Distance Plot for Multivariate Model Diagnosis

This plot, suggested by Rousseeuw & van Zomeren (1991), Rousseeuw et al. (2004) typically plots Mahalanobis distances (\(D\)) of the Y response variables against the distances of the X variables in a multivariate linear model (MLM). When applied to a multivariate linear model itself, it plots the distances of the residuals for the Y variables against the predictor terms in the model.matrix X. The motivation behind this plot was to contribute to understanding robust methods for multivariate linear models, such as implemented here in robmlm().

This diagnostic plot combines the information on regression outliers and leverage points, and often more useful than either distance separately, because it shows WHY observations are unusual, either in \(X\) or \(Y\) space or both. In spirit, it is similar to the ideas behind influence plots for univariate models, such as car::influencePlot() and the multivariate analogs provided in the mvinfluence package via influencePlot().

Usage

distancePlot(X, Y, ...)

# Default S3 method
distancePlot(
  X,
  Y,
  method = c("classical", "mcd", "mve"),
  level = 0.975,
  ids = rownames(X),
  pch = c(1, 16),
  col = c("black", "red"),
  label.pos = 2,
  xlab,
  ylab,
  verbose = FALSE,
  ...
)

# S3 method for class 'formula'
distancePlot(X, Y, data, ...)

# S3 method for class 'mlm'
distancePlot(X, ...)

Arguments

X: A multivariate linear model fit by lm, or a numeric data frame giving the predictors in the MLM
Y: A numeric data frame giving the responses in the MLM or the residuals
...: Other arguments passed to methods
method: Estimation method used for center and covariance, one of: "classical" (product-moment), "mcd" (minimum covariance determinant), or "mve" (minimum volume ellipsoid).
level: Lower-tail probability beyond which observations will be labeled.
ids: Labels for observations
pch: A vector of two point symbols, for the regular points and those beyond the cutoffs
col: A vector of two colors, for the regular points and those beyond the cutoffs
label.pos: Position of the label relative to the point; see text
xlab: Label stub for horizontal axis
ylab: Label stub for vertical axis
verbose: Logical; if TRUE print the cutoff values to the console
data: For the formula method, the dataset containing the variables

Value

   Returns invisibly a data frame containing the distances, `distX`, `distY`

Details

Observations with "large" distances on X or Y are labeled with their ids. The cutoffs are calculated as \(\sqrt{\chi^2_{k, \text{level}}}\).

The method argument provides a specification the covariance matrix used to calculate the Mahalanobis distances shown in this plot.

References

Rousseeuw, P. J., & van Zomeren, B. C. (1990). Unmasking multivariate outliers and leverage points. Journal of the American Statistical Association, 85(411), 633-639. doi:10.1080/01621459.1990.10474920 .

Rousseeuw P. J. & van Zomeren B. C. (1991). “Robust Distances: Simulation and Cutoff Values.” In W Stahel, S Weisberg (eds.), Directions in Robust Statistics and Diagnostics, Part II. Springer-Verlag, New York.

Rousseeuw, P. J., Van Driessen, K., Van Aelst, S., & Agullo, J. (2004). Robust multivariate regression. Technometrics, 46(3), 293–305. doi:10.1198/004017004000000329 .

Examples


if(require("robustbase")) {
  # Examples from Rousseeuw etal (2004)
  data(pulpfiber, package="robustbase")
  # Figure 1
  distancePlot(pulpfiber[, 1:4], pulpfiber[, 5:8])   
  # Figure 3
  pulp.mod <- lm(cbind(Y1, Y2, Y3, Y4) ~ X1 + X2 + X3 + X4, data = pulpfiber)
  distancePlot(pulp.mod, method = "mcd")
}
#> Loading required package: robustbase



# NLSY data
data(NLSY, package = "heplots")
NLSY.mlm <- lm(cbind(math, read) ~ income + educ + antisoc + hyperact,
               data = NLSY)

distancePlot(NLSY.mlm)


# gives the same result
distancePlot(NLSY[, 3:6], residuals(NLSY.mlm), level = 0.975)


distancePlot(NLSY.mlm, method ="mve")


# distancePlot(cbind(math, read) ~ income + educ + antisoc + hyperact,
#                data = NLSY)

# schooldata dataset
data(schooldata)
school.mod <- lm(cbind(reading, mathematics, selfesteem) ~ ., data=schooldata)
distancePlot(school.mod, cex = 1.5, cex.lab = 1.2)


data(Hernior)
Hern.mod <- lm(cbind(leave, nurse, los) ~
               age + sex +  pstat +  build + cardiac + resp, data=Hernior)
distancePlot(Hern.mod)