Study Questions, Chapter 14

1. Write out the CR-CF 2-factor model in means coding and in effects coding.

y_ijk = mu_ij + e_ijk

y_ijk = mu + alpha_i + beta_j + alpha-beta_ij + e_ijk

2. What is the formula for the residual in the CR-CF 2-way design?

e_ijk = y_ijk - y-bar_ij.

3. What four estimates of sigma-squared do we have in the CR-CF 2-way design? Which is/are unbiased? Which is/are unbiased only when H₀ is true?

MS_A - unbiased only when H₀ for A is true

MS_B - unbiased only when H₀ for B is true

MS_AB - unbiased only when H₀ for AB is true

MS_E - unbiased

4. What is/are the basic hypothesis/hypotheses in the CR-CF 2-way design, for means coding, and for effects coding?

Means coding:

no differences among A treatment levels

no differences among B treatment levels

no differences of differences among AB treatment levels

Effects coding

alpha_i = 0

beta_j = 0

alpha-beta_ij = 0

5. In a two-factor CRCF design, how can you use a graph of means to predict whether or not you will find a significant interaction?

Look for non-parallel lines

6. How does a significant interaction affect your assessment of the main effects?

If you find a significant interaction, you should really test the simple main effects (main effects limited to single levels of the other factor) rather than the main effects.

7. In a two-factor CRCF design, you can use a graph of means to predict whether or not you will find a significant interaction. Does this same method work for a three-factor CRCF design?

No, it's too complicated to reliably see visually.

8. How do we calculate the residuals for a three-way CR-CF design? (3 treatment factors)

e_ijkl = y_ijkl - y-bar_ijk

9. How do we calculate the residuals for a RCB-CF (n = 1), one blocking factor, 2 treatment factors design?

Because there is only one treatment factor, this is the same as the Latin Square RB-IF design we saw before:

e_ijk = y_ijk - y-bar_i.. - y-bar_.j. - y-bar_..k + 2 * y-bar_..