1. General Linear Model

2. Recall Dummy Coding Dummy coding 0s and 1s
k-1 predictors will go into the regression equation leaving out one reference category (e.g. control)
Coefficients will be interpreted as change with respect to the reference variable (the one with all zeros)
In this case group 3
Group D1 D2 1 2 3

3. General Equations When assumptions are met, average error, e = 0
In the end we can see the intercept as the mean for Group 3, the reference group; and the coefficients for the coded variables are the mean difference between that group’s mean and the reference group mean

4. The Structural Model
Generally we write the structural model for a One-way ANOVA where a person’s score in group j is a function of the grand mean of Y, the effect of being in group j, and error
The null hypothesis is that the coefficients are zero, which is equivalent to saying the means are equal

5. The Structural Model
Using deviation regressors, in which the coding weights are constrained to sum to 0, tests this explicitly
Here our reference group gets -1 for each variable
Group
(α1) D1
(α2) D2 1 2 3 -1

6. General Equations From before
The prediction for any group is equivalent to the grand mean of Y plus the effect of being in that group
In other words, for each group member the predicted value is the mean for that group

7. Factorial Design Recall for the general one-way anova Where:
μ = grand mean
 = effect of Treatment A (μa – μ)
ε = within cell error
So a person’s score is a function of the grand mean, the treatment mean, and within cell error

8. Effects for 2-way
Population main effect associated with the treatment Aj (first factor):
Population main effect associated with treatment Bk (second factor):
The interaction is defined as , the joint effect of treatment levels j and k (interaction of  and ) so the linear model is:
Each person’s score is a function of the grand mean, the treatment means, and their interaction (plus w/in cell error).

9. The general linear model
The interaction is a residual:
Plugging in  and  leads to:

10. GLM Factorial ANOVA Statistical Model: Statistical Hypothesis:
The interaction null is that the cell means do not differ significantly (from the grand mean) outside of the main effects present, i.e. that this residual effect is zero

11. Comparison to regression
Data using deviation coding
ANOVA output top with bold correlates to the regression output using an interaction product term1
F1 F2 DV
1. In a balanced design (true experiment) there is no need to center our variables before creating the product term to test moderating effects because the factors don’t correlate with one another, allowing for much easier interpretation (all effects add up to the total Between Groups variance).

12. Repeated Measures One way design for Repeated Measures has two effects
Effect of the treatment at a particular time
Effect of the between subjects factor

13. Repeated Measures
The basic linear model thus has 2 ‘main’ effects though typically only one is of interest
The interaction that would normally be present in such a situation is relegated to error variance
So the error variance equals the subject x treatment interaction + random error

14. Factorial Repeated Measures
With Factorial RM we have unique error for each main effect and interaction

15. Mixed Design β here is the repeated measure
The RM factor and interaction are tested on the same error term (in parentheses)