Sokal and Rohlf (1981) describe an experiment that utilized a multi-level nested design. Three treatments (Control, Compound 217, and Compound 217 plus sugar) were randomly assigned to six rats such that each treatment was administered to two rats. After a set period of time, multiple preparations were made on the liver of each rat (the data set you will use is a subset of the original data, in our data subset there are two preparations per rat). Two readings were then made on each preparation of the glycogen content, measured in what Sokal and Rohlf (1981) call "arbitrary units". Note that the rats are the experimental units; preparations and readings constitute two levels of subsampling.

The following table provides an EMS column appropriate for this experiment. Note that at the top level the treatments (Factor A) is a fixed effects factor.

(a) State a model compatible with the EMS table above. (Recall model = model equation +

model assumptions.)

(b)What set of null and alternative hypotheses would test for treatment effects? What set of null and alternative hypotheses would test if there is greater variation in glycogen content among rats than within rats? There is one more set of testable hypotheses; state this set.

(c) Using the data subset posted on our course's CourseLink website, obtain the appropriate ANOVA table for testing the three sets of hypotheses in (b) with the assistance of R's aov() function. The initial output may have some errors in how one or more F statistic values were calculated; if so, describe the nature of the errors and correct them for your final ANOVA table.

(d) Draw appropriate conclusions from the results of y

our tests above.

(e) Estimate the three variance components in the model, and calculate the percentage that each contributes to the sum of the variance components.

(f) Suppose the researchers who had conducted the experiment had published their test for treatment effects using the F statistic originally provided by R too. Subsequently, in a published Letter to the Editor, the researchers were accused of committing pseudoreplication in their test for treatment effects. Briefly explain why this is a valid criticism.

A B C D E 1 Treatmer Rat Preparati Reading Glycogen 2 1 136 1 1 1 2 142 4 1 1 2 1 131 5 1 2 125 160 1 150 140 9 143 10 147 11 153 12 154 13 142 14 162 15 2 152 16 147 17 147 18 135 19 3 136 20 3 138 21 3 2 138 22 3 N N N N PPP 1 134 23 3 127 24 3 139 25 3 2 138Source df EMS A a - 1 op + do, + cdo,+ bedEA/(a - 1) B (within A) a(b - 1) on + do? + cdo. C (within B) ab(c - 1) D (within C) abc(d - 1)