Factorial ANOVA is used to answer questions like this:
Are there any main effects or interactions between three independent variables - SES, ethnicity, and method - and job satisfaction?

A second factor is added to a one-way ANOVA design to increase power. We do not use covariates here. We may wish to explore interactions though.

Key points:

• Factorial ANOVA means you have 2 or more independent variables.
• Factorial ANOVA increases the power by adding another factor.
• Factorial ANOVA shows interactions between independent variables.
• It is very important to keep n the same in all the cells: this guarantees orthogonality.
• You cannot do a factorial ANOVA with less than 2 people per cell.
• A fully crossed design has all cells filled, e.g. 2 x 3 = 6 cells.

General Linear Model

• For a 1-way ANOVA (one independent variable), the general linear model is
  Xij = mu + group effect (alpha j or MSb) + error (eij or MSw)
• For a 2-way ANOVA on the same results (two independent variables), the general linear model becomes
  Xijk = mu + group effect #1 (alpha j) + group effect #2 (beta k) + interaction term + error (eijk)
• Note that the eijk error term is much smaller than the eij error term.
• This is because eij breaks up into group effect #2 + interaction + this much smaller error.
• Since the error goes down, the power goes up.

Hypotheses

• You increase the number of hypotheses: instead of Hzero = no significant difference between experimental and control, you add another factor like gender, and you get:
• Hzero: mu experimental = mu control
• Hzero: mu male = mu female
• Hzero: there is no interaction between method and gender.
• There is an F ratio for each of these null hypotheses.
• If you have 3 factors, then you get three main effects, three 2-way interctions, and one 3-way interaction - all with null hypotheses.

Interactions

• To find out if there is an interaction, graph the cell means, using one of the IVs on the x-axis and the DV on the y-axis. Use the other IV for a parameter (say one line for males, one line for females).
• If the lines are parallel, there is no interaction. (i.e. same slope, and we don't care about the separation).
• The separation simply says that there is a constant difference between the lines, e.g., experimental works better than control regardless of gender.
• Lack of interaction does increase the generalizability of the treatment design.

Variance

• You can tell how much of the variance in the scores is due to each factor.
• Look at the SS column and convert it to a percentage.
• In a 3-way ANOVA, variance is accounted for by all three factors.

Three-way designs

• Adding another factor increases power.
• Adding another factors increases complexity and problems in interpreting the results.
• Every single F is divided by the same MSe.
• To find the 2-way interactions, you need to draw three graphs, averaging over the factor that you are *not* including:
  • One for method x gender (gender on x-axis, method as parameter)
  • One for method x SES (SES on x-axis, method as parameter)
  • One for gender x SES (SES on x-axis, gender as parameter)
  • If the 3 graphs look alike, then there is no interaction.
  • If they don't, then you need to explore the possibility of interaction.
• To find the 3-way interaction, you need to split on one IV and draw two graphs - here, you do not average anything - use the cell means:
  • One for experimental method (SES on x-axis, gender as parameter)
  • One for control group (SES on x-axis, gender as parameter)
  • If these look alike, there is no interaction (i.e. you can overlay them)

Degrees of Freedom

• The group df goes across the top of the table; the dfw down the side.
• Main effect: df = J-1 (i.e., one less than the number of groups, for each factor) - just like in a 1-way ANOVA.
• Interaction: df = (j-1)*(k-1)
• Error term: dfw = N - jk (N is total number of subjects, jk is the product of the number of groups, say 2x2=4) - recall, for a 1-way ANOVA it used to be N - j.

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