Topic 13 Standard Post-Hoc anaylsis

Wherein Factor A has J levels and Factor B has K levels

begins with an overall test of the homogeneity hypothesis for the A main effect, B main Effect and a AB interaction effect
And followed with an F-STP and Scheffe SCI for contrast within each family for which the overall test is significant.

Write out the two-way ANOVA model?
What is the equation for a main effect
What is the equation for b main effect
What is the equation for AB interaction effect?

Explain the partition of variability?

What are the steps associated with a post-hox J x K analysis?
HOMOGENEITY HYPOTHESIS TEST FOR EACH FAMILY

1. Partition A main effect - Variation between rows.

For each level Mj-M across - A main effect
Then calculate SS(A)

2. Partition B main effect - Variation between columns

The calculate SS(B)

3. Partition AB interaction

Then calculate SS(AB)

4. Calculate SSW (sum of squares within (aka error))

Individual score minus specific cell mean

5. Calculate the SST

Individual score minus Grand Mean
Note that

Total degree of freedom for error can be cal as a function of SST df - number of participants per cell.

6. Calculate Mean Squares
   SS/df for everything

7. Calculate the Fa

Wherein F(v1)(v2)

8. Conclusions

How to proceed with follow up analysis of contrasts?

uses Scheffe Tests and SCI -> note that overall tests (such as the ANOVA F test ) may not be necessary, however, these tests help to determine if it is worthwhile continuing with the analysis for each family.

1. Calculate critical values (either F test or Scheffe)

Critical values are larger for INTERACTION effects as it is most difficult to prove an interaction effect as NOT due to chance

2. Based on contrast calculate A main effect contrast

If orthogonal - SS(A1) + SS(A2) = SS(A)

3. Based on contrast calculate B main effect

OR

o J coefficients for A main effect contrasts (applied to Row Means)

o K coefficients for B main effect contrasts (applied to Column Means).

What is the product contrast approach?

All factorial contrasts, such as main effect and interaction contrasts and simple effect contrasts, can be defined with a cell means model and expressed as product contrasts. Recall (from Topic 11), a product contrast has a contrast coefficient vector referring to cell means that can be expressed as a product of • a coefficient vector cA referring to levels of Factor A and • a coefficient vector cB referring to levels of Factor B.

Express A main effect as in the product contrast approach?

IF A contrast - contrast coefficient vector in COLUMN
IF B contrast - contrast coefficient vector in ROW
Express B main effect contrast as a product contrast?

What is the structure of the conclusion for main effects?

MAIN EFFECTS
The A main effect parameters (αj) are not all equal to 0, which implies population row means are not all the same. Averaged across levels of FACTOR A, average improvement differs between FACTOR B conditions.
AB INTERACTION
The AB effect parameters (αβjk) are heterogeneous (and not all equal to zero), which implies that variability among cell means due to the combined effect of Factor A and Factor B. The effect of the Treatment factor on average improvement depends upon the level of the Drug factor.

What is being estimated for mean squares?

The effect of a, b or AB interaction on an independent variable

What is the maximal interaction contrast?

Coefficients of maximal AB contrast are interaction means (Mjk – Mj – Mk + M)