Can One Cell Create an Interaction?
Dear hard-working students,
On some scratch paper, draw a square. Next, draw a horizontal line across the middle of your square, thereby creating two rows. Now, draw a vertical line through the middle of your initial square, thereby creating two columns. Finally, label the two rows "Males" and "Females" and the two columns "Tall" and "Short." What you have at this point is the "diagram" for a 2x2 ANOVA in which the two factors are "Gender" and "Height."
Now, imagine that there are 20 different people in each of the 4 cells. Further imagine that a competent psychologist is hired to measure the intelligence of all 80 people. Finally, imagine that the mean IQ of the 20 people in each cell turns out to be 100. Knowing this, you should put the number 100 into each of the 4 cells in your diagram,
Now, if someone were to ask you if there is any interaction in your 2x2 ANOVA, I hope you would quickly respond by saying "No!" (If you can't see why that's the proper answer, read pages 398-389 in the text.)
Grab your pen or pencil one last time, and now change the number in the lower right-hand cell of your 2x2 diagram from 100 to 115. This change might be necessary because the psychologist initially made a mistake in computing the mean for the 20 short females, or perhaps you didn't hear the psychologist collectly when she told you what that particular mean was. In any event, imagine now that the 4 means are 100, 100, 100, and 115.
Now, suppose that the same onlooker looks at your revised diagram and asks the question, "Now that you've changed that one cell mean, is there any interaction present in your 2x2 ANOVA?" This time, you would (I hope) respond quickly be saying "Yes!" (If you can't see why that's the proper answer, read pages 398-389 in the text.)
But let's suppose that you now get the 3rd and final question from the person who's seen your original and corrected 2x2 diagram of the cell means. And that question is simply, "Where is the interaction?"
In response to this query, lots of folks are tempted to think that the interaction "resides" in the cell that had its mean changed from 100 to 115. That's a tempting thought because there was no interaction to start with, then one cell mean was changed, with that one change causing interaction to appear. Even though it may SEEM logical to think that the interaction in the revised 2x2 diagram resides in the cell now having a mean of 115, that's not the right way to think about interaction.
If interaction exists in a 2x2 ANOVA, then it exists in the collection of cells. To prove this to yourself, imagine that you're given the power to change just one of the cell means in the revised 2x2 diagram along with the directive to eliminate the interaction. What would you do?
To eliminate the interaction, you could, of course, change the mean for the short females from 115 to 100. However, you could change any of the other cells instead and still accomplish your goal of eliminating the interaction. To be more specific, you could change the mean for the tall males from 100 to 85 or you could change the mean for the short males from 100 to 115 or you could change the mean for the tall females from 100 to 115. Any one of these 4 possible changes would cause the interaction to vanish entirely. And if you can eliminate the interaction by changing any of the 4 cell means, there's no logical defense for thinking that the interaction "resides" in any particular cell.
For those of you who go on to a more adanced course in which two-way ANOVAs are covered, you'll learn that each cell mean can be "decomposed" into three components: a part that corresponds to a "row effect," a part that corresponds to a "column effect," and a part that corresponds to an "interaction effect,". You'll also learn that if there's interaction present to any degree in a 2x2 ANOVA (as ememplified by the revised means in the gender-height-IQ example we looked at here), the estimated interaction effects are identical for all 4 cells!
I hope this message has helped dispell the incorrect notion that the interaction in a 2x2 ANOVA might reside inside just one of the 4 cells. It never does that! You might keep this fact in mind if you every read or hear someone argue that the interaction in such an ANOVA was caused by or is attributable to the "unusual" nature of the scores provided by just one of the samples.
Copyright © 2012
Schuyler W. Huck