STAT 512 Homework Assignment 7, Due in class Friday, Oct. 23
Daniel (1976) described an experiment carried out to investigate the effects of 4 experimental factors
on a process for drilling through stone (probably for application in the petrolium industry). The four
two level factors were type of ”mud” used (a slury of water an other material used for lubrication
and heat transfer), the flow rated (of the mud), the load, and the drilling speed. The response
variable of interest was labeled “advance”, a measure of the rate of progress under each of the 16
experimental conditions defined by the factors. The data, as repeated by Mee (2009) are as follows:
mud type
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
2
flow rate
1
1
1
1
2
2
2
2
1
1
1
1
2
2
2
2
load
1
1
2
2
1
1
2
2
1
1
2
2
1
1
2
2
speed
1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
advance
1.68
4.98
1.98
5.70
3.28
9.97
3.44
9.07
2.07
7.77
2.44
9.43
4.09
11.75
4.53
16.30
Mee noted that standard modeling assumptions are much better-met if the response (advance) is
transformed to a logarithmic scale, so use ln(advance) in this problem. For parts 1-5, assume that
the experiment was executed as a CRD.
1. Since the experiment was unreplicated, formal inference cannot be undertaken based on a
fit of the full model. Instead, use R (or another computer package if you prefer) to produce
a half-normal plot of the effect estimates, and indicate which factorial effects you think are
non-zero based on your plot. Include your code script, along with the plot you produce.
2. Use Lenth’s method to continue your analysis of the effect estimates from part 1. Show
how you calculate the pseudo standard error, and indicate which factorial effects should be
considered significant based on a t value of 1.9.
3. If each of the “significant” effects from part 2 are included in the model, what other effects
(if any) should also be included to satisfy the hierarchy principle? the heredity principle?
4. Suppose now, and in parts 5 and 6, that each data value listed is actually the average of 2
independent measurements (i.e. that the experimental design was actually a replicated full
factorial with a total of 32 observations), and that the residual sum-of-squares for the full
data set is 0.500. What is the value of the actual standard error against which each factorial
effect estimate should be compared?
5. Test the hypothesis that all interactions (of every order) are zero. Give values for the degrees
of freedom, the F -statistic, and the p-value.
6. Finally, suppose this experiment had actually been performed in the following way:
• The experiment was run over four days, two of the days usng mud 1 and two using mud
2.
• Each day was divided into four 2-hour sub-tests, with one of the four combinations of
flow rate and load randomly assigned to each 2-hour period.
• The measurement reflects progress in the drilling operation over the course of one hour,
with each drilling speed being randomly assigned to one hour of each 2-hour sub-test.
How many degrees of freedom would have been available to construct a standard error for
each of the 4 main effects in this study?
References:
• Mee, R.W. (2009). A Comprehensive Guide to Factorial Two-Level Experimentation, Springer,
New York.
• Daniel, C. (1976). Applications of Statistics to Industrial Experimentation, Wiley, New York.