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Rationale of ANOVA
Modeling the effect of an Independent Variable


Representational Convention

Here are several representations of 5 observations for one condition in an experiment. The bottom row of the table shows how I'm going to display the range and mean of a data set.

This represents a set of observations. Each square represents a number. Smaller values are to the left; larger values are towards the right.

 

 

To this we can add an indication of the range of the values.

 

 

And indicate the value of the "Sample Mean"

 

 

Finally, this abbreviates the representation. Remember there are separate observations which are hidden to clarify other relationships.

What should we expect an Independent Variable to Do?

Suppose we try 4 new techniques for teaching experimental psychology. We use a common final exam and the score on that exam as the dependent variable in our experiment. The Independent Variable (IV) is the teaching technique.

Description

Original Teaching Method

New Method "Red"
New Method "Blue"
New Method "Green"
New Method "Lavender"

Comprehension Check! Briefly describe how the four "new" methods are different from the Original data set. Your description should incorporate ways of describing observations you've learned in this class.

Write out your descriptive sentences on paper.

Stop! Don't continue until you have written out your answers!

Description
Original Teaching Method
New Method "Red"
New Method "Blue"
New Method "Green"
New Method "Lavender"

Each technique affects student performance in a different way.
Complete the following table. Do the values you calculate agree with the above statements? At the very end of the page I've provided a completed table if you feel the need to check it.

Condition
Black
Red
Blue
Green
Lavender
Transform
None
Add 0.3
Add - 0.3
X 0.5
X 2
Smallest z-score
-1.0
-0.7
2nd Smallest z-score
-0.5
2nd Largest z-score
0.8
Largest z-score
0.7
Sample Mean
0.0
Standard Deviation
0.89

Describe how each method affects student performance.

Stop. Write out answers to these questions.

Proceed after you've written out your thoughts.

Condition
Effect on Weaker Students
Effect on Stronger Students
Why
Black

The baseline. Changes are described in terms of how the new method would change these scores

Lavender
Hinders
Helps

Increases variability; the weaker students score lower and the stronger score higher

Green
Helps
Hinders

Decreases variability; weaker students score higher and the stronger score lower

Blue
Hinders
Hinders

Decreases the mean; both weaker and stronger students score lower. The entire data set is shifted

Red
Helps
Helps

Increases the mean; both weaker and stronger students score higher. The entire data set is shifted.

As you can see, if the independent variable changes the mean, all scores are affected in the same way. If the independent variable changes how variable the data set is, the scores are effected in two different ways. And, of course, an independent variable could effect both the mean and the variability.



Modeling the Effect of an Independent Variable

ANOVA Assumes:
  • The Independent Variable affects Mean Performance.
  • The Independent Variable Does NOT affect Variability.

Most often in research the effect of the independent variable is ssumed to change the central tendency rather than the dispersion of the sampled data. The assumption generally works well in "modeling" how changes in the independent variable affects the data.


An Exerimental Context

The simulated results are from a reaction time experiment similar to that first conducted by Shepard & Metzler (1971). The data are reaction time measures: how long did it take to press the "Same" key when both stimuli were the same though one was rotated 0, 45, or 90 degrees. A more detailed description of the "Mental Rotation" experiment can be read here.


Results of the Experiment when the Independent Variable is Effective

If the independent variable is effective then the observations in the groups will be offset from one another by some constant difference as shown in this picture of the results of collecting the data once.

  • The individual groups are all about equally variable (approximately the same range).
  • The overall variability is greater than the variability of the individual groups.

Here is a movie you can watch which shows many (simulated) replications of the same experiment where the independent variable is VERY effective. Only the means are shown. As you play the movie note how each of the three means jumps around--but each jumps around different "typical" values.


Results of the Experiment When the Independent Variable is NOT Effective?

If the independent variable is NOT effective then the observations in the groups will differ from one another only by random fluctuation. Of course with only one replication of the experiment you can't tell whether or not that ordering of means would repeat itself.

  • The three groups of subjects all have about the same variability (shown here as approximately equal ranges)
  • The variability of the total data set is about the same as the variability of an individual group.
  • Note: The ordering of the three means doesn't replicate itself. On each replication, the ordering can shift.

But here is a movie you can watch which shows many replications of this experiment where the independent variable was NOT effective. Only the means are shown. Note how the means all jump around the same typical value. As you watch the movie you'll notice there is no tendency for any one of the groups to be in the middle, left, or right of the RT dimension.


But now what?

This final movie shows you a situation where the independent variable may or may not be effective. Unlike the second movie, if the independent variable is effective, it is not a "whopper" effect.

Even after running the experiment many times, its hard to tell whether or not the independent variable is effective. Imagine trying to reach that decision after only running the experiment one time. There is a "hint" in the data as the movie plays through the replications. Do you see it?


What is the Next Step in Understanding Analysis of Variance?

The next step in understanding the conceptual basis of ANOVA is to see how the characteristics of the data can be used in an appropriate sampling distribution.

If the Independent Variable is INEFFECTIVE then

If the Independent Variable is Effective then


Key

Condition
Black
Red
Blue
Green
Lavender
Transform
None
Add 0.3
Add -0.3
X 0.5
X 2
Smallest z-score
-1.0
-0.7
-1.3
-0.5
-2.0
2nd Smallest z-score
-0.5
-0.2
-0.8
-0.25
-1.0
2nd Largest z-score
0.8
1.1
0.5
0.4
1.6
Largest z-score
0.7
1.0
0.4
0.35
1.4
Sample Mean
0.0
0.3
-0.3
0.0
0.0
Standard Deviation
0.89
0.89
0.89
0.45
1.78

The hint in the final movie is to notice the blue circle is never on the far left. If the independent variable were ineffective then the blue group should be on the left about one-third of the time.


©1997 - 2006 by Burrton Woodruff. All rights reserved. Last modified Tue, Sep 26, 2006