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We will consider two populations, but here we will call them responses
due to two different treatments. So suppose we have two treatments, say,
T_{1} and T_{2}. Let X be the response under T_{1} and Y be the response
under T_{2}. T_{1} may be a placebo (standard, control, old, etc.). It is easy
to think of examples. For instance, consider a new diet drink. Let X
be the reduction in weight following a low fat diet and Let Y be
the reduction in weight following a low fat diet and which uses the diet
drink. Or, let X be the durability of house paint XX and let Y
be the durability of house paint YY. You only have to paint a house once
to realize the importance of this experiment.
Now in a controlled experiment the treatments have an effect on the
response variables, but all other variables are kept in control (at the
same level), as much as possible. When investigators get ready to do a
controlled experiment they often sit and discuss all variables which could
have a bearing on the response. This is a very important part of the experiment.
For example consider the diet example. What else has a bearing on weight
reduction? Exercise, life style, age, heredity, sex, physical condition,
etc. There are many, many variables. These will have to be controlled as
well as possible. In certain cases, you may not be able to control a variable.
Such variables are called covariates and there are certain designs where
their effects are taken into account, but we will not consider these in
this course. But needless to say, uncontrolled variables may jeopardize
the experiment.
We will assume the location assumption of the last chapter, which we
rewrite as,
The distributions of Y and X differ by at most a shift
in locations (centers), say .
Again, we at least have visual checks, comparison boxplots, dotplots
with which to assess this assumption.
Our target parameter is ,
this is the effect. There is a
natural null hypothesis, i.e.,

H_{0}:
.
Alternatives may be one or two sided. For convenience, lets assume the
alternative is

H_{A}:
.
So we want to test hypotheses, estimate the effect, and find a confidence
interval for it.
In this section, we consider a completely randomized design , (CRD).

We randomly select N experimental units at random from our reference
population. We randomly assign m of these units to Treatment 1 and
the other n of these units to Treatment 2. The experiment (study)
is run for a pre assigned time and during this time all other variables
are kept under control. At the end of the assigned time, we measure the
responses for the
m units which were assigned to Treatment 1, call
them
X_{1}, ... , X_{m}. And we measure the responses
for the
n units which were assigned to Treatment 2, call them
Y_{1},
... , Y_{n}. It is assumed that these responses are independent
of one another.
Too wordy? Lets look at the experiment which produced the quail data discussed
in Chapter 1. Recall it was an experiment involving a drug which hopefully
reduces LDL cholesterol levels. There were two treatments: Placebo (Treatment 1) and
an active drug compound which hopefully reduces LDL cholesterol (Treatment
2). Let
denote the effect (i.e. typical quail's LDL level
on placebo minus typical quail's LDL level on treatment or the true mean
level of a quail on placebo minus the mean level of a quail on the treatment).
Our hypothesis of interest is H_{0}:
versus
H_{A}:
.
We will also estimate
and find a confidence
interval for it using the Wilcoxon analysis of the last chapter.
The Experiment: 30 quail were randomly selected (these are the experimental
units) from a reference population. 20 were randomly assign to Treatment
1 (a placebo) and the other 10 to Treatment 2. For those on Treatment 2
an active drug compound was mixed with their diet. Those on Treatment 1
had the same diet without the drug compound. Over the course of the experiment,
the quail were treated the same. Same amount of exercise, same types of
pens, etc. At the end of the time period their LDL cholesterol levels were
measured.
The CRD of course produces two samples. The statistical analyses described
in the last Chapter would be appropriate. Remember to do comparison dotplots
or boxplots to check on the location assumption. Lets look at the quail
experiment.
The data are:
Placebo: 64 49 54 64 97 66 76 44 71 89
70 72 71 55 60 62 46 77 86 71
Treatment2: 40 31 50 48 152 44 74 38 81 64
A comparison dotplot is:
. :
Placebo : . : ..: :... . . .
++++++C10
Treat. 2 . .. ... . . . .
++++++C10
25 50 75 100 125 150
There is not much data here. Ignoring the outlier, the scales do not seem
to be that different. The LDL levels of the treated quail seem to be shifted
lower.
The value of the Wilcoxon is
.
Using
class code (TwoSample hypothesis test and confidence interval for the location parameter based on the Wilcoxon), the pvalue is .055. Hence, there is evidence at the .05 level of
significance that the treated quail have lower LDL levels than the placebo
group. So the drug would be earmarked for further study. The estimate of
the effect
is 14 and a confidence interval is
(8.5, 25.5).
Notice that the confidence interval contains 0. This does not contradict
the test because it was a onesided test.
Exercise 11.2.1
 1.
 From Rasmussen, Statistics with Data Analyses, CA: BrooksCole. Investigators wanted to compare the drugs morphine and nalbuphine on their effect in
changing pupil size. So they selected 11 volunteers and randomly assigned 6 of them to several does of morphine and the other 5 to several doses of
nalbuphine. Before receiving the drug their pupil sizes were measured. After waiting a prescribe amount of time after the dosages of the drugs they
measured the change in diameters of the subjects pupils. The data are:
Treatment Change in pupil diameter
Morphine .08 .8 1.0 1.9 2.0 2.4
Nalbuphine .3 .0 .2 .4 .8
 (a)
 Obtain comparison dotplots of the data. Comment.
 (b)
 Let
be the effect of the different drugs on pupil size (morphine minus nalbuphine). We want to test
 H_{0}:
versus
 H_{A}:
Obtain the Wilcoxon test statistic and compare it to what we would expect under Ho. Use the class code (TwoSample Hypothesis and CI (Wilcoxon)) to determine the pvalue.
Conclude in terms of the problem.
 (c)
 Obtain the estimate of and a confidence interval for the effect, using the Wilcoxon. Conclude in terms of the problem.
 2.
 Suppose we wanted to investigate the difference in the thicknesses of a pages in two books, but all we had was a ruler with eighthsofinches. Set up an
experimental design to do this. (Note you can measure the thickness of a bunch of pages even though you cannot measure a page). What plots could you
use here? What are the parameters of interest? What are the hypotheses? What analysis would you use?
 3.
 From Rasmussen, Statistics with Data Analyses, CA: BrooksCole. Researchers wanted to study the effect of regular alcohol assumption on plasma estrogen.
The participants in the experiment were 20 adult male squirrel monkeys, of similar age and good health (What variables are being controlled here?). They
randomly divided the monkeys into two equal sized groups. Monkeys in the alcohol group consumed a steady diet of 12% ethyl alcohol while those in the
control group had the same diet with no alcohol. The results are:
Alcohol: 3.17 2.52 2.59 4.25 3.27 4.92
5.46 2.83 4.80 2.26
Control 6.57 5.81 5.63 5.75 4.54 5.35
4.16 5.12 4.69 4.52
 (a)
 Obtain comparison dotplots of the data. Comment.
 (b)
 Let
be the effect of the different drugs on pupil size (morphine minus nalbuphine). We want to test
 H_{0}:
versus
 H_{A}:
Obtain the Wilcoxon test statistic and compare it to what we would expect under H_{0}. Use the class code (TwoSample Hypothesis and CI (Wilcoxon)) to determine the pvalue.
Conclude in terms of the problem.
 (c)
 These sort of experiments produced what warning level?
Next: Randomized Paired Design
Up: Design of Experiments
Previous: Introduction
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