smspline {lmeSplines}

Smoothing splines in NLME

Description

Functions to generate matrices for a smoothing spline covariance structure, to enable fitting smoothing spline terms in LME/NLME.

Usage

  smspline(formula, data)
  smspline.v(time)

Arguments

formula

model formula with right hand side giving the spline covariate

data

optional data frame

time

spline ‘time’ covariate to smooth over

Details

A smoothing spline can be represented as a mixed model (Speed 1991, Verbyla 1999). The generated Z-matrix from smspline() can be incorporated in the users's dataframe, then used in model formulae for LME random effects terms at any level of grouping (see examples). The spline random terms are fitted in LME using an identity 'pdMat' structure of the form pdIdent(~Z - 1). The model formulation for a spline in time (t) is as follows (Verbyla 1999):

y = X_s \beta_s + Z_s u_s + e

where X_s = [1 | t] , Z_s = Q(t(Q) Q)^{-1} , and u_s ~ N(0,G_s), is a set of random effects. We transform the set of random effects u_s to independence with u_s = L v_s, where

v_s ~ N(0,I \sigma^2_s)

is a set of independent random effects. The Z-matrix is transformed accordingly to Z = Z_s L, where L is the lower triangle of the Choleski decomposition of G_s.

The function smspline.v() is called by smspline(), and can be used to access the matrices X_s, Q, G_s. See Verbyla (1999) for further information.

Value

For smspline(), a Z-matrix with the same number of rows as the data frame. After fitting, the LME model output gives a standard deviation parameter for the random effects, estimating \sigma_s. The smoothing parameter from the penalised likelihood formulation is

\lambda = \sigma^2/\sigma^2_s

For smspline.v(), a list of the form

Xs

X-matrix for fixed effects part of the model

Zs

Z-matrix for random effects part of the model

Q, Gs, R

Matrices Q, G_s, R associated to the mixed-model form of the smoothing spline.

Note

The time points for the smoothing spline basis are, by default, the unique values of the time covariate. This is the easiest approach, and model predictions at the fitted data points, can be obtained using predict.lme. By interpolation, using approx.Z, the Z-matrix can be obtained for any set of time points and can be used for fitting and/or prediction. (See examples). Synopsis:data$Z <- smspline(formula1, data); fit <-lme(formula2, data, random= ...)

Author(s)

Rod Ball [e-mail] https://www.scionresearch.com/

References

The correspondence between penalized likelihood formulations of smoothing splines and mixed models was pointed out by Speed (1991). The formulation used here for the mixed smoothing spline matrices are given in Verbyla (1999). LME/NLME modelling is introduced in Pinheiro and Bates (2000).

Pinheiro, J. and Bates, D. (2000) Mixed-Effects Models in S and S-PLUS Springer-Verlag, New York.

Speed, T. (1991) Discussion of “That BLUP is a good thing: the estimation of random effects” by G. Robinson. Statist. Sci., 6, 42–44.

Verbyla, A. (1999) Mixed Models for Practitioners, Biometrics SA, Adelaide.

See Also

approx.Z nlme

Examples

# smoothing spline curve fit
data(smSplineEx1)
# variable `all' for top level grouping
smSplineEx1$all <- rep(1,nrow(smSplineEx1))
# setup spline Z-matrix
smSplineEx1$Zt <- smspline(~ time, data=smSplineEx1)
fit1s <- lme(y ~ time, data=smSplineEx1,
    random=list(all=pdIdent(~Zt - 1)))
summary(fit1s)
plot(smSplineEx1$time,smSplineEx1$y,pch="o",type="n",
     main="Spline fits: lme(y ~ time, random=list(all=pdIdent(~Zt-1)))",
     xlab="time",ylab="y")
points(smSplineEx1$time,smSplineEx1$y,col=1)
lines(smSplineEx1$time, smSplineEx1$y.true,col=1)
lines(smSplineEx1$time, fitted(fit1s),col=2)

# fit model with cut down number of spline points
times20 <- seq(1,100,length=20)
Zt20 <- smspline(times20)
smSplineEx1$Zt20 <- approx.Z(Zt20,times20,smSplineEx1$time)
fit1s20 <- lme(y ~ time, data=smSplineEx1,
    random=list(all=pdIdent(~Zt20 - 1)))
# note: virtually identical df, loglik.
anova(fit1s,fit1s20)
summary(fit1s20)

# model predictions on a finer grid
times200 <- seq(1,100,by=0.5)
pred.df <- data.frame(all=rep(1,length(times200)),time=times200)
pred.df$Zt20 <- approx.Z(Zt20, times20,times200)
yp20.200 <- predict(fit1s20,newdata=pred.df)
lines(times200,yp20.200+0.02,col=4)


# mixed model spline terms at multiple levels of grouping
data(Spruce)
Spruce$Zday <- smspline(~ days, data=Spruce)
Spruce$all <- rep(1,nrow(Spruce))
# overall spline term, random plot and Tree effects
spruce.fit1 <- lme(logSize ~ days, data=Spruce,
                   random=list(all= pdIdent(~Zday -1),
                     plot=~1, Tree=~1))
# try overall spline term plus plot level linear + spline term
spruce.fit2 <- lme(logSize ~ days, data=Spruce,
                   random=list(all= pdIdent(~Zday - 1),
                     plot= pdBlocked(list(~ days,pdIdent(~Zday - 1))),
                     Tree = ~1))
anova(spruce.fit1,spruce.fit2)
summary(spruce.fit1)