# ANOVA Factorial Designs

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```					ANOVA: Factorial Designs
Experimental Design

 Choosing the appropriate statistic or
 The number of independent variables and levels
 The nature of assignment of subjects to
treatment levels
 http://psych.athabascau.ca/html/Validity/in
dex.shtml
Review of a Few Designs
 Subjects are completely randomly assigned to
treatments
Completely Randomized ANOVA: One treatment
with two or more levels
Completely Randomized Factorial ANOVA: Two or
more treatments each with two or more levels
 Treatments assigned to homogenous blocks of
experimental units or repeated measures used
to control for nuisance variation
Randomized Block ANOVA: One treatment with two
or more levels
The secret to mastering two-factor
analysis of variance
 In a two-factor analysis of variance, we look at
interactions along with main effects.
Interactions are the effect that one factor has on
another factor.
The main effect of a factor looks at the mean
response obtained by averaging each factor (IV) over
all levels.
 A two-factor ANOVA should begin with an
examination of the interactions. Interpretation of the
main effects changes according to whether
interactions are present.
Two-factor ANOVA
 Consists of three significance tests:
Each of the main effects (A and B)
Interaction of the two factors (AB)
 There is an F-test for each of the hypotheses: the
mean square for each main effect and the
interaction effect divided by the within-variance
(MSE).
Hypothesis testing
 The first hypothesis (main effect of factor A):
looks at the mean response for each level of A--that
is, the mean obtained by averaging over all levels of
B--and asks whether they are the same. This is the
case whether or not there is an interaction in the
underlying model.
 The second hypothesis (main effect of factor B):
looks at the mean response for each level of B--that
is, the mean obtained by averaging over all levels of
A--and asks whether they are the same. This is the
case whether or not there is an interaction in the
underlying model.
 The third hypothesis (interaction effect):
asks whether or not factor A has an effect on factor B.
A demonstration of a two-factor ANOVA

http://www.ruf.rice.edu/~lane/stat_sim/two
_way/index.html
Let’s use an example we’ve seen before

 In this study, interviewers telephone calls to
adults in randomly selected households to ask
opinions about the next election.
 Treatment A: introduction
A1: Gave name
A2: Identified the university he/she was representing
A3: Gave name and identified the university.
 Treatment B: copy of survey?
B1: Interviewer offered to send a copy of final results
B2: Interviewer did not offer to send a copy of the final
results
For default there is an effect of B but not
A. Make up data where…
…there is an effect of A but not B and no
interaction.
…there is an effect of B and an interaction,
but no effect of A.
…there is a cross-over interaction.
What part of the ANOVA summary table is
unaffected by group size?
Is it necessary for lines to cross to have a
significant interaction?
Conclusions
 If there is no significant interaction, the means for the
levels of factor A will behave like the expected values
from any of the individual levels of B.

 If there is a significant interaction, the hypotheses for the
main effects are the same (we still test whether the
means obtained from each level of A by averaging over
all levels of B are the same.) However, if the model
includes an interaction, this hypothesis might not be
useful.

 Look first at the interaction; never analyze or interpret
main effects in the presence of an interaction.
Never interpret main effects in the
presence of interactions
Survey response rate based on no compensation vs. \$5.
Suppose B1 (circles) were males and B2 (squares) were females.
 Graph 1: Since the means of the two levels of A are equal,
the main effect of A is 0. Yet, it would be a huge mistake to
say that A doesn't matter. Who benefits from what?
   Graph 2: How do males compare to females in terms of
response rate based on compensation? What would you do
with this information if you were advising how to reward
completed surveys?
   Graph 3: How do males compare to females in terms of
response rate based on compensation? What would you do
with this information if you were advising how to reward for
completed surveys?
   Graph 4: Do males and females differ in terms of response
rate? Explain.

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 views: 45 posted: 8/31/2010 language: English pages: 13