Completely Randomized Designs

We will consider two populations, but here we will call them responses due to two different treatments. So suppose we have two treatments, say, T₁ and T₂. Let X be the response under T₁ and Y be the response under T₂. T₁ 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 $\Delta$.

Again, we at least have visual checks, comparison boxplots, dotplots with which to assess this assumption.

Our target parameter  is $\Delta$, this is the effect. There is a natural null hypothesis, i.e.,

• H₀: $\Delta = 0$.

Alternatives may be one or two sided. For convenience, lets assume the alternative is

• H_A: $\Delta \neq 0$.

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 X₁, ... , X_m. And we measure the responses for the n units which were assigned to Treatment 2, call them Y₁, ... , 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 $\Delta$ 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 H₀: $\Delta = 0$ versus H_A: $\Delta > 0$. We will also estimate $\Delta$ 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 $T = \char93 \{ \textrm{Placebo }> T_2 \}= 134.5$. Using class code (Two-Sample hypothesis test and confidence interval for the location parameter based on the Wilcoxon), the p-value 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 $\Delta$ 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 one-sided test.

Exercise 11.2.1  

1.

From Rasmussen, Statistics with Data Analyses, CA: Brooks-Cole. 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 $\Delta$ be the effect of the different drugs on pupil size (morphine minus nalbuphine). We want to test

• H₀: $\Delta = 0$ versus
• H_A: $\Delta \ne 0$

Obtain the Wilcoxon test statistic and compare it to what we would expect under Ho. Use the class code (Two-Sample Hypothesis and CI (Wilcoxon)) to determine the p-value. 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 eighths-of-inches. 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: Brooks-Cole. 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 $\Delta$ be the effect of the different drugs on pupil size (morphine minus nalbuphine). We want to test

• H₀: $\Delta = 0$ versus
• H_A: $\Delta \ne 0$

Obtain the Wilcoxon test statistic and compare it to what we would expect under H₀. Use the class code (Two-Sample Hypothesis and CI (Wilcoxon)) to determine the p-value. Conclude in terms of the problem.

(c)

These sort of experiments produced what warning level?