STAT 415 - Midterm Examination (62 Points)
Owls, like many other birds, eat their food whole. Since birds do not have teeth, they can't chew their food. Therefore, they use their strong and sharp beaks to rip their prey apart and then swallow large chunks whole. The owl slowly digests its meal by separating the softer materials (such as meat) from the harder material (such as bones). It then regurgitates the harder material along with indigestible items such as feathers and fur in the form of a pellet. Owl pellets can be dissected and used to find out what and how much an owl has been eating. It can also used to determine in what areas they live etc. These data come from the dissection owl pellets from the Malay Barred Owl found in Malaysia. The measurements of
Owl Pellet
Your ultimate goal is to examine these data to determine what rodent skull characteristics seem to discriminate between the different species of rodent eaten by Malay Barred Owls. Complete each of the parts below using the data found in: OwlDiet.JMP, OwlDiet.lsp and OwlDiet in R. Some parts will require the use of specific software on others you can use whichever you feel is most appropriate.
a) Assess univariate and multivariate normality of these skull measurements. Be sure to take into account the fact that several species of rodent are represented in these data. Discuss your findings.
(14 pts. ~ 2 pts. for each species)
b) Examine comparative boxplots for each of these skull measurements across species. Which characteristics will be most useful for determining the species of the rodent?
(8 pts. ~ 2 pts. for each skull measurement)
c) Examine scatter plots with the data points colored by species. Which pairs of characteristics seem to discriminate between the different species? Explain. (4 pts.)
For parts (d) – (f) use the data frame OwlMini2 in R.
d) The data frame OwlMini2 in R contains the measurements for two of the rat species in the larger data set, the Red Spiny rat (Rattus surifer) and the Malayan Wood rat (Rattus tiomanicus).
1) Does the equality of the population variance-covariance matrices seem plausible here?
Explain. (2 pts.)
2) Use Hotelling’s statistic to test the equality of the population mean vectors of these skull
measurements for these two species of rats. Discuss the results. (3 pts.)
Note: Conduct the test assuming the variance-covariance matrices are equal regardless of your answer to part (1).
e) Construct appropriate 95% CI’s for the differences in the population means on each the skull measurements for the species comparisons in part (d). Discuss these intervals focusing on the practical/biological interpretation. (4 pts. ~ 1 for each skull measurement)
f) Two rat skulls are found in the owl pellet from a Malay Barred Owl with skull measurements of:
Skull LengthTeeth RowPalatine ForamenJaw Length
Skull 1 355.4658.16 64.89 162.06
Skull 2 409.0760.45 67.24 158.61
Calculate the value of the linear combination most responsible for rejection of the null hypothesis for the two unidentified skulls above and use the linear combination plot (lcp = T) along with these linear combination values to classify each skull according to species. Show you R output.
(4 pts.)
g) Perform a MANOVA comparing for testing the equality of the mean vectors across all the rodent species in these data. Be sure to examine and discussthe Centroid Plot of the results. Add the actual data points to this plot. Do the rat species seem well separated in this plot? Which rat species are most similar to one another? Characterize the differences between two species of rat using this plot. (8 pts.)
h) Read Section 6.5 of your text. Use Result 6.5 (pg. 305) to construct a simultaneous confidence interval for the difference in the mean teeth row distance for Rattus argentiventer vs. Rattus tiomanicus. (3 pts.)
i) Perform principal component analyses of both the unscaled and scaled skull measurements. Look at the principal component loadings of the first two PC’s from each.
1) Explain why the loadings on the first PC are different from one another and why you think they look the way the do. (3 pts.)
2) Which PC analysis do you feel is more appropriate, the one performed on the unscaled or
scaled data? Explain your answer. (2 pts.)
j) Explain what the 1st PC is measuring in the analysis performed on the scaled data. (1 pt.)
k) How much total and individual variation do the first two PC’s account for in the PC analysis of the scaled data. (3 pts.)
l) Construct a biplot for the first two principal components and use rat species to label the points. Do the species appear to be well separated in this plot? Should they be? Explain. (3 pts.)