Factorial Treatment Structure

We have been working with completely randomized designs, whereg treat-ments are assigned at random toNunits. Up till now, the treatments have had

no structure; they were justgtreatments. Factorial treatment structure ex- Factorials combine the levels of two or more factors to create treatments

ists when thegtreatments are the combinations of the levels of two or more factors. We call these combination treatments factor-level combinations or factorial combinations to emphasize that each treatment is a combination of one level of each of the factors. We have not changed the randomization; we still have a completely randomized design. It is just that now we are con-sidering treatments that have a factorial structure. We will learn that there are compelling reasons for preferring a factorial experiment to a sequence of experiments investigating the factors separately.

8.1 Factorial Structure

It is best to start with some examples of factorial treatment structure. Lynch and Strain (1990) performed an experiment with six treatments studying how milk-based diets and copper supplements affect trace element levels in rat livers. The six treatments were the combinations of three milk-based diets (skim milk protein, whey, or casein) and two copper supplements (low and high levels). Whey itself was not a treatment, and low copper was not a treatment, but a low copper/whey diet was a treatment. Nelson, Kriby, and Johnson (1990) studied the effects of six dietary supplements on the occur-rence of leg abnormalities in young chickens. The six treatments were the combinations of two levels of phosphorus supplement and three levels of calcium supplement. Finally, Hunt and Larson (1990) studied the effects of

166 Factorial Treatment Structure

Table 8.1:Barley sprouting data.

Age of Seeds (weeks)

ml H₂O 1 3 6 9 12

4 11

9 6

7 16 17

9 19 35

13 35 28

20 37 45 8

8 3 3

1 7 3

5 9 9

1 10

9

11 15 25

sixteen treatments on zinc retention in the bodies of rats. The treatments were the combinations of two levels of zinc in the usual diet, two levels of zinc in the final meal, and four levels of protein in the final meal. Again, it is the combination of factor levels that makes a factorial treatment.

We begin our study of factorial treatment structure by looking at two-factor designs. We may present the responses of a two-way two-factorial as a table

Two-factor

designs with rows corresponding to the levels of one factor (which we call factor A) and columns corresponding to the levels of the second factor (factor B). For example, Table 8.1 shows the results of an experiment on sprouting barley (these data reappear in Problem 8.1). Barley seeds are divided into 30 lots of 100 seeds each. The 30 lots are divided at random into ten groups of three lots each, with each group receiving a different treatment. The ten treatments are the factorial combinations of amount of water used for sprouting (factor A) with two levels, and age of the seeds (factor B) with five levels. The response measured is the number of seeds sprouting.

We use the notationy_ijkto indicate responses in the two-way factorial.

In this notation,y_ijkis thekth response in the treatment formed from theith

Multiple

subscripts denote factor levels and replication

level of factor A and thejth level of factor B. Thus in Table 8.1,y_2,5,3 = 25.

For a four by three factorial design (factor A has four levels, factor B has three levels), we could tabulate the responses as in Table 8.2. This table is just a convenient representation that emphasizes the factorial structure; treatments were still assigned to units at random.

Notice in both Tables 8.1 and 8.2 that we have the same number of re-sponses in every factor-level combination. This is called balance. Balance

Balanced data have equal replication

turns out to be important for the standard analysis of factorial responses. We will assume for now that our data are balanced with n responses in every factor-level combination. Chapter 10 will consider analysis of unbalanced factorials.

8.2 Factorial Analysis: Main Effect and Interaction 167

Table 8.2:A two-way factorial treatment structure.

B1 B2 B3

A1

y111

... y_11n

y121

... y_12n

y131

... y_13n

A2

y₂₁₁ ... y_21n

y₂₂₁ ... y_22n

y₂₃₁ ... y_23n

A3

y₃₁₁ ... y_31n

y₃₂₁ ... y_32n

y₃₃₁ ... y_33n

A4

y₄₁₁ ... y_41n

y₄₂₁ ... y_42n

y₄₃₁ ... y_43n

8.2 Factorial Analysis: Main Effect and Interaction

When our treatments have a factorial structure, we may also use a factorial analysis of the data. The major concepts of this factorial analysis are main effect and interaction.

Consider a two-way factorial where factor A has four levels and factor B has three levels, as in Table 8.2. There areg= 12treatments, with 11 degrees of freedom between the treatments. We usei andj to index the levels of factors A and B. The expected values in the twelve treatments may be denoted µ_ij, coefficients for a contrast in the twelve means may be denotedw_ij(where as usual^P_ijw_ij = 0), and the contrast sum is^P_ijw_ijµ_ij. Similarly,y_ij•

is the observed mean in the ij treatment group, andy_i•• andy_•j• are the Treatment, row, and column means

observed means for all responses having leveliof factor A or leveljof B, respectively. It is often convenient to visualize the expected values, means, and contrast coefficients in matrix form, as in Table 8.3.

For the moment, forget about factor B and consider the experiment to be a completely randomized design just in factor A (it is completely randomized in factor A). Analyzing this design with four “treatments,” we may compute

a sum of squares with 3 degrees of freedom. The variation summarized by Factor A ignoring factor B

this sum of squares is denotedSS_Aand depends on just the level of factor A.

The expected value for the mean of the responses in rowiisµ+α_i, where we assume that^P_iαi= 0.

168 Factorial Treatment Structure

Table 8.3:Matrix arrangement of (a) expected values, (b) means, and (c) contrast coefficients in a four by three factorial.

(a) µ₁₁ µ₁₂ µ₁₃ µ₂₁ µ₂₂ µ₂₃ µ₃₁ µ₃₂ µ₃₃ µ₄₁ µ₄₂ µ₄₃

(b)

y_11• y_12• y_13•

y_21• y_22• y_23•

y_31• y_32• y_33•

y_41• y_42• y_43•

(c) w₁₁ w₁₂ w₁₃ w₂₁ w₂₂ w₂₃ w₃₁ w₃₂ w₃₃ w₄₁ w₄₂ w₄₃

Now, reverse the roles of A and B. Ignore factor A and consider the periment to be a completely randomized design in factor B. We have an

ex-Factor B ignoring

factor A periment with three “treatments” and treatment sum of squaresSSBwith 2 degrees of freedom. The expected value for the mean of the responses in columnjisµ+β_j, where we assume that^P_jβ_j = 0.

The effects α_i and β_j are called the main effects of factors A and B, respectively. The main effect of factor A describes variation due solely to the

A main effect describes variation due to a single factor

level of factor A (row of the response matrix), and the main effect of factor B describes variation due solely to the level of factor B (column of the response matrix). We have analogously that SS_AandSS_Bare main-effects sums of squares.

The variation described by the main effects is variation that occurs from row to row or column to column of the data matrix. The example has twelve treatments and 11 degrees of freedom between treatments. We have de-scribed 5 degrees of freedom using the A and B main effects, so there must

Interaction is variation not described by main effects

be 6 more degrees of freedom left to model. These 6 remaining degrees of freedom describe variation that arises from changing rows and columns si-multaneously. We call such variation interaction between factors A and B, or between the rows and columns, and denote it bySS_AB.

Here is another way to think about main effect and interaction. The main effect of rows tells us how the response changes when we move from one row to another, averaged across all columns. The main effect of columns tells us how the response changes when we move from one column to an-other, averaged across all rows. The interaction tells us how the change in re-sponse depends on columns when moving between rows, or how the change in response depends on rows when moving between columns. Interaction be-tween factors A and B means that the change in mean response going from leveli₁ of factor A to leveli₂ of factor A depends on the level of factor B under consideration. We can’t simply say that changing the level of factor A changes the response by a given amount; we may need a different amount of change for each level of factor B.

8.2 Factorial Analysis: Main Effect and Interaction 169

Table 8.4:Sample main-effects and interaction contrast coefficients for a four by three factorial design.

A

-3 -3 -3 -1 -1 -1

1 1 1

3 3 3

1 1 1

-1 -1 -1 -1 -1 -1

1 1 1

-1 -1 -1

3 3 3

-3 -3 -3

1 1 1

B

-1 0 1

-1 0 1

-1 0 1

-1 0 1

1 -2 1

1 -2 1

1 -2 1

1 -2 1

AB

3 0 -3

1 0 -1

-1 0 1

-3 0 3

-1 0 1

1 0 -1

1 0 -1

-1 0 1

1 0 -1

-3 0 3

3 0 -3

-1 0 1

-3 6 -3

-1 2 -1

1 -2 1

3 -6 3

1 -2 1

-1 2 -1

-1 2 -1

1 -2 1

-1 2 -1

3 -6 3

-3 6 -3

1 -2 1

We can make our description of main-effect and interaction variation more precise by using contrasts. Any contrast in factor A (ignoring B) has four coefficientsw^⋆_i and observed valuew^⋆({y_i••}). This is a contrast in the four row means. We can make an equivalent contrast in the twelve treatment means by using the coefficientswij = w^⋆_i/3. This contrast just repeatsw^⋆_i across each row and then divides by the number of columns to match up with the division used when computing row means. Factor A has four levels,

so three orthogonal contrasts partition SS_A. There are three analogous or- Main-effects contrasts

thogonalwij contrasts that partition the same variation. (See Question 8.1.) Table 8.4 shows one set of three orthogonal contrasts describing the factor A variation; many other sets would do as well.

The variation in SS_B can be described by two orthogonal contrasts be-tween the three levels of factor B. Equivalently, we can describeSSB with orthogonal contrasts in the twelve treatment means, using a matrix of contrast coefficients that is constant on columns (that is,w_1j = w_2j = w_3j = w_4j for all columnsj). Table 8.4 also shows one set of orthogonal contrasts for factor B.

170 Factorial Treatment Structure

Inspection of Table 8.4 shows that not only are the factor A contrasts orthogonal to each other, and the factor B contrasts orthogonal to each other,

A contrasts orthogonal to B contrasts for balanced data

but the factor A contrasts are also orthogonal to the factor B contrasts. This orthogonality depends on balanced data and is the key reason why balanced data are easier to analyze.

There are 11 degrees of freedom between the twelve treatments, and the A and B contrasts describe 5 of those 11 degrees of freedom. The 6 addi-tional degrees of freedom are interaction degrees of freedom; sample inter-action contrasts are also shown in Table 8.4. Again, inspection shows that

Interaction

contrasts the interaction contrasts are orthogonal to both sets of main-effects contrasts.

Thus the 11 degrees of freedom between-treatment sum of squares can be partitioned using contrasts intoSS_A,SS_B, andSS_AB.

Look once again at the form of the contrast coefficients in Table 8.4.

Row-main-effects contrast coefficients are constant along each row, and add to zero down each column. Column-main-effects contrasts are constant down

Contrast coefficients satisfy zero-sum restrictions

each column and add to zero along each row. Interaction contrasts add to zero down columns and along rows. This pattern of zero sums will occur again when we look at parameters in factorial models.

8.3 Advantages of Factorials

Before discussing advantages, let us first recall the difference between facto-rial treatment structure and factofacto-rial analysis. Factofacto-rial analysis is an option

Factorial structure

versus analysis we have when the treatments have factorial structure; we can always ignore main effects and interaction and just analyze thegtreatment groups.

It is easiest to see the advantages of factorial treatment structure by com-paring it to a design wherein we only vary the levels of a single factor. This second design is sometimes referred to as “one-at-a-time.” The sprouting

One-at-a-time

designs data in Table 8.1 were from a factorial experiment where the levels of sprout-ing water and seed age were varied. We might instead use two one-at-a-time designs. In the first, we fix the sprouting water at the lower level and vary the seed age across the five levels. In the second experiment, we fix the seed age at the middle level, and vary the sprouting water across two levels.

Factorial treatment structure has two advantages:

1. When the factors interact, factorial experiments can estimate the inter-action. One-at-at-time experiments cannot estimate interinter-action. Use of one-at-a-time experiments in the presence of interaction can lead to serious misunderstanding of how the response varies as a function of the factors.

8.4 Visualizing Interaction 171

2. When the factors do not interact, factorial experiments are more ef-ficient than one-at-a-time experiments, in that the units can be used to assess the (main) effects for both factors. Units in a one-at-a-time experiment can only be used to assess the effects of one factor.

There are thus two times when you should use factorial treatment structure— Use factorials!

when your factors interact, and when your factors do not interact. Factorial structure is a win, whether or not we have interaction.

The argument for factorial analysis is somewhat less compelling. We usually wish to have a model for the data that is as simple as possible. When there is no interaction, then main effects alone are sufficient to describe the

means of the responses. Such a model (or data) is said to be additive. Additive model has only main effects

An additive model is simpler (in particular, uses fewer degrees of freedom) than a model with a mean for every treatment. When interaction is moderate compared to main effects, the factorial analysis is still useful. However, in some experiments the interactions are so large that the idea of main effects as the primary actors and interaction as fine tuning becomes untenable. For such experiments it may be better to revert to an analysis ofg treatment groups, ignoring factorial structure.

Pure interactive response Example 8.1

Consider a chemistry experiment involving two catalysts where, unknown to us, both catalysts must be present for the reaction to proceed. The response is one or zero depending on whether or not the reaction occurs. The four treat-ments are the factorial combinations of Catalyst A present or absent, and Catalyst B present or absent. We will have a response of one for the com-bination of both catalysts, but the other three responses will be zero. While it is possible to break this down as main effect and interaction, it is clearly more comprehensible to say that the response is one when both catalysts are present and zero otherwise. Note here that the factorial treatment structure was still a good idea, just not the main-effects/interactions analysis.

8.4 Visualizing Interaction

An interaction plot, also called a profile plot, is a graphic for assessing the

rel-ative size of main effects and interaction; an example is shown in Figure 8.1. Interaction plots connect-the-dots between treatment means

Consider first a two-factor factorial design. We construct an interaction plot in a “connect-the-dots” fashion. Choose a factor, say A, to put on the hori-zontal axis. For each factor level combination, plot the pair(i, y_ij•). Then

“connect-the-dots” corresponding to the points with the same level of factor

172 Factorial Treatment Structure

Table 8.5: Iron levels in liver tissue, mg/g dry weight.

Diet Control Cu deficient

Skim milk protein .70 1.28

Whey .93 1.87

Casein 2.11 2.53

B; that is, connect(1, y_1j•), (2, y_2j•), up to (a, y_aj•). In our four by three prototype factorial, the level of factor A will be a number between one and four; there will be three points plotted above one, three points plotted above two, and so on; and there will be three “connect-the-dots” lines, one for each level of factor B.

For additive data, the change in response moving between levels of factor A does not depend on the level of factor B. In an interaction plot, that simi-larity in change of level shows up as parallel line segments. Thus interaction

Interaction plot shows relative size of main effects and interaction

is small compared to the main effects when the connect-the-dots lines are parallel, or nearly so. Even with visible interaction, the degree of interaction may be sufficiently small that the main-effects-plus-interaction description is still useful. It is worth noting that we sometimes get visually different impressions of the interaction by reversing the roles of factors A and B.

Example 8.2 Rat liver iron

Table 8.5 gives the treatment means for liver tissue iron in the Lynch and Strain (1990) experiment. Figure 8.1 shows an interaction plot with milk diet factor on the horizontal axis and the copper treatments indicated by different lines. The lines seem fairly parallel, indicating little interaction.

Figure 8.1 points out a deficiency in the interaction plot as we have de-fined it. The observed means that we plot are subject to error, so the line

Interpret “parallel”

in light of variability

segments will not be exactly parallel—even if the true means are additive.

The degree to which the lines are not parallel must be interpreted in light of the likely size of the variation in the observed means. As the data become more variable, greater departures from parallel line segments become more likely, even for truly additive data.

Example 8.3 Rat liver iron, continued

The line segments are fairly parallel, so there is not much evidence of inter-action, though it appears that the effect of copper may be somewhat larger for milk diet 2. The mean square for error in the Lynch and Strain experiment was approximately .26, and each treatment had replicationn = 5. Thus the standard errors of a treatment mean, the difference of two treatment means,

8.4 Visualizing Interaction 173

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

1 2 3

Milk diet I

r o n

1

1

1

2

2

2

Figure 8.1:Interaction plot of liver iron data with diet factor on the horizontal axis, using MacAnova.

and the difference of two such differences are about .23, .32, and .46 respec-tively. The slope of a line segment in the interaction plot is the difference of two treatment means. The slopes from milk diet 1 to 2 are .23 and .59, and the slopes from milk diets 2 to 3 are 1.18 and .66; each of these slopes was calculated as the difference of two treatment means. The differences of the slopes (which have standard error .46 because they are differences of differences of means) are .36 and .48. Neither of these differences is large compared to its standard error, so there is still no evidence for interaction.

We finish this section with interaction plots for the other two nutrition experiments described in the first section.

Chick body weights Example 8.4

Figure 8.2 is an interaction plot of the chick body weights from the Nelson, Kriby, and Johnson (1990) data with the calcium factor on the horizontal axis and a separate line for each level of phosphorus. Here, interaction is clear. At the upper level of phosphorus, chick weight does not depend on calcium. At the lower level of phosphorus, weight decreases with increasing calcium. Thus the effect of changing calcium levels depends on the level of phosphorus.

174 Factorial Treatment Structure

1 2

3 2

1 600

550

500

450

Calcium

Phosphorus

Mean

Interaction plot --- Data means for weight

Figure 8.2:Interaction plot of chick body weights data with calcium on the horizontal axis, using Minitab.

Example 8.5 Zinc retention

Finally, let’s look at the zinc retention data of Hunt and Larson (1990). This is a three-factor factorial design (four by two by two), so we need to modify our approach a bit. Figure 8.3 is an interaction plot of percent zinc retention with final meal protein on the horizontal axis. The other four factor-level combinations are coded 1 (low meal zinc, low diet zinc), 2 (low meal zinc, high diet zinc), 3 (high meal zinc, low diet zinc), and 4 (high meal zinc, high diet zinc). Lines 1 and 2 are low meal zinc, and lines 3 and 4 are high meal zinc. The 1,2 pattern across protein is rather different from the 3,4 pattern across protein, so we conclude that meal zinc and meal protein interact.

On the other hand, the 1,3 pair of lines (low diet zinc) has the same basic pattern as the 2,4 pair of lines (high diet zinc), so the average of the 1,3 lines should look like the average of the 2,4 lines. This means that diet zinc and meal protein appear to be additive.

8.5 Models with Parameters 175

45 50 55 60 65 70 75 80 85

1 2 3 4

Meal protein Z

i n c r e t e n t i o

n 1

1

1

1 2

2 2 2

3

3

3

3 4

4

4

4

Figure 8.3:Interaction plot of percent zinc retention data with meal protein on the horizontal axis, using MacAnova.

8.5 Models with Parameters

Let us now look at the factorial analysis model for a two-way factorial

treat-ment structure. Factor A has alevels, factor B has b levels, and there are A hasalevels, B hasblevels,n replications

nexperimental units assigned to each factor-level combination. Thekth re-sponse at theith level of A andjth level of B isy_ijk. The model is

y_ijk=µ+α_i+β_j+αβ_ij +ǫ_ijk,

whereiruns from 1 toa,jruns from 1 tob,kruns from 1 ton, and theǫ_ijk’s Factorial model

are independent and normally distributed with mean zero and varianceσ². Theα_i,β_j, andαβ_ij parameters in this model are fixed, unknown constants.

There is a total ofN =nabexperimental units.

Another way of viewing the model is that the table of responses is broken down into a set of tables which, when summed element by element, give the response. Display 8.1 is an example of this breakdown for a three by two factorial withn= 1.

The term µ is called the overall mean; it is the expected value for the responses averaged across all treatments. The term α_i is called the main

effect of A at leveli. It is the average effect (averaged over levels of B) for Main effects

leveli of factor A. Since the average of all the row averages must be the overall average, these row effectsαi must sum to zero. The same is true for

176 Factorial Treatment Structure

responses overall mean row effects



 y₁₁₁ y₁₂₁ y₂₁₁ y₂₂₁ y₃₁₁ y₃₂₁



 =



 µ µ µ µ µ µ



 +



 α₁ α₁ α₂ α₂ α₃ α₃



 +

column effects interaction effects



 β1 β2

β₁ β₂ β₁ β₂



 +



 αβ11 αβ12

αβ₂₁ αβ₂₂ αβ₃₁ αβ₃₂



 +

random errors



 ǫ₁₁₁ ǫ₁₂₁ ǫ₂₁₁ ǫ₂₂₁ ǫ311 ǫ321





Display 8.1:Breakdown of a three by two table into factorial effects.

β_j, which is the main effect of factor B at levelj. The termαβ_ij is called the interaction effect of A and B in the ij treatment. Do not confuseαβ_ij with

Interaction effects

the product ofαi andβj; they are different ideas. The interaction effect is a measure of how far the treatment means differ from additivity. Because the average effect in theith row must beα_i, the sum of the interaction effects in theith row must be zero. Similarly, the sum of the interaction effects in the jth column must be zero.

The expected value of the response for treatmentij is E y_ijk=µ+αi+βj +αβij .

There are ab different treatment means, but we have 1 + a+b +ab

pa-Expected value

rameters, so we have vastly overparameterized. Recall that in Chapter 3 we had to choose a set of restrictions to make treatment effects well defined; we must again choose some restrictions for factorial models. We will use the following set of restrictions on the parameters:

Zero-sum restrictions on parameters

0 = Xa i=1

α_i= Xb j=1

β_j = Xa i=1

αβ_ij = Xb j=1

αβ_ij .

This set of restrictions is standard and matches the description of the param-eters in the preceding paragraph. The αi values must sum to 0, so at most

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