Crab Lab

I. Investigation
Examine the relationship between premolt and postmolt carapace size. Compare the two methods for predicting a crab's premolt size from its postmolt size:

• Fit local estimates
• Fit a line

A. How well does the best line approximate the best set of local estimates?
B. Does the line adequately describe the relationship between premolt and postmolt sizes?
Derive an expression for the average squared error you expect in the linear prediction.

C. Examine the crabs with carapace width between 147.5 and 152.5 mm. Compare the prediction estimates of premolt size for this subset with the actual premolt size distribution of the subset.

To address these questions, you will use the GUI found in the SimpleRegression package. Execute the following commands to run it:

• R
• library(SimpleRegrression)
• ptAvg()

Be sure to include a printout of your plot.

II. Investigation Next, consider more informative models for describing crab growth. Crabs, and their growth patterns, have long been studied by biologists and statisticians. In 1894, Pearson examined the size frequency distribution of crabs; Thompson, in his 1917 book On Growth and Form, compared the shape of the shell across species of crabs, and Huxley, in his 1932 book Problems of Relative Growth studied the changes over time in the relationship between the weight and carapace width of crabs. Assess the following models for crab growth. Use the data, other's findings, and biological arguments for growth to choose one of the models.

• Constant relative growth, i.e.
  molt increment/premolt = c.
• Constant growth increment, i.e.
  molt increment = c.
• Constant growth increment for weight and an allometric relationship between carapace width and weight, i.e.
  log(molt increment/premolt) = a - b*log(premolt).
• Linear extension of constant relative growth: i.e.
  molt increment = a + b*premolt
• Log-linear extension of constant relative growth: i.e.
  log(molt increment/premolt) = a+b*premolt

A. Begin your investigation of the relationship between molt increment and premolt size by examining a scatter plot of these two measurements. The plot rules out some of the models described above. Which are they?

B. With which models are the data consistent? Explain.

C. How would you choose among such models? Use the Dungeness crab data and take into account the biological explanations for growth in answering this question. Also keep in mind that it is reasonable to assume that the coefficients for the growth model may change for different stages of growth; that is, juvenile and adult crabs may have different growth rates.

To complete this part of the lab, you will use the the latest version of the descriptive stats GUI:

• R
• library(descriptiveStats2)
• descriptiveStats()

Be sure to provide between one and two scatterplot for each model. Also, when a regression line is fit to the data, please provide the correlation, coefficients of the line, and RMSE.

Note that when you regress one variable on another, two new variables are added tothe dataset -- the residuals and the fitted values from the regression. These will be at the bottom of the variable tree.

Note that you may want to read about model fit and residual plots in Chapter 9 (the snow gauge chapter).