Suggest you look at this link:

https://www.google.com/search?client=firefox-b-1-d&q=inference+in+generalized+additive+models

HTH, David Booth

Abraham Eustace

Interpreting the approximate significance of the smooth terms is as good as interpreting the edf in comparison to the basis dimension k-1. From your output, say s(dist_road_km), I can see that you you used the default k=10. This k is just saying or set as the maximum number of turning points to be used during the smoothing process. When the results give you edf, they are telling you how many turning points where actually found in the smoothing process. By the way these are usually referred to as the knots. The actual model fitting will use k-1 because the the other one is taken by the intercept. When some smoothing has been done, the edf gives an idea of how complex was the smoothing. If edf is close to k-1 it means there is potential to increase the basis dimension but watch out on your AIC if it is worth it. If edf is close to zero it is like no smoothing was done and the transformed response values are almost zero. To check this use the "inspect_random" function to plot the curves for that term. By the way, recall that GAM explore the response based on the mean-centered values. As such, the y-axis show values around zero (which you can call mean). The closer these new(transformed values) they are to this zero, the more it indicates that the smooth curve is basically not explaining the changes in the response.

NOW, instead of going through all this that I have explained, the p-value of smooth terms can give you a good sense of which smooth curves are indeed explaining your response...and it comes back to the same way you interpreted the GLM p-values. In your results: the smooth curves s() are showing significant changes in the response when the explanatory changes. The real nature of those changes can ONLY be visualized. GAM will have a horizontal line at zero (mean). When the curve is above that zero line, it means the original response value was above the average (intercept) and vice versa. NOTE: even if the p-value is insignificant, the response curves will still show patterns as usual but remember it will be just being sensitive to the very small values around zero...those trends mean nothing. I hope that will give you a footing going forward. Thank you for sharing your work.

Thank you very much for clarification Partson Tinarwo

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Article Statistical inference for generalized additive models: simul...