Homework 6
This homework is based on the
following experiment (data are taken from The
SAS System for Linear Models):
A balanced factorial experiment consists of all
possible combinations of levels of two or more factors. Factorial experiments are used to investigate
not only overall differences between levels of each factor (main effects), but
also how levels of one factor affect the response variable across levels of
another factor (interactions). Consider
the following example. Suppose that 3
seed growth promoting methods (METHOD) are applied to seeds from each of five
varieties (VARIETY) of turf grass. Six
pots are planted with seed from each METHODxVARIETY
combination. The resulting 90 pots are
randomly placed in a uniform growth chamber and the dry matter yields (YIELD)
are measured after clipping at the end of 4 weeks.
1.
This
experiment uses a factorial treatment design.
What are the 2 factors being studied in this experiment and how many
levels does each factor have? How many
total treatmentlevel combinations are being considered? How many times has each treatmentlevel
combination been replicated? Based on
the information that you’ve been given here, is the experimental design a
completely randomized design or a randomized complete block design?
The 2 factors are METHOD and VARIETY.
METHOD has 3 levels and VARIETY has 5 levels. The total number of treatmentlevel
combinations being considered is 15.
This experiment uses a completely randomized design.
2.
Construct
a 2way table of treatment means and generate plots of the treatment
means. Is there any evidence of
interaction? Explain.

Variety 1 
Variety 2 
Variety 3 
Variety 4 
Variety 5 
Method A 
21.8 
21.9 
23.1 
26.0 
22.3 
Method B 
15.1 
15.2 
15.5 
13.5 
19.2 
Method C 
18.4 
19.9 
17.3 
14.8 
12.6 
Because the lines are not parallel, there is evidence of potential
interaction between the seed growth promoting method and the variety of grass.
3.
Run
the 2way analysis of variance for these data.
First, conduct a hypothesis test to determine whether there is a
significant interaction between the 2 factors.
State the hypotheses, report the Fstatistic and pvalue for the test,
and state your conclusion. Is it
appropriate to also test and discuss the main effects? Why or why not?
Ho: no
interaction
Ha: interaction
Fvalue = 2.38 and pvalue = 0.0241
At the 5% significance level, reject Ho and conclude
that there is a significant interaction between method and variety. Therefore, since the response to method
depends on the variety of grass being tested, it is not appropriate to make
overall statements about the main effect of method or the main effect of
variety. The analysis must continue at
the simple effect level.