Re: st: SPLINE commands

[Roger Newson <r.newson@imperial.ac.uk>]

As it happens, linear splines are a special case of B-splines. The 
sensible basis for a linear spline (I think) is a basis of fuzzy 
interval membership functions, also known as tent functions. These fuzzy 
interval membership functions are both the B-spline basis and the 
reference-spline basis of the space of linear splines on a given set of 
knots. So, the corresponding parameters are the values of the spline at 
the knots. And the values of the spline between any 2 successive knots 
are linearly interpolated between the values of the spline at the knots. 
So, as Maarten says, linear splines are easy to interpret, although not 
expected to be literally true. For some illustrations see Newson (2011), 
which is the latest version of the -bspline- manual.

Best wishes

Roger


References

Newson R. B-splines and splines parameterized by their values at 
reference points. Updated 18 January 2011. Downloadable from
http://www.imperial.ac.uk/nhli/r.newson/papers.htm#program_manuals
as of 04 February 2011


Roger B Newson BSc MSc DPhil
Lecturer in Medical Statistics
Respiratory Epidemiology and Public Health Group
National Heart and Lung Institute
Imperial College London
Royal Brompton Campus
Room 33, Emmanuel Kaye Building
1B Manresa Road
London SW3 6LR
UNITED KINGDOM
Tel: +44 (0)20 7352 8121 ext 3381
Fax: +44 (0)20 7351 8322
Email: r.newson@imperial.ac.uk
Web page: http://www.imperial.ac.uk/nhli/r.newson/
Departmental Web page:
http://www1.imperial.ac.uk/medicine/about/divisions/nhli/respiration/popgenetics/reph/

Opinions expressed are those of the author, not of the institution.

On 04/02/2011 16:42, Maarten buis wrote:
> --- On Fri, 4/2/11, Ronald McDowell wrote:
> > I'm not familiar with the concept of splines, and am
> > looking for a gentle introduction to the area, in
> > order to move beyond using quadratic and cubic etc
> > terms in my models.
>
> You could look at Marsh, Lawrence C. and David R. Cormier
> (2002) "Spline Regression Models". Quantitative Applications
> in the Social Sciences, nr. 137. Thousand Oacks: Sage.
>
> I am actually moving back towards linear splines (from
> more smooth restricted cubic, B-splines, etc.), as I
> find linear splines to have a nicer balance between
> interpretability of the parameters and flexibility of the
> curve. Anyone who can interpret regular regression
> parameters can also interpret the parameters of a linear
> spline terms.
>
> Consider the example below:
> *--------------- begin example --------------------
> sysuse auto, clear
> mkspline mpg1 20 mpg2 = mpg
> reg price mpg1 mpg2 foreign
>
> // use adjust to predict price while keeping foreign at 0
> adjust foreign = 1, by(mpg) generate(yhat)
>
> // graph the predicted price against mpg
> twoway line yhat mpg, sort
> *---------------- end example ----------------------
>
> The graph illustrates what happend, we basically have
> two linear regression: one for cars with an mpg<  20
> and one for cars with an mpg>20, and the regression
> lines meet at mpg == 20. Moreover, the standard
> parameterization, as implemented by -mkspline-, lets
> you interpret the coefficients of these splines as
> regular regression coefficients. So, for cars with
> mpg<  20 and additional mile per gallon leads to a
> drop in price of 845 dollars, while for cars with
> mpg>  20 the drop in price is a insignificant 70
> dollars per mile per gallon.
>
> As always there is a price that needs to be paid for
> such convenient interpretability, and for linear
> splines it is that sudded change in direction at the
> knot and the linearity between the knots. Some people
> find this not smooth enough or not realistic enough.
> However, I am willing to sacrifice a lot of "realism"
> of my model if that helps me to get across what I
> have done to my data in order to arrive at my
> conclussions. With linear splines one often must view
> models as a useful summary/simplification of reality,
> but isn't that what a model is supposed to be anyhow?
>
> Having said all that, work has been done on making
> the coefficients of other types of splines more
> interpretable, but linear splines seems to me a
> logical place to start before entering into more
> complicated variations of it (and don't be afraid
> to move back to linear splines once you have looked
> at those variations).
>
> Hope this helps,
> Maarten
>
> --------------------------
> Maarten L. Buis
> Institut fuer Soziologie
> Universitaet Tuebingen
> Wilhelmstrasse 36
> 72074 Tuebingen
> Germany
>
> http://www.maartenbuis.nl
> --------------------------
>
>
>
>
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