# Penalized spline/mixed model

library(lmeSplines)
# 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)


This is the sample code from the help(smspline) documentation. I am have some trouble understanding the statistical model behind this example. What exactly is Spruce$Zday <- smspline(~ days, data=Spruce) accomplishing? Is it creating a set of random effects for the days variable? Can the model fit by spruce.fit1 be written out as:$Y_{ij} = \beta_1 + \beta_2 t_{ij} + \sum_{m=1}^{11} a_m(t_{ij} - \kappa_m)_+ + b_{1i}+b_{2i}t_{ij} + b_{3i}plot + b_{4i}tree + \epsilon_{ij}\$? because there is a random effect for plot and a random effect for tree. And how is this different from the formulation of spruce.fit2?

What does the below output from summary(spruce.fit1) mean?

 Formula: ~1 | plot %in% all
(Intercept)
StdDev:   0.0890616

Formula: ~1 | Tree %in% plot %in% all
(Intercept) Residual
StdDev:    0.621729 0.176505


Is this referring to the variance-covariance matrix of the plot and tree random effects? I am have a difficult time relating the code/output to the actual statistical concepts.