can i compute an interaction variable by multiplying a standardized variable and a mean centered variable?

Question

I need to do a multiple hierarchical regression analysis for my master thesis.

my continues independent variable is: online exposure to cannabis use. I used five items/questions/variables to compute a total score. However, the five variables have different scales (4 answer questions and 6 answer questions) thus, i need to standardize the variables in order to compute a total score of 'online exposure' with Z scores

i have done this in SPSS by discriptives: and checked the box 'save standardized values as variables'. Then i computed the mean of these standardized variables.

i am researching a moderation effect. So my moderator variables are 'peer use' (continues) and 'educational track (binary)'. Peer use has been mean centered (on insisting advice of my professor)

I think i standardized the right way. pls correct me if im wrong.

My question is:

1. can i calculate the interaction term by multiplying online exposure (standardized z scores) * peer use (mean centered)

Can i just do that? one is standardized with z scores and the other is mean centered.

Also, the second moderator is binary, not standardized and not mean centered, can i also do the same for this

2. calculate the interaction term by multiplying online exposure (standardized z scores) * educational track (binary, not mean centered, not standardized)

thank you for your time

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@EdM

Thank you so much for your answer!

i think i should a bit clearer with the regressions im analyzing. The dependant continues variable is in every mentioned model 'cannabis use in youth' (not centered, not standardized, scores 0-3)

the first hierarchical regression is the main link: model 1: age group (control variable, binary) model 2: online exposure to cannabis use (independent continues variable)

Then i need to explore 2 interactions in two seperate regressions

the second hierarchical regression: model 1: age group (control variable) model 2: online exposure to cannabis use, peer use (peer use is continues independent moderator variable) model 3: online exposure to cannabis use * peer use

hierarchical regression 3: model 1: age group (control variable) model 2: online exposure to cannabis use, educational track (education is categorial binary 0,1) model 3: interaction online exposure to cannabis use * educational track

so, online exposure to cannabis use is standardized with z scores, whereas peer use is only mean centered. The interaction in regression 2 is thus: standardized scores * mean centered scores.

The question was: is that correct? i tried to standardize peer use as wel in stead of mean center, and they almost gave me the same results. But im not sure if im suppose to do that

You told me to be consistent, so my guess is to standardize peer use with z scores as well, in stead of only mean center it. The reason why i standardized online exposure is because of the different scales, peer use has the same scales thus i only mean centered it at first) The interaction remains not significant, but for the thesis, it needs to be statistically right. So really want to know if what i am doing is okay. Obviously it makes no sense to standardize the binary variables, so i did not do that. I need to know if i must standardize peer use, because i standardized online exposure as well.

Thank you so much for your time, very appreciated!

Answer 1

An interaction term is just a product of the corresponding predictors in numeric form (e.g., the "dummy" 0/1 variables often used to encode categorical predictors). So you can specify interaction terms in any way that makes sense for your application.

You don't need to center at all, although that sometimes helps with numeric issues and can make some coefficients easier to interpret. Just make sure that you are consistent within your model. To that end, it's safest to do whatever centering/scaling you wish first and then let the software construct the interactions on its own.

What will change depending on your choice of centering of any one predictor will be, perhaps surprisingly, the lower level coefficients of other variables with which it interacts.

One example (probably the type that informs your professor) is when you have a continuous predictor that has values typically far from zero interacting with, say, a binary predictor. The coefficient for the binary predictor then represents its association with outcome when the continuous predictor has a value of 0. If you pre-center the continuous predictor around its mean, the coefficient for the binary predictor is its association with outcome at that original mean value. Predictions from the models are the same either way, but then the binary-predictor coefficient represents its association with outcome in a more "realistic" scenario.

When there is an interaction, however, it can be misleading or dangerous to interpret any lower-level coefficient without considering its interactions. In particular, don't jump to conclusions based on apparent "significance" of lower-level coefficients when there are interactions. That "significance" is just for the coefficient's difference from 0 when its interacting predictors are at 0 or reference levels. With interactions you need to take values of all interacting predictors into account.

In response to extended question:

First, if your dependent variable is just an integer score with values from 0 to 3, you shouldn't be treating it as a continuous variable. The assumptions about the distributions of error terms needed for interpreting standard linear regression almost certainly won't hold. You should do an ordinal regression. This page shows how to do that in SPSS. The handling of predictor variables is the same as with ordinary regression, but the associations with outcome are better suited to a scale of {0,1,2,3}.

Second, it doesn't seem that you are gaining anything by performing those separate regressions. From what you describe, I don't see a problem with a single model containing all of your predictors and interaction terms of interest: age group, online exposure to cannabis use, the age:exposure interaction, peer_use, the peer_use:exposure interaction, educational track, and the track:exposure interaction. That helps control for all of those predictors together and avoids the risk of omitted-variable bias in the individual models that omit some predictors. Also, you typically will do better if you model age continuously and flexibly (e.g., with regression splines) instead of breaking down into groups. From a single combined model you then can do specific post-hoc comparisons of scenarios of interest.

As the initial part of the answer says, you don't need to center or standardize any of your predictors. I don't find standardizing to be helpful, as that depends on the standard deviation of predictor values in the data set at hand, which might not hold for new data.

When I cautioned you to be "consistent" I meant that for any one predictor. That is, if you choose to standardize peer_use, make sure (if you calculate interactions yourself) that its interactions are also based on its standardized values. It's better to let the software calculate the interactions to avoid possible errors.

You certainly are welcome to center or standardize some predictors and handle others differently in this type of model. You just have to make sure that you remember what was done if you need to make predictions on new data from your model.