# Interaction/Moderation effect in ANOVA - variable is not one of the factors

I am running an univariate ANOVA in SPSS to test my hypotheses for my 3x2 between subjects design experiment. The factors for the ANOVA are review valence (3 levels) and presence of brand (2 levels). I want to see how these factors affect purchase probability.

I also gathered data for the brand commitment of each participant in the experiment. Brand commitment is measured by giving a score on a scale from 1-9(fully disagree/fully agree) for 3 statements (taken from previous paper). Brand commitment is high/low if the scores are in the upper/lower third of the scale.

I want to understand what is the interaction between brand commitment and review valence, and how it affects the DV. I predicted that brand commitment will moderate the effect of the review on the DV.

My question is how can I add this brand commitment "interaction or moderation" in the ANOVA or should I use a different method to analyse it. Also, should I compute a new variable that combines the 3 statements in one new variable for the analysis.

Any insights that you can provide me with, will be highly appreciated.

Stefan

• Maybe you can use Moderated Multiple Regression instead. Thanks. – user171576 Jul 28 '17 at 9:54

To test for interaction / moderation (basically synonymous), first add brand commitment as a predictor in your linear model, then multiply brand commitment by review valence, and add the product to the model too. It is important to have both the main effects in the model when including their interaction, and you may want to center the variables before multiplying them to handle nonessential multicollinearity (I've read in places that this is not much of a solution though, and haven't got a better one offhand).

Brand commitment is ordinal though, so ANOVA is not appropriate (I wonder if review valence might be ordinal too) – it wastes information by ignoring the order of ranks. There are a few options to consider here:

• If you think the three brand commitment items measure the same latent construct...
• ...and you have a lot of data and motivation to do things right, you can estimate a latent continuous construct with a rating scale model à la item response theory, then add the latent factor as a single continuous predictor. This will allow control of measurement error.
• ...and you want to go the quick and dirty route, you can adopt the usual classical test theory assumptions, take the average or sum of the three items' Likert ratings, and enter this as a single continuous predictor. This assumes measurement error balances out, but it may not.
• If you think the three brand commitment items measure different constructs, or want to estimate their relationships separately for some other reason (and are prepared to tangle with potential multicollinearity problems)...
• Penalized regression (see "Continuous dependent variable with ordinal independent variable") works well for ordinal predictors. It may give you a bit of a headache to have 8 dummy coefficients for each of your three brand commitment items though.
• Nine response options per item might be enough to justify treating responses as continuous, especially if their frequencies aren't distributed too bizarrely. See my discussion of Bollen and Barb (1981) here – basically, with more than five response options, binning continuous data into ordinal groups doesn't attenuate correlations too awfully much (i.e., less than 10%), so the logic is that data collected as ordinal ought to be continuous enough with > 5 ranks.
• Interactions will be tougher to interpret with this method. You'll end up with products of all your dummy coefficients, which will probably make for some terrifying output. AFAIK, you can test the overall significance of these interactions with an F-test comparing the model with all these interactions to the nested model without all of them, which will produce relatively succinct output...but IMO, it's still a lot more daunting than having an interaction between a single continuous variable and one other categorical factor with three levels. Even without the F-test, that's only two product terms (one for each dummy code of the discrete variable).

Another issue to consider is the reference distribution for your response variable. Purchase probability sounds binary, in which case you'd want to use logistic regression, but if you have actual percentages that you're modeling instead of just did/not purchase (0/1, or some such codes), you might want to go with beta regression, or maybe even a nonparametric model.

• Centring/scaling predictors isn't "much of a solution" because there isn't much of a problem. If three predictors take values of around one hundred, their three-way interaction takes values of around one million; it may therefore be convenient for calculation & interpretation to use linear transformations of some or all of the predictors that result in lower numbers to deal with. Such a change makes no difference whatsoever to predictions, likelihood, &c.; it's just that some coefficients are now measured in different units. – Scortchi Jun 3 '14 at 12:57
• That's true in general for maximum-likelihood estimation of linear models: when you add penalty terms to the likelihood that depend on the magnitude of the coefficient estimates, as in ridge regression or LASSO, then the scale does need to reflect your ideas of "big" and "small". – Scortchi Jun 3 '14 at 12:58
• Review valence and brand factors are the manipulations for the experiment - respondents read a piece of information with no/positive/negative review and not mentioned/mentioned brand. Purchase probability is measured as a percentage (1-100%). However, I am far from expert in statistics so most of the things that you mentioned sound alien to me. So to find out if there is interaction I should run a regression? I read somewhere that interactions can be added in the ANOVA somehow, if I am not mistaken. If you can suggest me some metod that I can look a tutorial for somewhere will be great. – Stefan Jun 3 '14 at 16:59
• T-tests of the same null hypotheses give the same results whether you centre predictors or not. Sometimes people forget that even if you use the same symbols for all coefficients after centring, some now represent different population parameters (see here) and therefore a test that says $\beta_2=0$ isn't necessarily the same test in models with & without centred predictors. – Scortchi Jun 4 '14 at 10:54
• There are lots of questions about centring & scaling on the site, but it's not always very clear when the answers deal with (1) the theoretical (un)equivalence of, (2) the computational (dis)advantages of fitting, or (3) the ease (difficulty) of interpreting different models. A canonical answer - rigorous & general, with a clear scope - might be useful. – Scortchi Jun 4 '14 at 10:55