gsorth: Orthogonalize successive columns of a data frame or matrix

Description

gsorth uses sequential, orthogonal projections, as in the Gram-Schmidt method, to transform a matrix or numeric columns of a data frame into an uncorrelated set, possibly retaining the same column means and standard deviations as the original.

In statistical applications, interpretation depends on the order of the variables orthogonalized. In multivariate linear models, orthogonalizing the response, Y variables provides the equivalent of step-down tests, where Y1 is tested alone, and then Y2.1, Y3.12, etc. can be tested to determine their additional contributions over the previous response variables.

Similarly, orthogonalizing the model X variables provides the equivalent of Type I tests, such as provided by anova.

Usage

gsorth(y, order, recenter = TRUE, rescale = TRUE, adjnames = TRUE)

Arguments

A numeric data frame or matrix

order

An integer vector specifying the order of and/or a subset of the columns of y to be orthogonalized. If missing, order=1:p where p=ncol(y).

recenter

If TRUE, the result has same column means as original; else means = 0 for cols 2:p.

rescale

If TRUE, the result has same column standard deviations as original; else sd = residual variance for cols 2:p

adjnames

If TRUE, the column names of the result are adjusted to the form Y1, Y2.1, Y3.12, by adding the suffixes '.1', '.12', etc. to the original column names.

Value

Returns a matrix or data frame with uncorrelated columns. Row and column names are copied to the result.

Details

The method is equivalent to setting each of columns 2:p to the residuals from a linear regression of that column on all prior columns, i.e.,

z[,j] <- resid( lm( z[,j] ~ as.matrix(z[,1:(j-1)]), data=z) )

However, for accuracy and speed the transformation is carried out using the QR decomposition.

Examples

Run this code

# NOT RUN {
GSiris <- gsorth(iris[,1:4])
GSiris <- gsorth(iris, order=1:4)   # same, using order
str(GSiris)
zapsmall(cor(GSiris))
colMeans(GSiris)
# sd(GSiris) -- sd(<matrix>) now deprecated
apply(GSiris, 2, sd)

# orthogonalize Y side
GSiris <- data.frame(gsorth(iris[,1:4]), Species=iris$Species)
iris.mod1 <- lm(as.matrix(GSiris[,1:4]) ~ Species, data=GSiris)
Anova(iris.mod1)

# orthogonalize X side
rohwer.mod <- lm(cbind(SAT, PPVT, Raven) ~ n + s + ns + na + ss, data=Rohwer)
Anova(rohwer.mod)

# type I tests for Rohwer data
Rohwer.orth <- cbind(Rohwer[,1:5], gsorth(Rohwer[, c("n", "s", "ns", "na", "ss")], adjnames=FALSE))

rohwer.mod1 <- lm(cbind(SAT, PPVT, Raven) ~ n + s + ns + na + ss, data=Rohwer.orth)
Anova(rohwer.mod1)
# compare with anova()
anova(rohwer.mod1)

# compare heplots for original Xs and orthogonalized, Type I
heplot(rohwer.mod)
heplot(rohwer.mod1)

# }