9  Descriptive Statistics

flowchart LR
des[describe Function] --> ss[Statistical Summary] & dsh[Spike Histogram]
cat[Categorical Variables] --> dot[Frequency Dot Charts]
con[Continuous Variables] --> sh[Spike Histograms] & ebp[Extended Box Plots]
lon[Longitudinal Data] --> sg[Spaghetti Graphs] & rc[Representative Curves] & ol[Ordinal Transitions<br>and States]
ev[Events] --> mc[Multi-category Event Charts] & tl[Timelines]
rel[Relationships] --> cm[Graphical Correlation Matrix] & vc[Variable Clustering]


The Hmisc describe function is my main tool for getting initial descriptive statistics and quality controlling the data in a univariate fashion. Here is an example. The Info index is a measure of the information content in a numeric variable relative to the information in a continuous numeric variable with no ties. A very low value of Info will occur when a highly imbalanced variable is binary. Clicking on Glossary on the right will pop up a browser window with a more in-depth glossary of terms used in Hmisc package output. It links to hbiostat.org/R/glossary.html which you can link from your reports that use Hmisc.

Info comes from the approximate formula for the variance of a log odds ratio for a proportional odds model/Wilcoxon test, due to Whitehead.
Glossary

Gmd in the output stands for Gini’s mean difference—the mean absolute difference over all possible pairs of different observations. It is a very interpretable measure of dispersion that is more robust than the standard deviation.

require(Hmisc)
getRs('reptools.r')
hookaddcap()   # make knitr call a function at the end of each chunk
# to try to automatically add to list of figure

getHdata(stressEcho)
d <- stressEcho
# The callout was typed manually; could have run
#  makecnote(~ html(describe(d)), wide=TRUE)
w <- describe(d)
html(w)
d Descriptives
d

31 Variables   558 Observations

bhr: Basal heart rate bpm
nmissingdistinctInfoMeanGmd.05.10.25.50.75.90.95
5580690.99975.2916.57 54.0 58.0 64.0 74.0 84.0 95.3102.0
lowest : 42 44 45 46 47 , highest: 108 115 116 127 210
basebp: Basal blood pressure mmHg
nmissingdistinctInfoMeanGmd.05.10.25.50.75.90.95
5580940.998135.323.35104.0110.0120.0133.0150.0162.3170.1
lowest : 85 88 90 97 98 , highest: 192 194 195 201 203
basedp: Basal Double Product bhr*basebp bpm*mmHg
nmissingdistinctInfoMeanGmd.05.10.25.50.75.90.95
55804411101812813 6607 7200 8400 9792116631361014770
lowest : 5000 5220 5280 5400 5460 , highest: 17604 17710 17748 21082 27300
pkhr: Peak heart rate mmHg
nmissingdistinctInfoMeanGmd.05.10.25.50.75.90.95
55801051120.625.36 81.85 90.70106.25122.00135.00147.00155.15
lowest : 52 61 62 63 66 , highest: 170 171 176 182 210
sbp: Systolic blood pressure mmHg
nmissingdistinctInfoMeanGmd.05.10.25.50.75.90.95
55801421146.940.72 96102120141170200210
lowest : 40 60 70 79 80 , highest: 240 250 274 283 309
dp: Double product pkhr*sbp bpm*mmHg
nmissingdistinctInfoMeanGmd.05.10.25.50.75.90.95
5580508117634576510256113411403317060206442453626637
lowest : 5100 5940 7490 8100 8360 , highest: 32518 33400 33840 38205 45114
dose: Dose of dobutamine given mg
nmissingdistinctInfoMeanGmd
558070.8433.758.334
lowest : 10 15 20 25 30 , highest: 20 25 30 35 40
 Value         10    15    20    25    30    35    40
Frequency      2    28    47    56    64    61   300
Proportion 0.004 0.050 0.084 0.100 0.115 0.109 0.538


maxhr: Maximum heart rate bpm
nmissingdistinctInfoMeanGmd.05.10.25.50.75.90.95
55801031119.424.64 82.0 91.0104.2120.0133.0146.0154.1
lowest : 58 62 63 66 67 , highest: 170 171 176 182 200
pctMphr: Percent maximum predicted heart rate achieved %
nmissingdistinctInfoMeanGmd.05.10.25.50.75.90.95
5580780.99978.5716.86 53 60 69 78 88 97104
lowest : 38 39 40 41 42 , highest: 116 117 126 132 133
mbp: Maximum blood pressure mmHg
nmissingdistinctInfoMeanGmd.05.10.25.50.75.90.95
55801320.99915635.03110.0120.0133.2150.0175.8200.0211.1
lowest : 84 90 92 93 96 , highest: 240 250 274 283 309
dpmaxdo: Double product on max dobutamine dose bpm*mmHg
nmissingdistinctInfoMeanGmd.05.10.25.50.75.90.95
5580484118550538511346128651526018118212392489327477
lowest : 7130 8100 8360 9240 9280 , highest: 32518 33400 33840 38205 45114
dobdose: Dobutamine dose at max double product mg
nmissingdistinctInfoMeanGmd
558080.94130.2410.55
lowest : 5 10 15 20 25 , highest: 20 25 30 35 40
 Value          5    10    15    20    25    30    35    40
Frequency      7     7    55    73    71    78    62   205
Proportion 0.013 0.013 0.099 0.131 0.127 0.140 0.111 0.367


age: Age years
nmissingdistinctInfoMeanGmd.05.10.25.50.75.90.95
5580620.99967.3413.4146.8551.0060.0069.0075.0082.0085.00
lowest : 26 28 29 30 33 , highest: 89 90 91 92 93
gender
nmissingdistinct
55802
 Value        male female
Frequency     220    338
Proportion  0.394  0.606


baseEF: Baseline cardiac ejection fraction %
nmissingdistinctInfoMeanGmd.05.10.25.50.75.90.95
5580540.99455.610.7132405257626566
lowest : 20 21 22 23 25 , highest: 74 75 77 79 83
dobEF: Ejection fraction on dobutamine %
nmissingdistinctInfoMeanGmd.05.10.25.50.75.90.95
5580600.99265.2412.3840.049.762.067.073.076.080.0
lowest : 23 25 26 27 28 , highest: 86 87 89 90 94
chestpain: Chest pain
nmissingdistinctInfoSumMeanGmd
558020.641720.30820.4272

restwma: Resting wall motion abnormality on echocardiogram
nmissingdistinctInfoSumMeanGmd
558020.7452570.46060.4978

posSE: Positive stress echocardiogram
nmissingdistinctInfoSumMeanGmd
558020.5531360.24370.3693

newMI: New myocardial infarction
nmissingdistinctInfoSumMeanGmd
558020.143280.050180.09549

newPTCA: Recent angioplasty
nmissingdistinctInfoSumMeanGmd
558020.138270.048390.09226

newCABG: Recent bypass surgery
nmissingdistinctInfoSumMeanGmd
558020.167330.059140.1115

death
nmissingdistinctInfoSumMeanGmd
558020.123240.043010.08247

hxofHT: History of hypertension
nmissingdistinctInfoSumMeanGmd
558020.6253930.70430.4173

hxofDM: History of diabetes
nmissingdistinctInfoSumMeanGmd
558020.6992060.36920.4666

hxofCig: History of smoking
nmissingdistinct
55803
 Value           heavy   moderate non-smoker
Frequency         122        138        298
Proportion      0.219      0.247      0.534


hxofMI: History of myocardial infarction
nmissingdistinctInfoSumMeanGmd
558020.5991540.2760.4004

hxofPTCA: History of angioplasty
nmissingdistinctInfoSumMeanGmd
558020.204410.073480.1364

hxofCABG: History of coronary artery bypass surgery
nmissingdistinctInfoSumMeanGmd
558020.399880.15770.2661

any.event: Death, newMI, newPTCA, or newCABG
nmissingdistinctInfoSumMeanGmd
558020.402890.15950.2686

ecg: Baseline electrocardiogram diagnosis
nmissingdistinct
55803
 Value         normal equivocal        MI
Frequency        311       176        71
Proportion     0.557     0.315     0.127


# To create a separate browser window:
cat(html(w), file='desc.html')
browseURL('desc.html', browser='firefox -new-window')

Better, whether using RStudio or not:

htmlView(w, contents(d))  # or htmlView(describe(d1), describe(d2), ...)
# Use htmlViewx to use an external browser window (see above)

There is also a plot method for describe output. It produces two graphics objects: one for categorical variables and one for continuous variables. The default is to use ggplot2 to produce static graphics. The result can be fed directly into maketabs described earlier. results='asis' must appear in the chunk header.

cap <- 'Regular plot(describe) output'
maketabs(plot(w, bvspace=2.5), basecap=cap, cap=1)

By specifying grType option you can instead get plotly graphics that use hover text to show more information, especially when hovering over the leftmost dot or tick mark for a variable.

options(grType='plotly')
cap <- 'plotly plot(describe) output'
maketabs(plot(w, bvspace=2.5), wide=TRUE, cap=1, basecap=cap)

See this for other Hmisc functions for descriptive graphics and tables, especially for stratified descriptive statistics for categorical variables. The summaryM function prints a tabular summary of a mix of continuous and categorical variables. Here is an example where stratification is by history of myocardial infarction (MI).

require(data.table)
setDT(d)   # turn d into a data table
# tables() with no arguments will give a concise summary of all active data tables
w <- d
w[, hxofMI := factor(hxofMI, 0 : 1, c('No history of MI', 'History of MI'))]
vars <- setdiff(names(d), 'hxofMI')
form <- as.formula(paste(paste(vars, collapse='+'), '~ hxofMI'))
print(form)

bhr + basebp + basedp + pkhr + sbp + dp + dose + maxhr + pctMphr + mbp + dpmaxdo + dobdose + age + gender + baseEF + dobEF + chestpain + restwma + posSE + newMI + newPTCA + newCABG + death + hxofHT + hxofDM + hxofCig + hxofPTCA + hxofCABG + any.event + ecg ~ hxofMI

s <- summaryM(form, data=d, test=TRUE)
# Note: there is a problem with the width of the categorical variable
# plot.  Neither fig.size() nor options(plotlyauto=FALSE) fixed it.
source('~/R/rscripts/reptools.r')
maketabs(
                          ~  ,   # empty tab
Table 1                   ~ html(s, exclude1=TRUE, npct='both', digits=3, middle.bold=TRUE),
Categorical Variable Plot ~ plot(s, which='categorical', vars=1 : 4) +
caption('summaryM plots') +
fig.size(width=6),
Continuous Variable Plot  ~ plot(s, which='continuous',  vars=1 : 4),
wide=TRUE)
 No history of MIN=404 History of MIN=154 Test Statistic Descriptive Statistics (N=558). Basal heart ratebpm 65 74 85 63 72 84 F1 556=1.41, P=0.2351 Basal blood pressuremmHg 120 134 150 120 130 150 F1 556=1.39, P=0.2381 Basal Double Product bhr*basebpbpm*mmHg 8514 9874 11766 8026 9548 11297 F1 556=3.32, P=0.0691 Peak heart ratemmHg 107 123 136 104 120 132 F1 556=2.35, P=0.1261 Systolic blood pressuremmHg 124 146 174 115 134 158 F1 556=12.1, P<0.0011 Double product pkhr*sbpbpm*mmHg 14520 17783 21116 13198 15539 18885 F1 556=15, P<0.0011 Dose of dobutamine givenmg : 10 0.00 2⁄404 0.00 0⁄154 χ26=8.77, P=0.1872 15 0.05 21⁄404 0.05 7⁄154 20 0.10 40⁄404 0.05 7⁄154 25 0.11 45⁄404 0.07 11⁄154 30 0.11 43⁄404 0.14 21⁄154 35 0.11 45⁄404 0.10 16⁄154 40 0.51 208⁄404 0.60 92⁄154 Maximum heart ratebpm 107 122 134 102 118 130 F1 556=4.05, P=0.0451 Percent maximum predicted heart rate achieved% 69.0 78.0 89.0 70.0 77.0 87.5 F1 556=0.5, P=0.4791 Maximum blood pressuremmHg 138 154 180 130 142 162 F1 556=13, P<0.0011 Double product on max dobutamine dosebpm*mmHg 15654 18666 21664 14489 16785 19680 F1 556=16.1, P<0.0011 Dobutamine dose at max double productmg : 5 0.01 4⁄404 0.02 3⁄154 χ27=8.5, P=0.292 10 0.01 6⁄404 0.01 1⁄154 15 0.11 43⁄404 0.08 12⁄154 20 0.14 58⁄404 0.10 15⁄154 25 0.13 54⁄404 0.11 17⁄154 30 0.13 51⁄404 0.18 27⁄154 35 0.10 40⁄404 0.14 22⁄154 40 0.37 148⁄404 0.37 57⁄154 Ageyears 59.0 68.0 75.0 63.2 71.0 76.8 F1 556=9.75, P=0.0021 gender : female 0.68 273⁄404 0.42 65⁄154 χ21=30, P<0.0012 Baseline cardiac ejection fraction% 55 59 63 40 54 60 F1 556=56.4, P<0.0011 Ejection fraction on dobutamine% 65.0 70.0 74.2 50.0 64.5 70.0 F1 556=50.3, P<0.0011 Chest pain 0.29 119⁄404 0.34 53⁄154 χ21=1.29, P=0.2572 Resting wall motion abnormality on echocardiogram 0.57 230⁄404 0.18 27⁄154 χ21=69.7, P<0.0012 Positive stress echocardiogram 0.21 86⁄404 0.32 50⁄154 χ21=7.56, P=0.0062 New myocardial infarction 0.03 14⁄404 0.09 14⁄154 χ21=7.4, P=0.0072 Recent angioplasty 0.02 10⁄404 0.11 17⁄154 χ21=17.8, P<0.0012 Recent bypass surgery 0.05 21⁄404 0.08 12⁄154 χ21=1.35, P=0.2462 death 0.04 15⁄404 0.06 9⁄154 χ21=1.23, P=0.2672 History of hypertension 0.69 280⁄404 0.73 113⁄154 χ21=0.89, P=0.3462 History of diabetes 0.36 147⁄404 0.38 59⁄154 χ21=0.18, P=0.6742 History of smoking : heavy 0.21 83⁄404 0.25 39⁄154 χ22=3.16, P=0.2062 moderate 0.24 96⁄404 0.27 42⁄154 non-smoker 0.56 225⁄404 0.47 73⁄154 History of angioplasty 0.04 15⁄404 0.17 26⁄154 χ21=28.4, P<0.0012 History of coronary artery bypass surgery 0.08 34⁄404 0.35 54⁄154 χ21=59.6, P<0.0012 Death, newMI, newPTCA, or newCABG 0.12 48⁄404 0.27 41⁄154 χ21=18.1, P<0.0012 Baseline electrocardiogram diagnosis : normal 0.59 240⁄404 0.46 71⁄154 χ22=8, P=0.0182 equivocal 0.29 117⁄404 0.38 59⁄154 MI 0.12 47⁄404 0.16 24⁄154 a b c represent the lower quartile a, the median b, and the upper quartile c for continuous variables. Tests used: 1Wilcoxon test; 2Pearson test .

Semi-interactive stratified spike histograms are also useful descriptive plots. These plots also contain a superset of the quantiles used in box plots, and the legend is clickable, allowing any of the statistical summaries to be turned off.

d[, histboxp(x=maxhr, group=ecg, bins=200)]

9.1 Longitudinal Continuous Y

For a continuous response variable measured longitudinally, two of the ways to display the data are

• “spaghetti plots” showing all data for all individual subjects, if the number of subjects is not very large
• finding clusters of individual subject curve characteristics and plotting a random sample of raw data curves within each cluster if the sample size is large

The Hmisc package curveRep function facilitates the latter approach using representative curves. It considers per-subject sample sizes and measurement time gaps as these will not only limit how we look at the data but may be informative, e.g., subjects who are failing may start to get more frequent measurements.

To demonstrate we simulate data in the following way.

• Simulate 200 subjects (curves) with per-curve sample sizes ranging from 1 to 10
• Make curves with odd-numbered IDs have a measurement time distribution that is random uniform [0,1] and those with even-numbered IDs have a time distribution that is half as wide but still centered at 0.5. Shift y values higher with increasing IDs
• Make 1/3 of subjects have a flat trajectory, 1/3 linear and not flat, and 1/3 quadratic

Request curveRep to cluster the 200 curves on the following characteristics:

• kn=3 sample size groups (curveRep actually used one more than this)
• kxdist=2 time point distribution groups based here on the earliest and latest measurement times and the longest gap between any two measurements within a subject
• k=4 trajectory clusters. Trajectories are determined by loess-smoothing each subject’s curve and linearly interpolating the estimates to the same evenly-spaced grid of time points, then clustering on these 10 y-estimates. This captures intercepts, slopes, and shapes.

All subjects within each cluster are shown; we didn’t need to take random samples for 200 subjects. Results for the 4 per-curve sample size groups are placed in separate tabs.

set.seed(1)
N <- 200
nc <- sample(1:10, N, TRUE)
id <- rep(1:N, nc)
x  <- y <- id
# Solve for coefficients of quadratic function such that it agrees with
# the linear function at x=0.25, 0.75 and is lower by -delta at x=0.5
delta    <- -3
cof      <- list(c(0, 0, 0), c(0, 10, 0), c(delta, 10, -2 * delta / (2 * 0.25^2)))

for(i in 1 : N) {
x[id==i] <- if(i %% 2) runif(nc[i]) else runif(nc[i], c(.25, .75))
shape    <- sample(1:3, 1)
xc       <- x[id == i] - 0.5
k        <- cof[[shape]]
y[id == i] <- i/20 + k[1] + k[2] * xc + k[3] * xc ^ 2 +
runif(nc[i], -2.5, 2.5)
}
require(cluster)
w   <- curveRep(x, y, id, kn=3, kxdist=2, k=4, p=10)
gg  <- vector('list', 4)
nam <- rep('', 4)
for(i in 1 : 4) {
z <- plot(w, i, method='data')  # method='data' available in Hmisc 4.7-1
z <- transform(z,
distribution = paste('Time Distribution:', distribution),
cluster      = paste('Trajectory Cluster:', cluster))
gg[[i]] <-
if(i == 1)
ggplot(z, aes(x, y, color=curve)) + geom_point() +
facet_grid(distribution ~ cluster) +
theme(legend.position='none')
else
ggplot(z, aes(x, y, color=curve)) + geom_line() +
facet_grid(distribution ~ cluster) +
theme(legend.position='none')
nam[i] <- z\$ninterval[1]
}
names(gg) <- nam
maketabs(gg, basecap='Representative curves determined by curveRep stratified by per-subject sample size ranges', cap=1)

9.2 Longitudinal Ordinal Y

Continuous longitudinal Y may be analyzed flexibly using semiparametric models while being described using representative curves as just discussed. Now suppose that Y is discrete. There are three primary ways of describing such data graphically.

1. Show event occurrences/trajectories for a random sample of subjects (Hmisc::multEventChart)
2. Show all transition proportions if time is discrete (Hmisc::propsTrans)
3. Show all state occupancy proportions if time is discrete (Hmisc::propsPO)
Thanks to Lucy D’Agostino McGowan for writing most of the code for multEventChart

Starting with a multiple event chart, simulate data on 5 patients, then display all the raw data.

pt1 <- data.frame(pt=1, day=0:3,
status=.q(well, well, sick, 'very sick'))
pt2 <- data.frame(pt=2, day=c(1,2,4,6),
status=.q(sick, 'very sick', coma, death))
pt3 <- data.frame(pt=3, day=1:5,
status=.q(sick, 'very sick', sick, 'very sick', 'discharged'))
pt4 <- data.frame(pt=4, day=c(1:4, 10),
status=.q(well, sick, 'very sick', well, discharged))
d <- rbind(pt1, pt2, pt3, pt4)
d <- upData(d,
status = factor(status, .q(discharged, well, sick,
'very sick', coma, death)),
labels = c(day = 'Day'), print=FALSE)
kabl(pt1, pt2, pt3, pt4)
multEventChart(status ~ day + pt, data=d,
absorb=.q(death, discharged),
colorTitle='Status', sortbylast=TRUE) +
theme_classic() + theme(legend.position='bottom')

For an example of displaying transition proportions, whereby all the outcome information in the raw data is shown, simulate some random data. The result is a plotly graphic over which you hover the pointer to see details. The size of the bubbles is proportional to the proportion in that transition.

set.seed(1)
d <- expand.grid(id=1:30, time=1:7)
setDT(d)   # convert to data.table
d[, sex   := sample(c('female', 'male'), .N, replace=TRUE)]
d[, state := sample(LETTERS[1:4], .N, replace=TRUE)]
ggplotlyr(propsTrans(state ~ time + id, data=d))

The final display doesn’t capture the within-subject correlation as done with the transition proportions, but is the most familiar display for longitudinal ordinal data as it shows proportions in the current states, which are cumulative incidence estimates for absorbing states.

Should absorbing states have occurred in the data you would need to carry these forward to the end of follow-up for propsPO to work properly, even though the real data file would terminate follow-up at an absorbing event.
ggplotlyr(propsPO(state ~ time + sex, data=d))

When there is a potentially large number of event types, such as adverse events (AEs) in a clinical trial, and the event timing is not considered, a dot chart is an excellent way to present the proportion of subjects suffering each type of AE. The AEs can be sorted in descending order by the difference in proportions between treatments, and plotly hover text can display more details. Half-width confidence intervals are used (see Section 15.6). An AE chart is easily produced using the aePlot function in the qreport repository on github. aePlot expects the dataset to have one record per subject per AE, so the dataset itself does not define the proper denominator and this must be specified by the user (see denom below). The color coded needles in the right margin are guideposts to which denominators are being used in the analysis (details are here).

getRs('misc.r',   grepo='qreport')
getRs('aePlot.r', grepo='qreport')
getHdata(aeTestData)  # original data source: HH package
# One record per subject per adverse event

# For this example, the denominators for the two treatments in the
# pop-up needles will be incorrect because the dataset did not have
# subject IDs.

ae <- aeTestData
# qreport requires us to define official clinical trial counts and
# name of treatment variable
denom <- c(enrolled   = 1000,
randomized =  400,
a=212, b=188)

setqreportOption(tx.var='treat', denom=denom)
aePlot(event ~ treat, data=ae, minincidence=.05, size='wide')

Category N Used
enrolled 1000 400
randomized 400 400
a 212 212
b 188 188

9.4 Continuous Event Times

For time-to-event data with possibly multiple types of events, an event chart is a good way to show the raw outcome data for a sample of up to perhaps 40 subjects. The Hmisc package offers the event.chart function written by Jack Lee, Kenneth Hess, and Joel Dubin. Here is an example they provided. Patients are sorted by diagnosis date.

getHdata(cdcaids)
event.chart(cdcaids,
subset.c=.q(infedate, diagdate, dethdate, censdate),
x.lab = 'Observation Dates',
y.lab='Patients',
titl='AIDS Data Calendar Event Chart',
point.pch=c(1,2,15,0), point.cex=c(1,1,0.8,0.8),
legend.plot=TRUE, legend.location='i', legend.cex=0.8,
legend.point.text=.q(transfusion,'AIDS diagnosis',death,censored),
legend.point.at = list(c(7210, 8100), c(35, 27))) 

9.5 Describing Variable Interrelationships

The most basic way to examine interrelationships among variables is to graphically depict a correlation matrix. Below is an example on support using the Spearman’s $$\rho$$ rank correlation coefficient. Another good descriptive analysis to help understand relationships among variables and redundancies/collinearities is variable clustering. Here one clusters on variables instead of observations, and instead of a distance matrix we have a similarly matrix. A good default similarity matrix is based on the square of $$\rho$$. In order to restrict ourselves to unsupervised learning, also called data reduction, we restrict attention to non-outcome variables in both displays. The vClus function in reptools runs the dataset, after excluding some variables, through the Hmisc dataframeReduce function to eliminate variables that are missing more than 0.4 of the time and to ignore character or factor variables having more than 10 levels. Binary variables having prevalence < 0.05 are dropped, and categorical variables having < 0.05 of their non-missing values in a category will have such low-frequency categories combined into an “other” category for purposes of computing all the correlation coefficients.

Normally one would omit the fracmiss, maxlevels, minprev arguments as the default values are reasonable.
getHdata(support)
outcomes <- .q(slos,   charges, totcst,   totmcst, avtisst,
label='fig-varclus')
The most strongly related variables are competing indicator variables for categories from the same variable, scoma vs. dzgroup “Coma”, and dzgroup “Cancer” vs. dzclass “Cancer”. The first type of relationship generates a strong negative correlation because if you’re in one category you can’t be in the other.