2No main effects but interaction present Can I have a significant interaction without significant main effects?YesConsider the following table of meansB1B2A1102015A2
3No main effects but interaction present We can see from the marginal means that there is no difference in the levels of A, nor difference in the levels of factor BHowever, look at the graphical displayB1B2A1102015A2
4No main effects but interaction present In such a scenario we may have a significant interaction without any significant main effectsAgain, the interaction is testing for differences among cell means after factoring out the main effectsInterpret the interaction as normal
5RobustitudeAs in most statistical situations there would be a more robust method for going about factorial anovaInstead of using means, we might prefer trimmed means or medians so as to have tests based on estimates not so heavily influenced by outliers.And, there’s nothing to it.
6Between subjects factorial using trimmed means InteractionMain effects
7Robustitude Or using R you type in something along the lines of t2way(A, B, x, grp=c(1:p), tr=.2, alpha=.05)Again, don’t be afraid to try robust methods as they are often easily implemented with appropriate software.
8Unequal sample sizesAlong with the typical assumptions of Anova, we are in effect assuming equal cell sizes as wellIn non-experimental situations, there will be unequal numbers of observations in each cellSemester/time period for collection ends and you need to graduateQuasi-experimental designParticipants fail to arrive for testingData are lost etc.In factorial designs, the solution to this problem is not simpleFactor and interaction effects are not independentDo not total up to SSb/tInterpretation can be seriously compromisedNo general, agreed upon solution
9The problem (Howell example) MichiganArizonaColumnmeansNon-drinkingDrinkingRow meansDrinking participants made on average 6 more errors, regardless of whether they came from Michigan or ArizonaNo differences between Michigan and Arizona participants in that regard
10ExampleHowever, there is a difference in the row means as if there were a difference between StatesOccurs because there are unequal number of participants in the cellsIn general, we do not wish sample sizes to influence how we interpret differences between meansWhat can be done?
11Another exampleHow men and women differ in their reports of depression on the HADS (Hospital Anxiety and Depression Scale), and whether this difference depends on ethnicity.2 independent variables--Gender (Male/Female) and Ethnicity (White/Black/Other), and one dependent variable-- HADS score.
12Note the difference in gender 2.47 vs. 4.73A simple t-test would show this difference to be statistically significant and noticeable effect
13Unequal sample sizesNote that when the factorial anova is conducted, the gender difference disappearsIt’s reflecting that there is no difference by simply using the cell means to calculate the means for each gender( )/3 vs. ( )/3
14Unequal sample sizes What do we do? One common method is the unweighted (i.e. equally weighted)-means solutionAverage means without weighting them by the number of observationsNote that in such situations SStotal is usually not shown in ANOVA tables as the separate sums of squares do not usually sum to SStotal
15Unequal sample sizesIn the drinking example, the unweighted means solution gives the desired resultUse the harmonic mean of our sample sizesNo state difference17 v 17With the HADS data this was actually part of the problemThe t-test would be using the weighted means, the anova the ‘unweighted’ meansHowever, with the HADS data the tests of simple effects would bear out the gender difference and as these would be part of the analysis, such a result would not be missedIn fact the gender difference is largely only for the white categoryi.e. there really was no main effect of gender in the anova design
16A note about proportionality Unequal cells are not always a problemConsider the following tables of sample sizesB1B2B3A151020A240B1B2B3A152010A250
17Table 2 is not proportional The cell sizes in the first table are proportional b/c their relative values are constant across all rows (1:2:4) and columns (1:2)Table 2 is not proportionalRow 1 (1:4:2)Row 2 (1:1:5)B1B2B3A151020A240B1B2B3A152010A250
18ProportionalityEqual cells are a special case of proportional cell sizesAs such, as long as we have proportional cell sizes we are ok with traditional analysisWith nonproportional cell sizes, the factors become correlated and the greater the departure from proportional, the more overlap of main effects
19More complex design: the 3-way interaction Before we had the levels of one variable changing over the levels of anotherSo what’s going on with a 3-way interaction?How would a 3-way interaction be interpreted?
202 X 2 X 2 Example*Sometimes you will see interactions referred to as ordinal or disordinal, with the latter we have a reversal of treatment effect within the rangeof some factor being considered (as in the left graph).
23InterpretationAn interaction between 2 variables is changing over the levels of another (third) variableInteraction is interacting with another variableAB interaction depends on CRecall that our main effects would have their interpretation limited by a significant interactionMain effects interpretation is not exactly clear without an understanding of the interactionIn other words, because of the significant interaction, the main effect we see for a factor would not be the same over the levels of anotherIn a similar manner, our 2-way interactions’ interpretation would be limited by a significant 3-way interaction
24Simple effects Same for the 2-way interactions However now we have simple, simple main effects (differences in the levels of A at each BC) and simple interaction effects
25Simple effectsIn this 3 X 3 X 2 example, the simple interaction of BC is nonsignificant, and that does not change over the levels of A (nonsig ABC interaction)Consider these other situations
26Simple effectsAs mentioned previously, a nonsignificant interaction does not necessarily mean that the simple effects are not significant as simple effects are not just a breakdown of the interaction but the interaction plus main effectIn a 3-way design, one can test for simple interaction effects in the presence of a nonsignificant 3-way interactionThe issue now arises that in testing simple, simple effects, one would have at minimum four comparisons (for a 2X2X2),Some examples are provided on the website using both the GLM and MANOVA procedures. Here is another from our backyard: