Influence Measures and Diagnostic Plots for Multivariate Linear Models

Version 0.9.3

Functions in this package compute regression deletion diagnostics for multivariate linear models following methods proposed by Barrett & Ling (1992) and provide some associated diagnostic plots. The diagnostic measures include hat-values (leverages), generalized Cook’s distance, and generalized squared ‘studentized’ residuals. Several types of plots to detect influential observations are provided.

In addition, the functions provide diagnostics for deletion of subsets of observations of size m>1. This case is theoretically interesting because sometimes pairs (m=2) of influential observations can mask each other, sometimes they can have joint influence far exceeding their individual effects, as well as other interesting phenomena described by Lawrence (1995). Associated methods for the case m>1 are still under development in this package.

Documentation

Documentation for the package is now available at https://friendly.github.io/mvinfluence/.

Installation

Get the released CRAN version or the development version, here or R-universe

Goals

The design goal for this package is that, as an extension of standard methods for univariate linear models, you should be able to fit a linear model with a multivariate response,

mymlm <- lm( cbind(y1, y2, y3) ~ x1 + x2 + x3, data=mydata)

and then get useful diagnostics and plots with:

influence(mymlm)
hatvalues(mymlm)
cooks.distance(mymlm)
influencePlot(mymlm, ...)

As is done in comparable univariate functions in the car package, noteworthy points are identified in printed output and graphs.

Examples

The Rohwer data contains data on kindergarten children designed to examine how well performance on a set of paired-associate (PA) learning tasks can predict performance on some measures of aptitude and achievement— SAT (a scholastic aptitude test), PPVT (Peabody Picture Vocabulary Test), and Raven ( Raven Progressive Matrices Test). The PA tasks differ in how the stimulus item was presented: n (named), s (still), ns (named still), na (named action) and ss (sentence still).

Here, we fit a MLM to a subset of the Rohwer data (the Low SES group).

data(Rohwer, package="heplots")
Rohwer2 <- subset(Rohwer, subset=group==2)
rownames(Rohwer2)<- 1:nrow(Rohwer2)
Rohwer.mod <- lm(cbind(SAT, PPVT, Raven) ~ n + s + ns + na + ss, data=Rohwer2)
car::Anova(Rohwer.mod)
#> 
#> Type II MANOVA Tests: Pillai test statistic
#>    Df test stat approx F num Df den Df Pr(>F)   
#> n   1     0.202     2.02      3     24 0.1376   
#> s   1     0.310     3.59      3     24 0.0284 * 
#> ns  1     0.358     4.46      3     24 0.0126 * 
#> na  1     0.465     6.96      3     24 0.0016 **
#> ss  1     0.089     0.78      3     24 0.5173   
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Influence plots

The default influence plot (type="stres") shows the squared standardized residual against the Hat value. The areas of the circles representing the observations are proportional to generalized Cook’s distances.

(infl <-influencePlot(Rohwer.mod, id.n=4, type = "stres"))

#>        H      Q  CookD     L      R
#> 5  0.568 0.3439 0.8467 1.316 0.7964
#> 10 0.452 0.0324 0.0634 0.824 0.0591
#> 14 0.126 0.2997 0.1643 0.145 0.3431
#> 15 0.332 0.0105 0.0152 0.498 0.0158
#> 25 0.157 0.3820 0.2601 0.186 0.4532
#> 27 0.367 0.2128 0.3387 0.580 0.3363
#> 29 0.304 0.2295 0.3026 0.437 0.3299

As you can see above, the function returns a data frame of the influence statistics for the identified points. “Noteworthy” points are those that are unusual on either Hat value (H) or the squared studentized residual (Q), so more points will be shown than the id.n value. It is often more useful to sort these in descending order by one of the influence measures.

infl |> dplyr::arrange(desc(H))
#>        H      Q  CookD     L      R
#> 5  0.568 0.3439 0.8467 1.316 0.7964
#> 10 0.452 0.0324 0.0634 0.824 0.0591
#> 27 0.367 0.2128 0.3387 0.580 0.3363
#> 15 0.332 0.0105 0.0152 0.498 0.0158
#> 29 0.304 0.2295 0.3026 0.437 0.3299
#> 25 0.157 0.3820 0.2601 0.186 0.4532
#> 14 0.126 0.2997 0.1643 0.145 0.3431

An alternative (type="LR") plots residual components against leverage components, both on log scales. Because influence is a product of residual