Explaining gam model output with a reference

Question

I'm seeking a good reference to explain the gam model output to colleague. Here is my model.

mod1 <- gam(severity ~  s(mean_rh, k = 8) + s(mean_temp, k = 10) + s(mean_ws, k =7) + s(avg_daily_rain, k = 7), family = betar(),  data = dat_seasonal)

summary(mod1)

Here is the output:

Formula:
disease_severity ~ s(mean_rh, k = 8) + s(mean_temp, k = 10) + 
    s(mean_ws, k = 7) + s(avg_daily_rain, k = 7)

Parametric coefficients:
            Estimate Std. Error z value Pr(>|z|)
(Intercept)  -0.1687     0.1374  -1.228    0.219

Approximate significance of smooth terms:
                    edf Ref.df Chi.sq  p-value    
s(mean_rh)        1.000  1.000   2.76 0.096633 .  
s(mean_temp)      4.231  4.598  74.22  < 2e-16 ***
s(mean_ws)        2.461  2.673  17.53 0.000669 ***
s(avg_daily_rain) 1.000  1.000  49.89  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R-sq.(adj) =  0.847   Deviance explained = 91.8%
-REML = -29.205  Scale est. = 1         n = 37

Here is the raw data plot (left) and the model output plot (right) for the predictor mean_temp.

My collaborator is not aware of GAMS and expects linear/continuous/higher disease severity with increasing temperatures OR a bell shaped curve (he doesn't expect less disease at 14 & 16 degree Celsius and higher disease at 12 degree Celcius). Here is his specific comment. "Does the figure show greater disease severity at 11 and 17 degree Celsius than the temperature in between? That is what I saw in this subfigure and that is against current knowledge and indefensible." I need to explain to him with a reference that the model has captured the raw data pattern very well, and the whole point behind a GAM is that it models complex behavior between predictors and outcomes. Terms with EDFs higher than one are not supposed to have a singular explanation. The response variable is not linear, DHARMa residuals are fine and the gam.check() output is also good. Thanks

Answer 1

An Introduction to Statistical Learning covers the principles of GAMs in Section 7.7, with Section 7.7.1 specifically on regression models like yours. I don't think that the particular smooths described there are identical to the s() smooths in the gam() function of mgcv that you used, but the general principle of smooth nonlinear additive modeling is the same.

Note that comparison to the raw data plot can be misleading, as it doesn't take into account the other predictor variables. I'll admit that I didn't at first see how the curve on the right could reasonably have represented the raw data on the left, until I remembered what I was looking at.

As mean_rh and avg_daily_rain ended up with what seem to be linear fits, you might display the residuals around what you would have predicted based on those linear fits, instead of showing raw data values. That might make the curvilinear association with mean_temp more obvious.

Answer 2

You could cite the famous book Generalized Additive Models, An introduction with R. It include a lot of explanations and several examples.