Class Notes, Chapter 14

Factorial Experiments: Two or more factors

Reading: 14.1 - 14.4

(7) CRCF two factors

Completely Randomized Complete Factorial Design

Fixed Effects (Model I)

2 factors/manipulations/independent variables (e.g. time, temperature)

1 measure/dependent variable (e.g. yield)

Complete factorial means all levels of factor A crossed with all levels of factor B: all combinations (no empty cells)

Traditional set of questions (effects):

Does treatment factor A affect the outcome?

Does treatment factor B affect the outcome?

Is there an interaction between factors A and B?

Traditional set of questions (means):

Is there a difference among the A marginal means?

Is there a difference among the B marginal means?

In the joint table, are the differences of cell means different?

The Major Effects:

The main effect of A

The main effect of B

The interaction of A and B (notated A x B)

Model using Effects Coding

model is: effects

the major effects are encoded in the model, so hypotheses are effects = 0:

Model using Means Coding

model is: group membership

questions, effects or other comparisons, are encoded in the Hypotheses

Here is a useful matrix operator:

Kronecker product

Example - suppose there are 2 levels of treatment A and 2 levels of treatment B

The main effect of A - a test on the A marginals

the hypothesis for means coding could be:

But using the cells in the two-factor design, we have to combine across levels of B:

The main effect of B would be:

The interaction would be:

Testing the hypotheses - setting up the F ratios

Calculation formulas:

Anova Table

Source	SS	df	MS	F
Treatment, A	SSA	dfA	MSA	F = MSA/MSE
Treatment, B	SSB	dfB	MSB	F = MSB/MSE
Interaction, A x B	SSAB	dfAB	MSAB	F = MSAB/MSE
Error, E	SSE	dfE	MSE

Interpretation:

If the interaction is significant, then (properly) the A and B conditionals should be tested, instead of the main effects. This is done by restricting tests to single levels of one factor.

Can do contrasts, as with the single-factor design

Graphs of means for the 2-way design:

Parallel lines means no interaction

(8) CRCF three factors

Completely Randomized Complete Factorial Design

Fixed Effects (Model I)

3 factors/manipulations/independent variables (e.g. time, temperature, amount of catalyst)

1 measure/dependent variable (e.g. yield)

Complete factorial means all levels of factor A crossed with all levels of factor B crossed with all levels of factor C: all combinations

Traditional set of questions (effects):

Does treatment factor A affect the outcome?

Does treatment factor B affect the outcome?

Does treatment factor C affect the outcome?

Is there an interaction between factors A and B?

Is there an interaction between factors A and C?

Is there an interaction between factors B and C?

Is there an interaction between factors A, B and C?

Translate to means:

Is there a difference among the A marginal means?

Is there a difference among the B marginal means?

Is there a difference among the C marginal means?

In the joint AB table, are the differences of cell means different?

In the joint AC table, are the differences of cell means different?

In the joint ABC table, are the differences of differences of cell means different?

The Major Effects:

The main effect of A

The main effect of B

The main effect of C

The A x B interaction

The A x C interaction

The B x A interaction

The A x B x C interaction

Model using Effects Coding

y_ijkl = mu + alpha_i + beta_j + gamma_k + alpha-beta_ij + alpha-gamma_ik + beta-gamma_jk + alpha-beta-gamma_ijk + epsilon_ijkl

model is: effects

the major effects are encoded in the model, so hypotheses are effects = 0:

H₀: alpha_i = 0

H₀: beta_j = 0

H₀: gamma_k = 0

H₀: alpha-beta_ij = 0

H₀: alpha-gamma_ik = 0

H₀: beta-gamma_jk = 0

H₀: alpha-beta-gamma_ijk = 0

Model using Means Coding

y_ijkl = mu_ijk + epsilon_ijkl

model is: group membership

questions, effects or other comparisons, are encoded in the Hypotheses

Example of testing the major effects: suppose there are 2 levels of each treatment, A, B, and C

Test the main effect of A (test of the A marginals)

the hypothesis for means coding could be:

But using the cells in the three-factor design, we have to combine across levels of B and C:

Hypotheses of the major effects for a 3-way CRCF design:

Setting up the F-tests: Each E(MS) = sigma-squared + a term specific to that effect

Under H₀ for that effect, the term = zero

So under H₀, E(MS) = sigma-squared

and as before, you build an F-test of the effect.

Calculation formulas:

Anova table for 3-way design:

Source	SS	df	MS	F
Treatment, A
Treatment, B
Treatment, C
Interaction, A x B
Interaction, A x C
Interaction, B x C
Interaction, A x B x C
Error, E

Can we use the plot of an ABC design to judge the presence of an interaction: Does there seem to be one in this plot?

The AxB interaction pattern is certainly different from level C1 to level C2; maybe there is one. Let's do the tests and find out:

Formulas for the SS of the major effects:

Anova Table

Source	SS	df	MS	F
Treatment, A	1600	1	1600	72.7
Treatment, B	2500	1	2500	114
Treatment, C	100	1	100	4.55
Interaction, A x B	6400	1	6400	291
Interaction, A x C	1600	1	1600	72.7
Interaction, B x C	2500	1	2500	114
Interaction, A x B x C	0	1	0	0
Error, E	176	8	22

The three-way interaction is zero! So the point is made: visually examining the plot of a 3-way interaction will not necessarily give you a good idea about the magnitude of the effect. Non-parallel lines are important for the 2-way in predicting an effect, but the 3-way is more abstract: differences of differences of differences.

Further::

Can do contrasts, as with the single-factor design

Can do certain random effects designs (Model II or Mixed Model), but not others

Can have blocking factor(s) as before - see below

Two treatment factors and one blocking factor

Suppose A is the blocking factor and B and C are treatment factors

Model is: y_ijk = alpha_i + beta_j + gamma_k + beta-gamma_jk + epsilon_iijk

Added assumption: No alpha-beta, no alpha-gamma, and no alpha-beta-gamma interactions.

Calculate SSA, SSB, SSC, dfA, dfB, dfC as in three-way

Calculate SS(BC) and df(BC) as in three-way

Calculate SST and dfT as in three-way

Calculate SSE = SST - SSA - SSB - SSC - SS(BC)

Calculate dfE = dfT - dfA - dfB - dfC - df(BC)

Calculate MSB, MSC, MS(BC), MSE

Calculate f for B, C, and BC

Purpose of this design: It's really a 2-way design, but with increased statistical precision, because the variability due to the blocks is "removed" from the error variation, making MSE smaller.

Example 14.5 in your text (p 584) is 3 treatment factors and 1 blocking factor - we'll discuss briefly in class.

One treatment factor and two blocking factors - often an incomplete design

A common example of this type of design is a Latin Square design as in the class notes (#20), where in the example fertilizers and years were the blocking factors, and the varieties of wheat were the treatment factor. The design was incomplete, i.e. not fully crossed.

Note about y-hats and residuals: if there is more than one observation per cell, the y-hat is the y-bar is the group mean, and the residual is the observation minus the group mean. If there is only one observation per cell, there is no y-bar, so the y-hat has to be calculated in a different way.