In Chapter 12 we focused on "one-way" analysis of variance which is the appropriate analysis when you have only one variable (or factor) with multiple levels
In the current chapter, we will focus instead on situations where we have multiple variables, each with multiple levels
For example, "fairness" of midterm as a function of gender and year:
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Can ask three questions � (1) does opinions of fairness differ across the genders?, (2) does opinions of fairness differ across the years?, (3) is the effect of year on opinions different for the different genders?
Terminology
Simple Effects. We will also be talking about simple effects. Simple effects relate to questions like .. if we only consider second year students, are opinions concerning the exam different depending gender.
In class example with memory for words of different imageability and frequency.
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Plotting the Data
Main Effects:
- Is there an overall effect of frequency
- Is there an overall effect of imageability
- Is the effect of imageability significant for low frequency items?
- what about high frequency items?
What that means is that we included all combinations of different levels of our two variables (sometimes called a fully crossed design)
We could have tested all subjects in all conditions, that would be called a complete within-subjects design because all the variables were manipulated within subjects
Finally, we could have a mixed design in which one (or more) variables are within subjects, and one (or more) other variables are between subjects
Chapter 13 only considers between-subject designs, Chapter 14 will consider within and mixed designs
- how much of the total variance is due to the effect of the first variable? - SSA
- how much of the total variance is due to the effect of the second variable? - SSB
- how much is due to the combined effects of the two variables (including the interaction) - SScells
- how much of the variance is due to the interaction? - SSAB = SScells - SSA - SSB
- calculate SSwithin and SStotal
- usual ANOVA logic from there
Thus, for each variable we are going to compute an SS which is simply the sum of squares representing the degree to which the means at each level of the variable deviate from the grand mean, multiplied by the n per cell (because of CLT .. remember?)
The grand mean for our data is 2.75
For the frequency variable, the high freq mean is 2.76, the low freq mean is 2.75. So:
To get that we first calculate the SS for all of the cells in the design around the grand mean (multiplied by n per cell)
This "variance" is due to the interaction plus the two main effects, so by subtracting the main effects we are left with the SS for the interaction. So:
Degrees of Freedom
dfwithin = dftotal - (dffreq + dfimg + dfFxI) = 31 - 3 = 28
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Freq | ||||
image | ||||
F x I | ||||
Within | ||||
Total |
In the case of 2 by 2 ANOVAs there are actually three null effects; one of each main effect and one for the interaction
For example:
To properly interpret an interaction we usually need more specific information than that.
For example, consider the following interactions:
In order to accurately describe these interactions, we have to know whether the effect of variable B is significant at each level of variable A
This involves simple effects tests
SS freq at hi image:
For the above two examples Fobtained = 0.36
This is not significant implying that there was no frequency effect at either level of imageability
- Say that I am interested in understanding phobias and, as a first step, I want to see if fear builds over time when a phobic is put in a feared situation.
So, I get 24 clausteraphobics and 24 control subjects and randomly assign 8 of each to stand in a closed elevator for 2, 5 or 10 mins, then to rate there fear on a 10 point scale with 1 being fearless and 10 being terrified.
Say that I give you the following information:
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We will do the latter as it seems to make the most sense .. so:
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- So, we could describe the interaction by saying that fear increased over time for phobics, but fear did not change at all over time for the controls
- Notice that the simple effects test gives you more information about the interaction but it still doesn�t tell you specific information about which means are different from which other means
For that kind of information, you can use all of the same multiple comparison techniques described in Chapter 12 with a factorial design as well, and you use them in the exact same way
For example, if we did a Tukey test on the "Phobics in Elevators" dataset �
C-T1 C-T2 C-T3 Ph-T1 Ph-T2 Ph-T3
5 5 5 7 8 9
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C-T1 C-T2 C-T3 Ph-T1 Ph-T2 Ph-T3
5 5 5 7 8 9
-------------------- ----- ----- -----
First, for control subjects time had no effect at all as their mean fear level was not different across the three times examined
At all of the times tested, the phobic subject showed more fear than the control subjects
Each additional amount of time significantly increased the fear level of the phobic subjects such that they were more scared at 5 mins than 2 mins, and even more scared at 10 mins than 5 mins
The moral of the multiple comparisons part of this chapter is that when you do multiple comparisons in a factorial design, you basically act like it is a single factor design with each cell of the multi-factor design being like a level of the single factor design.
Magnitude of the Effect
This is also true in factorial designs with the only difference being that you know multiple effects that can be quanitified
Once again, one can use h 2 (The SS relevant to the effect divided by the SStotal) as a quick and dirty way of calculating how much of the total variation in the data was due the variable of interest
However, as mentioned, h 2 is biased in that in overestimates the true magnitude of an effect
The textbook goes into a description of a revised w 2 estimate that can be calculated for factorial designs
However, for our purposes, you don�t have to worry about understanding that
Instead, know why you would want to calculate the magnitude of an effect, know how to do so via h 2, know that h 2 is a biased estimator and that w 2 is better, and know that if you ever need to calculate w 2 the text shows you how
Power depends on the size of the effect you expect AND the number of subjects you plan to run
In Chapter 11 we said that to calculate power in a one-way ANOVA, we do the following:
So, lets say we are using a 2-way factorial design � now we have 3 null hypothesis � 1) the main effect of A, the main effect of B, and the interaction of A & B.
Assuming you have some estimate of the mean squared error �
All you need to do to find the power associated with these nulls is to estimate (based on past research or an educated guess) what you think our final means will look like. With those estimates in combination with our intended n, you can compute sum of squares and use the exact logic as we did before
The only real difference is that we now have 3 power analyses we could do (assuming 2 variables)
Note: Read the meat of these sections in the text (ignoring their computations if you like)
Consider the following example from the text:
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The apparent effect of state is due the "drinking" effect and the unequal ns in the various cells
We could similarly compute an "unweighted" column mean which would simple be the mean of the cell means, as opposed to the mean of all the numbers that went into the cell means
Note that when ns are equal, the weighted and unweighted means are the same
However, if we calculate unweighted means in the previous example, notice that they seem to provide a better depiction of the cell data (means of 17 for both states)
We could then do our analysis using the unweighted means instead
However, in order to do this we have to "act as though" we were in an equal n condition with those row and cell means .. but what n do we use?
Say we have three variables .. then we actually have 3 main effects, 3 two-way interactions, and one three-way interaction
So, he chooses to teach 4 versions of his class next year which represent the cells of a textbook (old vs new) by quiz (have vs not have) design. However, he also splits performance by mark in B07 (B or better vs. less than B)
Assume he gets the following data:
Less than B B or better
Text Text
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P.S. - forget about the computations for now .. just worry about being able to interpret the data.
For example, on a test you might get something like we have been discussing along with the following source table:
Note: I made up the entire source table below � if you did the computations on the above you would not get these numbers
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B07 x T x Q | ||||
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