ANALYSIS OF VARIANCEANALYSIS OF VARIANCEAnalysis of VarianceI.A. CAN I TEST FOR SIMPLE EFFECTSIN THE PRESENCE OF ANINSIGNIFICANT INTERACTION?On more than a few occasions I have encountered the situa-tion in which a hypothesized two-way interaction was notsignificant according to the analysis of variance (ANOVA),and yet simple effects tests yielded results consistent withthe expected interaction (e.g., a significant simple effect inone condition but an insignificant simple effect in the othercondition). What can account for this apparent inconsis-tency? Moreover, if an interaction is hypothesized, is it ap-propriate to proceed with the simple effects tests eventhough the interaction term fails to achieve significance inthe ANOVA? I have seen this done in published articles andbelieve that it is appropriate when a priori expectations exist,but I have also heard others (including reviewers) argue thatsimple effects tests are appropriate only if the ANOVA inter-action term is significant.Editor: I see two elements in this question—namely, (a) Whymight a simple effect be significant when the overall interac-tion was not?, and (b) Must the overall interaction be signifi-cant to conduct tests of the simple effects?To answer to the first question, consider the apparent inter-action depicted in Figure 1. One’s MSerrorterm may be largeenough that the interaction would not be significant. The sim-ple effect of Factor A at Level b = 1 may also be insignificant.However, a test of the simple effect of Factor A at Level b = 2could be significant. In the assessment of the overall interac-tion, which is, after all, an aggregate of these components, thesignificant simple effect may be washed out by the insignifi-cant one.It would be a superiorly clean result if both simple effectswere significant—for example, either a qualitative differencein that the slopes were going in opposite directions, or a quan-titative difference with both slopes going in the same direc-tion, but one being steeper than the other. Either of thesesituations capture what is meant intuitively by an “interac-tion” (that the effect of one factor is contingent on the other).When one slope, or difference between means, is significant,but the other is not, the researcher is left to conclude that onehalf of the data are interesting, and one half yield the philo-sophically and statistically troublesome null result.Why is such a result, like that depicted in the figure,suboptimal? The analysis of “simple effects” breaks downsums of squares, not just those attributable to the A × B inter-action, but also the A main effect for the simple effects of A ateach level of B (or the B main effect for the simple effects of Bat each level of A). Most researchers will test all four simpleeffect combinations—SSA@b1, SSA@b2, SSB@a1, SSB@a2—andthen write up the two that illuminate the theorizing mostclearly. Let us say we focus on the simple effects of Factor Aat each level of Factor B. The sum of those simple effects’sums of squares, SSA@b1+ SSA@b2, will equal not SSA × B, butSSA+ SSA × B(cf. Keppel, 1991, pp. 241–242). The interactionin the typical 2 × 2 design has a single degree of freedom.Each simple effect also has one numerator degree of freedom.Testing two simple effects then uses two degrees of freedom,one more than may be derived from the interaction. The sec-ond degree of freedom is borrowed from the main effect ofFactor A—that is, the simple effects will in part reflect themain effects; the simple effects do not reflect only the interac-tion. Therefore, for example, an SSA@b1effect is partly deter-mined by the A × B interaction effect (which we want) andpartly capitalizes on the A main effect in the presence or evenabsence of an interaction (which is not good, and mathemati-cally the only thing we can do about that is to verify that theinteraction itself is significant).An intuitive way to understand these relations is again byexamining the plot. One pattern of data that is consistent withJOURNAL OF CONSUMER PSYCHOLOGY, 10(1&2), 5–35Copyright © 2001, Lawrence Erlbaum Associates, Inc.FIGURE 1 Hypothetical two-way interaction plot of means.the scenario under discussion would be if there was a strongmain effect for Factor A. If there were, even in the absence ofa significant interaction, the simple effect of A at b = 2 couldbe significant. As depicted, the a = 1 mean is lower than the a= 2 mean, both for the main effect (i.e., collapsing over b = 1and b = 2), and specifically also for the b = 2 case, the focalsimple effect.If one’s data looked like those in the figure, and one con-cluded that there was a significant simple effect of Factor A atLevel b = 2, but not for b = 1, it is somewhat misleading to sug-gest that there is an element of interaction in one’s data. Thesignificant simple effect states that the top line’s slope is sig-nificantly different from (greater than) zero. However, the in-significant interaction indicates that it is nevertheless notsignificantly different from the slope of the line beneath it.Abstractly, the second part of the question addresses theappropriate relation between an omnibus, overall F test of ahypothesis and tests for follow-up comparison of means thatcomprise a part of that global hypothesis. Overall F tests inour typical experimental world consist of two kinds—thosefor main effects and those for interactions. The statisticstexts that address this issue tend to present the argument inthe simpler, main effects context. We begin with main ef-fects, but subsequently will address the case of interactions.One may expect that the argument for the interaction is asimple extension of that for the main effect, but stay tuned,because it is not.Within the main effects context, we can further distinguishtwo classes of questions: What to do when the factor has twolevels, and what to do when the factor has three or more levels.Even after exposure to the most elemental ANOVA material,researchers know that factors with two levels are mathemati-cally simple; if a 2 × 2 factorial yields significant main effectsfor Factors A or B, there is no further analytical work to bedone to understand the nature of those main effects. The re-searcher knows the null hypothesis, H0: m1= m2, is to be re-jected, and taken together with the means on the two levels ofthe factor, the interpretation of such a finding is unambiguous.(The big mean is significantly larger than the small one.)