Assignment 10

Due Date: Friday, April 14, 2006

A Multilevel Model of Pelagic Larval Duration Time

Data 

The file 74species.csv contains the data of invertebrate development times that we've been analyzing in class. This is a comma-delimited text file in which the variable names appear in the first row.

Background 

As has been discussed in class, the data set contains the pelagic larval duration times for 74 marine vertebrate and invertebrate species obtained at different ambient temperatures. Multiple records (between two and six) exist for each species. The variables of interest to us are the following.

  1. "lnPLD" denotes the natural log of pelagic larval duration time.
  2. "lntemp " denotes the natural log of temperature (measured on the Celsius scale). Multiple temperatures are available for each species. This variable is a level-1 predictor.
  3. "species" is an identifier of the species.
  4. "climate.3" denotes the region of the world that the species inhabits: polar, temperate, or tropics. This is a species characteristic and hence a level-2 predictor.
  5. "feeding.type" denotes the larval feeding type, P for planktotrophic and L for lecithotrophic. This is a species characteristic and hence a level-2 predictor.

The goal of this assignment is to discover the role feeding type plays in larval development.

The Questions

Question 1  In Wednesday's class we found it necessary when adding the variable climate.3 to center the variable lntemp in order to get the optimization algorithm to converge. To simplify interpretation we then used as our level-1 predictor, lntemp–log(15).

  1. Fit a multilevel model with random slopes and intercepts in which the response is lnPLD and the only predictor is the level-1 predictor lntemp. This is model2 from class.
  2. Next fit a model that is identical in every respects to this model except that the level-1 predictor is lntemp-log(15), a centered version of log temperature. This is model2.5 from class.
  3. Show with minimal algebraic manipulation that the fixed effect estimates you get back (both the intercept and slope) are numerically exactly the same in the two models. This proves that centering is just a numerical trick to achieve better numerical stability. It does not actually change the model being fit.

Question 2  Assess whether the variable feeding.type affects the linear relationship between lnPLD and lntemp. Decide if its primary effect is on the intercept, slope, both the intercept and slope, or neither the intercept or slope.

Question 3  For the model you chose to be best in Question 2, do the following.

  1. Write the model as a two-stage model, i.e., write down the level-1 equation and the level-2 equations.
  2. Write the model as a composite model, i.e., write down the model as a single equation in which the fixed effect and random effects are segregated into different portions of the equation.
  3. Write down the equation of the population-averaged (marginal) model and the subject-specific (conditional) model.
  4. Repeat the last step but this time separately write the marginal and conditional models as two equations, one for planktotrophic larvae and the other for lecithotrophic larvae.

Question 4  Calculate an appropriate pseudo-R2 to quantify the importance of the variable feeding.type for the model you chose in Question 2.

Question 5  Plot the equation of the population-averaged model over the range of the data using the model you chose as being best in Question 2. Clearly indicate the two larval feeding types on your graph. In words, what does your model say about the differences in the relationship between temperature and pelagic larval duration time for the two feeding types? Be as specific as you can be.

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Jack Weiss
Phone: (919) 962-5930
E-Mail: jack_weiss@unc.edu
Address: Curriculum in Ecology, Box 3275, University of North Carolina, Chapel Hill, 27516
Copyright © 2006
Last Revised--April 7, 2006
URL: http://www.unc.edu/courses/2006spring/ecol/145/001/docs/assignments/assign10.htm