STAT6950 - River Nitrogen Data - Closing Force of Crab Claws - Survival of Sparrows - Statistics Assignment Help

Assignment Task:

STAT*6950 - River Nitrogen Data Statistics Assignment Help

Part I: River Nitrogen Data

The rise in the abundance of algae in coastal waters is thought to be due to increases in nutrients such as nitrates and other forms of nitrogen, and these increases may be (are) due to human influences.

Cole (1993) gathered the data stored in 6950_F19_RiverData.txt, to gauge the evidence that nitrates in the discharges of rivers around the world are associated with human population density. (N.B. You MUST use the data in 6950_F19_RiverData.txt, which is a modified version of the original data.)

There are eight variables in the data set. The response variable is nitrate concentration (in µM/l), and there are 7 explanatory variables:

• Discharge: the estimated annual average discharge of the river into an ocean in (m3/sec)
• Runoff: the estimated annual average runoff from the watershed disturbance in (litres/(sec ? km2))
• Area: area of the watershed in (km2)
• Precipitation: annual precipitation in (cm/yr)
• Density: population density in (people/km2)
• Deposition: calculated atmospheric nitrate deposition
• Nitrate precipitation: concentration of nitrate in wet precipitation at sites located near the watersheds, in (µmol NO3 / (sec ? km2))

a) Run a model including all explanatory variables. Is human population density associated with nitrate concentration, after adjusting for the other covariates? Give an appropriate conclusion to the hypothesis test that addresses this question.

b) Now suppose we want to model nitrate concentrate with these explanatory variables. Using an appropriate method, come up with a reasonable final model relating nitrate concentration to these explanatory variables. (There is no one right answer, as there are many reasonable final models.) Justify your choice of the final model. (Explain how you arrived at your final model, and what factors you considered. e.g. Was a transformation necessary? Did you need to include any higher-order terms? Include a check of model assumptions for your final model.)

Part II: Closing force of crab claws

Behrens Yamada and Boulding (1998) investigated various characteristics of shell-breaking crab species. In one aspect of the study, they investigated the relationship between propal height of claw and closing force for three crab species: Hemigrapsus nudus, Lophopanopeus Bellus, and Cancer products. Here we’ll look at just L. Bellus and C. productus. Let X1 be an indicator variable representing whether the crab is an L. Bellus (X1 = 1 for L. Bellus and X1 = 0 otherwise). Let X2 represent claw propal height (in mm). The response variable Y is a closing force (N). The data for this problem is contained in the data set 6950_F19_crab. Import this data into R.

a) Fit the following model in R:

F orce = 0 + 1X1 + 2X2 + 3X1X2 + ?

Draw a coded scatterplot of force vs height, with different symbols for the two crab species. Superimpose the least-squares lines for the two species on the scatterplot. Make sure your plot is well labelled and easy to interpret.

b) Student #1: “Wait a minute – this model allows for different slopes and intercepts between the species, so it’s exactly the same as running two separate simple linear regressions.”

Student #2: “Nah, the lines would be similar of course, but a little different, because the sample sizes are different.”

Student #3: “Of course we’re going to get the same two lines as if we ran two simple linear regressions, but here we’d be pooling the residuals together, so that will affect the estimate of the variance and various inference procedures.”

Who’s right?

c) Consider again the full model with the interaction term. Give an appropriate conclusion to the hypothesis test of the null hypothesis that the slope of height is the same for both species of crab.

d) Suppose that we feel it’s best to remove the interaction term, so we have this model:

F orce = 0 + 1X1 + 2X2 + ?

Give an appropriate interpretation of the estimate of 1 for this model.

e) Consider the following regression models:

I. A model with separate slopes and intercepts.
II. A model with separate slopes, but common intercept.
III. A model with a common slope, but separate intercepts.
IV. A model with a common slope and common intercept.

Which one do you feel is the most appropriate model? Briefly justify your response.

f) Suppose we feel that it’s best to log transform both force and height, and use the model:

ln(Force) = 0 + 1X1 + 2ln(X2) + ?

Fit this model in R, and give appropriate interpretations of the estimates of 1 and 2 on the original scale of measurement.

Survival of sparrows in a harsh winter storm

In an in-class example earlier this semester, we looked at an example involving sparrows that perished or survived a harsh winter storm. In this assignment, we will model the probability of survival as a function of total length (TL), weight (WT), humerus length (HL), the width of the skull (SK), and length of the keel of the sternum (KL).

Fit a logistic regression model to this data, modelling the logit of the probability of survival as a linear function of the explanatory variables. Use the backwards selection procedure (using Wald tests, with ↵ = 0.10) to eliminate variables.

a) What is the estimated model equation of your final model?
b) Give an interpretation of the parameter estimate of weight in your final model.
c) Based on your final model, find a 95% confidence interval for the odds ratio for two birds that differ in weight by 0.1 gram, but have the same value of all other explanatory variables. (Odds of survival, heavier bird relative to the lighter.)

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