Stat 502
Homework 7
Assigned 11/18/08
Due 11/25/08
1. Crops: An experiment was performed to analyze the effect of nitrogen density (0, 50, 100 parts
per million) on nitrogen uptake by four crops: alfalfa, soybean, guar, and mungbean (a,s,g,m).
Four replications were made for each of the 12 nitrogen*crop treatment combinations, and
the Acetylene reduction was measured for each replication when the plants were flowering.
Acetylene reduction is generally low if the amount of nitrogen fixed by the plant is large.
(a) Make some exploratory plots and tables describing the data.
(b) Write a full model for the experiment, explain the terms, and compute the ANOVA
table.
(c) Perform diagnostic tests and find an appropriate transformation of the data. Compute
the ANOVA table for the transformed data and perform diagnostic checks.
(d) After applying a transformation, test for significant additive crop and nitrogen effects,
as well as crop*nitrogen interaction effects.
(e) Compute the LSD for each factor, and make LSD groupings as described in class. De-
scribe any interaction in terms of how the effects of nitrogen differ across the levels of
crop. Feel free to perform t-tests to analyze any interaction, but you aren’t required to
do so for this homework.
(f) Comment on the responses from treatment combinations (a,50) and (a,100). Such a
phenomenon is possible if the presence of a threshold level of nitrogen in the plant
completely inhibits Acetylene reduction. How does this affect the assumptions about
homogeneity of the variances? Can you think of a way to analyze the data to get around
this problem? (Hint: think of analyzing a subset of the treatment combinations).
2. Orthogonality: Consider analysis of a two-factor design.
(a) Recall the decomposition based on the additive model:
y−1¯y··· =ˆ
a+ˆ
b+ˆ
Show that the vectors on the right-hand side of the equation are orthogonal, and that
SSTotal=SSA+SSB+SSE.
(b) Show the analogous result for the decomposition based on the interaction model.
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