Index of heterogeneity

[jlaufenb]

In several papers using Pledger closed population models, I have seen the use of the CV of the mean capture probability (mean P) as an index of the amount of unidentifiable heterogeneity in the capture data. Is there a way to determine which contributes most to the degree of heterogeneity: the difference in capture probabilities (Pa and Pb) between mixtures or the difference in mixture proportions (PIa and 1-PIa)? Below are the equations I've seen.

mean P = PIa*Pa + (1-PIa)Pb

CV(mean P-hat) = (SQRT(PIa*(1-PIa))*ABS(Pa-Pb))/mean P

Thanks
Jared

[cooch]

In several papers using Pledger closed population models, I have seen the use of the CV of the mean capture probability (mean P) as an index of the amount of unidentifiable heterogeneity in the capture data. Is there a way to determine which contributes most to the degree of heterogeneity: the difference in capture probabilities (Pa and Pb) between mixtures or the difference in mixture proportions (PIa and 1-PIa)? Below are the equations I've seen.

mean P = PIa*Pa + (1-PIa)Pb

CV(mean P-hat) = (SQRT(PIa*(1-PIa))*ABS(Pa-Pb))/mean P

Thanks
Jared

Difference in estimated mixture proportions (Pi) in Pledger models is not informative. See section 14.6.1

[jlaufenb]

If the Pi estimates from those models are uniformative, is the use of this index invalid?

What about reporting capture probabilities? Do the mixture-specific estimates depend on where the 'best estimate' of the 'breakpoint' falls? How valid is the mean P calculation?

Also, does that mean that nothing informative can be gained from the Pi values in the example below?

Group 1: PIa = 0.8, Pa = 0.05 and Pb = 0.30
mean P = 0.1 CV = 1.0

Group 2: PIa = 0.5, Pa = 0.05 and Pb = 0.30
mean P = 0.175 CV = 0.71


One last question. Although Ch14 warns against 'post hoc story-telling', is there no value in suggesting plausible sources of heterogeneity that may have caused the patterns observed in the parameter estimates?

Despite all of the praise mixture models have been given for their utility in helping deal with unidentifiable heterogeneity, I have not found much guidance on what and how estimates should be reported and interpreted (aside from the Interpreting Pi section in Ch14). I will gladly accept any "Source programmable guidance!" (If you get that movie quote you're alright in my book)

[cooch]

One last question. Although Ch14 warns against 'post hoc story-telling', is there no value in suggesting plausible sources of heterogeneity that may have caused the patterns observed in the parameter estimates?

In my opinion, no. If you had a plausible argument, you probably could have figured out appropriate covariates to partition heterogeneity a priori. I'll let others weigh in here, but in general.

Despite all of the praise mixture models have been given for their utility in helping deal with unidentifiable heterogeneity, I have not found much guidance on what and how estimates should be reported and interpreted (aside from the Interpreting Pi section in Ch14). I will gladly accept any "Source programmable guidance!" (If you get that movie quote you're alright in my book)

You're missing the point - mixture models improve estimates of abundance, which invariably is the point of the exercise. Finite mixtures are a kludge - a kludge that has proven remarkably good (and relatively easy to implement), but the very fact that the mixture is discrete means its already an approximation for many situations. I suppose you could concoct scenarios where the major axes affecting heterogeneity are discrete (e.g., unknown age, unknown sex...), but in general...

ps.s Spies Like Us was a terrible movie.

[murray.efford]

I agree with Evan. I also like the idea of reporting the estimated mean and CV of capture probability, as suggested by Jared. The CV documents the all-important heterogeneity without getting distracted by whether the (unseen) variation is discrete or continuous. You could also use the coefficient of heterogeneity from Dorazio & Royle, or Pledger 2005 Biometrics 61, 868–876.

[egc]

In my opinion, no. If you had a plausible argument, you probably could have figured out appropriate covariates to partition heterogeneity a priori. I'll let others weigh in here, but in general.

To finish the thought I started here - post hoc story-telling is analogous to inferring process from pattern - something we wish to avoid (we prefer a priori predictions based on theory/prior information, which provides the null against which we test observation).