For your homework for this week, please complete the following by the beginning of class next Tuesday.
The file “hw04_data_v2.inp” contains information from a live-recaptures study of male and females of a species of your choosing(!) collected annually over 8 years. Data are entered for 2 groups: group 1 = males; group 2 = females. There are 2 individual covariates provided: the 1st is body length, \(length\), and the 2nd is standardized body length (\(z.length\), where \(z.length_i\) for individual \(i\) is \(\frac{(length_i - \overline{length})}{SD_{length}}\), where \(\overline{length}\) is the average value for \(length\).
When you set the problem up in MARK, you need to tell it that you have data for 2 groups (males and females) and that there are 2 individual covariates in the data set (\(length\) and \(z.length\)). If you name the groups and the covariates when you set the problem up, it will be easier to work with the output, so please take the time to do that.
How many PIMs are there for this problem?
How many columns are there in each of the PIMs?
For \(\phi\), please run 4 base models: \(\phi_.\), \(\phi_t\), \(\phi_g\), and \(\phi_{g*t}\). Next, please run \(\phi_{length}\), \(\phi_{g + length}\), which has 2 intercepts and 1 slope, and \(\phi_{g * length}\), which has 2 intercepts and 2 slopes. For each of those 6 models, run 2 versions: one with \(p_.\) and a 2nd with \(p_g\).
Number the PIMs as you see fit and feel free to simplify them for the models with length so that you only have 1 row for \(\phi_{male}\) and 1 row for \(\phi_{female}\) in your design matrix for those models. It’s probably simplest to just always leave the PIMs for \(p_{male}\) and \(p_{female}\) the same for all models, and I recommend numbering each so it uses a single number that is not used in any other PIM. Also, you might wish to run many of the models using the pre-defined models tool; if you do, just be sure you know what each model looks like and how you would set it up.
You can check out the short videos that are linked on today’s list of materials to see some details of how to set some of the length models up and how to move data to Excel and R for further work.
Provide an AIC table that is ordered by AICc (lowest at the top). You can obtain a copy using the “Output” menu when the Results Browser is active. If you output the table to Excel, you can work on it in Excel pretty efficiently to prepare it for copying to WORD, printing, etc. E.g., you can delete the time-stamp column, format the data portion as numbers with 2 decimal places, change column widths, etc.
Based on the AIC table in the results browser, which model structure(s) was(were) most supported by the data and what general statements can you make about sources of variation in \(\phi\) and \(p\)?
For the top model, what are the \(\hat\beta_i\) and how would they be used to obtain values for \(\hat\phi\) and \(\hat{p}\).
Write out the equations for obtaining point estimates of (a) \(ln \Bigl(\frac{\hat\phi}{1 - \hat\phi}\Bigl)\) and (b) \(\hat\phi\) for a male and for a female of length 115.
What are your point estimates of (a) \(ln \Bigl(\frac{\hat\phi}{1 - \hat\phi}\Bigl)\) and (b) \(\hat\phi\) for a male and for a female of length 115? Be sure you know how to use qlogis
and plogis
to move back and forth between those values and that you understand what those 2 functions are doing in terms of the logit and inverse-logit transformations.
For model \(\phi_{g * length}, p_g\), use the individual-covariate plotting tool to examine estimated survival values as a function of length for each sex (will need to build a separate curve for each sex). How would you describe the interaction between sex and length in model \(\phi_{g * length}\) based on the estimates and curves?
View the video linked on today’s list of materials that shows an alternate design matrix for \(\phi_{g * length}\). Then build that version of the model yourself. What are some advantages and disadvantages of using the alternative method (Hint: advantages might regard interpreting the \(\hat{\beta_i}\)).
Which model has the lowest deviance? How does that compare to the deviance for model \(\phi_{g*t},p_g\)? How is it possible for a model with no time variation and many fewer parameters to have a lower deviance than model \(\phi_{g*t},p_g\)?