariates) and the data support this higher model structure.

For fish populations, contributions of single year classes can be big and generate lambdas that are quite large (say, > 3.0 in some cases). And, model selection might often indicate less than time-specific structure on parameters, perhaps due to low recaptures, potentially leading to a constant lambda model.

I'm still not clear on whether the estimate of lambda from a dot model would be closer to the geometric mean, the arithmetic mean, or perhaps an estimate from a random effects model. The differences can be big in terms of characterizing what the status has been over a reasonable length of time, say 15 years.

Given the definition of realized lambda (successive changes in population size), we clearly need a geometric mean to generate a time series of population sizes (in the background) that matches the overall lambda across that period. But is this what a dot model gets you (ignoring the issues with the earlier and later estimates being trash)? My gut says it is not, but I'm not an expert on likelihood. If it isn't equivalent to a geometric mean, what is a dot model estimate good for?

Depends on who you ask. Personally, I think putting a constraint on lambda rarely makes sense. If lambda is a straight function of survival and recruitment (in the simplest formulation), then if you make lambda constant (or anything other than time-dependent - not that constraining to a time-varying covariate would be equivalent), then you necessarily force a covariance between survival and recruitment. For the obvious reason that if lambda is a constant (as in a dot model), then if survival goes up, recruitment must go down.

If you want geometric means, just do it by hand, and apply the Delta method to get the SE. Its relatively straightforward to do this. You don't use dot models to generate averages for lambda.

Such models can be used *with care* to estimate the mean where the standard arithmetic mean is appropriate, but there are drawbacks - see the sidebar beginning on p. 81 in Chapter 6). The estimate of such an arithmetic mean is generally unbiased, but the SE is wrong. The more appropriate approach would be to use the random effects VC approach - the noted sidebar is the only place this is mentioned in the book (so far...).

[darryl]

But is this what a dot model gets you (ignoring the issues with the earlier and later estimates being trash)? My gut says it is not, but I'm not an expert on likelihood. If it isn't equivalent to a geometric mean, what is a dot model estimate good for?

My (likely imperfect) view of it goes something like this...

If we were omniscient, we'd know what the true value of lambda, phi, S, psi or whatever, actually was at each point in time. But we're not (at least I'm not) so instead we have to do this whole mark-recapture thing, fit a bunch of models, and estimate these darned demographic parameters. If we have a time effect, that's allowing these true values to vary among years, which more often than not is likely the biological reality. Instead of having 15 lambda though, we might want to summarize those values in terms of a mean and process variance or standard deviation. We could either do this with the estimates from a fixed-effect time model, or assume a distribution for these values and move to a random-effects time model. (I agree that for lamda you're best working with the geometric mean which is equivalent to the arithmetic mean on the log-scale as lambdas are on the range 0-inf).

Now where does the dot-model come in? That assumes that all these true (unknown) values are exactly equal. Probably unlikely in practice, but it might be a reasonable approximation if the true process variance is small, or if the system is at some point of equilibrium (depending upon the demographic parameter). It might also end up being a parsimonious model if your sample sizes are small. While the estimate from a dot-model might be close to a mean value from a more complicated model, it's not a mean in any shape or form. It's the estimated value for that parameter assuming that that value is constant in time. Nothing more, nothing less.

[cooch]

But is this what a dot model gets you (ignoring the issues with the earlier and later estimates being trash)? My gut says it is not, but I'm not an expert on likelihood. If it isn't equivalent to a geometric mean, what is a dot model estimate good for?

My (likely imperfect) view of it goes something like this...

If we were omniscient, we'd know what the true value of lambda, phi, S, psi or whatever, actually was at each point in time. But we're not (at least I'm not) so instead we have to do this whole mark-recapture thing, fit a bunch of models, and estimate these darned demographic parameters. If we have a time effect, that's allowing these true values to vary among years, which more often than not is likely the biological reality. Instead of having 15 lambda though, we might want to summarize those values in terms of a mean and process variance or standard deviation. We could either do this with the estimates from a fixed-effect time model, or assume a distribution for these values and move to a random-effects time model. (I agree that for lamda you're best working with the geometric mean which is equivalent to the arithmetic mean on the log-scale as lambdas are on the range 0-inf).

As per my first reply... 1/t x sum (log(lambda)) - essentially the arithmetic mean over t time intervals.

I'm fooling with an MCMC-based approach to this - which is analogous to the random effects model Darryl refers to.

But - independent of other things, putting a logical 'dot' constraint on one parameter out of a set of parameters which are effectively collinear makes little sense to me, since it necessarily forces a negative covariance amongst the other parameters.

Now where does the dot-model come in? That assumes that all these true (unknown) values are exactly equal. Probably unlikely in practice, but it might be a reasonable approximation if the true process variance is small, or if the system is at some point of equilibrium (depending upon the demographic parameter).

This is the point some hack named Nichols (who has forgotten more about most subjects than I'll ever know) makes periodically, but the problem is that while it is perhaps reasonable to test a model that implies explicitly that (as Darryl puts it) that variance is small, or lambda is approximately stationary, or both, there are still challenges. To cut a long story short, I'm still a bit uncomfortable putting 'dot' constraints on a parameter which will necessarily impart a covariance structure on remaining parameters (subtle little issues that involve whether or not, then, the other parameters are in fact separately estimated, given that the constraint on one forces the other parameters to be estimated in a particular fashion...hmmm.)

It might also end up being a parsimonious model if your sample sizes are small. While the estimate from a dot-model might be close to a mean value from a more complicated model, it's not a mean in any shape or form. It's the estimated value for that parameter assuming that that value is constant in time. Nothing more, nothing less.

An excellent point, and one I need to add to the sidebar I mentioned in Chapter 6.

My personal approach to this (reflecting my own biases) is to (i) model S and f in the usual fashions, (ii) dump out model-specific time-specific lambdas, and (iii) model average them. If I want an overall estimate of lambda over all sampling periods, you could generate the geometric mean lambda and SE for each model, then model average those.

[dhewitt]

I'll start by agreeing whole-heartedly with Darryl RE: modeling, estimation, and parsimonious explanations of observed (imperfect) data. I'd also agree with Evan that the variance components approach to determining central tendency is clearly better than a dot model estimate (excellent sidebar, by the way). And that, of course, constraints on one parameter imply constraints on others in the Pradel models (analogous to the problem of developing "uninformative" priors in Bayesian analyses -- Jeffrey's priors, etc.). I will note that the suggestion is in the literature in multiple places that when the interest is in lambda, to proceed by allowing time-specific Phi and p and then placing constraints on lambda. I assume this is to reduce problems that result from tangling up constraints.

But, if I can be permitted to dig a little deeper on the challenges that Evan mentions (my thoughts might be related to the details in Evan's shortened long story, I'm not sure). ... I'm not even convinced that the target of the random effects approach for a time series mean from these models is the right one (not that anyone but me cares if I'm convinced, mind you).

Because I'm not smart enough to dig this out of the likelihood components directly, I'm thinking of a test via simulation that goes something like this:
(1) we have a series of lambdas that includes some big values (say, > 2);
(2) we pick a starting popn size, which doesn't matter much, and generate the truth in the popn trajectory;
(3) we choose some reasonable series of Phi and p with no other structure;
(4) we simulate a sufficient number of capture histories that jives with these conditions (the hard part?);
(5) we fit a variance components model and see what the estimate for the overall lambda is;
(6) do 4 and 5 a lot of times

Would the estimates from the VC approach be close to the truth -- the geometric mean of our series of lambdas? Intuitively, given the definition of lambda, we want our target for this estimate to be a value that gets us from starting popn size to ending popn size, right? Order matters, and I'm just not sure the VC approach accounts for the history of things as we intuitively desire. I'll reiterate that this isn't an issue when lambdas are nicely dist'd around 1, but I think we'd like a modeling approach that covers all bases.

Evan and I seem to end up in the same place -- you best get time-specific estimates -- but I think we might be getting there from slightly different angles.

[dhewitt]

As per my first reply... 1/t x sum (log(lambda)) - essentially the arithmetic mean over t time intervals.

Just a clarification that you need to exponentiate the result of Evan's calculation to get the geometric mean of lambda. One more step.

[cooch]

As per my first reply... 1/t x sum (log(lambda)) - essentially the arithmetic mean over t time intervals.

Just a clarification that you need to exponentiate the result of Evan's calculation to get the geometric mean of lambda. One more step.

Yes, but the expression I give is actually what you want for your estimate of stochastic growth rate. From basic stochastic theory, the time-averaged growth rate from 0 to t is (from the Heyde-Cohen estimator)

N(t)-N(0)/t = 1/t sum(log(lambda)) = E(log lambda), where the time-averaged growth rate over a single realization of the stochastic process is the log mean E(log lambda). This is different than the growth rate of the mean population size that you'd see if you had replicate populations. By Jensen's inequality, these two different measures of average growth rate are not equivalent.

Thus, we can see (with a bit of thought) that it is E(log lambda) which is the relevant measure to characterize the growth rate of a stochastic population - it is the rate at which almost every realization of the process will grow. The growth rate of the mean overestimates the growth rate of almost every realization (as Hal Caswell has noted, this is just like in Lake Wobegon - where “all children are above average” - except that here, most realizations in the stochastic process are below average).

But I digress - the main point is that our best estimate of the stochastic growth rate is, in fact, 1/t X sum (log(lambda)), which is related to the geometric mean as Dave points out. I had lapsed into the standard convention which is to report stochastic growth rate on the log scale (given the log-normal distribution of abundances in a stochastic projection).

[dhewitt]

But I digress

Indeed