Statistics 579
Applied Multivariate Methods
Exam 3
1. The file FoodNutrient.txt contains nutritional data on 27 different food items.
Nutritionists desire to find clusters that contain food items that are similar enough to
warrant a recommendation of interchangeability when planning menus.
a) Perform a Principal Components Analysis of these data, and determine the
number of principal components that would be needed to explain most of the
variation in these data. Plot the first three principal components.
b) Using the Average Linkage Method and the Centroid Method, do a cluster
analysis of these data, and produced the corresponding dendrogram. Using the
plot of the first three principal components, the cubic clustering criterion, the
pseudo F statistic, and the pseudo T2 statistic, determine the number of clusters
that exist in this data set.
c) Perform a k-Means clustering of these data, using the number of clusters
determined in part (b).
d) Identify which food items belong in each of the clusters. Do the 3 different
clustering methods produce the same results?
e) For each of the three clustering methods used, produce a 3-D scatter plot of the
first three principal components, and annotate each point with the cluster
number.
2. The file Milk.txt contains data from the first phase of a study of the cost of
transporting milk from farms to dairy plants. Cost information was obtained on Fuel,
Repair, and Capital, all measured on a per-mile basis, for 2 Types of trucks:
Gasoline and Diesel.
a) Create a 3-D plot of the data, using different symbols for the 2 different types of
trucks.
b) I have identified 2 data points that are clear outliers. Find them and eliminate
them before you do any of the analyses below. Make sure you clearly identify
which 2 points you are eliminating, and state your reasons why.
c) For each of the two types of trucks separately, determine if it is reasonable to
conclude that the data arose from a multivariate normal distribution.
d) Test for the equality of the population variance-covariance matrices for the 2
types of trucks. Use α = .01.
e) Assuming that the variance-covariance matrices are equal, test whether the
mean vectors are the same for both types of trucks.
f) If you found, in part (b), that the variance-covariance matrices are in fact
different, test for the equality of the mean vectors assuming unequal variance-
covariance matrices. Compare these results with the results from part (c).
g) Using Bonferroni t-tests, construct simultaneous 95% confidence intervals for the
difference between the 2 Types of truck, for each of the 3 dependent variables
separately.
