aomisc: Statistical methods for the agricultural sciences

Documented in AMMI

## ADDITIVE MAIN EFFECTS MULTIPLICATIVE INTERACTION MODEL FOR GENOTYPE TRIALS
## REFERENCE: H. F. Gollob, 1968. A statistical model which combine features of factor analytic and
## analysis of variance techniques. Psychometrika, 33, 1, 73-114.
## Update: 26/05/2023  

## INPUTS:
## genotype is a vector of genotype codes (factor)
## env is a vector of environment codes (factor)
## block is a vector of block codes (factor)
## yield is a vector of yields to be analysed (numerical)
## PC is the number of PC to be considered (default is 2)

AMMI <- function(yield, genotype, environment, block = NULL,
                 PC = 2, MSE = NULL, dfr = NULL) {

## 1 - Model fitting and descriptive statistics
  
  envir <- as.factor(environment)
  variety <- as.factor(genotype)
  
  # Verify whether it is balanced and replicated
  conTab <- table(envir, variety)
  if(any(conTab == 0)) stop("Missing cells were found. Imputing values is necessary")
  balanced <- ifelse(length(unique(conTab)) == 1, TRUE, FALSE)
  unreplicated <- FALSE
  if(balanced) unreplicated <- ifelse(unique(conTab) == 1, TRUE, FALSE)
  numReps <- as.numeric(unique(conTab))
  if(!is.null(MSE) & !is.null(dfr)) externalError <- TRUE else externalError <- FALSE
  
  # Estimate the MSE if not passed in...
  add.anova <- NULL
  
  if(!is.null(block) & is.null(MSE) & !unreplicated) {
    # With blocks, it includes the block %in% environments in the model
    add.anova <- lm(yield ~ variety*envir + block %in% envir)
    suppressMessages( tab <- as.data.frame(emmeans::emmeans(add.anova, ~variety:envir)) )
    MSE <- summary(add.anova)$sigma^2
    MSE <- MSE/numReps
    df.res <- add.anova$df.residual
  } else if(is.null(block) & is.null(MSE) & !unreplicated) {
    # It does not include the blocks
    add.anova <- lm(yield ~ variety*envir)
    suppressMessages( tab <- as.data.frame(emmeans::emmeans(add.anova, ~variety:envir)) )
    # tab <- data.frame(variety, envir, emmean = yield)
    MSE <- summary(add.anova)$sigma^2
    MSE <- MSE/numReps
    df.res <- add.anova$df.residual
  } else if(is.null(MSE) & unreplicated) {
    add.anova <- lm(yield ~ variety*envir)
    suppressWarnings(suppressMessages( tab <- as.data.frame(emmeans::emmeans(add.anova, ~variety:envir)) ))
    MSE <- 0
    df.res <- 0
    
  } else {
    add.anova <- lm(yield ~ variety*envir)
    suppressWarnings(suppressMessages( tab <- as.data.frame(emmeans::emmeans(add.anova, ~variety:envir)) ))
    df.res <- add.anova$df.residual
  }
  
  if(is.null(dfr)) dfr <- df.res
  
  # Calculate AMMI matrix (int.eff) and means for main effects
  int.mean <- unstack(tab, form = emmean ~ envir)
  rownames(int.mean) <- unique(tab$variety)
  
  int.eff <- as.data.frame(t(scale( t(scale(int.mean, center=T,
  scale=F)), center=T, scale=F)))
  
  suppressWarnings(suppressMessages({ 
    overall.mean <- as.data.frame(emmeans(add.anova, ~1))$emmean
    envir.mean <- as.data.frame(emmeans::emmeans(add.anova, ~envir))$emmean
    var.mean <- as.data.frame(emmeans::emmeans(add.anova, ~variety))$emmean
  }))
  names(var.mean) <- rownames(int.mean)
  names(envir.mean) <- colnames(int.mean)
  envir.num <- length(envir.mean)
  var.num <- length(var.mean)
  
## 2 - Variance partitioning (additive model)
if(!unreplicated) tab <- anova(add.anova) else tab <- NULL
  
## 3 - SVD
  scale <- 0.5 # if necessary, can be user-defined
  dec <- svd(int.eff, nu = PC, nv = PC)
  if (PC > 1){ 
    D <- diag(dec$d[1:PC]^scale)
    Dbis <- diag(dec$d[1:PC]^(1-scale)) 
  } else {
    D <- dec$d[1:PC]^scale
    Dbis <- dec$d[1:PC]^(1-scale)
  }
  G<-dec$u %*% D
  E<-dec$v %*% Dbis
  
  Ecolnumb <- c(1:PC)
  Ecolnames <- paste("PC", Ecolnumb, sep = "")
  dimnames(E) <- list(levels(envir), Ecolnames)
  dimnames(G) <- list(levels(variety), Ecolnames)
  
## 4 - Mean Squares of PCs
  int.SS <- sum(int.eff^2)
  PC.SS <-  (dec$d[1:PC]^2) # * numblock 
  PC.DF <- var.num + envir.num - 1 - 2*Ecolnumb
  residual.SS <- int.SS - sum(PC.SS)
  residual.DF <- ((var.num - 1)*(envir.num - 1)) - sum(PC.DF)
  PC.SS[PC + 1] <- residual.SS  
  PC.DF[PC + 1] <- residual.DF
  MS <- PC.SS/PC.DF
  
  # if(!is.null(block)) MS <- PC.SS/PC.DF * numblock else MS <- PC.SS/PC.DF
  # 
  # if(!is.null(MSE)){
  #   sigma2res <- MSE
  # } else {
  #   if(!is.null(add.anova$sigma)){
  #     sigma2res <- add.anova$sigma^2
  #   } else {
  #     sigma2res <- NA
  #   }
  # }
  
  if(balanced | externalError){
    Fcomp <- MS/MSE
    probab <- pf(Fcomp, PC.DF, dfr, lower.tail = FALSE)
    percSS <- PC.SS/int.SS
    rowlab <- c(Ecolnames, "Residuals")
    mult.anova <- data.frame(Effect = rowlab, SS = PC.SS, DF = PC.DF, MS = MS, F = Fcomp, Prob. = probab, "Perc_of_Total_SS"=percSS)
  } else {
    percSS <- PC.SS/int.SS
    rowlab <- c(Ecolnames, "Residuals")
    mult.anova <- data.frame(Effect = rowlab, SS = PC.SS, DF = PC.DF, MS = MS, "Perc_of_Total_SS"=percSS)
    MSE <- NULL
    dfr <- NULL
  }
  
  if(PC >=2) stability <- sqrt((PC.SS[1]/PC.SS[2]*G[,1])^2+G[,2]^2) else stability = NA
  
## 6 - Results  
  result <- list(genotype_means = var.mean, environment_means = envir.mean,  
       interaction_means = int.mean,
       interaction_effect=int.eff, # additive_ANOVA = tab, 
       mult_Interaction = mult.anova, MSE = MSE, dfr = dfr,
       environment_scores = E, genotype_scores = G, stability = stability)
       class(result) <- "AMMIobject"    
  return(invisible(result))
}

# AMMImeans <- function(yield, genotype, environment, PC = 2,
#                       MSE = NULL, dfr = NULL) {
# 
#   variety <- genotype; envir <- environment
#   add.anova <- aov(yield ~ envir * variety)
#   int.eff <- model.tables(add.anova, type = "effects", cterms = "envir:variety")$tables$"envir:variety"
#   int.mean <- model.tables(add.anova, type = "means", cterms = "envir:variety")$tables$"envir:variety"
#   envir.mean <- model.tables(add.anova, type = "means", cterms = "envir")$tables$"envir"
#   var.mean <- model.tables(add.anova, type = "means", cterms = "variety")$tables$"variety"
#   overall.mean <- model.tables(add.anova, type = "means")$tables[[1]]
#   
# ## 3 - SVD
#   dec <- svd(int.eff, nu = PC, nv = PC)
#   if (PC > 1){ 
#     D <- diag(dec$d[1:PC]) 
#   } else {
#     D <- matrix(dec$d[1:PC],1,1)
#   }
#   E <- dec$u %*% sqrt(D)
#   G <- dec$v %*% sqrt(D)
#   Ecolnumb <- c(1:PC)
#   Ecolnames <- paste("PC", Ecolnumb, sep = "")
#   dimnames(E) <- list(levels(envir), Ecolnames)
#   dimnames(G) <- list(levels(variety), Ecolnames)
#   #stability <- sqrt(G[,1]^2+G[,2]^2)
# 
# ## 4 - Significance of PCs
#   svalues<-dec$d
#   PC.n<-c(1:length(svalues))
#   PC.SS<-svalues^2
#   percSS<-PC.SS/sum(PC.SS)*100
#   GGE.table<-data.frame("PC"=PC.n,"Singular_value"=svalues,"PC_SS"=PC.SS, "Perc_of_Total_SS"=percSS,
#                         cum_perc = cumsum(percSS))
#   GGE.SS <- (t(as.vector(int.eff))%*%as.vector(int.eff))
#     
#   
#   if(!is.null(MSE) & !is.null(dfr)){
#     ngen <- length(int.mean[1,])
#     nenv <- length(int.mean[,1])
#     df_n <- ngen + nenv - 1 - 2 * GGE.table[,1]
#     ss <- GGE.table$PC_SS
#     ms <- ss/df_n
#     f <- ms/MSE
#     pf <- pf(f, df_n, dfr, lower.tail = F)
#     aov.tab <- data.frame(PC = GGE.table[,1], SS = ss, DF = df_n, MS = ms,
#                           "F" = f, "P value" = pf)
#   } else {
#     aov.tab <- NA
#   }
#     
# ## 6 - Other results  
#   result <- list(means_table = int.mean,
#        interaction_effect=int.eff, "AMMI_SS"=GGE.SS, 
#        summary = GGE.table, anova = aov.tab,
#        environment_scores = E, genotype_scores = G,
#        "genotype_means"=var.mean, "environment_means"=envir.mean)#,Stability = stability)
#   # cat(paste("Result of AMMI Analysis", "\n", "\n"))
#   # class(result) <- "AMMIobject"
#   # print(E)
#   # cat(paste("\n","Genotype Scores", "\n"))
#   # print(G)
#   return(invisible(result))
# }
# 
# AMMI_old <- function(variety, envir, block, yield, PC=2, biplot=1) {
# 
# ## 1 - Descriptive statistics
#   overall.mean <- mean(yield)
#   envir.mean <- tapply(yield, envir, mean)
#   var.mean <- tapply(yield, variety, mean)  
#   int.mean <- tapply(yield, list(variety,envir), mean)
#   envir.num <- length(envir.mean)
#   var.num <- length(var.mean)
#   
# ## 2 - ANOVA (additive model) 
#   variety <- factor(variety)
#   envir <- factor(envir)
#   block <- factor(block)  
#   add.anova <- aov(yield ~ envir + block %in% envir + variety + envir:variety)
#   modelTables <- model.tables(add.anova, type = "effects", cterms = "envir:variety")
#   int.eff <- modelTables$tables$"envir:variety"
#   add.anova.residual.SS <- deviance(add.anova)
#   add.anova.residual.DF <- add.anova$df.residual
#   add.anova.residual.MS <- add.anova.residual.SS/add.anova.residual.DF
#   anova.table <- summary(add.anova)
#   row.names(anova.table[[1]]) <- c("Environments", "Genotypes", "Blocks(Environments)","Environments x Genotypes", "Residuals")
#   
# ## 3 - SVD
#   dec <- svd(int.eff, nu = PC, nv = PC)
#   if (PC > 1){ 
#     D <- diag(dec$d[1:PC]) 
#   } else {
#     D <- dec$d[1:PC]
#   }
#   E <- dec$u %*% sqrt(D)
#   G <- dec$v %*% sqrt(D)
#   Ecolnumb <- c(1:PC)
#   Ecolnames <- paste("PC", Ecolnumb, sep = "")
#   dimnames(E) <- list(levels(envir), Ecolnames)
#   dimnames(G) <- list(levels(variety), Ecolnames)
#   
# ## 4 - Significance of PCs
#   numblock <- length(levels(block))
#   int.SS <- (t(as.vector(int.eff)) %*% as.vector(int.eff))*numblock
#   PC.SS <- (dec$d[1:PC]^2)*numblock  
#   PC.DF <- var.num + envir.num - 1 - 2*Ecolnumb
#   residual.SS <- int.SS - sum(PC.SS)
#   residual.DF <- ((var.num - 1)*(envir.num - 1)) - sum(PC.DF)
#   PC.SS[PC + 1] <- residual.SS  
#   PC.DF[PC + 1] <- residual.DF
#   MS <- PC.SS/PC.DF
#   F <- MS/add.anova.residual.MS
#   probab <- pf(F, PC.DF, add.anova.residual.DF, lower.tail = FALSE)
#   percSS <- PC.SS/int.SS
#   rowlab <- c(Ecolnames, "Residuals")
#   mult.anova <- data.frame(Effect = rowlab, SS = PC.SS, DF = PC.DF, MS = MS, F = F, Prob. = probab, "% of Total SS"=percSS)
#   stability <- sqrt(G[,1]^2+G[,2]^2)
# 
# ## 5 - Biplots
#   if (biplot == 1) {
#     plot(1, type = 'n', xlim = range(c(envir.mean, var.mean)), ylim = range(c(E[,1], G[,1])), xlab = "Yield",
#     		 ylab = "PC 1")
#     points(envir.mean, E[,1], "n", col = "red", lwd = 5)
#     text(envir.mean, E[,1], labels = row.names(envir.mean), adj = c(0.5, 0.5), col = "red")
#     points(var.mean, G[,1], "n", col = "blue", lwd = 5)
#     text(var.mean, G[,1], labels = row.names(var.mean), adj = c(0.5, 0.5), col = "blue")
#     abline(h = 0, v = overall.mean, lty = 5)
#   } else {
#     plot(1, type = 'n', xlim = range(c(E[,1], G[,1])), ylim = range(c(E[,2], G[,2])), xlab = "PC 1", 
#     			ylab = "PC 2")
#     points(E[,1], E[,2], "n",col = "red", lwd = 5)
#     text(E[,1], E[,2], labels = row.names(E),adj = c(0.5,0.5),col = "red")
#     points(G[,1],G[,2], "n", col = "blue", lwd = 5)
#     text(G[,1], G[,2], labels = row.names(G),adj = c(0.5, 0.5), col = "blue")
#   }
#     
# ## 6 - Other results  
#   result <- list(Genotype_means = var.mean, Environment_means = envir.mean,  Interaction_means = int.mean,
#        Interaction_effect=int.eff, Additive_ANOVA = anova.table, Mult_Interaction = mult.anova, 
#        Environment_scores = E, Genotype_scores = G, Stability = stability)
#        class(result) <- "AMMIobject"
#   return(result)
# }