Residual plots of a regression model in ggplot2 - r

Question

I'm trying to transform two residual plots performed below into ggplot2.
As a description, in order to perform these graphs, it is necessary to previously define some functions associated with the specifics of the class of the adopted model, which I am providing below.
The model is in the fit argument whose data is from the nlme library, and the graphs are plotted at the end of the code using the qqPlot2 function.
rm(list = ls()); cat('\014')
library(ggplot2)
library(dplyr)
library(plotly)
library(nlme)
library(lme4)
library(MASS)
library(tidyverse)
library(splines)
library(gamlss)
library(gridExtra)
library(hnp)
library(car)
extract.lmeDesign2 <- function(m){
start.level = 1
data <- getData(m)
grps <- nlme::getGroups(m)
n <- length(grps)
X <- list()
grp.dims <- m$dims$ncol
Zt <- model.matrix(m$modelStruct$reStruct, data)
cov <- as.matrix(m$modelStruct$reStruct)
i.col <- 1
n.levels <- length(m$groups)
Z <- matrix(0, n, 0)
if (start.level <= n.levels) {
for (i in 1:(n.levels - start.level + 1)) {
if (length(levels(m$groups[[n.levels - i + 1]])) != 1)
{
X[[1]] <- model.matrix(~m$groups[[n.levels - i +
1]] - 1,
contrasts.arg = c("contr.treatment",
"contr.treatment"))
}
else X[[1]] <- matrix(1, n, 1)
X[[2]] <- as.matrix(Zt[, i.col:(i.col + grp.dims[i] -
1)])
i.col <- i.col + grp.dims[i]
Z <- cbind(mgcv::tensor.prod.model.matrix(X),Z)
}
Vr <- matrix(0, ncol(Z), ncol(Z))
start <- 1
for (i in 1:(n.levels - start.level + 1)) {
k <- n.levels - i + 1
for (j in 1:m$dims$ngrps[i]) {
stop <- start + ncol(cov[[k]]) - 1
Vr[ncol(Z) + 1 - (stop:start),ncol(Z) + 1 - (stop:start)] <- cov[[k]]
start <- stop + 1
}
}
}
X <- if (class(m$call$fixed) == "name" && !is.null(m$data$X)) {
m$data$X
} else {
model.matrix(formula(eval(m$call$fixed)),data)
}
y <- as.vector(matrix(m$residuals, ncol = NCOL(m$residuals))[,NCOL(m$residuals)] +
matrix(m$fitted, ncol = NCOL(m$fitted))[,NCOL(m$fitted)])
return(list(
Vr = Vr,
X = X,
Z = Z,
sigmasq = m$sigma ^ 2,
lambda = unique(diag(Vr)),
y = y,
k = n.levels
)
)
}
fit = lme(distance ~ age, method="REML",data = Orthodont)
data.fit <- extract.lmeDesign2(fit)
data <- getData(fit)
y <- data.fit$y
X <- data.fit$X
N <- length(y)
id <- sort(as.numeric(getGroups(fit, level = 1)), index.return = TRUE)$x
n <- length(as.numeric(names(table(id))))
vecni <- (table(id))
p <- ncol(X)
n.levels <- length(fit$groups)
start.level <- 1
Cgrps <- nlme::getGroups(fit, level = start.level)
CCind <- levels((Cgrps))
sigma2 <- fit$sigma^2
obs <- numeric()
for (i in 1:n)
{
obs <- append(obs,1:vecni[i])
}
if (n.levels > 1) {
lZi <- list()
lgi <- list()
numrow <- numeric()
mgroups <- fit$groups
for (n in 1:length(CCind)) {
dgi <- data.frame(as.matrix(mgroups[mgroups == CCind[n], ]))
nrowzi <- dim(dgi)[1]
ncolzi <- 0
girep <- as.numeric(length(levels(dgi[,1])))
for (k in 2:n.levels) {
girep <- c(girep,as.numeric(length(levels(dgi[,k]))))
}
for (k in 1:n.levels) {
ncolzi <- ncolzi + as.numeric(length(levels(dgi[,k])))
}
auxi <- as.vector(table(dgi[,1]))
for (i in 2:n.levels) {
auxi <- c(auxi,as.vector(table(dgi[,i])))
}
l <- 1
Zi <- matrix(0,nrowzi,ncolzi)
for (j in 1:ncolzi) {
Zi[l:(l + auxi[j] - 1),j] <- rep(1,auxi[j])
l <- l + auxi[j]
if (l == (nrowzi + 1)) l <- 1
}
lZi[[n]] <- Zi
numrow[n] <- dim(Zi)[1]
comp.var <- as.matrix(fit1$modelStruct$reStruct)
auxg <- rep(as.numeric(comp.var[1])*sigma2,girep[1])
for (i in 2:length(girep)) {
auxg <- c(auxg,rep(as.numeric(comp.var[i])*sigma2,girep[i]))
}
lgi[[n]] <- diag(auxg)
}
q <- dim(lgi[[1]])[1]
for (h in 2:length(CCind)) {
q <- c(q,dim(lgi[[h]])[1])
}
Z <- lZi[[1]]
for (k in 2:length(CCind)) {
Z <- bdiag(Z,(lZi[[k]]))
}
Z <- as.matrix(Z)
nrowZi <- lZi[[1]]
for (h in 2:length(CCind)) {
nrowZi <- c(nrowZi,dim(lZi[[h]])[1])
}
Gam <- lgi[[1]]
for (k in 2:length(CCind)) {
Gam <- bdiag(Gam,(lgi[[k]]))
}
Gam <- as.matrix(Gam)
}else{
mataux <- model.matrix(fit$modelStruct$reStruct,data)
mataux <- as.data.frame(cbind(mataux,id))
lZi <- list()
lgi <- list()
for (i in (as.numeric(unique(id)))) {
lZi[[i]] <- as.matrix((subset(split(mataux,id == i,
drop = T)$`TRUE`,select = -id)))
lgi[[i]] <- getVarCov(fit,type = "random.effects")
}
Z <- as.matrix(bdiag(lZi))
g <- getVarCov(fit,type = "random.effects")
q <- dim(g)[1]
Gam <- as.matrix(kronecker(diag(length(as.numeric(unique(id)))),g))
}
if (n.levels > 1) {
if (!inherits(fit, "lme"))
stop("object does not appear to be of class lme")
grps <- nlme::getGroups(fit)
n <- length(grps)
n.levels <- length(fit$groups)
if (is.null(fit$modelStruct$corStruct))
n.corlevels <- 0
else n.corlevels <- length(all.vars(nlme::getGroupsFormula(fit$modelStruct$corStruct)))
if (n.levels < n.corlevels) {
getGroupsFormula(fit$modelStruct$corStruct)
vnames <- all.vars(nlme::getGroupsFormula(fit$modelStruct$corStruct))
lab <- paste(eval(parse(text = vnames[1]), envir = fit$data))
if (length(vnames) > 1)
for (i in 2:length(vnames)) {
lab <- paste(lab, "/", eval(parse(text = vnames[i]),
envir = fit$data), sep = "")
}
grps <- factor(lab)
}
if (n.levels >= start.level || n.corlevels >= start.level) {
if (n.levels >= start.level)
Cgrps <- nlme::getGroups(fit, level = start.level)
else Cgrps <- grps
Cind <- sort(as.numeric(Cgrps), index.return = TRUE)$ix
rCind <- 1:n
rCind[Cind] <- 1:n
Clevel <- levels(Cgrps)
n.cg <- length(Clevel)
size.cg <- array(0, n.cg)
for (i in 1:n.cg) size.cg[i] <- sum(Cgrps == Clevel[i])
}
else {
n.cg <- 1
Cind <- 1:n
}
if (is.null(fit$modelStruct$varStruct))
w <- rep(fit$sigma, n)
else {
w <- 1/nlme::varWeights(fit$modelStruct$varStruct)
group.name <- names(fit$groups)
order.txt <- paste("ind<-order(data[[\"", group.name[1],
"\"]]", sep = "")
if (length(fit$groups) > 1)
for (i in 2:length(fit$groups)) order.txt <- paste(order.txt,
",data[[\"", group.name[i], "\"]]", sep = "")
order.txt <- paste(order.txt, ")")
eval(parse(text = order.txt))
w[ind] <- w
w <- w * fit$sigma
}
w <- w[Cind]
if (is.null(fit$modelStruct$corStruct))
lR <- array(1, n)
else {
c.m <- nlme::corMatrix(fit$modelStruct$corStruct)
if (!is.list(c.m)) {
lR <- c.m
lR <- lR[Cind, ]
lR <- lR[, Cind]
}
else {
lR <- list()
ind <- list()
for (i in 1:n.cg) {
lR[[i]] <- matrix(0, size.cg[i], size.cg[i])
ind[[i]] <- 1:size.cg[i]
}
Roff <- cumsum(c(1, size.cg))
gr.name <- names(c.m)
n.g <- length(c.m)
j0 <- rep(1, n.cg)
ii <- 1:n
for (i in 1:n.g) {
Clev <- unique(Cgrps[grps == gr.name[i]])
if (length(Clev) > 1)
stop("inner groupings not nested in outer!!")
k <- (1:n.cg)[Clevel == Clev]
j1 <- j0[k] + nrow(c.m[[i]]) - 1
lR[[k]][j0[k]:j1, j0[k]:j1] <- c.m[[i]]
ind1 <- ii[grps == gr.name[i]]
ind2 <- rCind[ind1]
ind[[k]][j0[k]:j1] <- ind2 - Roff[k] + 1
j0[k] <- j1 + 1
}
for (k in 1:n.cg) {
lR[[k]][ind[[k]], ] <- lR[[k]]
lR[[k]][, ind[[k]]] <- lR[[k]]
}
}
}
if (is.list(lR)) {
for (i in 1:n.cg) {
wi <- w[Roff[i]:(Roff[i] + size.cg[i] - 1)]
lR[[i]] <- as.vector(wi) * t(as.vector(wi) * lR[[i]])
}
}
else if (is.matrix(lR)) {
lR <- as.vector(w) * t(as.vector(w) * lR)
}
else {
lR <- w^2 * lR
}
if (is.list(lR)) {
R <- lR[[1]]
for (k in 2:n.cg) {
R <- bdiag(R,lR[[k]])
}
R <- as.matrix(R)
}
else{
R <- diag(lR)
}
}else{
R <- getVarCov(fit,type = "conditional",individual = 1)[[1]]
for (i in 2:length(as.numeric(unique(id)))) {
R <- as.matrix(bdiag(R,getVarCov(fit,
type = "conditional",individual = i)[[1]] ) )
}
}
sqrt.matrix <- function(mat) {
mat <- as.matrix(mat)
singular_dec <- svd(mat,LINPACK = F)
U <- singular_dec$u
V <- singular_dec$v
D <- diag(singular_dec$d)
sqrtmatrix <- U %*% sqrt(D) %*% t(V)
}
V <- (Z %*% Gam %*% t(Z)) + R
iV <- solve(V)
varbeta <- solve((t(X) %*% iV %*% X))
Q <- (iV - iV %*% X %*% (varbeta) %*% t(X) %*% iV )
zq <- t(Z) %*% Q
norm.frob.ZtQ <- sum(diag(zq %*% t(zq)))
eblue <- as.vector(fixef(fit))
eblup <- Gam %*% t(Z) %*% iV %*% (y - X %*% eblue)
predm <- X %*% eblue
predi <- X %*% eblue + Z %*% eblup
resm <- (y - predm)
resc <- (y - predi)
var.resm <- V - X %*% solve(t(X) %*% iV %*% X) %*% t(X)
var.resc <- R %*% Q %*% R
ident <- diag(N)
auxnum <- (R %*% Q %*% Z %*% Gam %*% t(Z) %*% Q %*% R)
auxden <- R %*% Q %*% R
CF <- diag(auxnum)/diag(auxden)
rescp <- resc/sqrt(diag(var.resc))
R.half <- sqrt.matrix(R)
auxqn <- eigen((R.half %*% Q %*% R.half), symmetric = T, only.values = FALSE)
lt <- sqrt(solve(diag((auxqn$values[1:(N-p)])))) %*% t(auxqn$vectors[1:N,1:(N-p)]) %*% solve(sqrt.matrix(R[1:N,1:N]))
var.resmcp <- lt %*% var.resc[1:N,1:N] %*% t(lt)
resmcp <- (lt %*% resc[1:N] )/sqrt(diag(var.resmcp))
if (n.levels > 1) {
aux <- Gam %*% t(Z) %*% Q %*% Z %*% Gam
qm <- q - 1
dm <- matrix(0,length(CCind),1)
gbi <- aux[1:(q[1]),(1:q[1])]
eblupi <- eblup[1:(q[1]),]
dmi <- t(eblupi) %*% ginv(gbi) %*% eblupi
dm[1] <- dmi
for (j in 2:length(CCind)) {
gbi <- aux[((j - 1)*q[(j - 1)] + 1 ):(q[j] + q[(j - 1)]),((j - 1)*q[(j - 1)] + 1 ):(q[j] + q[(j - 1)])]
eblupi <- eblup[((j - 1)*q[(j - 1)] + 1 ):(q[j] + q[(j - 1)]),]
dmi <- t(eblupi) %*% ginv(gbi) %*% eblupi
dm[j] <- dmi
}
}else{
aux <- Gam %*% t(Z) %*% Q %*% Z %*% Gam
qm <- q - 1
dm <- matrix(0,n,1)
for (j in 1:length(CCind))
{
if (q == 1)
{
gbi <- aux[j,j]
eblupi <- eblup[(q*j - qm):(q*j)]
dmi <- t(eblupi) %*% ginv(gbi) %*% eblupi
dm[j] <- dmi
}
else
{
gbi <- aux[(q*j - qm):(q*j),(q*j - qm):(q*j)]
cat(gbi,'\n','\t')
eblupi <- eblup[(q*j - qm):(q*j)]
dmi <- t(eblupi) %*% ginv(gbi) %*% eblupi
dm[j] <- dmi
}
}
}
qqPlot2 <- function(x, distribution="norm", ..., ylab=deparse(substitute(x)),
xlab=paste(distribution, "quantiles"), main = NULL,
las = par("las"),
envelope = .95,
col = palette()[1],
col.lines = palette()[2], lwd = 2, pch = 1, cex = par("cex"),
cex.lab = par("cex.lab"), cex.axis = par("cex.axis"),
line = c("quartiles", "robust", "none"),
labels = if (!is.null(names(x))) names(x) else seq(along = x),
id.method = "y",
id.n = if (id.method[1] == "identify") Inf else 0,
id.cex = 1, id.col=palette()[1], grid = TRUE)
{
line <- match.arg(line)
good <- !is.na(x)
ord <- order(x[good])
ord.x <- x[good][ord]
ord.lab <- labels[good][ord]
q.function <- eval(parse(text = paste("q", distribution, sep = "")))
d.function <- eval(parse(text = paste("d", distribution, sep = "")))
n <- length(ord.x)
P <- ppoints(n)
z <- q.function(P, ...)
plot(z, ord.x, type = "n", xlab = xlab,
ylab = ylab, main = main,
las = las,cex.lab = cex.lab, cex.axis = cex.axis)
if (grid) {
grid(lty = 1, equilogs = FALSE)
box()}
points(z, ord.x, col = col, pch = pch, cex = cex)
if (line == "quartiles" || line == "none") {
Q.x <- quantile(ord.x, c(.25,.75))
Q.z <- q.function(c(.25,.75), ...)
b <- (Q.x[2] - Q.x[1])/(Q.z[2] - Q.z[1])
a <- Q.x[1] - b*Q.z[1]
abline(a, b, col = col.lines, lwd = lwd)
}
if (line == "robust") {
coef <- coef(rlm(ord.x ~ z))
a <- coef[1]
b <- coef[2]
abline(a, b)
}
conf <- if (envelope == FALSE) .95 else envelope
zz <- qnorm(1 - (1 - conf)/2)
SE <- (b/d.function(z, ...))*sqrt(P*(1 - P)/n)
fit.value <- a + b*z
upper <- fit.value + zz*SE
lower <- fit.value - zz*SE
if (envelope != FALSE) {
lines(z, upper, lty = 2, lwd = lwd, col = col.lines)
lines(z, lower, lty = 2, lwd = lwd, col = col.lines)
}
}
x11()
qqPlot2(resmcp, ylab = "Resíduos",
xlab = "Quantil N(0,1)", pch = 20)
qqPlot2(dm, distribution = 'chisq', df = q, pch = 20,
ylab = expression(paste("Quantis de Mahalanobis")),
xlab = "Quantis da Qui-quadrado")
My attempt to reproduce them in ggplot2 was as follows:
P1 = qqPlot2(resmcp, ylab = "Resíduos",
xlab = "Quantil N(0,1)", pch = 20)
PP1 = ggplot(data = P1, aes(resmcp)) +
geom_point(aes(y = resmcp), show.legend = FALSE)
P2 = qqPlot2(dm, distribution = 'chisq', df = q, pch = 20,
ylab = expression(paste("Quantis de Mahalanobis")),
xlab = "Quantis da Qui-quadrado")
PP2 = ggplot(data = P2, aes(dm)) +
geom_point(aes(y = dm), show.legend = FALSE)
x11()
gridExtra::grid.arrange(PP1,PP2, ncol = 2)
However, something is happening, as I have gotten the following result:
See my attempt below for the quantile mahalanobis distance graph vs. chi-square quantiles:
gVals <- function(y, dist, conf){ # distribution; confidence interval
y <- sort(y) # make sure they're in order
p <- ppoints(length(y))
if(dist == "chisq") {
zi <- qchisq(p, df = length(p) - 1)
zd <- dchisq(zi, df = length(p) - 1)
qz <- qchisq(c(.25, .75), length(p) - 1)
} else {
zi <- qnorm(p)
zd <- dnorm(zi)
qz <- qnorm(c(.25, .75))
}
# if quartiles preferred
qx <- quantile(y, c(.25, .75))
b <- (qx[2] - qx[1]) / (qz[2] - qz[1])
a <- qx[1] - b * qz[1]
# if robust preferred
# coef <- coef(rlm(y~zi))
# a <- coef[1]
# b <- coef[2]
z <- qnorm(1 - (1 - conf)/2) # z = 1.96 for 95%...
se <- (b / zd) * sqrt(p * (1 - p)/length(p))
ft <- a + b * zi
uc <- ft + z * se
dc <- ft - z * se
dff = data.frame(z = zi, y = y, uc = uc, dc = dc)
list(a = a, b = b, dff = dff) # returns intercept, slope, and data frame
}
cdf <- gVals(dm, "chisq", .95) # dm is defined in the previous code above
ggplot(cdf$dff, aes(x = z, y = y)) +
geom_point() +
geom_abline(intercept = cdf$a[[1]], slope = cdf$b[[1]]) +
annotate("line", x = cdf$dff$z, y = cdf$dff$uc, color = "red", lty = 2) +
annotate("line", x = cdf$dff$z, y = cdf$dff$dc, color = "red", lty = 2)
Note that the x axis should go from 0 to 8, and the y axis should go from 0 to 14. Also, the shape of the simulation envelope is not similar. I am not able to fix this problem.

Answer 1

Update
Instead of having the code for the option quartile commented out, I have commented out the code for the option robust in the function. Additionally, instead of returning a data frame, it returns a list. FYI, you only need the MASS package if you use the robust option (for the function rlm).
This function is based on the code used in qqPlot2 in your question. However, it doesn't return a plot; it returns data.
library(car)
library(MASS)
library(tidyverse)
gVals <- function(y, dist, conf){ # distribution; confidence interval
y <- sort(y) # make sure they're in order
p <- ppoints(length(y))
if(dist == "chisq") {
zi <- qchisq(p, df = length(p) - 1)
zd <- dchisq(zi, df = length(p) - 1)
qz <- qchisq(c(.25, .75), length(p) - 1)
} else {
zi <- qnorm(p)
zd <- dnorm(zi)
qz <- qnorm(c(.25, .75))
}
# if quartiles preferred
qx <- quantile(y, c(.25, .75))
b <- (qx[2] - qx[1]) / (qz[2] - qz[1])
a <- qx[1] - b * qz[1]
# if robust preferred
# coef <- coef(rlm(y~zi))
# a <- coef[1]
# b <- coef[2]
z <- qnorm(1 - (1 - conf)/2) # z = 1.96 for 95%...
se <- (b / zd) * sqrt(p * (1 - p)/length(p))
ft <- a + b * zi
uc <- ft + z * se
dc <- ft - z * se
dff = data.frame(z = zi, y = y, uc = uc, dc = dc)
list(a = a, b = b, dff = dff) # returns intercept, slope, and data frame
}
Here is a comparison with some arbitrary data.
data(mtcars)
qqPlot2(mtcars$mpg)
qqPlot2(mtcars$mpg, dist = "chisq", df = 31)
ndf <- gVals(mtcars$mpg, "norm", .95)
ggplot(ndf$dff, aes(x = z, y = y)) +
geom_point() +
geom_abline(intercept = ndf$a[[1]], slope = ndf$b[[1]]) +
annotate("line", x = ndf$dff$z, y = ndf$dff$uc, color = "red", lty = 2) +
annotate("line", x = ndf$dff$z, y = ndf$dff$dc, color = "red", lty = 2)
cdf <- gVals(mtcars$mpg, "chisq", .95)
ggplot(cdf$dff, aes(x = z, y = y)) +
geom_point() +
geom_abline(intercept = cdf$a[[1]], slope = cdf$b[[1]]) +
annotate("line", x = cdf$dff$z, y = cdf$dff$uc, color = "red", lty = 2) +
annotate("line", x = cdf$dff$z, y = cdf$dff$dc, color = "red", lty = 2)

Answer 2

I managed to solve it through the library qqplotr.
library(qqplotr)
dist <- "chisq"
dpar <- list(df = q)
QT <- data.frame(QUANTIS = dm); ggplot(QT, aes(sample = QUANTIS)) +
stat_qq_band(distribution = dist, dparams = dpar) +
stat_qq_point(distribution = dist, dparams = dpar) +
stat_qq_line(distribution = dist, dparams = dpar, color = "blue");
qqPlot2(dm, distribution = 'chisq', df = q, pch = 20,
ylab = expression(paste("Quantis de Mahalanobis")),
xlab = "Quantis da Qui-quadrado")