mod_gam1 <- gam(Overall ~ s(Income, bs="cr"), data=d)
## Family: gaussian
## Link function: identity
## Formula:
## Overall ~ s(Income, bs = "cr")

## Approximate significance of smooth terms:
## edf Ref.df F p-value
## s(Income) 6.9 7.74 16.4 2e-14 ***

It significant p-value<0.05 mean that smooth component use for income was correct or that independent variable income had significant effect on overall?

I'm new to GAM. I read several comment, paper and lecture not on it but i'm still confuse.


It means that given the smoothing function that was applied, there is a significant association between $income$ and $Overall$. There is no way to judge from the output whether the specific spline you chose via bs="cr" is "correct". This rather depends on which type of model works best for your goal and application.

You can compare the performance of different models (GAMs and non-GAMs) and then base your inference about the influence of income on the best one, but the reported p-value and coefficients will always refer to the effect of income on the outcome given the specified model.

You can check the spline that was fit via plot(model_gam1) and employ some critical reasoning whether this makes sence. If the only "problem" with $income$ is that it is right-skewed, which is quite usual, a simple log-transformation might be sufficient, which avoids some interpretability issues of GAMs.


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