Compare different models via p-value, AIC and BIC

Question

A sensor with the response Sw shall be investigated if it is affected by external influences like Temperature Tu and relative humidity rH.

m1<- lm(Sw ~ Tu + rH + Tu:rH, data=data)

Coefficients:
(Intercept)           Tu           rH        Tu:rH  
  9.927e-01    2.805e-04    9.455e-04    2.264e-05  

summary.lm(rH_Dat_fit3)

Call:
lm(formula = Sw ~ Tu + rH + Tu:rH, data = rH_Dat_ges)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.028543 -0.003187  0.001420  0.005486  0.018200 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 9.927e-01  4.443e-04 2234.38   <2e-16 ***
Tu          2.805e-04  7.961e-06   35.24   <2e-16 ***
rH          9.455e-04  8.649e-06  109.33   <2e-16 ***
Tu:rH       2.264e-05  1.591e-07  142.32   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.009133 on 8996 degrees of freedom
Multiple R-squared:  0.9817,    Adjusted R-squared:  0.9817 
F-statistic: 1.608e+05 on 3 and 8996 DF,  p-value: < 2.2e-16

> AIC(m1)
[1] -58978
> BIC(m1)
[1] -58942.47

First, is that a correct expression to study these influences? Second, to study whether e.g. rH is necessary, the following model comes into mind:

m2<- lm(Sw ~ Tu, data=data)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.046917 -0.020902 -0.003081  0.012441  0.073881 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) 1.009e+00  5.202e-04  1939.5   <2e-16 ***
rH          2.036e-03  1.029e-05   197.8   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.02918 on 8998 degrees of freedom
Multiple R-squared:  0.8131,    Adjusted R-squared:  0.813 
F-statistic: 3.913e+04 on 1 and 8998 DF,  p-value: < 2.2e-16

> AIC(m2)
[1] -38069.12
> BIC(m2)
[1] -38047.81

A colleague of mine asked me to have a look at (t)his analysis and I'm used to perform that a bit different but I'm not sure whether this makes a difference or not.

For example, when I want to know whether I need another parameter, I would run above models and compare them via an anova and the AIC, BIC and p-value would tell me whether I need this parameter. Is this different then when I look at these parameters only in regards to the single models?

Furthermore, when having a look at model m2: Since there is only one explanatory variable isn't it obvious that it is necessary (due to the p-value)?

Answer 1

First, is that a correct expression to study these influences?

It is if you can rule out non-linear, especially U-shaped influences, in which case you could consider quadrativ terms, e. g.

m1<- lm(Sw ~ Tu*rH + I(Tu^2) + I(rH^2), data = data)

but that will depend on your knowledge about how the sensor works.

Second, to study whether e.g. rH is necessary

rHis super-significant in the statistical meaning of the term "significance" with $p<10^{-15}$, so we know that rH definitively plays a role. Models without it will fit worse. However, you have to define what "necessary" means in your specific context. Maybe the influence is so small that it does not matter, even though it is statistically significant. The estimated coefficient for rHis 9.455e-04 = .0009455. Is it necessary to know when using the sensor?