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I have a question about multiple regression model. May I kindly ask how should I interpret the conditional main effect when there is interaction and the interaction is 0, more specifically if one of the interaction terms contains zero such as angle or time. Please see an example representing my problem below,

z= b0+b1*X+b2*Y-b3*X*Y    
Y (time) and X are continues variables

If I am not mistaken, we cannot interpret an isolated main effect, when there is a significant interaction; however, we can interpret in a whole model conditionally. If we check the effect of X on z when the time is different than 0 and and equals to zero, the equations then becomes as follow,

z1= b0+b1*X+b2*Time-b3*X*Time 
z1=b0+b1*X+b2-b3*X  -> y1=b0+(b1-b3*Time)*X+b2*Time

The effect of X, while increasing by one unit, on z1 is b1-b3*Time. So as time increases the effect is moving to more negative side.

z2= b0+b1*X+b2*Time(0)-b3*X*Time (0) 
z2=b0-b1*X

The effect of X, while increasing by one unit, on z2 is positive b1 and it actually increase the z1. What does this practically mean? Or, should we conclude that as the Time increases the effect of X on Z1 is decreasing b1-b3*Time and you cannot individually check each time point but treat it as continues variable?

Last question, is it important if main effects here X and Time are significant or not if interaction is significant.

Can someone please help me to understand this kind of equation?

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You are correct that we cannot interpret the main effects of variables that are interacted with another variable, or at least not in the usual way.

When a variable is interacted with another variable, the interpretation is conditional on the other variable that it is being interacted with, being held at zero (or in the case of a categorical variable, at it's reference level). For this reason, it is often a good idea to centre numeric variables around their mean, so that it then become conditional on the other variable being held at it's mean.

My opinion on the questions around statistical significance in models that have an interaction, is to not rely on significance at all. If your interpretation of a model completely changes when an estimate's p-value is 0.04999 compared to when it is 0.05001 (or using whatever level of significance you choose) then I believe you are making a mistake. If you think that an interaction should be important for your study, then include it. As for interpretation, the interaction itself is interpreted as the estimated change in the outcome for a unit change in a variable, when the other variable it is interacted with also changes. If you want to say something about the p-value then just make sure you use the correct definition: that it is the probability of obserbving these data, or data more extreme, if the null hypothesis is true (that is, in regression, if the true parameters are really zero).

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  • $\begingroup$ thank you for your kind reply. If I centre the numeric variables, should I centre all the variables, in this case both X and Time? How this would effect my result? If I am not mistaken, if I center time points 0,1,2,3, does it mean, it wont contain any more 0 and it means I can interpret only this b1-b3*Time as there is no zero in the Time? $\endgroup$
    – Tim
    Commented May 31, 2021 at 13:02
  • $\begingroup$ You're welcome :) Centring both means that each main effect is conditional on the other being at it's mean value. This is useful for things like height ot weight where it does not make sense to think of them as being zero. But since you have time and presumably time=0 makes sense in your model, there is no need to centre it. The same is true of X, if X=0 makes sense. $\endgroup$
    – Robert Long
    Commented May 31, 2021 at 13:57
  • $\begingroup$ thank you for your valuable explanation :). I gave an example of 'Time', however, I have actually used angle in my model (0, 30, 60, 90, 120). The complication is that when angle is 0 degree, the effect of X becomes b1-b3*0 and the effect of X on z becomes positive b1. However, if angle is different than 0, i.e. 120 degrees, the effect of X on z becomes b1-b3*120. So we see that, as time increases the effect is becoming more negative. Technically, increasing X should have decreasing effect on Y, which we cant say when angle is 0 since b1 is positive. What am I missing here? $\endgroup$
    – Tim
    Commented May 31, 2021 at 14:18

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