# Smoothing splines in NLME Functions to generate matrices for a smoothing spline covariance structure, to enable fitting smoothing spline terms in LME/NLME. ```r 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. ## Returns 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$Z <- smspline(formula1, data); fit <-lme(formula2, data, random= `...`) ## References The correspondence between penalized likelihood formulations of smoothing splines and mixed models was pointed out by Speed (1991). The formulation used here for the mixed smoothing spline matrices are given in Verbyla (1999). LME/NLME modelling is introduced in Pinheiro and Bates (2000). Pinheiro, J. and Bates, D. (2000) Mixed-Effects Models in S and S-PLUS Springer-Verlag, New York. Speed, T. (1991) Discussion of ``That BLUP is a good thing: the estimation of random effects'' by G. Robinson. Statist. Sci., 6, 42--44. Verbyla, A. (1999) Mixed Models for Practitioners, Biometrics SA, Adelaide. ## Author(s) Rod Ball rod.ball@scionresearch.com [https://www.scionresearch.com/](https://www.scionresearch.com/) ## See Also `approx.Z` `nlme` ## Examples ```r # smoothing spline curve fit data(smSplineEx1) # variable `all' for top level grouping smSplineEx1$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) ```

smspline function