## Deep Thought and the ANOVA tables: the Answer to Life, the Universe, and Everything

"Yes," said Deep Thought. "Life, the Universe, and Everything. There is an answer. But," he added, "I'll have to think about it." -- The Hitchhiker's Guide to the Galaxy, Douglas Adams, 1983
Exercise 1:

#### Comparison of Regression Analysis and Analysis of Variance

Consider the data set: "Trout.txt", it contains a pair of original variables: Pond: [1,2,3] - the particular pond the trout came from; stLength - the standard length of the trout [cm]

As an exercise into the underlying math - compare ANOVA tables from:

• one-way ANOVA of standard length (stLength) of trout as a function of the pond (Pond) used to raise them
• linear regression of length as a function of pond while treating pond as a continuous variable (I agree, this analysis makes little sense with the previous setup. Consider the numbers in "Pond" a measure of the depth of the pond in meters). Easiest way is to use the lm(Y~X) command
• second order polynomial regression of length as a function of pond (as a continuous variable). Easiest way is to use the lm(Y~poly(X,2)) command.

Examine the ANOVA tables and understand how all terms are derived. Are any terms the same across ANOVA and 2nd order regression? Examine the formula for the second order equation, plug in values for x, and examine the resulting y values. Do you recognize these numbers? If so, understand why some ANOVA tables are identical.

Exercise 2 - optional:

Does eyesize differ among lineages of cichlids?

1. Download the file "CichlidEyesize.txt" at <http://caspar.bgsu.edu/~courses/Stats/Labs/Datasets/CichlidEyeSize.txt>. The first line contains the names of the three variables - Lineage, EyeDiameter in mm and BodyVolume in mm3.
2. Examine whether the size of the eye differs among lineages.
3. Consider that fish differ in body size, which presumably also impacts eye size.
4. Can we distinguish between the effects of body size and lineage? How would you go about answering this question?