9 – Introduction to Factorial Experiments
Example - Capsule Dissolving Experiment (Capsule.JMP)
In this experiment researchers are interested in studying the effect of two factors or treatments on the time to begin dissolving a capsule which is recorded as the time until bubbles first appear (seconds). The factors of interest to the researchers are digestive juice type - gastric or duodenal (Factor A) and capsule type - C or V (Factor B).
To conduct the experiment 5 capsules of each type are randomly assigned to each juice giving us 5 observations or replicates for each of the four treatment combinations
(Gastric & C, Gastric & V, Duodenal & C, Duodenal & V). The data obtained from the experiment are shown below:
Juice Type Means C V Type of Digestive Juice
45.7 43.5 Y？45.71：49.8 39.8 Gastric 50.2 36.1
36.7 41.2 Y？40.22：42 47.3 Duodenal 38.1 45.3
Capsule Type Means Grand Mean Y？43.05Y？42.85：1：2 X？42.95：：
We can construct plots to visualize the effects of each factor.
Digestive Juice Capsule Type
By plotting the mean time until By plotting the mean time until bubbles for both digestive juices, bubbles for both capsule types we see we can see that mean dissolution that the mean dissolution times for
time for duodenal juice is slightly the capsule types are approximately
smaller than that for gastric (about equal.
5 seconds). 127
Our preliminary conclusions would be first that fluid type has a small effect on the dissolution time with duodenal juice dissolving capsules about 5 seconds quicker on average, and secondly that capsule type has little or no effect.
These conclusions are completely WRONG!! Why?
When considering the effect of two factors on the response we cannot do so marginally, i.e. individually. It is possible, for example, that the effect of digestive juice is not the same for both capsule types. If we consider the means for each of the treatment
combinations above we see that for type C capsules the duodenal juice dissolves the capsule quicker, while the exact opposite is true for type V capsules, gastric juice dissolves the capsules faster.
A better display shows the means for each treatment combination. Here we have a separate profile for each digestive juice showing how the capsule effect depends on the type of digestive juice we are using. This is what we call an interaction.
Questions of Interest in Two-way ANOVA:
1) Is there a significant interaction between the two factors being studied?
This question needs to answered first, because if we conclude there is a
significant interaction then both effects are important and there effects can
not be discussed individually. If we conclude there isn’t a significant
interaction between the factors being studied then we can test the effects
2) Is there a significant Factor A effect?
3) Is there a significant Factor B effect?
As always it is important to quantify any significant differences using pair-wise comparisons and CI’s for the differences in the population/treatment means.
Analysis in JMP
To fit the two-way model for these data select Fit Model from the Analyze menu and put
the response Time to Bubbles in the Y box and then highlight both Fluid & Capsule and select Full Factorial from the Macros pull-down menu as shown below.
Then click Run Model to obtain the results on the next page.
These sections of output can be shut
off as our interest is in primarily
identifying which effects are
significant. These results are in the
Effect Tests box.
The Fluid*Capsule interaction is
significant (p=.0049), so we know
both fluid and capsule type
significantly effect the response.
The p-values for the effects suggests that the Fluid*Capsule interaction is significant (p = .0049), which implies the main effect tests for Fluid and Capsule are of little interest.
It is interesting to note that the main effect of Capsule is not significant (p = .9361). This
happens because the presence of the Fluid*Capsule interaction "masks" the main effect of
capsule as we have seen in marginal effect plots above. The main effect of fluid is only partially masked by the Fluid*Capsule interaction and so it still tests as significant.
Because the interaction is significant we can ONLY MEANINGFULLY COMPARE LEVELS OF ONE FACTOR WHILE HOLDING THE OTHER FACTOR FIXED! For
example, in this experiment we can compare juice types for a given capsule type or conversely we can compare capsule types for a given digestive juice type. To do this select LSMeans Tukey HSD from the Fluid*Capsule interaction pull-down menu.
Results of the treatment mean comparisons are shown below.
Here we see that Gastric,C and Duodenal,C mean dissolution times significantly differ. In particular we estimate that the type C capsules in gastric fluid take between 3.57 and
23.429 seconds longer to dissolve on average. In contrast type V capsules appear to dissolve equally well in either digestive juice.
If the interaction between the two factors is NOT significant we can use the Tukey’s
procedure to compare the means across the levels of each factor individually. This means that we can use Tukey’s pair-wise comparisons to compare the mean response across the
levels of factor A and factor B individually.
Checking Two-way ANOVA Assumptions (Normality and constant variance)
1. The observations between and within the treatment combinations are
2. The response is normally distributed for each treatment combination.
3. The variance of the response is the same for each treatment combination.
To check the constant variance assumption we can examine the residuals plotted vs. the
fitted values and each factor. The fitted values are simply the observed mean response at
each of the four treatment combinations and the residuals are the deviations from the treatment combination means. The spread of the residuals, i.e. the spread of the observed response values about their respective treatment combination means, should be uniform indicating constant response variation for the different treatment combinations.
A plot of the residuals vs. the fitted values is given each time we fit a model in JMP. The resulting plot is shown below:
There appears to be a potential outlier in this plot, otherwise this plot looks fine.
To examine the normality assumption we assess the normality of the residuals. Save the
residuals to the spreadsheet as shown below and use Analyze > Distribution to examine
them. With the exception of two mild outliers, normality seems satisfied.
STATISTICAL DETAILS (FYI, equal sample size case only)
Two-way ANOVA Model
Y？(;，;！;(，！);， i？1,...,a j？1,...,b k？1,...,nijkijijijk
th k observed response value when level i of factor A and level j of factor B is used. Y？ijk
effect due to the fact level i of Factor A was used. ，？i
effect due to the fact level j of Factor B was used. ！？j
thth effect due to the interaction of i level of Factor A and the j level of Factor B. (，！)？ij
ththe random error, represents the variation in the response values when the i level of Factor A and ，？ijkththe j level of Factor B are used.
2We assume that , i.e. the errors are normal and their variation is constant. ，~N(0,；)ijk
See your text for formulae used to estimate these quantities and those used to test the
hypotheses. The three questions of interest in a two-way ANOVA can be formulated in
terms of these parameter values.
1. For testing the interaction between Factors A and B we have:
，！H:()？0 for all treatment combinationsoij H:(，！)；0 for all treatment combinationsaij
2. For testing the Factor A effect we have:
，H:？0 for all ioi H:，；0 for all iai
3. For testing the Factor B effect we have:
！H:？0 for all joj H:！；0 for all jaj
As in one-way ANOVA the test procedures decomposes total response variation into components that measure how much variation in the response is due to Factor A, Factor B, the interaction between Factors A & B, and random error.
SS？SS;SS;SS;SS Sum of Squares: TotalABA，BError
N；1？(a；1);(b；1);(a；1)(b；1);ab(n；1)Degrees of Freedom:
SUM OF SQUARES FORMULAE:
aabnb222 = + + nb(X；X)(X；X)an(X；X)?：：：：：????：：：：：：：：iijkj？1？？？111？1iijkj
ababn22 + n(X；X；X;X)(X；X)?????：ij：i：：：j：：：：ijkij？？？111i？？j11ijk
MEAN SQUARES (measures of variation)
The mean square for an effect is the effect sum of squares divided by the degrees of freedom.
2When the null hypothesis of “no effect” is true the mean squares are all estimates of , ；
the common response variance for all treatment combinations. If there is a significant effect then we expect the ; (within treatment combination variation). MS？？MSeffectError
Testing Effect Significance
，For testing the main effects (A & B) and the interaction effect (AB) we simply compare
the size of the to the . If the >> we have evidence that MSMSMSMSeffecteffectErrorError
the effect is significant. If then we have little evidence that the effect is MS~MSeffectError
significant. This is analogous to the comparison of the between group variation to the within group variation in One-way ANOVA.
To compare the mean squares we use the ratio, which has an F-distribution.
MSeffectF？ F-distribution (numerator df = df for the effect , denominator df = df for error) ~oMSError
>> 1 will lead to the conclusion that the effect in question significantly impacts the Fo
response. Large values lead to small p-values which support effect significance. Fo
Example 2 – Comparing the Effectiveness of Three Forms of
Psychotherapy for Alleviating Depression
Suppose that a clinical psychologist is interested in comparing the relative effectivenss of three forms of psychotherapy for alleviating depression. Fifteen individuals are randomly assigned to each of three treatment groups: cognitive-behavioral, Rogerian, and assertiveness training. The Depression Scale of MMPI serves as the response. The psychologist also wished to incorporate information about the patients severity of depression, so all subjects in the study were classified as having mild, moderate, or severe depression. Thus we have two factor of interest in this study: the treatment they received and the initial severity of their depression. It is possible some forms of therapy may be more effective for certain levels of depression so a two-way ANOVA would be an appropriate method of analysis. The results are presented in the table below.
Therapy Mild Moderate Severe
Cognitive-Behavioral 41 51 45
(CB) 43 43 55
50 53 56
Rogerian (R) 56 58 59
47 54 55
45 49 68
46 61 63
Assertiveness Training 43 59 55
(A) 56 46 69
48 58 63
46 54 56
The data in JMP is entered as shown on the
left. The first column contains the Therapy
grown, the second column contains the
Degree of Severity of their depression, and
the last column denotes the MMPI
Data File: MMPI Depression
In JMP select Analyze > Fit Model and set up the dialog box as shown below. First highlight both Therapy and Degree of Severity by holding down the
select Full Factorial from the Macro pull-down menu as shown below.
The output is shown on the next page.
Degree of Severity
The interaction plot shown on the
left shows no signs of non-
parallelism and hence interaction,
and the p-value in the ANOVA
table suggests we have absolutely
no evidence for its significance