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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.

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