Diabetes data: heplots and candisc examples

Setting up

Load the required packages and the data

library(heplots)
library(candisc)
library(car)

data(Diabetes, package="heplots")
str(Diabetes)
## 'data.frame':    145 obs. of  6 variables:
##  $ relwt  : num  0.81 0.95 0.94 1.04 1 0.76 0.91 1.1 0.99 0.78 ...
##  $ glufast: int  80 97 105 90 90 86 100 85 97 97 ...
##  $ glutest: int  356 289 319 356 323 381 350 301 379 296 ...
##  $ instest: int  124 117 143 199 240 157 221 186 142 131 ...
##  $ sspg   : int  55 76 105 108 143 165 119 105 98 94 ...
##  $ group  : Factor w/ 3 levels "Normal","Chemical_Diabetic",..: 1 1 1 1 1 1 1 1 1 1 ...

The variables are:

• relwt: relative weight, expressed as the ratio of actual weight to expected weight, given the person’s height
• glufast: fasting plasma glucose level
• glutest: test plasma glucose level, a measure of glucose intolerance,
• instest: plasma insulin during test, a measure of insulin response to oral glucose,
• sspg: steady state plasma glucose, a measure of insulin resistance
• group: diagnostic group

relwt is not used in this example, but that is not for any principled reason.

Box’s M test

Box’s M test confirms that there is substantial heterogeneity of covariance matrices. The plot method for the result gives a convenient and informative display of how the groups differ in the components (\(log | S_i |\)) that go into Box’s M test.

diab.boxm <- boxM(Diabetes[,2:5], Diabetes$group)
diab.boxm
## 
##  Box's M-test for Homogeneity of Covariance Matrices
## 
## data:  Diabetes[, 2:5]
## Chi-Sq (approx.) = 383, df = 20, p-value <2e-16

op <- par(mar=c(4,6,1,1)+.5)
plot(diab.boxm, cex.lab=1.5)

The log determinants are ordered according to the sizes of the data ellipses we saw in the covariance ellipse plots.

Examine residuals in a QQ plot

Another assumption of MANOVA is that the residuals have a multivariate normal distribution. This can be assessed visually with a Chi-square QQ plot. This plots squared Mahalanobis distances (\(D^2\)) of the residuals vs. corresponding quantiles of the \(\chi^2\) distribution with \(p\) degrees of freedom. The cqplot function in the heplots package is designed to do this from a fitted MLM object. A confidence envelope makes it easy to see when the departure is large enough to cause concern.

It can be seen here that the points differ substantially from the red line of equality. There are far too many points with large \(D^2\) values.

cqplot(diab.mlm, id.n = 3)

Canonical discriminant plot

The plot method for a candisc object plots the scores on the canonical dimensions and overlays 60% data ellipses for each group. The positions of the group means show the the means on the canonical dimensions.

The relationship of the response variables to the canonical dimensions is shown by vectors (similar to a biplot). Each vector is defined by the correlations (structure coefficients) it has with the canonical dimensions.¹

plot(diab.can, ellipse=TRUE, var.lwd=2)

In this plot it can be seen that:

• Dimension 1 is highly related to glufast, glutest and sspg, and the groups are ordered Normal < Chemical < Overt.

• Dimension 2 distinguishes the Chemical_Diabetic group from the other two. This dimension is largely related to instest, the plasma insulin during test. This confirms what we saw in the HE plot matrix for all the responses.

• The relative sizes of the data ellipses for the canonical scores is another visual indication of the lack of heterogeneity of variances.

Canonical HE plot

The heplot method for candisc objects gives an HE plot summary of the canonical discriminant analysis. The variable vectors are the same as in the canonical structure plot. The size of the \(H\) ellipse relative to that of the \(E\) ellipse is an indication of the magnitude of the multivariate effect for group.

heplot(diab.can, fill=c(TRUE, FALSE), fill.alpha=0.1, var.lwd=2)

## Vector scale factor set to  12.06

Linear and quadratic discriminant analysis

Linear discriminant analysis (LDA) is similar in spirit to MANOVA, but the emphasis is on classification rather than on testing differences among means. As well, LDA allows specification of the prior probabilities of group membership to make classification error rates comparable to what would be obtained in a population of interest. Quadratic discriminant analysis allows for differences in the covariance matrices among groups, and gives quadratic rather then linear classification boundaries.

LDA is in some sense equivalent to our canonical discriminant analysis. A simple function for this is lda in the MASS package. This gives the same class means and the same variance proportions as candisc. However, here, the prior probabilities are taken from the proportions of the groups in the data.

library(MASS)
diab.lda <- lda(group ~ glufast + glutest + instest + sspg, data = Diabetes)
diab.lda
## Call:
## lda(group ~ glufast + glutest + instest + sspg, data = Diabetes)
## 
## Prior probabilities of groups:
##            Normal Chemical_Diabetic    Overt_Diabetic 
##            0.5241            0.2483            0.2276 
## 
## Group means:
##                   glufast glutest instest  sspg
## Normal              91.18   350.0   172.6 114.0
## Chemical_Diabetic   99.31   493.9   288.0 209.0
## Overt_Diabetic     217.67  1043.8   106.0 318.9
## 
## Coefficients of linear discriminants:
##                LD1       LD2
## glufast  0.0326457 -0.038928
## glutest -0.0119958  0.006560
## instest  0.0003149  0.006202
## sspg    -0.0056269  0.003231
## 
## Proportion of trace:
##    LD1    LD2 
## 0.8958 0.1042

From the LDA perspective, one goal in visualizing the results is to see the boundaries of classification based on the LD1 and LD2 scores. The plot shown here was produced with ggplot2, following a method described in this stackoverflow question. What is missing here is any indication of how this relates to the response variables.

LDA classification for the diabetes data

Quadratic discriminant analysis can be carried out in a similar way, using the qda function. The results are not shown.

Recursive partitioning

Recursive partitioning is a method that creates a decision tree designed to classify the members of a population. It recursively splits the data into sub-populations using dichotomous intervals of the predictors. One strength of the method is that it is nicely tied to cross-validation, where a sample is repeatedly split into training and test samples, and the model fit to each training sample is assessed on the test sample.

For the diabetes data, the result is very simple: A value of glutest < 420 distinguishes normals from the two clinical groups. For the latter, glufast < 117 classifies an individual as chemical diabetic rather than overt diabetic.

library(rpart)
diab.part <- rpart(group ~ glufast + glutest + instest + sspg, data=Diabetes)
diab.part
## n= 145 
## 
## node), split, n, loss, yval, (yprob)
##       * denotes terminal node
## 
## 1) root 145 69 Normal (0.52414 0.24828 0.22759)  
##   2) glutest< 420.5 76  1 Normal (0.98684 0.01316 0.00000) *
##   3) glutest>=420.5 69 34 Chemical_Diabetic (0.01449 0.50725 0.47826)  
##     6) glufast< 117 36  1 Chemical_Diabetic (0.02778 0.97222 0.00000) *
##     7) glufast>=117 33  0 Overt_Diabetic (0.00000 0.00000 1.00000) *

A nice plot of the partition tree is produced using the rpart.plot package. The numbers in the nodes give the proportions classified in each of the three groups.

library(rpart.plot)
rpart.plot(diab.part, box.palette=list("Blues", "Reds",  "Greens"))

Recursive partition tree for diabetes data

How well does that do? We can examine a table of predicted group membership vs. actual and calculate the error rate. Not too bad!

(class.pred <- table(predicted=predict(diab.part, type="class"), actual=Diabetes$group))
##                    actual
## predicted           Normal Chemical_Diabetic Overt_Diabetic
##   Normal                75                 1              0
##   Chemical_Diabetic      1                35              0
##   Overt_Diabetic         0                 0             33

# error rate
1 - sum(diag(class.pred))/sum(class.pred)
## [1] 0.01379