ANOVA: Factorial Designs

Experimental Design

Choosing the appropriate statistic or design involves an understanding of

The number of independent variables and levels The nature of assignment of subjects to

treatment levels

http://psych.athabascau.ca/html/Validity/index.shtml

Review of a Few Designs

Subjects are completely randomly assigned to treatmentsCompletely Randomized ANOVA: One treatment

with two or more levelsCompletely Randomized Factorial ANOVA: Two or

more treatments each with two or more levelsTreatments 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 testingThe 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: introductionA1: Gave nameA2: Identified the university he/she was representingA3: Gave name and identified the university.

Treatment B: copy of survey? B1: Interviewer offered to send a copy of final resultsB2: 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.