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, yij•). Then
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, y1j•), (2, y2j•), up to (a, yaj•). 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
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