Solution to Problem 4; Sample Final Exam from Spring 2011
4. [25 pts] Rosmarinic acid is an antioxidant found in herbs of the mint family. Ursolic
acid, found in apples and other fruits and herbs, is capable of inhibiting growth of
some cancer cells. An experiment is performed to see the effects of concentration
levels of Rosmarinic acid, RA and Ursolic acid, UA on cells. Sixteen rectangular
plates, each with six wells, will be used. The experimenter wants to avoid crosscontamination of compounds so only one compound will be used on a plate. There
are six concentration levels: 1, 2, 5, 10, 20 and 50 M . The experiment is done as a
split plot design. Each acid is randomly assigned to eight of the rectangular plates.
The six concentration levels are randomly assigned to the six wells within each plate
for each compound. The response is the logarithm of the number of viable cells at the
end of the incubation time. Refer to the JMP output “Effects of acids on cell
numbers.”
a) [2] What is the whole plot?
A whole plot is a rectangular plate.
b) [2] What is the whole plot factor?
The whole plot factor is the type of acid, Rosmarinic acid or Ursolic acid.
c) [2] What is the sub plot?
A sub plot is a well in the rectangular plate.
d) [2] What is the sub plot factor?
The sub plot factor is the concentration of acid, 1, 2, 5, 10, 20 and 50 M .
e) [4] The two compounds are statistically different. Give the appropriate F statistic
that supports this result.
1
f) [3] There are some statistically significant concentration effects. Below are the
means for the six concentrations. Calculate the estimated effects of each
concentration level.
Concentration
1
Mean
12.32
Estimated Effect
1.59
2
12.00
1.27
5
11.56
0.83
10
10.58
–0.15
20
9.26
–1.47
50
8.66
–2.07
Overall
10.73
g) [5] Compute the value of the HSD (use q* = 2.93014) and indicate which
concentration means have statistically significant differences.
√
√
(
)
Concentrations 1 and 2 are not statistically different, difference in means =
0.32 < 0.52. Concentrations 2 and 5 are not statistically different, difference
in means = 0.44 < 0.52.
All other comparisons between pairs of
concentrations have statistically significant differences, difference in means >
0.52.
h) [5] There is a statistically significant interaction between Compound and
Concentration. Below is the interaction plot. Describe the plot and comment on
the nature of the interaction.
2
Both acids have a decreasing trend between the mean Ln(Cell) and
concentration. As concentration increases the log of the number of cells
tends to decrease. Rosmarinac acid starts off lowere and decreases more
slowly. Ursolic acid starts off higher and decreases more rapidly. At lower
concentrations there is a big difference between the means for the two acids.
For higher concentrations there is very little difference between the means
for the two acids.
Effects of Acids on Cell Numbers
Response Ln(Cell)
Summary of Fit
RSquare
RSquare Adj
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
0.935289
0.912178
0.504229
10.73068
96
Analysis of Variance
Source
DF Sum of Squares
Model
25
257.23058
Error
70
17.79726
C. Total
95
275.02784
Effect Tests
Source
Compound
Concentration
Compound*Concentration
Plate[Compound]
Mean Square
10.2892
0.2542
DF Sum of Squares
1
45.83500
5
181.18668
5
23.58829
14
6.62061
F Ratio
40.4695
Mean Square
45.83500
36.23734
4.71766
0.47290
Prob > F
<.0001*
F Ratio
3