Psy 232 Notes GW15 Factorial AOV

Psy 232 Class notes GW15: Factorial Analysis of Variance

Thus far: analyses of univariate, single-factor designs

t-test: one-group, two-group

F-test: one IV, with three or more groups (One-way AOV)

But life is complex -- usually multiple factors (IVs) simultaneously affect behaviors
Research designs intended to address this complexity entail statistical analyses that are also complex

Multi-factor analysis of variance - usually "factorial" AOV

Remember that in AOV, IV is called “factor.”
In AOV, designs with multiple IV’s are called “factorial designs.”

For example, if one were interested in Response as the DV, and two IV’s:
       Treatment, with 3 groups, as one IV
       Age, with 4 groups, as a second IV
That is, one would have two factors:
     Treatment - 3 groups
     Age - 4 groups

In factorial AOV, the groups within each factor are called levels.
Thus in the preceding example,
Treatment has 3 levels and Age has 4 levels.

We refer to such designs symbolically.
In the forgoing example, it would be a 3 X 4 factorial design.

The first factor, Treatment, has 3 levels,
and the second factor, Age, has 4 levels.

In the design schematic, each numeral refers to an IV, and each number itself refers to the # of levels within each IV.
So the first numeral (3) refers to the first IV - Treatment, and the second numeral(4) refers to the 2nd IV - Age.
The specific number "3" tells us that there are 3 treatments, and the specific number "4" tells us that there are 4 ages.

How many groups in total would there be in this design?
Let A represent Age, and B represent treatment:

               A1        A2       A3        A4
B1        B1A1   B1A2   B1A3   B1A4
B2        B2A1   B2A2   B2A3   B2A4
B3        B3A1   B2A2   B3A3   B4A4     =   12 groups for all possible combinations of A X B

That is, 3 x 4 = 12 groups

In-class practice of the foregoing . . .

The logic for the factorial AOV is exactly the same as for the one-way AOV:

F = (variance between sample means) / (variance within groups i.e., sampling error)
or

F = (variance between treatment groups) / (variance within treatment groups)

For the purposes of explanation, well limit the discussion to 2-factor problems,
but the same principles apply to 3 or more factors.

Since there is more than one factor (IV), we speak of the effects of each separate factor.

Called Main Effects. The factors are labeled alphabetically: Factor A, Factor B, etc.

That is, Factor A = IV(A), Factor B = IV(B), etc.
Main Effects refer to possible diffs among the means in each separate factor (IV).

There are now three pairs of null versus alternative hypothesis tests rather than just one pair:

The null hypothesis for Factor A & for Factor B taken separately would be

(1) The means of the DV for the groups in Factor A do not reliably differ.

(2) The means of the DV for the groups in Factor B do not reliably differ.

So what's the third hypothesis? Its called an Interaction.

The effects of one factor change across its levels, depending on its relation to the other factor.
We call this the interaction of A by B, and symbolically, its A X B.

Thus the third null hypothesis refers to the interaction:

(3) The effects of factors A and B on the DV do not reliably interact with one another.

In-class examples via graphs of Main effects versus Interactions . . .

For ALL F-test formulae (both single-factor & factorial designs), the structure is the same.

F = MS_B/MS_W

Between-treatments effects (numerator of the F-formula) are caused by three things:
The actual treatment effects (the IVs)
Individual differences
Experimental error

Within-treatment effects (denominator of the F-formula) are caused by only two things:
Individual differences
Experimental error

Thus, the formula is a ratio of the three numerator effects divided by the two denominator effects:
F = (tx effects + Individual diffs + Exper error) / (Individual diffs + Exper error)

If tx effects are large, then the numerator will be greater than the denominator, and thus, F > 1.

If tx effects are small, then the numerator will approximately equal the denominator, and thus, F = 1.

NOTE: Recall that in stats, we often refer to both items in the denominator collectively as Error variance.
Don't confuse this with Experimental error, which is only one part of the total Error variance.

Partitioning of the variance:
Total variance in the DV = Factor A variance + Factor B variance + A X B variance + Error variance

Based on this, we can compute the percentage of variance accounted for by each component:
100% total var = % due to Factor A + % due to Factor B + % due to A X B interaction + % due to error
We use η2 as our measure of % of variance explained by each component

In the social sciences, and especially in psychology, Error variance usually accounts for most of the variance.
That is, differences among the scores usually reflect the effects of individual differences and experimental error
far more than the effects of the IVs and their interaction.
Examples

Assumptions - same as for one-way AOV & t-test

Higher-order designs: i.e., three or more factors.

Great advantage:

They account for more of the variance in the DV, since they include more info
That is, they allow us to explain & predict behaviors more accurately.

Great disadvantage:
Interactions involving several IVs can be impossible to interpret.

Pragmatic disadvantages:

Number of participants, resources, time etc are usually greatly increased. That is, as # of groups increases,

so does the # of participants; more resources are needed, etc.

Overall, regarding factorial AOV

Very similar to one-way AOV

Only difference is that two or more IV’s are examined simultaneously,

so there are multiple hypotheses rather than just one

Major advantage over one-way AOV & t-test:

Can see interactions between IV’s