# [Rd] Factor correlations in factanal

John Fox jfox at mcmaster.ca
Sun Jun 26 16:00:35 CEST 2005

```Dear R-devel list members,

Ben Fairbank draw it to my attention that factanal() (in the stats package)
doesn't report factor correlations for oblique rotations. Looking at the
source, I see that factanal also doesn't save the factor-transformation
(rotation) matrix from which these correlations can be computed. I've
modified the source, attached below, so that the transformation matrix is
saved if its available; as well, I've modified print.factanal() to print the
factor-correlations for oblique solutions.

Regards,
John

------------ snip------------

# modifications by J. Fox, 26 June 2005 marked in source

## Hmm, MM thinks diag(.) needs checking { diag(vec) when length(vec)==1 !}
## However, MM does not understand that factor analysis
##   is a *multi*variate technique!
factanal <-
function (x, factors, data = NULL, covmat = NULL, n.obs = NA,
subset, na.action, start = NULL,
scores = c("none", "regression", "Bartlett"),
rotation = "varimax",
control = NULL, ...)
{
{
cn <- colnames(Lambda)
Phi <- attr(Lambda, "covariance")
ssq <- apply(Lambda, 2, function(x) -sum(x^2))
Lambda <- Lambda[, order(ssq), drop = FALSE]
colnames(Lambda) <- cn
neg <- colSums(Lambda) < 0
Lambda[, neg] <- -Lambda[, neg]
if(!is.null(Phi)) {
unit <- ifelse(neg, -1, 1)
attr(Lambda, "covariance") <-
unit %*% Phi[order(ssq), order(ssq)] %*% unit
}
Lambda
}
cl <- match.call()
na.act <- NULL
if (is.list(covmat)) {
if (any(is.na(match(c("cov", "n.obs"), names(covmat)))))
stop("'covmat' is not a valid covariance list")
cv <- covmat\$cov
n.obs <- covmat\$n.obs
have.x <- FALSE
}
else if (is.matrix(covmat)) {
cv <- covmat
have.x <- FALSE
}
else if (is.null(covmat)) {
if(missing(x)) stop("neither 'x' nor 'covmat' supplied")
have.x <- TRUE
if(inherits(x, "formula")) {
## this is not a `standard' model-fitting function,
## so no need to consider contrasts or levels
mt <- terms(x, data = data)
if(attr(mt, "response") > 0)
stop("response not allowed in formula")
attr(mt, "intercept") <- 0
mf <- match.call(expand.dots = FALSE)
names(mf)[names(mf) == "x"] <- "formula"
mf\$factors <- mf\$covmat <- mf\$scores <- mf\$start <-
mf\$rotation <- mf\$control <- mf\$... <- NULL
mf[[1]] <- as.name("model.frame")
mf <- eval.parent(mf)
na.act <- attr(mf, "na.action")
if(any(sapply(mf, function(x) is.factor(x) || !is.numeric(x))))
stop("factor analysis applies only to numerical variables")
z <- model.matrix(mt, mf)
} else {
z <- as.matrix(x)
if(!is.numeric(z))
stop("factor analysis applies only to numerical variables")
if(!missing(subset)) z <- z[subset, , drop = FALSE]
}
covmat <- cov.wt(z)
cv <- covmat\$cov
n.obs <- covmat\$n.obs
}
else stop("'covmat' is of unknown type")
scores <- match.arg(scores)
if(scores != "none" && !have.x)
stop("requested scores without an 'x' matrix")
p <- ncol(cv)
if(p < 3) stop("factor analysis requires at least three variables")
dof <- 0.5 * ((p - factors)^2 - p - factors)
if(dof < 0)
stop(gettextf("%d factors is too many for %d variables", factors,
p),
domain = NA)
sds <- sqrt(diag(cv))
cv <- cv/(sds %o% sds)

cn <- list(nstart = 1, trace = FALSE, lower = 0.005)
cn[names(control)] <- control
more <- list(...)[c("nstart", "trace", "lower", "opt", "rotate")]
if(length(more)) cn[names(more)] <- more

if(is.null(start)) {
start <- (1 - 0.5*factors/p)/diag(solve(cv))
if((ns <- cn\$nstart) > 1)
start <- cbind(start, matrix(runif(ns-1), p, ns-1, byrow=TRUE))
}
start <- as.matrix(start)
if(nrow(start) != p)
stop(gettextf("'start' must have %d rows", p), domain = NA)
nc <- ncol(start)
if(nc < 1) stop("no starting values supplied")

best <- Inf
for (i in 1:nc) {
nfit <- factanal.fit.mle(cv, factors, start[, i],
max(cn\$lower, 0), cn\$opt)
if(cn\$trace)
cat("start", i, "value:", format(nfit\$criteria[1]),
"uniqs:", format(as.vector(round(nfit\$uniquenesses, 4))),
"\n")
if(nfit\$converged && nfit\$criteria[1] < best) {
fit <- nfit
best <- fit\$criteria[1]
}
}
if(best == Inf) stop("unable to optimize from these starting value(s)")
if(rotation != "none") {
# the following lines modified by J. Fox, 26 June 2005
if (is.list(rot)){
fit\$rotmat <- rot\$rotmat
}
# end modifications J. Fox, 26 June 2005
}
fit\$na.action <- na.act # not used currently
if(have.x && scores != "none") {
zz <- scale(z, TRUE, TRUE)
switch(scores,
regression = {
sc <- zz %*% solve(cv, Lambda)
if(!is.null(Phi <- attr(Lambda, "covariance")))
sc <- sc %*% Phi
},
Bartlett = {
d <- 1/fit\$uniquenesses
tmp <- t(Lambda * d)
sc <- t(solve(tmp %*% Lambda, tmp %*% t(zz)))
})
rownames(sc) <- rownames(z)
colnames(sc) <- colnames(Lambda)
if(!is.null(na.act)) sc <- napredict(na.act, sc)
fit\$scores <- sc
}
if(!is.na(n.obs) && dof > 0) {
fit\$STATISTIC <- (n.obs - 1 - (2 * p + 5)/6 -
(2 * factors)/3) * fit\$criteria["objective"]
fit\$PVAL <- pchisq(fit\$STATISTIC, dof, lower.tail = FALSE)
}
fit\$n.obs <- n.obs
fit\$call <- cl
fit
}

factanal.fit.mle <-
function(cmat, factors, start=NULL, lower = 0.005, control = NULL, ...)
{
FAout <- function(Psi, S, q)
{
sc <- diag(1/sqrt(Psi))
Sstar <- sc %*% S %*% sc
E <- eigen(Sstar, symmetric = TRUE)
L <- E\$vectors[, 1:q, drop = FALSE]
load <- L %*% diag(sqrt(pmax(E\$values[1:q] - 1, 0)), q)
}
FAfn <- function(Psi, S, q)
{
sc <- diag(1/sqrt(Psi))
Sstar <- sc %*% S %*% sc
E <- eigen(Sstar, symmetric = TRUE, only.values = TRUE)
e <- E\$values[-(1:q)]
e <- sum(log(e) - e) - q + nrow(S)
##        print(round(c(Psi, -e), 5))  # for tracing
-e
}
FAgr <- function(Psi, S, q)
{
sc <- diag(1/sqrt(Psi))
Sstar <- sc %*% S %*% sc
E <- eigen(Sstar, symmetric = TRUE)
L <- E\$vectors[, 1:q, drop = FALSE]
load <- L %*% diag(sqrt(pmax(E\$values[1:q] - 1, 0)), q)
diag(g)/Psi^2
}
p <- ncol(cmat)
if(is.null(start))
start <- (1 - 0.5*factors/p)/diag(solve(cmat))
res <- optim(start, FAfn, FAgr, method = "L-BFGS-B",
lower = lower, upper = 1,
control = c(list(fnscale=1,
parscale = rep(0.01, length(start))), control),
q = factors, S = cmat)
Lambda <- FAout(res\$par, cmat, factors)
dimnames(Lambda) <- list(dimnames(cmat)[[1]],
paste("Factor", 1:factors, sep = ""))
p <- ncol(cmat)
dof <- 0.5 * ((p - factors)^2 - p - factors)
un <- res\$par
names(un) <- colnames(cmat)
ans <- list(converged = res\$convergence == 0,
correlation = cmat,
criteria = c(objective = res\$value, counts = res\$counts),
factors = factors, dof = dof, method = "mle")
class(ans) <- "factanal"
ans
}

print.loadings <- function(x, digits = 3, cutoff = 0.1, sort = FALSE, ...)
{
Lambda <- unclass(x)
p <- nrow(Lambda)
factors <- ncol(Lambda)
if (sort) {
mx <- max.col(abs(Lambda))
ind <- cbind(1:p, mx)
mx[abs(Lambda[ind]) < 0.5] <- factors + 1
Lambda <- Lambda[order(mx, 1:p),]
}
fx <- format(round(Lambda, digits))
names(fx) <- NULL
nc <- nchar(fx[1], type="c")
fx[abs(Lambda) < cutoff] <- paste(rep(" ", nc), collapse = "")
print(fx, quote = FALSE, ...)
vx <- colSums(x^2)
if(is.null(attr(x, "covariance"))) {
varex <- rbind(varex, "Proportion Var" = vx/p)
if(factors > 1)
varex <- rbind(varex, "Cumulative Var" = cumsum(vx/p))
}
cat("\n")
print(round(varex, digits))
invisible(x)
}

print.factanal <- function(x, digits = 3, ...)
{
cat("\nCall:\n", deparse(x\$call), "\n\n", sep = "")
cat("Uniquenesses:\n")
print(round(x\$uniquenesses, digits), ...)
# the following lines added by J. Fox, 26 June 2005
if (!is.null(x\$rotmat)){
tmat <- solve(x\$rotmat)
R <- tmat %*% t(tmat)
factors <- x\$factors
rownames(R) <- colnames(R) <- paste("Factor", 1:factors, sep="")

if (TRUE != all.equal(R, diag(factors))){
cat("\nFactor Correlations:\n")
print(R, digits=digits, ...)
}
}
# end additions J. Fox, 23 June 2005
if(!is.null(x\$STATISTIC)) {
factors <- x\$factors
cat("\nTest of the hypothesis that", factors, if(factors == 1)
"factor is" else "factors are", "sufficient.\n")
cat("The chi square statistic is", round(x\$STATISTIC, 2), "on",
x\$dof,
if(x\$dof == 1) "degree" else "degrees",
"of freedom.\nThe p-value is", signif(x\$PVAL, 3), "\n")
} else {
cat(paste("\nThe degrees of freedom for the model is",
x\$dof, "and the fit was", round(x\$criteria["objective"],
4),
"\n"))
}
invisible(x)
}

varimax <- function(x, normalize = TRUE, eps = 1e-5)
{
nc <- ncol(x)
if(nc < 2) return(x)
if(normalize) {
sc <- sqrt(drop(apply(x, 1, function(x) sum(x^2))))
x <- x/sc
}
p <- nrow(x)
TT <- diag(nc)
d <- 0
for(i in 1:1000) {
z <- x %*% TT
B  <- t(x) %*% (z^3 - z %*% diag(drop(rep(1, p) %*% z^2))/p)
sB <- La.svd(B)
TT <- sB\$u %*% sB\$vt
dpast <- d
d <- sum(sB\$d)
if(d < dpast * (1 + eps)) break
}
z <- x %*% TT
if(normalize) z <- z * sc
dimnames(z) <- dimnames(x)
}

promax <- function(x, m = 4)
{
if(ncol(x) < 2) return(x)
dn <- dimnames(x)
xx <- varimax(x)
Q <- x * abs(x)^(m-1)
U <- lm.fit(x, Q)\$coefficients
d <- diag(solve(t(U) %*% U))
U <- U %*% diag(sqrt(d))
dimnames(U) <- NULL
z <- x %*% U
U <- xx\$rotmat %*% U
dimnames(z) <- dn