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 treatment-level combinations are being considered?  How many times has each treatment-level 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 treatment-level combinations being considered is 15.  This experiment uses a completely randomized design.

2.     Construct a 2-way 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 2-way 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 F-statistic and p-value 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

F-value = 2.38 and p-value = 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.