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st: Restricted cubic splines


From   "Dupont, William" <william.dupont@vanderbilt.edu>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: Restricted cubic splines
Date   Fri, 12 Nov 2004 14:41:54 -0600

Statalisters

We have recently posted an ado file on SSC to calculate restricted cubic
splines.

Description
-----------

rc_spline creates variables that can be used for regression models in
which the linear predictor f(xvar) is assumed to equal a restricted
cubic spline function of an independent variable xvar.  In these
regressions, the user explicitly or implicitly specifies k knots located
at xvar = t1, t2, ..., tk.  f(xvar) is defined to be a continuous smooth
function that is linear before t1, is a piecewise cubic polynomial
between adjacent knots, and is linear after tk.  See Harrell (2001) for
additional details.

rc_spline creates variables called _Sxvar1, _Sxvar2, ..., _Sxvar(k-1),
where "xvar" is the input variable name.  There are always one fewer
variables created than there are knots. If the model has k parameters
beta0, beta1, ... , beta(k-1) then

f(xvar) = beta0 + beta1*_Sxvar1 + beta2*_Sxvar2 + ... +
beta(k-1)*_Sxvar(k-1).  

An important aspect of restricted cubic splines is that the variables
_Sxvar1, ... , _Sxvar(k-1) are functions of xvar and the knots only and
are not affected by the response variable.  This means that we can use
rc_spline to define the _Sxvar* variables before specifying the response
variable or the type of regression model.  

Restricted cubic splines are also called natural splines.

Options
-------

nknots specifies the number of  knots.

knots specifies the exact location of the  knots.  The values of these
knots must be given in increasing order.


If both of these options are given they must both specify the same
number on knots.  When knots is omitted the default knot values are
chosen according to Table 2.3 of Harrell (2001) with the additional
restriction that the smallest knot may not be less than the 5th smallest
value of xvar and the largest knot may not be greater than the 5th
largest value of xvar.  The values of the all knots are displayed.  When
knots is omitted the number of knots specified by nknots must be between
3 and 7.  The default number of knots when neither nknots nor knots is
given is 5.

Frequency weights are allowed.

Examples
--------

    *
    * Perform a linear regression of y against a restricted  
    * cubic spline (RCS) function of x with 5 knots.
    *
    . rc_spline x
    . regress y _Sx1 _Sx2 _Sx3 _Sx4
    *
    * Perform a logistic regression of fate against
    * the RCS function of x defined above.
    *
    . logistic fate _S*
    
    *
    * Perform a linear regression of y against a RCS of x with 3 knots
chosen
    * at their default values according to Harrell (2001).  Graph the
observed
    * and expected values of y against x
    *
    . drop _S*
    . rc_spline x, nknots(3)
    . regress y _S*
    . predict yhat
    . scatter y x || line yhat x
    
    *
    * Perform a proportional hazard regression analysis of fate against
a RCS 
    * function of x with four knots specified at x =  2, 4, 6 and 8.
    *
    . drop _S*
    . stset time, failure(fate)
    . rc_spline x, knots(2 (2) 8)
    . stcox _S*
    
Remarks
-------

Restricted cubic splines provide a fairly general and robust approach
for adapting linear methods to model non-linear relationships between a
response variable and one or more continuous covariates.  They can often
be used effectively as an alternative to converting continuous to
categorical variables, which results in the discarding of information.
See Harrell (2001) for arguments in favor of this approach and guidance
on how to build models with RCSs.

This program is similar to spline (Sasieni 1994).  It differs in the
choice of default knots and in its output.  spline requires the user to
specify a response and independent variable.  It then allows the user to
specify a number of different regression models and version 7 graphs.
In contrast, rc_spline only calculates the RCS covariates.  However,
this allows the use of the full range and power of Stata's regression,
post-estimation and v.8 graph commands.  In particular, more
sophisticated residual analyses and graphs can be generated as well as
multiple regression models involving more than one independent variable.

See also mkspline for fitting models involving linear splines.

Authors
-------

William D. Dupont 
W. Dale Plummer, Jr.
Department of Biostatistics
Vanderbilt University School of Medicine
Nashville, TN 37232-2158

e-mail:  william.dupont@vanderbilt.edu
         dale.plummer@vanderbilt.edu

References
----------

Harrell, F.E: Regression Modeling Strategies with Applications to Linear
Models, Logistic
Regression and Survival Analysis. New York: Springer-Verlag 2001.

Sasieni, P: Natural cubic splines STB reprints. 1994; 4: 19-22.  See
also STB reprints 
1995; 4:174, and  package snp7_1 from http://www.stata.com/stb/stb24.




William D. Dupont          phone: 615-322-2001          URL
http://biostat.mc.vanderbilt.edu/twiki/bin/view/Main/WilliamDupont

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