Yes - you can still interpret the main the effects. The problem is that the main effects mean something different in a main effects only model versus a model with an interaction (unless the interaction accounts for no variance in the outcome Y at all).

Assuming a linear regression model:

Y = b0 + b1*X1 + b2*X2 + e

Here the effects of X1 and X2 on Y are independent and thus additive. This means that an effect of X1 can be interpreted in isolation from X2 when X2 is held constant.

In the interaction model:

Y = b0 + b1*X1 + b2*X2 + b3X1*X3 + e

Even if X2 is held constant it is not possible to know what effect X1 has on Y without also taking to account the value of X2. So if b1 is positive in the first equation one can interpret this as saying that increases in X1 are associated with increases in Y when X2 is constant, but in the second equation this is true for some values of X2 only.

That said if you know the ranges of X1 and X2 that you are interested in one can parameters the model in a way that allows you to interpret the main of effect of X1 as an average or typical effect (e.g., it might be that X1 always increases Y within the range of interest). In more complex cases it will nearly always be better to graph the whole pattern over the range of interest and see how the main effects and interaction effects combine to influence the outcome.

Yes, you should interpret the full set of effects as a package, because the interaction term basically expresses differences in how the variables involved affect the dependent variable.

I get results that don't make sense. I have a variable that is supposed to have a negative effect on the dependent variable. When I test main effects only, it has a negative effect, but when I enter an interaction term b/w this variable and a moderator, this variable's main effect becomes positive (and significant). How can I explain that?

Dear Marina,

I've recently looked at this for binary logistic regression based on the advice in Andy Field's book. I don't know if this is the same as scale variable regression but the advice seems to be:

Yes - you can still interpret the main the effects. The problem is that the main effects mean something different in a main effects only model versus a model with an interaction (unless the interaction accounts for no variance in the outcome Y at all).

Assuming a linear regression model:

Y = b0 + b1*X1 + b2*X2 + e

Here the effects of X1 and X2 on Y are independent and thus additive. This means that an effect of X1 can be interpreted in isolation from X2 when X2 is held constant.

In the interaction model:

Y = b0 + b1*X1 + b2*X2 + b3X1*X3 + e

Even if X2 is held constant it is not possible to know what effect X1 has on Y without also taking to account the value of X2. So if b1 is positive in the first equation one can interpret this as saying that increases in X1 are associated with increases in Y when X2 is constant, but in the second equation this is true for some values of X2 only.

That said if you know the ranges of X1 and X2 that you are interested in one can parameters the model in a way that allows you to interpret the main of effect of X1 as an average or typical effect (e.g., it might be that X1 always increases Y within the range of interest). In more complex cases it will nearly always be better to graph the whole pattern over the range of interest and see how the main effects and interaction effects combine to influence the outcome.

If you are trying to interpret a main effect in an interaction model it isn't a main effect in the classical sense and its sign could be very misleading if interpreted that way. One way to aid interpretations is to centre the predictors before computing the interaction term (or before having the software compute it for you). This means that subtract the mean from each predictor so that its new mean is zero. Thus you evaluate its effects in the model relative to the mean of other variables (rather than when they are zero) and this tends to make the model more interpretable (because you are evaluating its effects at typical values of your data set).

Even so you really need to plot the interaction if it not obvious from the coefficients.

In terms of interpreting the results, it would help to know if your two predictors are continuous or dichotomous, or a mix of both.

I attach a file with the results and interaction plots. 

You can see the variables in my model. I have a problem with the effect of academic and decisional procrastination, and LD in Model 2 (incl. interaction): they become positive, and I don't know how to explain it.

Thank you very much!

Hi.  u could use any model or change scale of depended variable

Plot your data. Assuming two qualitative factors A and B with a and b levels, compare the axb fitted values from the Y=A+B model to the fits from Y=A+B+A.B (both averaged over C, D, X1, X2, etc) to see what effect the interaction is having. (Put Y on y-axis and A or B on x-axis (highest of a or b usually) using b or a different lines. If one of A or B is quantitative, as if X, then X goes on x-axis, with as many lines as the qualitative factor has.

The interaction may be statistically real but scientifically tiny compared to main effects. Conversely you may have crossover interaction such that the average main effect of A does not make sense unless you specify over which level of B. E.g. Y = A + X + A.X where A is gender of parent, X=weight at offspring conception, Y = weight at child's birth. Silly example but still silly to talk of average parental wt change. Your own answer will be context dependent.

In your case the interactions look potentially substantial. I think the interpretation will be cleaner if you centre the predictors before computing the interaction terms. The positive coefficients are really just a quirk of the algebra here because the ranges of the outcome is a large positive number. So the coefficients in the interaction model have to be positive for the main effects as the interaction term is superadditive (well actually 'super-subtractive' because of the direction of the effect).

In your case the pattern is more complicated because you've simultaneously added two interaction terms and they may be somewhat collinear. So the model has probably also changed because some of the original main effect in LD is accounted for by the interaction of decisional procrastination and academic procrastination.

In other words the change in the model is not solely due to a single interaction effect.

A rough summary: In your final model it looks as though academic procrastination has little effect for the non-LD group but a substantial negative effect for the LD group. In addition academic and decisional procrastination is very negative for all groups - but particularly so for the when people are high in both. (All these effects should be obvious from the graphs and come out clearer in a centred model).

It sounds to me like you are running into an issue with VIF. If some factors in your data set are highly correlated to each other, when you add or subtract variables, the coefficients can change significantly.

I would center each of your variables and look for the interaction again. By subtracting the mean of each factor from every factor in that column, (X1i-X1bar), X2i -X2bar,...) and looking at interactions again, the VIF for each interaction term will be significantly lower. This will give you a better idea if the interaction really is real, or not.

As a general rule, an interaction occurs when the effect of one factor is dependent upon another factor. I also believe they are more prevalent that most believe because without centering a variable, the VIF is so high, you can get a false positive and false negative significance from your factors.   

Dear all, thank you very much! Centering variables solved the problem!

It's good Marina, that you arrived at a better model. I just want to add that IF INTERACTIONS REA REALLY REAL and we have significant interaction term,  we should still interpret main effects.

It is sometimes the case that an interaction term has a very small p-value, but

the associated main effects do not.

The hierarchical principle states that if we include an interaction in a model, we

should also include the main effects, even if the p-values associated with principle

their coefficients are not significant. In other words, if the interaction between X1 and X2 seems important, then we should include both X1 and X2 in the model even if their coefficient estimates have large p-values.

The rationale for this principle is that if X1 × X2 is related to the response, then whether or not the coefficients of X1 or X2 are exactly zero is of little interest. Also X1 × X2 is typically correlated with X1 and X2, and so leaving them out tends to alter the meaning of the interaction.

Reference: Introduction to Statistical Learning with applications in R