General Linear Model Introduction to ANOVA

Questions (1) What does it mean to pick parameter estimates by least squares? Why are least squares estimates desirable, that is, in what sense are they good from an estimation or decision standpoint? What is a treatment effect in ANOVA? (write the equation and explain the terms) What is a fixed-effects model?

Questions (2) What is error in a fixed-effects ANOVA model? Describe in words the partitioning of variance in a one-way ANOVA.

Linear models Line Y = a+bX or Y =mX+b Or y=a0x0+a1x1+…+anxn+e X can take on values or just indicate group membership (0,1) ANOVA model if group membership Regression model if X takes on scale values X and Y are variables; a and b are coefficients to be estimated.

ANOVA models People X0 X1 X2 X3 Compositon - effects 1 Y1=a0+a1+e1 2 Y1=a0+a1+e1 2 Y2=a0+a1+e2 3 Y3=a0+a2+e3 4 Y4=a0+a2+e4 5 Y5=a0+a3+e5 Shows group membership – IVs in ANOVA

Effects & Least Squares (1) Linear model applies to everyone. How to pick the best weights? Minimize the errors. Make the error variance small. Find the least squared error. Suppose we pick for each cell the value of the mean as the weight aj. Then the errors within each cell will be deviations from the cell mean. The average of the errors (deviations) within cells will be zero, and the variance of the deviations will be minimum.

Population Effects Make it Greek. Sample Population Population effect of treatment j= If there is no effect of a treatment, If there is no effect, then Absence of treatment effects is equivalent to equality of all population means. ANOVA is used to test for differences in means.

Fixed vs. Random Effects Fixed: Treatments in the study are the only ones of interest. Computer tutor vs. control. 4 types of psychotherapy. 3 antidepressant drugs. Treatment is fixed in that it is whole and presented to all. Random: Treatments selected to represent larger population of treatments. Presentation order on Rorschach. Representative drug dosages, therapies, or antidepressant drugs. Schools as stimuli. Random in that treatment is sampled from a population of treatments.

Fixed Effects Model Cell 1 Cell 2 Cell 3 40-2+5=43 40+6-5=41 40-4+3=39 Population model: 3 treatments: pop data: Sample (Observed) Data Cell 1 Cell 2 Cell 3 40-2+5=43 40+6-5=41 40-4+3=39 40-2+2=40 40+6+1=47 40-4-2=34 40-2-3=35 40+6+8=54 40-4+1=37 = 39.3 =47.3 =36.7

Fixed Effects Model (2) The graph shows the terms in the equation. There are three cells or levels in this study. The IV effect and error for the highest scoring cell is shown.

Review What does it mean to pick parameter estimates by least squares? What is a treatment effect in ANOVA? (write the equation and explain the terms) What is a fixed-effects model? Describe a fixed-effects study. Describe a Random-effects study.

Partition of the Sum of Squares (SS) We are interested in variance, but can often work with SS because N cancels out. Bottom Line: SS total = SS within + SS between Deviation of a score from the grand mean: Deviation made of two parts: Deviation = treatment+error Total variance = treatment variance+error variance

SS 2 Add the squared deviations over people and treatment groups. With a little algebra, we can show : SS total = SS within cells + SS between cells. SS total = SS error + SS treatment. Variance is additive. We break the total variance into components or analyze the variance, hence ANOVA

Numerical Example Cell 1 4 9 2 3 5 6 7 Total 28 24 Total = within + between Cell 1 4 9 2 3 5 6 7 Total 28 24

Review What is error in a fixed-effects ANOVA model? What is a treatment effect in fixed-effects ANOVA? Describe in words the partitioning of variance in a one-way ANOVA.