Smooth terms are like a simple line in a generalized linear model (package mgcv)

Question

I have fit a generalized additive model with a binary response using the code below:

library(mgcv)
attach(mydata)
m = gam(y ~ dm  + af + s(BMI) + s(sleepworkday), family=binomial(logit), data=mydata, method="REML")
summary(m)

Not that dm and af are also binary. The results are:

Family: binomial 
Link function: logit 

Formula:
y ~ dm + af + s(BMI) + s(sleepworkday)

Parametric coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)  -1.3683     0.1291 -10.600  < 2e-16 ***
dmyes        -1.0930     0.3019  -3.621 0.000294 ***
afyes        -2.5139     1.0209  -2.462 0.013801 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
              edf Ref.df Chi.sq p-value  
s(BMI)          1.000  1.000  4.834  0.0279 *
s(sleepworkday) 1.112  1.216  5.700  0.0305 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R-sq.(adj) =  0.0447   Deviance explained = 6.58%
-REML = 247.49  Scale est. = 1         n = 606

It seems that both of smooth terms are significant. But they ate just a simple line as I use:

plot(m)

Here is the plot:

The other smooth term is also like a simple line. Why the smooth terms are significant yet are not curved? Should I change any options?

Answer 1

The p-values are for testing whether the curve is equal to zero, rather than whether it is equal to a straight line (the latter being not the most natural test when smoothing parameters are estimated automatically).