two-way anova and interactions

8/2/2019 Two-Way ANOVA and Interactions

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Multivariate Statistics: Concepts, Models, and Applications

David W. Stockburger

Two Way ANOVA and Interactions

The Design

Suppose a statistics teacher gave an essay final to his class. He randomly divides the classes in half such

that half the class writes the final with a blue-book and half with notebook computers. In addition the

students are partitioned into three groups, no typing ability, some typing ability, and highly skilled at

typing. Answers written in blue-books will be transcribed to word processors and scoring will be done

blindly. Not with a blindfold, but the instructor will not know the method or skill level of the student

when scoring the final. The dependent measure will be the score on the essay part of the final exam.

The first factor will be called Method and will have two levels, blue-book and computer. The second

factor will be designated as Ability and will have three levels: none, some, and lots. Each subject will be

measured a single time. Any effects discovered will necessarily be between subjects or groups and hence

the designation "between groups" designs.

The Data

In the case of the example data, the Ability factor has two levels while the Method factor has three. The

X variable is the score on the final exam. The example data file appears below.

Example Program using General Linear Model in SPSS/WIN


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The analysis is done in SPSS/WIN by selecting "Statistics", "General Linear Model", and then "GLM -

General Factorial." In the next screen, the Dependent Variable is X and the Fixed Factors are Ability and

Method. The screen will appear as follows.

The only "Options" that will be selected in this example is the "Descriptive Statistics" option under

"Display." This will produce the table of means and standard deviations.

Interpretation of Output

The interpretation of the output from the General Linear Model command will focus on two parts: the

table of means and the ANOVA summary table. The table of means is the primary focus of the analysis

while the summary table directs attention to the interesting or statistically significant portions of the table

of means.

Often the means are organized and presented in a slightly different manner than the form of the output

from the GENERAL LINEAR MODEL command. The table of means may be rearranged and presented


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as follows:

None

SomeLots

blue-book 26.67 31.00 33.33 30.33

computer 28.00 36.67 27.00 30.56

27.33 33.83 30.17 30.44

The means inside the boxes are called cell means, the means in the margins are called marginal means,

and the number on the bottom right-hand corner is called the grand mean. An analysis of these means

reveals that there is very little difference between the marginal means for the different levels ofMethod

across the levels ofAbility (30.31 vs. 30.56). The marginal means ofAbility over levels ofMethod are

different (27.33 vs. 33.83 vs. 30.17) with the mean for "Some" being the highest. The cell means show anincreasing pattern for levels ofAbility using a blue-book (26.67 vs. 31.00 vs. 33.33) and a different

pattern for levels ofAbility using a computer (28.00 vs. 36.67 vs. 27.00).

Graphs of Means

Graphs of means are often used to present information in a manner that is easier to comprehend than the

tables of means. One factor is selected for presentation as the X-axis and its levels are marked on that

axis. Separate lines are drawn the height of the mean for each level of the second factor. In the following

graph, the Ability, or keyboard experience, factor was selected for the X-axis and the Method, factor was

selected for the different lines.

Presenting the information in an opposite fashion would be equally correct, although some graphs are

more easily understood than others, depending upon the values for the means and the number of levels of

each factor. The second possible graph is presented below.


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It is recommended that if there is any doubt that both versions of the graphs be attempted and the one

which best illustrates the data be selected for inclusion into the statistical report. In this case it appears

that the graph with Ability on the X-axis is easier to understand than the one with Method on the X-axis.

The ANOVA Summary Table

The results of the analysis are presented in the ANOVA summary table, presented below for the example

data.

The items of primary interest in this table are the effects listed under the "Source" column and the valuesunder the "Sig." column. As in the previous hypothesis test, if the value of "Sig" is less than the value of a

as set by the experimenter, then that effect is significant. If a =.05, then the Ability main effect and the

Ability BY Method interaction would be significant in this table.

Main Effects

Main effects are differences in means over levels of one factor collapsed over levels of the other factor.

This is actually much easier than it sounds. For example, the main effect ofMethod is simply the

difference between the means of final exam score for the two levels of Method, ignoring or collapsing

over experience. As seen in the second method of presenting a table of means, the main effect ofMethod

is whether the two marginal means associated with the Method factor are different. In the example case

these means were 30.33 and 30.56 and the differences between these means was not statistically

significant.


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As can be seen from the summary table, the main effect ofAbility is significant. This effect refers to the

differences between the three marginal means associated with Ability. In this case the values for these

means were 27.33, 33.83, and 30.17 and the differences between them may be attributed to a real effect.

Simple Main Effects

A simple main effect is a main effect of one factor at a given level of a second factor. In the example data

it would be possible to talk about the simple main effect ofAbility at Method equal blue-book. That

effect would be the difference between the three cell means at level a1

(26.67, 31.00, and 33.33). One

could also talk about the simple main effect ofMethod at Ability equal lots (33.33 and 27.00). Simple

main effects are not directly tested in this analysis. They are, however, necessary to understand an

interaction.

Interaction Effects

An interaction effect is a change in the simple main effect of one variable over levels of the second. An A

X B or A BY B interaction is a change in the simple main effect ofB over levels ofA or the change in the

simple main effect ofA over levels ofB. In either case the cell means cannot be modeled simply by

knowing the size of the main effects. An additional set of parameters must be used to explain the

differences between the cell means. These parameters are collectively called an interaction.

The change in the simple main effect of one variable over levels of the other is most easily seen in the

graph of the interaction. If the lines describing the simple main effects are not parallel, then a possibility

of an interaction exists. As can be seen from the graph of the example data, the possibility of a significant

interaction exists because the lines are not parallel. The presence of an interaction was confirmed by the

significant interaction in the summary table. The following graph overlays the main effect ofAbility on

the graph of the interaction.

Two things can be observed from this presentation. The first is that the main effect ofAbility is possibly

significant, because the means are different heights. Second, the interaction is possibly significant because

the simple main effects ofAbility using blue-book and computer are different from the main effect of

Ability.

One method of understanding how main effects and interactions work is to observe a wide variety of data

and data analysis. With three effects, A, B, and A x B, which may or may not be significant there are eight

possible combinations of effects. All eight are presented on the following pages.

Example Data Sets, Means, and Summary Tables

No Significant Effects


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Main Effect of A


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Main Effect of B


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A x B Interaction


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Main Effects of A and B


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Main effect of A, A x B Interaction


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Main Effect of B, A x B Interaction


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Main Effects of A and B, A x B Interaction


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No Significant Effects


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Note that the means and graphs of the last two example data sets were identical. The ANOVA table,

however, provided a quite different analysis of each data set. The data in this final set was constructed

such that there was a large standard deviation within each cell. In this case the marginal and cell means


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were not different enough to warrant rejecting the hypothesis of no effects, thus no significant effects

were observed

two-way anova and interactions

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