Variable Clustering

Does a hierarchical cluster analysis on variables, using the Hoeffding D statistic, squared Pearson or Spearman correlations, or proportion of observations for which two variables are both positive as similarity measures. Variable clustering is used for assessing collinearity, redundancy, and for separating variables into clusters that can be scored as a single variable, thus resulting in data reduction. For computing any of the three similarity measures, pairwise deletion of NAs is done. The clustering is done by hclust(). A small function naclus is also provided which depicts similarities in which observations are missing for variables in a data frame. The similarity measure is the fraction of NAs in common between any two variables. The diagonals of this sim matrix are the fraction of NAs in each variable by itself. naclus also computes na.per.obs, the number of missing variables in each observation, and mean.na, a vector whose ith element is the mean number of missing variables other than variable i, for observations in which variable i is missing. The naplot function makes several plots (see the which argument).

So as to not generate too many dummy variables for multi-valued character or categorical predictors, varclus will automatically combine infrequent cells of such variables using combine.levels.

plotMultSim plots multiple similarity matrices, with the similarity measure being on the x-axis of each subplot.

na.pattern prints a frequency table of all combinations of missingness for multiple variables. If there are 3 variables, a frequency table entry labeled 110 corresponds to the number of observations for which the first and second variables were missing but the third variable was not missing.

Usage

varclus(x, similarity=c("spearman","pearson","hoeffding","bothpos","ccbothpos"),
        type=c("data.matrix","similarity.matrix"), 
        method="complete",
        data=NULL, subset=NULL, na.action=na.retain,
        trans=c("square", "abs", "none"), ...)
# S3 method for class 'varclus'
print(x, abbrev=FALSE, ...)
# S3 method for class 'varclus'
plot(x, ylab, abbrev=FALSE, legend.=FALSE, loc, maxlen, labels, ...)

naclus(df, method)
naplot(obj, which=c('all','na per var','na per obs','mean na',
                    'na per var vs mean na'), ...)

plotMultSim(s, x=1:dim(s)[3],
            slim=range(pretty(c(0,max(s,na.rm=TRUE)))),
            slimds=FALSE,
            add=FALSE, lty=par('lty'), col=par('col'),
            lwd=par('lwd'), vname=NULL, h=.5, w=.75, u=.05,
            labelx=TRUE, xspace=.35)

na.pattern(x)

Arguments

x

a formula, a numeric matrix of predictors, or a similarity matrix. If x is a formula, model.matrix is used to convert it to a design matrix. If the formula excludes an intercept (e.g., ~ a + b -1), the first categorical (factor) variable in the formula will have dummy variables generated for all levels instead of omitting one for the first level. For plot and print, x is an object created by varclus. For na.pattern, x is a data table, data frame, or matrix.

For plotMultSim, is a numeric vector specifying the ordered unique values on the x-axis, corresponding to the third dimension of s.

df

a data frame

s

an array of similarity matrices. The third dimension of this array corresponds to different computations of similarities. The first two dimensions come from a single similarity matrix. This is useful for displaying similarity matrices computed by varclus, for example. A use for this might be to show pairwise similarities of variables across time in a longitudinal study (see the example below). If vname is not given, s must have dimnames.

similarity

the default is to use squared Spearman correlation coefficients, which will detect monotonic but nonlinear relationships. You can also specify linear correlation or Hoeffding's (1948) D statistic, which has the advantage of being sensitive to many types of dependence, including highly non-monotonic relationships. For binary data, or data to be made binary, similarity="bothpos" uses as a similarity measure the proportion of observations for which two variables are both positive. similarity="ccbothpos" uses a chance-corrected measure which is the proportion of observations for which both variables are positive minus the product of the two marginal proportions. This difference is expected to be zero under independence. For diagonals, "ccbothpos" still uses the proportion of positives for the single variable. So "ccbothpos" is not really a similarity measure, and clustering is not done. This measure is useful for plotting with plotMultSim (see the last example).

type

if x is not a formula, it may be a data matrix or a similarity matrix. By default, it is assumed to be a data matrix.

method

see hclust. The default, for both varclus and naclus, is "compact" (for R it is "complete").

data

a data frame, data table, or list

subset

a standard subsetting expression

na.action

These may be specified if x is a formula. The default na.action is na.retain, defined by varclus. This causes all observations to be kept in the model frame, with later pairwise deletion of NAs.

trans

By default, when the similarity measure is based on Pearson's or Spearman's correlation coefficients, the coefficients are squared. Specify trans="abs" to take absolute values or trans="none" to use the coefficients as they stand.

...

for varclus these are optional arguments to pass to the dataframeReduce function. Otherwise, passed to plclust (or to dotchart or dotchart2 for naplot).

ylab

y-axis label. Default is constructed on the basis of similarity.

legend.

set to TRUE to plot a legend defining the abbreviations

loc

a list with elements x and y defining coordinates of the upper left corner of the legend. Default is locator(1).

maxlen

if a legend is plotted describing abbreviations, original labels longer than maxlen characters are truncated at maxlen.

labels

a vector of character strings containing labels corresponding to columns in the similar matrix, if the column names of that matrix are not to be used

obj

an object created by naclus

which

defaults to "all" meaning to have naplot make 4 separate plots. To make only one of the plots, use which="na per var" (dot chart of fraction of NAs for each variable), ,"na per obs" (dot chart showing frequency distribution of number of variables having NAs in an observation), "mean na" (dot chart showing mean number of other variables missing when the indicated variable is missing), or "na per var vs mean na", a scatterplot showing on the x-axis the fraction of NAs in the variable and on the y-axis the mean number of other variables that are NA when the indicated variable is NA.

abbrev

set to TRUE to abbreviate variable names for plotting or printing. Is set to TRUE automatically if legend=TRUE.

slim

2-vector specifying the range of similarity values for scaling the y-axes. By default this is the observed range over all of s.

slimds

set to slimds to TRUE to scale diagonals and off-diagonals separately

add

set to TRUE to add similarities to an existing plot (usually specifying lty or col)

lty, col, lwd

line type, color, or line thickness for plotMultSim

vname

optional vector of variable names, in order, used in s

h

relative height for subplot

w

relative width for subplot

u

relative extra height and width to leave unused inside the subplot. Also used as the space between y-axis tick mark labels and graph border.

labelx

set to FALSE to suppress drawing of labels in the x direction

xspace

amount of space, on a scale of 1:n where n is the number of variables, to set aside for y-axis labels

Value

for varclus or naclus, a list of class varclus with elements call (containing the calling statement), sim (similarity matrix), n (sample size used if x was not a correlation matrix already - n is a matrix), hclust, the object created by hclust, similarity, and method. naclus also returns the two vectors listed under description, and naplot returns an invisible vector that is the frequency table of the number of missing variables per observation. plotMultSim invisibly returns the limits of similarities used in constructing the y-axes of each subplot. For similarity="ccbothpos" the hclust object is NULL.

na.pattern creates an integer vector of frequencies.

Details

options(contrasts= c("contr.treatment", "contr.poly")) is issued temporarily by varclus to make sure that ordinary dummy variables are generated for factor variables. Pass arguments to the dataframeReduce function to remove problematic variables (especially if analyzing all variables in a data frame).

Author

Frank Harrell
Department of Biostatistics, Vanderbilt University
fh@fharrell.com

Side Effects

plots

References

Sarle, WS: The VARCLUS Procedure. SAS/STAT User's Guide, 4th Edition, 1990. Cary NC: SAS Institute, Inc.

Hoeffding W. (1948): A non-parametric test of independence. Ann Math Stat 19:546–57.

Examples

set.seed(1)
x1 <- rnorm(200)
x2 <- rnorm(200)
x3 <- x1 + x2 + rnorm(200)
x4 <- x2 + rnorm(200)
x <- cbind(x1,x2,x3,x4)
v <- varclus(x, similarity="spear")  # spearman is the default anyway
v    # invokes print.varclus
#> varclus(x = x, similarity = "spear")
#> 
#> 
#> Similarity matrix (Spearman rho^2)
#> 
#>      x1   x2   x3   x4
#> x1 1.00 0.00 0.26 0.00
#> x2 0.00 1.00 0.26 0.42
#> x3 0.26 0.26 1.00 0.12
#> x4 0.00 0.42 0.12 1.00
#> 
#> No. of observations used for each pair:
#> 
#>     x1  x2  x3  x4
#> x1 200 200 200 200
#> x2 200 200 200 200
#> x3 200 200 200 200
#> x4 200 200 200 200
#> 
#> hclust results (method=complete)
#> 
#> 
#> Call:
#> hclust(d = as.dist(1 - x), method = method)
#> 
#> Cluster method   : complete 
#> Number of objects: 4 
#> 
print(round(v$sim,2))
#>      x1   x2   x3   x4
#> x1 1.00 0.00 0.26 0.00
#> x2 0.00 1.00 0.26 0.42
#> x3 0.26 0.26 1.00 0.12
#> x4 0.00 0.42 0.12 1.00
plot(v)


# Convert the dendrogram to be horizontal
v <- as.dendrogram(v$hclust)
plot(v, horiz=TRUE, axes=FALSE, xlab=expression(paste('Spearman ', rho^2)))
rh <- seq(0, 1, by=0.1)  # re-label x-axis re:similarity not distance
axis(1, at=1 - rh, labels=format(rh))


# plot(varclus(~ age + sys.bp + dias.bp + country - 1), abbrev=TRUE)
# the -1 causes k dummies to be generated for k countries
# plot(varclus(~ age + factor(disease.code) - 1))
#
#
# use varclus(~., data= fracmiss= maxlevels= minprev=) to analyze all
# "useful" variables - see dataframeReduce for details about arguments


df <- data.frame(a=c(1,2,3),b=c(1,2,3),c=c(1,2,NA),d=c(1,NA,3),
                 e=c(1,NA,3),f=c(NA,NA,NA),g=c(NA,2,3),h=c(NA,NA,3))
par(mfrow=c(2,2))
for(m in c("ward","complete","median")) {
  plot(naclus(df, method=m))
  title(m)
}
#> The "ward" method has been renamed to "ward.D"; note new "ward.D2"
naplot(naclus(df))

n <- naclus(df)
plot(n); naplot(n)


na.pattern(df)
#> pattern
#> 00000111 00011101 00100100 
#>        1        1        1 

# plotMultSim example: Plot proportion of observations
# for which two variables are both positive (diagonals
# show the proportion of observations for which the
# one variable is positive).  Chance-correct the
# off-diagonals by subtracting the product of the
# marginal proportions.  On each subplot the x-axis
# shows month (0, 4, 8, 12) and there is a separate
# curve for females and males
d <- data.frame(sex=sample(c('female','male'),1000,TRUE),
                month=sample(c(0,4,8,12),1000,TRUE),
                x1=sample(0:1,1000,TRUE),
                x2=sample(0:1,1000,TRUE),
                x3=sample(0:1,1000,TRUE))
s <- array(NA, c(3,3,4))
opar <- par(mar=c(0,0,4.1,0))  # waste less space
for(sx in c('female','male')) {
  for(i in 1:4) {
    mon <- (i-1)*4
    s[,,i] <- varclus(~x1 + x2 + x3, sim='ccbothpos', data=d,
                      subset=d$month==mon & d$sex==sx)$sim
    }
  plotMultSim(s, c(0,4,8,12), vname=c('x1','x2','x3'),
              add=sx=='male', slimds=TRUE,
              lty=1+(sx=='male'))
  # slimds=TRUE causes separate  scaling for diagonals and
  # off-diagonals
}
par(opar)