After the linear regression on the main predictors, how to include the interactions of them?

Question

I'm currently using R to do a multiple linear regression with 7 main predictors. I've already completed the first step of regressing the dependent variable onto those main predictors.

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -17.218435   4.644294  -3.707  0.00024 ***
cylinders     -0.493376   0.323282  -1.526  0.12780    
displacement   0.019896   0.007515   2.647  0.00844 ** 
horsepower    -0.016951   0.013787  -1.230  0.21963    
weight        -0.006474   0.000652  -9.929  < 2e-16 ***
acceleration   0.080576   0.098845   0.815  0.41548    
year           0.750773   0.050973  14.729  < 2e-16 ***
origin         1.426141   0.278136   5.127 4.67e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The result shows that 3 of them have their p-value higher than 0.1, which means that they are not significantly important to dependent variable. However, when came to consider the interaction terms, should I neglect those insignificant ones, or still keep them for regression?

If includes, the result will be as follows:

Coefficients:
                            Estimate Std. Error t value Pr(>|t|)   
(Intercept)                3.548e+01  5.314e+01   0.668  0.50475   
cylinders                  6.989e+00  8.248e+00   0.847  0.39738   
displacement              -4.785e-01  1.894e-01  -2.527  0.01192 * 
horsepower                 5.034e-01  3.470e-01   1.451  0.14769   
weight                     4.133e-03  1.759e-02   0.235  0.81442   
acceleration              -5.859e+00  2.174e+00  -2.696  0.00735 **
year                       6.974e-01  6.097e-01   1.144  0.25340   
origin                    -2.090e+01  7.097e+00  -2.944  0.00345 **
cylinders:displacement    -3.383e-03  6.455e-03  -0.524  0.60051   
cylinders:horsepower       1.161e-02  2.420e-02   0.480  0.63157   
cylinders:weight           3.575e-04  8.955e-04   0.399  0.69000   
cylinders:acceleration     2.779e-01  1.664e-01   1.670  0.09584 . 
cylinders:year            -1.741e-01  9.714e-02  -1.793  0.07389 . 
cylinders:origin           4.022e-01  4.926e-01   0.816  0.41482   
displacement:horsepower   -8.491e-05  2.885e-04  -0.294  0.76867   
displacement:weight        2.472e-05  1.470e-05   1.682  0.09342 . 
displacement:acceleration -3.479e-03  3.342e-03  -1.041  0.29853   
displacement:year          5.934e-03  2.391e-03   2.482  0.01352 * 
displacement:origin        2.398e-02  1.947e-02   1.232  0.21875   
horsepower:weight         -1.968e-05  2.924e-05  -0.673  0.50124   
horsepower:acceleration   -7.213e-03  3.719e-03  -1.939  0.05325 . 
horsepower:year           -5.838e-03  3.938e-03  -1.482  0.13916   
horsepower:origin          2.233e-03  2.930e-02   0.076  0.93931   
weight:acceleration        2.346e-04  2.289e-04   1.025  0.30596   
weight:year               -2.245e-04  2.127e-04  -1.056  0.29182   
weight:origin             -5.789e-04  1.591e-03  -0.364  0.71623   
acceleration:year          5.562e-02  2.558e-02   2.174  0.03033 * 
acceleration:origin        4.583e-01  1.567e-01   2.926  0.00365 **
year:origin                1.393e-01  7.399e-02   1.882  0.06062 . 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The term acceleration is previously insignificant but now turns to be significant instead, does it mean that it is influenced by the interaction terms so that it conceals its contribution to the dependent variable?

Answer 1

Yes (although I would not call it "concealing"). The point is that if an interaction is relevant, the main effect may or may not be significant in itself. Thus, it is generally considered a bad idea to start with an all-additive, all-linear model, since effects may be non-linear and non-additive.

There are literally tons of posts on this. I recommend those answered by Frank Harrell, who has a very explicit view on how to identify a model. Maybe start here: Should covariates that are not statistically significant be 'kept in' when creating a model?

The longer answer is that it all depends on the purpose of your model (again I recommend Frank Harrell's book Regression Modeling Strategies). For inference, you should avoid model selection, while for prediction it is fine. There are several places to look for guidance, e.g. under the topic of Shmueli's (2010, Stat Sci) paper "To explain or predict?".