robmlm function

Robust Fitting of Multivariate Linear Models

Fit a multivariate linear model by robust regression using a simple M estimator.

robmlm(X, ...)

## Default S3 method:
robmlm(
  X,
  Y,
  w,
  P = 2 * pnorm(4.685, lower.tail = FALSE),
  tune,
  max.iter = 100,
  psi = psi.bisquare,
  tol = 1e-06,
  initialize,
  verbose = FALSE,
  ...
)

## S3 method for class 'formula'
robmlm(
  formula,
  data,
  subset,
  weights,
  na.action,
  model = TRUE,
  contrasts = NULL,
  ...
)

## S3 method for class 'robmlm'
print(x, ...)

## S3 method for class 'robmlm'
summary(object, ...)

## S3 method for class 'summary.robmlm'
print(x, ...)

Arguments

• X: for the default method, a model matrix, including the constant (if present)

• ...: other arguments, passed down. In particular relevant control arguments can be passed to the to the robmlm.default method.

• Y: for the default method, a response matrix

• w: prior weights

• P: two-tail probability, to find cutoff quantile for chisq (tuning constant); default is set for bisquare weight function

• tune: tuning constant (if given directly)

• max.iter: maximum number of iterations

• psi: robustness weight function; psi.bisquare is the default

• tol: convergence tolerance, maximum relative change in coefficients

• initialize: modeling function to find start values for coefficients, equation-by-equation; if absent WLS (lm.wfit) is used

• verbose: show iteration history? (TRUE or FALSE)

• formula: a formula of the form cbind(y1, y2, ...) ~ x1 + x2 + ....

• data: a data frame from which variables specified in formula

  are preferentially to be taken.

• subset: An index vector specifying the cases to be used in fitting.

• weights: a vector of prior weights for each case.

• na.action: A function to specify the action to be taken if NAs are found. The 'factory-fresh' default action in R is na.omit, and can be changed by options``(na.action=).

• model: should the model frame be returned in the object?

• contrasts: optional contrast specifications; see lm for details.

• x: a robmlm object

• object: a robmlm object

Returns

An object of class "robmlm" inheriting from c("mlm", "lm").

This means that the returned "robmlm" contains all the components of "mlm" objects described for lm, plus the following:

• weights: final observation weights
• iterations: number of iterations
• converged: logical: did the IWLS process converge?

The generic accessor functions coefficients, effects, fitted.values and residuals extract various useful features of the value returned by robmlm.

Details

These S3 methods are designed to provide a specification of a class of robust methods which extend mlms, and are therefore compatible with other mlm extensions, including Anova and heplot.

Fitting is done by iterated re-weighted least squares (IWLS), using weights based on the Mahalanobis squared distances of the current residuals from the origin, and a scaling (covariance) matrix calculated by cov.trob. The design of these methods were loosely modeled on rlm.

An internal vcov.mlm function is an extension of the standard vcov method providing for observation weights.

Examples

##############
# Skulls data

# make shorter labels for epochs and nicer variable labels in heplots
Skulls$epoch <- factor(Skulls$epoch, labels=sub("c","",levels(Skulls$epoch)))
# variable labels
vlab <- c("maxBreadth", "basibHeight", "basialLength", "nasalHeight")

# fit manova model, classically and robustly
sk.mod <- lm(cbind(mb, bh, bl, nh) ~ epoch, data=Skulls)
sk.rmod <- robmlm(cbind(mb, bh, bl, nh) ~ epoch, data=Skulls)

# standard mlm methods apply here
coefficients(sk.rmod)

# index plot of weights
plot(sk.rmod$weights, type="h", xlab="Case Index", ylab="Robust mlm weight", col="gray")
points(sk.rmod$weights, pch=16, col=Skulls$epoch)
axis(side=1, at=15+seq(0,120,30), labels=levels(Skulls$epoch), tick=FALSE, cex.axis=1)

# heplots to see effect of robmlm vs. mlm
heplot(sk.mod, hypotheses=list(Lin="epoch.L", Quad="epoch.Q"), 
    xlab=vlab[1], ylab=vlab[2], cex=1.25, lty=1)
heplot(sk.rmod, hypotheses=list(Lin="epoch.L", Quad="epoch.Q"), 
    add=TRUE, error.ellipse=TRUE, lwd=c(2,2), lty=c(2,2), 
    term.labels=FALSE, hyp.labels=FALSE, err.label="")

##############
# Pottery data

data(Pottery, package = "carData")
pottery.mod <- lm(cbind(Al,Fe,Mg,Ca,Na)~Site, data=Pottery)
pottery.rmod <- robmlm(cbind(Al,Fe,Mg,Ca,Na)~Site, data=Pottery)
car::Anova(pottery.mod)
car::Anova(pottery.rmod)

# index plot of weights
plot(pottery.rmod$weights, type="h")
points(pottery.rmod$weights, pch=16, col=Pottery$Site)

# heplots to see effect of robmlm vs. mlm
heplot(pottery.mod, cex=1.3, lty=1)
heplot(pottery.rmod, add=TRUE, error.ellipse=TRUE, lwd=c(2,2), lty=c(2,2), 
    term.labels=FALSE, err.label="")

###############
# Prestige data
data(Prestige, package = "carData")

# treat women and prestige as response variables for this example
prestige.mod <- lm(cbind(women, prestige) ~ income + education + type, data=Prestige)
prestige.rmod <- robmlm(cbind(women, prestige) ~ income + education + type, data=Prestige)

coef(prestige.mod)
coef(prestige.rmod)
# how much do coefficients change?
round(coef(prestige.mod) - coef(prestige.rmod),3)

# pretty plot of case weights
plot(prestige.rmod$weights, type="h", xlab="Case Index", ylab="Robust mlm weight", col="gray")
points(prestige.rmod$weights, pch=16, col=Prestige$type)
legend(0, 0.7, levels(Prestige$type), pch=16, col=palette()[1:3], bg="white")

heplot(prestige.mod, cex=1.4, lty=1)
heplot(prestige.rmod, add=TRUE, error.ellipse=TRUE, lwd=c(2,2), lty=c(2,2), 
    term.labels=FALSE, err.label="")

References

A. Marazzi (1993) Algorithms, Routines and S Functions for Robust Statistics. Wadsworth & Brooks/Cole.

See Also

rlm, cov.trob

Author(s)

John Fox; packaged by Michael Friendly