gsorth: Orthogonalize successive columns of a data frame or matrix

In heplots: Visualizing Hypothesis Tests in Multivariate Linear Models

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.

Usage

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

Arguments

y	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.

Details

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.

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.

Value

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

Author(s)

Michael Friendly

See Also

qr,

Examples

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)
car::Anova(iris.mod1)

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

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

heplots documentation built on Sept. 8, 2023, 5:32 p.m.