## Theorem 6.4(for balanced data & variable order p): call the functions of theorem 5.6 (a)(b)(c)

test.r.order<-function(data, covariate, p0, p1){
  data <- as.matrix(data)
  Z <- as.matrix(covariate)
  N <- nrow(data)
  n <- ncol(data)
  m <- ncol(Z)
  p <- p0
  q <- p1-p0

  ##theorem 5.6(c)   
  partial <- partial.inter(cov.cor(REMLE.cov(data, Z, p+q, REMLE=TRUE)))

  sum <- 0
  sum.M <- 0
  for(i in 2:n){
    
    sum1 <- 0
    sum1.M <- 0
    if(q[i]!=0){
      for(j in 1:q[i]){
        r <- partial[i,i-p[i]-j]  
        sum1 <- sum1+log(1-r^2)
        sum1.M <- sum1.M+(N-p[i]-j)*log(1-r^2)
      }
    }
    sum <- sum+sum1
    sum.M <- sum.M+sum1.M
  }
    
  LRT <- -N*sum
  MLRT <- -sum.M

  df <- sum(q[2:n])
  pvalue.LRT <- 1-pchisq(LRT, df)
  pvalue.MLRT <- 1-pchisq(MLRT, df)
  
  list(LRT=LRT, pvalue.LRT=pvalue.LRT, MLRT=MLRT, pvalue.MLRT=pvalue.MLRT)    
}



####################################################################################
## the following functions are of theorem 5.6 (a)(b)(c)

###############Theorem 5.6(a)###############
## set REMLE=TRUE for REML estimation
## set REMLE=FALSE for ML estimation

REMLE.para<-function(data, covariate, p, REMLE){
  data <- as.matrix(data)
  Z <- as.matrix(covariate)
  N <- nrow(data)
  n <- ncol(data)
  m <- ncol(Z)
  V <- matrix(rep(0, n*n), nrow = n)

  if(length(p)==1)  p <- rep(0:p, c(rep(1,p),(n-p)))    

  if(m==1){
    beta.hat <- as.vector(apply(data,2, mean))
    A <- as.matrix(cov(data))*(N-m)/N
  }

  else{
    Y <- NULL
    for(i in 1:N)
      Y <- cbind(Y, t(data[i,]))
    Y <- t(Y) 
    A <- matrix(rep(0, n*n), nrow = n)
 
    ## (5.30) p.134
    beta.hat <- kronecker(solve(t(Z)%*%Z)%*%t(Z), diag(n))%*%Y
  
    for(i in 1:N){
      M <- matrix(data[i,], nr=n)-kronecker(matrix(Z[i,], nr=1), diag(n))%*%beta.hat
      A <- A+M%*%t(M)
    }
    A <- A/N 
  }

  ## (5.31, 5.32)
  if(REMLE==TRUE)  cov <- (N/(N-m))*A       
  else  cov <- A 
         
  for(i in 1:n){
    for(j in (i-p[i]):i)
      V[j,i] <- V[i,j] <- cov[i,j]
  }

  if(m==1)   
    list(muhat=beta.hat, cov.p=V)
  else   
    list(betahat=beta.hat, cov.p=V)
}



###############Theorem 5.6(b)###############   

## calculate the correlations on off diagonals
cov.cor<-function(V){
  for(i in 2: nrow(V)){
    for(j in 1:(i-1))
      V[j,i] <- V[i,j] <- V[i,j]/sqrt(V[i,i]*V[j,j])
  }
  V
}

## recursive formula for mle in(2.32)
recur.mle<-function(i, p, cov){
  kasi <- matrix(t(cov[c((i-p):(i-1)), c(1:(i-p-1))]), nr=(i-p-1)) %*%
    solve(t(cov[c((i-p):(i-1)), c((i-p):(i-1))])) %*% matrix(t(cov[i, c((i-p):(i-1))]), nr=p)
  kasi
}

REMLE.cov<-function(data, covariate, p, REMLE){
  n <- ncol(data)

  if(length(p)==1)  p <- rep(0:p, c(rep(1,p),(n-p)))  

  cov <- REMLE.para(data, covariate, p, REMLE=REMLE)$cov.p

  for(i in 2:n){
    if((i-p[i]) > 1 && (i-p[i]) < i){
      kasi <- recur.mle(i,p[i], cov) ## recursive formula in (2.44)
      for(j in 1:(i-p[i]-1))
        cov[j,i] <- cov[i,j] <- as.vector(kasi[j])
    }
  }     
  cov
}


###############Theorem 5.6(c)###############

## inverse
inv <- function(cov){
  solve(cov)
}

## pi>0 for i>1
## intervenor-adjusted partial correlation

## recursion formula for partial correlations in (2.6)
recur.pcor<-function(index1, index2, index3, V){
  index1 <- index1
  index2 <- index2
  index3 <- index3
  index <- c(index1, index2, index3)

  if(length(index3)==1){
    rxy.z <- (V[index[1], index[2]]-V[index[1], index[3]]*V[index[2], index[3]])/
            (sqrt(1-V[index[1], index[3]]^2)*sqrt(1-V[index[2], index[3]]^2))
    return(rxy.z)
  }
  else{
    index1 <- index[1]
    index2 <- index[2]
    index30 <- index[3]    
    index3c <- index[-c(1,2,3)]

    rxy.zc <- recur.pcor(index1, index2, index3c, V)
    rxz0.zc <- recur.pcor(index1,index30,index3c, V)
    ryz0.zc <- recur.pcor(index2,index30,index3c, V)
    rxy.z <- (rxy.zc - rxz0.zc*ryz0.zc)/(sqrt(1-rxz0.zc^2)*sqrt(1-ryz0.zc^2))
    return(rxy.z)
  }                       
}  

partial.inter<-function(cor){
  V <- cor
  n <- ncol(cor)
  m <- n-2
  for(k in 1:m){
    if(k==m)
       V[1, n] <- V[n, 1] <- recur.pcor(1, n, c(2:(n-1)), cor)

     else
       diag(V[1:(n-k-1), (k+2):n]) <- diag(V[(k+2):n, 1:(n-k-1)]) <- sapply(1:(m+1-k), function(i) if(k==1) 
                    recur.pcor(i, i+k+1, i+1, cor) else recur.pcor(i, i+k+1, c((i+1):(i+k)), cor))
  }
  V
}