Use the corn borer data in Exercise 9.4. Make the data unbalanced by assuming the first two observations in Days = 3 (the 17 and 22) and in Days = 6 (the 37 and 26) are missing. Analyze the data using (a) unweighted analysis of cell means, (b) weighted analysis of cell means, and (c) a general linear models procedure such as PROC GLM. Obtain the simple treatment means and the least squares treatment means. Do they differ? Why or why not?
Exercise 9.4
The accompanying table gives survival data for tropical corn borer under field conditions in Thailand (1974). Researchers inoculated 30 experimental plots with egg masses of the corn borer on the same date by placing egg masses on each corn plant in the plot. After each of 3, 6, 9, 12, and 21 days, the plants in 6 random plots were dissected and the surviving larvae were counted. This gives a completely random experimental design with the treatments being “days after inoculation.” (Data are used with permission of Dr. L. A. Nelson, North Carolina State University.)
(a) Do the classical analysis of variance by hand for the completely random design. Include in your analysis a partitioning of the sum of squares for treatments to show the linear regression on “number of days” and deviations from linearity.
(b) Regard “days after inoculation” as a class variable. Define Y, X, and β so that the model for the completely random design Yij
= µ + τi
+ ϵij
can be represented in matrix form. Show enough of each matrix to make evident the order in which the observations are listed. Identify the singularity that makes X not of full rank. \
(c) Show the form of X and β for each of the three reparameterizations—the means model, the τi = 0 constraint, and the τ5
= 0 constraint.
(d) Choose one of the reparameterizations to compute R(τ |µ) and SS(Res). Summarize the results in an analysis of variance table and compare with the analysis of variance obtained under (a).
(e) Use SAS PROC GLM, or a similar program for the analysis of less than full-rank models, to compute the analysis of variance. Ask for the solution to the normal equations so that “estimates” of β are obtained. Compare these sums of squares and estimates of β with the results from your reparameterization in Part (d). Show that the unbiased estimates of µ + τ1
and τ1
− τ2
are the same from both analyses.
(f) Now regard X as a quantitative variable and redefine X and β so that Y = Xβ +
expresses Y as a linear function of “number of days.” Compute SS(Regr) and compare the result with that under Part (a). Test the null hypothesis that the linear regression coefficient is zero. Test the null hypothesis that the linear function adequately represents the relationship.
(g) Do you believe the assumptions for least squares are valid in this example? Justify.