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I'm regressing a model $Y = X_1 + X_2 + X_1X_2$ and the result turns out that none of them are significant. However, if I drop the interaction term, $X_1$ becomes significant.

Is it ok to drop the interaction term? What could be the reason behind this issue?

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    $\begingroup$ It might help if you add more information. E.g. what is the range of the Y, X1, and X2 variables and is the coefficient on X1 very different for both models. In the model without interactions, the coefficient on X1 indicates the effect of X1 holding X2 constant. In the model with interactions, the coefficient on X1 reflects the effect of X1 with X2 = 0. I.e. your testing different hypotheses. Another possibility is that by adding another explanatory variable (the interaction), you are decreasing the degrees of freedom in the p-value calculations, hence p-values increase. $\endgroup$
    – stijn
    Apr 28 '15 at 9:29
  • $\begingroup$ Thank you for your answers. The magnitude of the coefficients are similar in both models. The P value of the interaction term is very high (0.84). X1 in the model with interaction term has a p-value of 0.18 when it was 0.05 in the model without interaction term. I also have the constant term (intercept) in my estimations. $\endgroup$ Apr 29 '15 at 4:46
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    $\begingroup$ Based on your answers, I would guess you do not have that many observations and the interaction term absorbs some of the main effect (are X1 and X1X2 correlated?). A change from p=0.05 and p=0.18 is not that unexpected with few observations. At this point, I would say report both models if that is possible and if both models make sense a priori. $\endgroup$
    – stijn
    Apr 29 '15 at 4:55
  • $\begingroup$ There are 405 observation in my study. I do believe that the interaction term absorbs some of the main effect as well. Thank you very much. $\endgroup$ Apr 29 '15 at 5:03
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Since you thought apriori that interaction might exist, statistical performance of estimators and confidence intervals will work better if you stay with the pre-specified model and form meaningful contrasts from that 3 parameter (+ intercept) model. You can get confidence limits for the effect of raising X1 by so many units if you hold X2 to a pre-specified constant (then vary that constant).

There are some occasions where removal of a term is OK as far as preserving statistical inference when the $P$-value is high enough (say 0.4) so that bootstrap repetition of the entire process selects the same model every time.

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