[R] which coefficients for a gam(mgcv) model equation?

[Simon Wood]

Darren, 

Sorry for the slow reply. This is probably much to late to be of use to you 
now. I think that the referee is being unreasonable here. There are many 
perfectly respectable ways of estimating GAMs for which no explicit 
expression for the estimated smooth terms is available (See Hastie and 
Tibshirani's GAM book). mgcv actually  does use smoothers that can be 
represented explicitly, but it is completely un-illuminating to do so. 

I've attached a paper describing exactly how the smooths you have used are 
represented. You could provide the referee with expression (7), and the 
description of how \delta_i and \alpha_i are obtained from the coefficients 
actually estimated, but it's not going to help the paper's readers much.

best,
Simon

On Thursday 11 March 2010 16:22, Darren Norris wrote:
> I have spent a few days trying to figure this from the reply out but am
> still stuck!
> I need the equation to reply to a request from a referee that was to: "show
> the specific estimating equation associated with the fitted line".
> the model I am running is (I hope the data frame is not necessary as I
> think I am just not getting some basic concept, but it can be provided off
> list): gam1<-gam(LR~s(Property_lg),data=property) ## use default family
> gaussian(link = "identity")
>
> coef(gam1) returns:
>      (Intercept) s(Property_lg).1 s(Property_lg).2 s(Property_lg).3
> s(Property_lg).4 s(Property_lg).5
>       44.1777350       -9.4673457       -1.5743877        0.5658906
> 2.2219434        0.4118942
> s(Property_lg).6 s(Property_lg).7 s(Property_lg).8 s(Property_lg).9
>        2.4477335       -0.6590291       14.6142365        3.4184510
>
> so is the "estimating equation":
> E(y_i) ~ 44.1777 + f_1(-9.467) + f_2(-1.574) + f_3(0.565) +f_4(2.221) +
> f_5(0.411) + f_6(2.447) + f_7(-0.659) +f_8(14.614)+f_9(3.418)
>
> >From the “predict” function I know the fitted value at x = 0 (intercept)
> > is
>
> -25.5256255 and at x = 1 is -1.3417508. How do I calculate these values
> from the “estimating equation” above?
> For x=1, I am doing the calculation below which is obviously incorrect, but
> how do I calculate the predicted values by hand, I must be missing
> something incredibly obvious?
>  44.1777 + (1*-9.467) + (1*-1.574) + (1*0.565) + (1*2.221) + (1*0.411) +
> (1*2.447) + (1*-1*0.659) + (1*4.614)+ (1*3.418)
>
> Many thanks for any further guidance,
> Darren
>
>
>
> coef(b)
>
> will give you the coefficients for the smooth terms + the intercept for
> that model. ?gamObject describes thus a little.
>
> The coefficients for the smooth/spline terms here are of length 9 each,
> and each set of 9 coefficients pertains to an f(), so the model in the R
> code you gave would be something like
>
> E(y_i) ~ alpha + f_1(x0_i) + f_2(x1_i) + f_3(x2_i) +f_4(x3_i)
>
> or
>
> E(y_i) = alpha + f_1(x0_i) + f_2(x1_i) + f_3(x2_i) +f_4(x3_i) + e, where
> e ~ N(0, sigma)
>
>
> If so, perhaps you could provide more details on why you want the
> equation for the model?

-- 
> Simon Wood, Mathematical Sciences, University of Bath, Bath, BA2 7AY UK
> +44 1225 386603  www.maths.bath.ac.uk/~sw283 
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