For your homework for this week, please complete the following by the beginning of class next Tuesday. The file “hw06_data.inp” contains information on live recaptures of adult females obtained during a 6-occasion study conducted over 2 weeks when the study population could reasonably be assumed to be closed to changes due to births, immigration, deaths or emigration. Upon initial capture, each individual was given a unique mark such that it’s individual encounter history over the 6 occasions could be recorded. Use Program MARK and the “Closed Captures Data Type” to estimate abundance for the study species using full-likelihood modeling approaches.

Setting up the problem in Program MARK

In Program MARK, begin by choosing the “Closed Captures Data Type” and then select the 1st option, which is ‘Full Likelihood p and c’ as discussed in section 14.3 of CW.

1. How many PIMs are there for this problem and what parameters are they for?

2. How many columns are there in each of the PIMs and why is that so?

Models to evaluate using ‘Full Likelihood p and c’

Please run models (1) $$M(0)$$ (constant $$p$$ and $$c = p$$), (2) $$M(t)$$ ($$p_i$$ is allowed to vary across the $$i$$ occasions but $$p_i$$ = $$c_i$$ on each occasion), (3) $$M(b)$$ (constant $$p$$ and constant $$c$$ but $$c$$ allowed to be different from $$p$$), (4) $$M(t*b)$$ ($$p_i$$ is allowed to vary across the $$i$$ occasions, $$c_i$$ is also allowed to vary by occasion, and temporal changes in $$p$$ are independent of the temporal changes in $$c$$, i.e., it’s a truly interactive model of $$t$$ and $$b$$), and (5) $$M(t+b)$$ ($$p_i$$ is allowed to vary across the $$i$$ occasions and $$c_i$$ is also allowed to vary by occasion but a constant offset between the 2 exists on the logit scale, i.e., it’s an additive model). You can use the Run menu’s “pre-defined” models option to obtain all the models in this week’s lab if you like. But, if you do so, be sure you look at the PIMs and design matrices and understand what’s happening.

1. What is $$M_{t+1}$$ for the study? If you look at the full output for the model, you can see the value you listed. To obtain the full output, while in the Results Browser, click on the icon that is just to the right of the trash-can icon in MARK. You should be able to find the value $$M_{t+1}$$ listed about halfway through the output.

2. For $$M(0)$$,

• what is your estimate of $$p$$?
• Given that estimate, what is the probability that an individual would never be caught during the study? Hint: the probability of being caught at least once during the study, is $$\hat p^{*} = 1 - \displaystyle\prod_{i=1}^{6} (1 - \hat p_i)$$. For $$M(0)$$, because $$p$$ is constrained to be constant, you can simplify this to $$\hat p^{*} = 1 - (1 - \hat p)^6$$. And the probability of never being caught is simply $$(1 - \hat p)^6$$.
• What value do you obtain when you multiply the probability of never being caught times $$\hat N$$ and how does that relate to $$\hat f0$$?
• What value do you obtain when you multiply $$\hat{p}^{*}$$ times $$\hat N$$ and how does that relate to $$M_{t+1}$$?
1. For $$M(b)$$, what are your estimates for $$p$$, $$c$$, and $$N$$? How does your estimate of $$\hat N$$ for this model compare to the value you obtained from $$M(0)$$. Why do you think that this is so based on your estimates of $$p$$ and $$c$$ for model $$M(b)$$?

2. For $$M(t*b)$$, what are your estimates of $$p_6$$, $$f0$$, and $$N$$? Why? Hint: review the material in section 14.3.1 of CW.

3. For $$M(t+b)$$, what are your estimates of $$p_6$$, $$f0$$, and $$N$$? Do you see any temporal pattern in the values for $$p_i$$ or $$c_i$$ and if so what are they? If you have individual heterogeneity in capture probabilities, how could that cause such a temporal pattern in the estimated $$p_i$$?

4. Use the PIM menu and change the data type to “Full Likelihood Heterogeneity pi, p and c” and select “2 mixtures” on the window that is spawned. Then use the Run menu and “pre-defined models” to run “Mh2” and “Mbh2”. After they are run, retrieve each, and explain what each model is doing with regards to modeling $$p$$ and $$c$$ .

5. For model “Mbh2”, what is your estimate of $$\pi$$, what does that parameter refer to, and how do the 2 values of $$\hat p$$ relate to $$\hat \pi$$?

6. Delete $$M(t*b)$$ from your Results Browser. Provide a model-selection table for the 6 models that remain.

7. Use the Output menu to obtain a model-averaged estimate of the derived parameter, $$N$$. On the window that is spawned and titled “Model Averaging Parameter Selection”, (1) check the sole box for your derived parameter (there’s only a single $$N$$ being estimated here by each of these models), (2) uncheck the box that’s labeled, “Only select models for the current data type” (here, the models are okay to compare but that wouldn’t be the case if your set included some models that weren’t full likelihood models), and (3) click “OK” on the warning box after you read it.

• Report what you obtain from the model averaging routine.

• Given the model-specific estimates of $$p$$ and $$c$$, can you explain why some models produce values for $$\hat N$$ that are lower or higher than others?