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I have full model summary:

lm(formula = Swir2ref ~ x1 + y1 + x2 + y2)

Residuals:
       Min         1Q     Median         3Q        Max 
-0.0219319 -0.0061738 -0.0000579  0.0060872  0.0197778 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    

(Intercept)  0.637722   0.029059  21.946  < 2e-16 ***

x1           0.040600   0.004534   8.954 5.24e-13 ***

y1          -0.012630   0.032577  -0.388  0.69950    

x2          -4.438403   1.642197  -2.703  0.00874 ** 

y2          -1.502393   0.724077  -2.075  0.04190 *  
---

Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.009531 on 66 degrees of freedom
Multiple R-squared:  0.579, Adjusted R-squared:  0.5535 
F-statistic: 22.69 on 4 and 66 DF,  p-value: 8.043e-12

I have summary of interaction model:

    lm(formula = Swir2ref ~ x1 + y1 + x2 + y2 + x1 * y1)

Residuals:
       Min         1Q     Median         3Q        Max 
-0.0199133 -0.0059021 -0.0008173  0.0061518  0.0204091 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.72335    0.05054  14.313  < 2e-16 ***
x1           0.19106    0.07361   2.596  0.01166 *  
y1          -0.25164    0.12098  -2.080  0.04146 *  
x2          -4.52826    1.60445  -2.822  0.00632 ** 
y2          -1.47253    0.70732  -2.082  0.04130 *  
x1:y1       -0.41058    0.20050  -2.048  0.04462 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.009309 on 65 degrees of freedom
Multiple R-squared:  0.6045,    Adjusted R-squared:  0.5741 
F-statistic: 19.87 on 5 and 65 DF,  p-value: 5.767e-12

My question is in the full model there is y2 is insignificant, but for the interaction model it is significant. What would be the reason? Help will be appreciated. Thank you!

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  • $\begingroup$ Do you mean y1? Because y2 looks to be significant in both models... $\endgroup$
    – Pyll
    Sep 11, 2017 at 17:03
  • $\begingroup$ yes, I am sorry. It should be y1. $\endgroup$
    – ychaulagain
    Sep 11, 2017 at 17:04
  • $\begingroup$ This is not a programming question. If you need help with the interpretation of statistical models, you should be asking such questions over at Cross Validated, not Stack Overflow. $\endgroup$
    – MrFlick
    Sep 11, 2017 at 17:31

1 Answer 1

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I started typing a response, but @Glen_b said it really well...

https://stats.stackexchange.com/questions/160026/why-does-an-insignificant-regressor-become-significant-if-i-add-some-significant

There are two main reasons this can happen.

Adding a significant regressor - whether related to dummies or not - reduces the mean square residual. By reducing estimate of error variance, other regressors may become more significant even if the parameter estimate hardly changes, by reducing the denominator of their t-statistic (equivalently, their partial F). This one needn't involve any dependence between regressors at all.

Adding a regressor can also change the numerator of the t-statistic, by changing the parameter estimate, due to dependence between regressors, which can move coefficients either toward or away from zero; it will also alter the denominator (so it's not as simple as just considering the numerator). Sometimes the overall effect can be to make a previously insignificant regressor significant.

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