Biometry Assignment #6 – Spring 2014 (34 pts.) – Due 03/28/14
Chi-square Tests of Independence & Applications of the Normal Distribution
Problems 1 and 2 are Chi-square Tests of Independence
Problem 1 - Effects of Environment on the Claw Development in Lobsters
Data File: Lobster-claws.JMP
The two claws of the lobster (Homarus americanus) are identical in the juvenile stages.
By adulthood, however, the two claws normally have differentiated into a stout claw
called a "crusher" and slender claw called a "cutter". In a study of the differentiation
process, 26 juvenile animals were reared in smooth plastic trays and 18 were reared in
trays containing oyster chips (which they could use to exercise their claws.) Another 23
animals were reared in trays containing only one oyster chip. The claw configurations of
all animals as adults are summarized in these data.
CLAW CONFIGURATION
RCrush,LCut
RCut,LCrush
RCut,LCut
Row Totals
7
9
7
23
Oyster Chips
8
9
1
18
Smooth Plastic
2
4
20
26
Column Totals
17
22
28
n = 67
TREATMENT
One oyster chip
(𝐸11 =
)
(𝐸22 =
)
a) Which variable is the response? Which variable is the explanatory or predictive
variable? (2 pts.)
b) Use JMP to construct a mosaic plot for the relationship between treatment and claw
configuration. Also calculate the conditional probabilities of the form P(claw
configuation | treatment) using the Row % option and discuss the results. (4 pts.)
c) Fill in the expected frequencies (𝐸11 𝑎𝑛𝑑 𝐸22 ) in the table above. (2 pts.)
c) Perform a test to determine whether there is a relationship between the environment
the lobster was reared in and the claw configuration when it reaches adulthood. Discuss
your findings. (4 pts.)
Problem 2 – Comparison of Tooth Wear between Big Cats, Hyaenas, Wolves,
and Wild Dogs
Excerpt from the paper “Incidence of Tooth Breakage Amongst Large, Predatory
Animals” by Valkenburgh in the The American Naturalist is given below:
The sample is described as…
In this analysis you will be comparing wear stage across species, where wear stage is
categorized as follows:
Question and Tasks for Problem 3 (Data File: Skull Tooth Wear)
a) Is there evidence to suggest that the species differ in terms of tooth wear? Use an
appropriate test to answer this research question. Obtain the results for both the broad
species and individual species categorization and then summarize your findings from
each test. (8 pts.)
b) Use correspondence analysis to construct a display showing the difference in tooth
wear across individual species. Discuss what this plot, along with the mosaic plot, show
regarding differences in wear between the individual species examined. (4 pts.)
Application of the Normal Distribution (see Spina Bifida example in notes)
Problem 3 - Diabetes Screening Using Fasting Glucose Levels (12 pts.)
A standard test for diabetes is based on glucose levels in the blood after fasting
for prescribed period. For healthy people the mean fasting glucose level is found
to be 5.31  mole/liter with a standard deviation of 0.58  mole/liter. For
untreated diabetics the mean is 11.74, and the standard deviation is 3.50. In both
groups the levels appear to be approximately normally distributed.
To operate a simple diagnostic test based on fasting glucose levels we need to set
a threshold value, T, so that if a patient’s fasting glucose level is at least T we say
they have diabetes. If it is lower, we say they do not have diabetes. Suppose we
use T = 6.5.
a) What is the probability that a diabetic is correctly diagnosed as having
diabetes, i.e. what is the sensitivity of the test? (2 pts.)
b) What is the probability that a nondiabetic is correctly diagnosed as not having
diabetes, i.e. what is the specificity? (2 pts.)
Suppose we lower the cutoff value to T = 5.7.
c) What is the sensitivity now? (2 pts.)
d) What is the specificity now? (2 pts.)
e) In deciding what T to use, we have to trade off sensitivity for specificity. To
do so in a reasonable way, some assessment is required of the relative “costs” of
misdiagnosing a diabetic and misdiagnosing a non-diabetic. Suppose we wish to
have 98% sensitivity. What value of T would need to be used to for the test to
have a sensitivity of .98 or 98%? How specific is the test when T has this value,
i.e. what is specificity if you were to use this new value of T? (4 pts.)