Background
Reaven and Miller (1979) examined the relationship among blood chemistry measures of glucose tolerance and insulin in 145 nonobese adults. They used the PRIM9 system at the Stanford Linear Accelerator Center to visualize the data in 3D, and discovered a peculiar pattern that looked like a large blob with two wings in different directions.
After further analysis, the subjects were classified as sub-clinical (chemical) diabetics, overt diabetics and normals. This study was influential in defining the stages of development of Type 2 diabetes. Overt diabetes is the most advanced stage, characterized by elevated fasting blood glucose concentration and classical symptoms. Preceding overt diabetes is the latent or chemical diabetic stage, with no symptoms of diabetes but demonstrable abnormality of oral or intravenous glucose tolerance.
The Diabetes data set is contained in the heplots package. This vignette uses these data to illustrate various graphical methods for multivariate linear models. As we’ll see, the data is peculiar in several respects and a standard MANOVA is problematic because some assumptions are violated. We treat this as a learning opportunity, because it allows us to illustrate several diagnostic plots. Some of the content here serves as an additional example for a recent paper on visualizing tests for equality of variance in MANOVA (Friendly & Sigal, 2017).
Setting up
Load the required packages and the data
The variables are:
relwt: relative weight, expressed as the ratio of actual weight to expected weight, given the person’s heightglufast: fasting plasma glucose levelglutest: 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 resistancegroup: diagnostic group
relwt is not used in this example, but that is not for any principled reason.
Data ellipsoids and homogeneity of variance
We start by plotting covariance ellipses for three of the variables in the data set. heplots::covEllipses is similar to dataEllipse in the car package, but tuned to plots designed to assess homogeneity of covariance matrices in MANOVA designs.
It is clear from this that there is a problem of heterogeneity of variance-covariance matrices here. The normal group shows the smallest variances and the overt diabetic group the largest.
As well, the plot suggests something more interesting about the difference between the overt and chemical diabetes groups. In the panel for glufast and glutest, there appears to be a direct progression from normal to chemical to overt. However, this is not true in the other panels, where the chemical diabetes group differs from the normals in one direction, while the overt group differs in a different direction and has a within-group correlation opposite in sign to the others. This can be seen more easily in individual scatterplots, such as that below (using car::scatterplot).
scatterplot( instest ~ glutest | group, data=Diabetes,
pch=c(16,15,17),
col=c("blue", "red", "darkgreen"),
smooth=FALSE,
grid=FALSE,
legend=list(coords="topright"),
lwd=2,
ellipse=list(levels=0.5))
This insight was part of the reason that Reaven and Miller concluded that chemical diabetes and overt diabetes reflected different disease regimes, rather than an increasing progression of severity.
As an aside, we note that the differences among the groups can be readily seen in 3D using scatter3d in the car package.
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.
The log determinants are ordered according to the sizes of the data ellipses we saw in the covariance ellipse plots.
Fit the MLM
We ignore this problem for the sake of this example, and proceed to fit the MANOVA model for the mean differences among groups.
The MANOVA shows a highly significant effect of group on the collection of response variables.
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.
HE plots
An HE plot shows the \(H\) ellipse for differences among means together with the \(E\) ellipse for error. In the default scaling, such plots have the visual property that a given effect in the MLM is significant (by Roy’s maximum root test) iff the \(H\) ellipse projects anywhere outside the \(E\) ellipse.
By default, the first two response variables are plotted. The result shows that the means on both glufast and glutest are ordered Normal < Chemical < Overt.
The pairs method for an MLM gives a scatterplot matrix of HE plots for all pairs of response variables. The pattern for instest seems to differ from the other variables.
Canonical discriminant analysis
Canonical discriminant analysis effectively projects the data into the space of linear combinations of the responses that accounts for the greatest proportion of the between-group variance relative to within-group variance.
In this case, with \(g=3\) groups, this space is two-dimensional, so all differences can be seen in a 2D plot. From the output below, nearly 90% of between-group mean differences are accounted for by the first canonical dimension. Both canonical dimensions are significant, by a likelihood ratio stepdown test.
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.
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.
## Vector scale factor set to 12.06
Other analyses
There are other methods that could be used to analyze these data and display the results visually. We merely sketch some alternatives here.
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.
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.
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.
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.
How well does that do? We can examine a table of predicted group membership vs. actual and calculate the error rate. Not too bad!
References
Friendly, M. & Sigal, M. (2017) Visualizing Tests of Equality of Covariance Matrices. Submitted for publication.
Reaven, G. M. & Miller, R. G. (1979). An Attempt to Define the Nature of Chemical Diabetes Using a Multidimensional Analysis Diabetologia, 16, 17-24.
