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#' @title Example 8.3 from Generalized Linear Mixed Models: Modern Concepts, Methods and Applications by Walter W. Stroup (p-255)
#' @name Exam8.3
#' @description Exam8.3 explains Response surface design with incomplete blocking
#' @author \enumerate{
#' \item Muhammad Yaseen (\email{myaseen208@@gmail.com})
#' \item Adeela Munawar (\email{adeela.uaf@@gmail.com})
#' }
#' @references \enumerate{
#' \item Stroup, W. W. (2012).
#' \emph{Generalized Linear Mixed Models: Modern Concepts, Methods and Applications}.
#' CRC Press.
#' }
#'
#' @seealso
#' \code{\link{DataSet8.3}}
#'
#' @import lmerTest lattice
#'
#' @examples
#'
#' ## Response Surface Design with incomplete blocking (page 255)
#' data(DataSet8.3)
#' DataSet8.3$block <- factor(x = DataSet8.3$block)
#' DataSet8.3$aa <- factor(x = DataSet8.3$a)
#' DataSet8.3$bb <- factor(x = DataSet8.3$b)
#' DataSet8.3$cc <- factor(x = DataSet8.3$c)
#'
#' library(lmerTest)
#' library(lattice)
#'
#' Exam8.3.fm1 <-
#' lmer(
#' y ~ aa:bb:cc + a + b + c +
#' I(a^2) + I(b^2) + I(c^2) +
#' a*b + a*c + b*c + (1|block)
#' , data = DataSet8.3
#' )
#'
#' ##--- page 256
#' anova(Exam8.3.fm1, ddf = "Kenward-Roger", type = 1)
#'
#'
#' Exam8.3.fm2 <-
#' lmer(
#' y ~ a + b + c +
#' I(a^2) + I(b^2) + I(c^2) +
#' a*b + a*c + b*c + (1|block)
#' , data = DataSet8.3
#' )
#' ##--- page 257
#' anova(Exam8.3.fm2, ddf = "Kenward-Roger", type = 1)
#'
#' ##--- page 257
#' Exam8.3.fm3 <-
#' lmer(
#' y ~ a + b + c +
#' I(a^2) + I(b^2) +
#' a*c + b*c + (1|block)
#' , DataSet8.3
#' )
#' anova(Exam8.3.fm3, ddf = "Kenward-Roger", type = 1)
#'
#' ##--- scatter plot with regression plane by using Hoerl function ( page#233)
#' a <- seq(from = -1, to = 1, by = 1)
#' b <- seq(from = -1, to = 1, by = 1)
#' c <- seq(from = -1, to = 1, by = 1)
#' abc <- expand.grid(a = a, b = b, c = c)
#'
#' Yhat <- NULL
#' for(i in 1:nrow(abc)) {
#' Yhat[i] <- 50.08500 + 1.6*abc$a[i] + 1.69375*abc$b[i] + 0.51875*abc$c[i]-
#' 3.30250*I((abc$a[i])^2)-3.51500*I((abc$b)^2)[i] -
#' 0.52500*(abc$a)[i]*(abc$c)[i]-1.16250*(abc$b)[i]*(abc$c)[i]
#' }
#'
#' Newdata <- data.frame(abc, Yhat)
#' Plot1 <-
#' wireframe(Yhat ~ b*a, data = subset(Newdata,c==-1),
#' xlab = "b", ylab = "a",
#' main = "Predicte response surface at C=-1", colorkey = FALSE,
#' drape = TRUE, scales = list(arrows = FALSE),xlim=c(max(b),(min(b))),
#' screen = list(z = -50, x =-70)
#' )
#'
#' Plot2 <-
#' wireframe(Yhat ~ b*a, data = subset(Newdata,c==0),
#' xlab = "b", ylab = "a",
#' main = "Predicte response surface at C=0", colorkey = FALSE ,
#' drape = TRUE, scales = list(arrows = FALSE),xlim=c(max(b),(min(b))),
#' screen = list(z = -50, x =-70)
#' )
#'
#' Plot3 <-
#' wireframe(Yhat ~ b*a, data = subset(Newdata,c==1),
#' xlab = "b", ylab = "a",
#' main = "Predicte response surface at C=1", colorkey = FALSE,
#' drape = TRUE, scales = list(arrows = FALSE),xlim=c(max(b),(min(b))),
#' screen = list(z = -50, x =-70)
#' )
#'
#' print(Plot1)
#' print(Plot2)
#' print(Plot3)
#'
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