[R] A question about using “by” in GAM model fitting of interaction between smooth terms and factor

[willow1980]

I am a little bit confusing about the following help message on how to fit a
GAM model with interaction between factor and smooth terms from
http://rss.acs.unt.edu/Rdoc/library/mgcv/html/gam.models.html:
“Sometimes models of the form: 
E(y)=b0+f(x)z
need to be estimated (where f is a smooth function, as usual.) The
appropriate formula is:
y~z+s(x,by=z)
- the by argument ensures that the smooth function gets multiplied by
covariate z, but GAM smooths are centred (average value zero), so the z+
term is needed as well (f is being represented by a constant plus a centred
smooth). If we'd wanted: 
E(y)=f(x)z
then the appropriate formula would be: y~z+s(x,by=z)-1.”
When I tried two scripts, I found they gave the same results. That is, the
codes “y~z+s(x,by=z)” and “y~z+s(x,by=z)-1” gave the same results. The
following is my result:
###########################################################################
“anova(model1,model2,test="Chisq")
Analysis of Deviance Table

Model 1: FLBS ~ SES + s(FAFR, by = SES) + s(byear, by = SES) + s(FAFR,
    byear, by = SES)
Model 2: FLBS ~ SES + s(FAFR, by = SES) + s(byear, by = SES) + s(FAFR,
    byear, by = SES) - 1
   Resid. Df Resid. Dev         Df  Deviance P(>|Chi|)
1 1.2076e+03     1458.4                               
2 1.2076e+03     1458.4 1.9099e-11 5.030e-10 2.074e-10”
###########################################################################
Is this in conflict with above statement that “If we'd wanted: E(y)=f(x)z
then the appropriate formula would be: y~z+s(x,by=z)-1.”? Also, if you are
familiar with GAM modelling, please have a look at my modelling process.
That is, I want to study how one factor together with two smooth terms will
influence the response. In model2, I also fitted the interaction between two
smooth terms, together with the interaction of this interaction with factor.
Is model 2 reasonable? I find it is rather complicated to interpret the plot
of model 2.
Thank you very much for helping!

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