Description
1) (20 points) Download the “kellog.dat” data file which contains data on 22 cereals from Kellog.
Each cereal has 9 metric values that measure various aspects of the cereal. We are not given
the meanings of these variables, but in spite of this use the data to classify the cereals:
a. Read the data into a data.frame in R. Note that the data file has two extra rows, you
can ignore these with the “skip=2” parameter in read.table, or you can manually delete
them. Also, you will want to put the cereal names in the row.names with
“row.names=1” which indicates to use the first column as the row names.
b. Compute the distance matrix with “dist”. Just treat the ordinal and binary categorical
variables as metric variables (this is actually ok here because they are either interval
variables), or binary variables encoded as [0, 1].
c. Run multidimensional scaling on the distance matrix with the “isoMDS” command from
the MASS library. This computes MDS and provides a bit more and as its output,
providing both an array of “$points” to plot and a stress value. Plot the points from c)
and report the stress value. How faithfully does the plot reproduce the distances in the
data according to the stress value (remember the stress value from R is actually
multiplied by 100 so it is a percentage)?
d. How many clusters or groups does the data fall into? Can you identify some distinct
groupings? Interpret at least two of the groupings of cereals based on their names in
the data file.
e. Run an agglomerative hierarchical clustering on the dataset and plot the result as a
dendogram.
f. At a level of 3 clusters in the dendogram, use the cutree(h, k=3) command to evaluate
the clusters and then replot the MDS using these categories to color the data. Interpret
the results.
g. (Extra Credit, 3 points) Give a practical interpretation for at least one of the two
dimensions in the MDS.
2) Problem #2 (Canonical Correlation Analysis – 10 points): Water, soil, and mosquito fish samples
were collected at n = 165 sites/stations in the marshes of southern Florida. The following water
variables were measured:
MEHGSWB Methyl Mercury in surface water, ng/L
TURB in situ surface water turbidity
DOCSWD Dissolved Organic Carbon in surface water, mg/L
SRPRSWFB Soluble Reactive Phosphorus in surface water,mg/L or ug/L
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THGFSFC Total Mercury in mosquitofish (Gambusia affinis), average of 7 individuals, ug/kg
In addition, the following soil variables were measured:
THGSDFC Total Mercury in soil, ng/g
TCSDFB Total Carbon in soil, %
TPRSDFB Total Phosphorus in soil, ug/g
Perform a canonical correlation analysis, describing the relationships between the soil and water
variables using the data1
found in data_marsh_cleaned.csv.
1) Answer the following questions regarding the canonical correlations. (Note that a, b
and c can all be done directly from the output of canonical correlation)
a. Test the null hypothesis that the canonical correlations are all equal to zero. Give
your test statistic, d.f., and p-value.
b. Test the null hypothesis that the second and third canonical correlations equal
zero. Give your test statistic, d.f., and p-value.
c. Test the null hypothesis that the third canonical correlation equals zero. Give
your test statistic, d.f., and p-value.
d. Present the three canonical correlations and list any conclusions that you can
draw.
2. Answer the following questions regarding the canonical variates.
a. Give the formulae for the first canonical variate for the soil and water variables.
b. Give the correlations between the significant canonical variates for soils and the
soil variables, and the correlations between the significant canonical variates for
water and the water variables and use these to interpret the variates (do this as
best as you can. Even with a lack of domain knowledge you should be able to
describe the relationship in more general terms given the variables involved and
the correlations.)
1 http://www.epa.gov/region4/sesd/reports/epa904r07001.html
