Introduction

Experiments that involve more than two treatment factors are designed and analyzed using many of the same principles that were discussed previously for two-factor experiments. We continue to label the factors with uppercase Latin letters and their numbers of levels with the corresponding lowercase letters. An experiment that involves four factors, \(A, B, C\), and \(D\), having \(a, b, c\), and \(d\) levels, respectively, for example, is known as an ” \(a \times b \times c \times d\) factorial experiment” (read ” \(a\) by \(b\) by \(c\) by \(d\) “) and has a total of \(v=a b c d\) treatment combinations. A completely randomized design is conducted in the same way as an \(a\times b\) factorial design.

However, for the analysis, the high order interactions may not exist or be ignored. Therefore a set of models that may be all appropriate for the design.

Statistical Models

We will consider models involving three treatment factors \(A\), \(B\) and \(C\). The following cell-means model is just a rewrite of one-way model \[ \begin{gathered} Y_{i j k t}=\mu+\tau_{i j k}+\epsilon_{i j k t}, \\ \epsilon_{i j k t} \sim N\left(0, \sigma^2\right),\\ \epsilon_{i j k t} \text{'s mutually independent},\\ t=1, \ldots, r_{i j k} ; \quad i=1, \ldots, a ; ~j=1, \ldots, b ;~ k=1, \ldots, c . \end{gathered} \] where \(\tau_{ijk}\) are treatment effects and \(Y_{ijkt}\) represents the \(t\)th observation for the treatment \(ijk\).

We can use this cell-means model if our objective is to compare some specific treatment means.

The cell-means model does not reveal how each factor affects the treatment means or how the factors interact. To answer those questions, we use multi-way models. For example the three-way complete model is

\[ \begin{gathered} Y_{i j k t}=\mu+\alpha_i+\beta_j+\gamma_k+(\alpha \beta)_{i j}+(\alpha \gamma)_{i k}+(\beta \gamma)_{j k}+(\alpha \beta \gamma)_{i j k}+\epsilon_{i j k t}, \\ \epsilon_{i j k t} \sim N\left(0, \sigma^2\right), \\ \epsilon_{i j k t} \text { 's mutually independent }, \\ t=1, \ldots, r_{i j k} ; \quad i=1, \ldots, a ; ~j=1, \ldots, b ;~ k=1, \ldots, c . \end{gathered} \]

If prior to the experiment certain interaction effects are known to be negligible, these interaction terms shall be excluded in the model. For example, if the factors \(A\) and \(B\) are known not to interact in a three-factor experiment, then the \(A B\) and \(A B C\) interaction effects are negligible, so the terms \((\alpha \beta)_{i j}\) and \((\alpha \beta \gamma)_{i j k}\) are excluded from model.

However, when a model includes an interaction between a specific set of \(m\) factors, then all interaction terms involving subsets of those \(m\) factors should be included in the model. For example, a model that includes the effect of the three-factor interaction \(A B C\) would also include the effects of the \(A B, A C\), and \(B C\) interactions as well as the main effects \(A, B\), and \(C\).

Use of a submodel or reduced model, when appropriate, is advantageous, because simpler models generally yield tighter confidence intervals and more powerful tests of hypotheses. However, if interaction terms are removed from the model when the factors do, in fact, interact, then the resulting analysis and conclusions may be totally incorrect.

We use the three-way complete model if we are interested in investigating the main effects of all factors and all interaction effects. If for example, it is not our objectives to investigate the main effects of A and B, and the AB interaction effects, we can combine the two factors A and B into one. Then the three-way model then becomes a two-way model.

High Order oInteractions

What does the three way ABC interaction mean? If the interaction effects of any pair (say, A and B) depends on the level of the third factor, then ABC interaction effects do exist.

The above plots show the three way ABC-interaction does exist because the way A and B interacts depends on the level of C!

Analysis and SAS Code

The contrast is defined the same way as before: A linear combination of the treatment means with the contrast coefficients adding up to be 0. We may also define specific contrasts such as main effects contracts for each factor, two-way interaction effects, three-way interaction effects, etc.

The least squares estimate of the treatment means are the sampling means. Therefore, a contrast can be estimated by a corresponding linear combination of sample means, whose variance and standard derivation can be easily obtained. The error variance \(\sigma^2\) is estimated by the mean squares for error (MSE). Therefore the confidence interval for each individual contrast can be constructed.

The simultaneous confidence intervals can be obtained by applying Bonferroni, Scheffé, Tukey, and Dunnett methods similar to in the two-way models.

The test of hypothesis about the mean effects or interaction effects is conducted similarly by applying an F-test.

The following ANOVA table is for the three-way complete model.

Source of variation Degrees of freedom Sum of squares Mean square F-Ratio p-value
A a-1 SSA MSA MSA/MSE
B b-1 SSB MSB MSB/MSE
C c-1 SSC MSC MSC/MSE
AB (a-1)(b-1) SSAB MSAB MSAB/MSE
AC (a-1)(c-1) SSAC MSAC MSAC/MSE
BC (b-1)(c-1) SSBC MSBC MSBC/MSE
ABC (a-1)(b-1)(c-1) SSABC MSABC MSABC/MSE
Error n-abc SSE MSE
Total n-1

A Real Experiment-Popcorn-Microwave Experiment

The experiment described section 7.4 was compare brands of microwave popcorn. The objective of the experiment was to find out which brand gives rise to the best popcorn in terms of the proportion of popped kernels. The experiment was restricted to popcorn produced in a microwave oven.

The first treatment factor was “brand.”’ Three levels were selected, including two national brands (levels 1 and 2) and one local brand (level 3). All three brands are packaged for household consumers in boxes of \(3.5\) ounce packages, and a random selection of packages was used in this experiment.

Power of the microwave oven was identified as a possible major source of variation and was included as a second treatment factor. Three available microwave ovens had power ratings of 500,600 , and 625W. The experimenters used only one oven for each power level. This means that their conclusions could be drawn only about the three ovens in the study and not about power levels in general.

Popping time was taken as a third treatment factor. The usual instructions provided with microwave popcorn are to microwave it until rapid popping slows to 2 to 3 seconds between pops. Five preliminary trials using brand 3, a 600W microwave oven, and times equally spaced from 3 to \(5 \mathrm{~min}\) suggested that the best time was between 4 and \(5 \mathrm{~min}\). Hence, time levels of \(4,4.5\), and \(5 \mathrm{~min}\) were selected for the experiment and coded 1-3, respectively.

Read Section 7.4 for detail about the measurements to be taken and how the sample size was determined.

We use the three-way complete model to anlyze the data.

SAS code:

data popmic;
  input brand power @@;
  do time=1 to 3;
    do rep=1 to 2; drop rep;
      input y @@;
      output;
  end; end;
  lines;
  1 1  73.8 65.5  70.3 91.0  72.7 81.9
  1 2  70.8 75.3  78.7 88.7  74.1 72.1
  2 1  73.7 65.8  93.4 76.3  45.3 47.6
  2 2  79.3 86.5  92.2 84.7  66.3 45.7
  3 1  62.5 65.0  50.1 81.5  51.4 67.7
  3 2  82.1 74.5  71.5 80.0  64.0 77.0
;
run;

proc print data=popmic;
run;

proc glm data=popmic;
class brand power time;
model y=brand power time brand*power brand*time power*time brand*power*time;
run;

Partial outcome is repreduced below.

Source DF Sum of Squares Mean Square F Value Pr > F
Model 17 4065.728889 239.160523 2.73 0.0206
Error 18 1577.870000 87.659444
Corrected Total 35 5643.598889
Source DF Type III SS Mean Square F Value Pr > F
brand 2 331.100556 165.550278 1.89 0.1801
power 1 455.111111 455.111111 5.19 0.0351
time 2 1554.575556 777.287778 8.87 0.0021
brand*power 2 196.040556 98.020278 1.12 0.3485
brand*time 4 1433.857778 358.464444 4.09 0.0157
power*time 2 47.708889 23.854444 0.27 0.7648
brand*power*time 4 47.334444 11.833611 0.13 0.9673

We see that power is significant and does not interact with brand and time. Therefore it makes sense to compare the main effects of power. We also like to know how the significant interaction effects of brank and time look like.

lsmeans power/cl pdiff;
lsmeans brand*time;

We see Time level 2 has the highest mean for Brand 1 and 2 but not for Brand 3.