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] & mlong[Multiple<br>Longitudinal<br>Variables]
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)
options(prType='html')   # have certain Hmisc functions render in html
require(qreport)
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(~ describe(d), wide=TRUE)
w <- describe(d)
w
d Descriptives
d

31 Variables   558 Observations

bhr: Basal heart rate bpm
image
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
image
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
image
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
image
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
image
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
image
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
image
nmissingdistinctInfoMeanGmd
558070.8433.758.334
 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 
For the frequency table, variable is rounded to the nearest 0
maxhr: Maximum heart rate bpm
image
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 %
image
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
image
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
image
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
image
nmissingdistinctInfoMeanGmd
558080.94130.2410.55
 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 
For the frequency table, variable is rounded to the nearest 0
age: Age years
image
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 %
image
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 %
image
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
image
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
image
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)

Or just type contents(d) with options(prType='html') in effect. This will launch a browser window if you are not in RStudio.

Nicer output from describe is obtained by separating categorical and continuous variables. The special print method used here, with options(prType='html') in effect, creates interactive sparklines so that variable values and frequencies can be looked up in spike histograms without using plotly graphics but using more basic javascript. Hover over the lowest and highest points on the histograms to see the 5 most extreme data values. Note that to initialize javascript jQuery dependencies we must include sparkline::sparkline(0) somewhere in the report before creating the first sparkline output. The qreport maketabs function is helpful here.

sparkline::sparkline(0)
maketabs(print(w, 'both'), wide=TRUE, initblank=TRUE)
d Descriptives
13 Continous Variables of 31 Variables, 558 Observations
Variable Label Units n Missing Distinct Info Mean Gini |Δ| Quantiles
.05 .10 .25 .50 .75 .90 .95
bhr Basal heart rate bpm 558 0 69 0.999 75.29 16.57 54.0 58.0 64.0 74.0 84.0 95.3 102.0
basebp Basal blood pressure mmHg 558 0 94 0.998 135.3 23.35 104.0 110.0 120.0 133.0 150.0 162.3 170.1
basedp Basal Double Product bhr*basebp bpm*mmHg 558 0 441 1.000 10181 2813 6607 7200 8400 9792 11663 13610 14770
pkhr Peak heart rate mmHg 558 0 105 1.000 120.6 25.36 81.85 90.70 106.25 122.00 135.00 147.00 155.15
sbp Systolic blood pressure mmHg 558 0 142 1.000 146.9 40.72 96 102 120 141 170 200 210
dp Double product pkhr*sbp bpm*mmHg 558 0 508 1.000 17634 5765 10256 11341 14033 17060 20644 24536 26637
maxhr Maximum heart rate bpm 558 0 103 1.000 119.4 24.64 82.0 91.0 104.2 120.0 133.0 146.0 154.1
pctMphr Percent maximum predicted heart rate achieved % 558 0 78 0.999 78.57 16.86 53 60 69 78 88 97 104
mbp Maximum blood pressure mmHg 558 0 132 0.999 156 35.03 110.0 120.0 133.2 150.0 175.8 200.0 211.1
dpmaxdo Double product on max dobutamine dose bpm*mmHg 558 0 484 1.000 18550 5385 11346 12865 15260 18118 21239 24893 27477
age Age years 558 0 62 0.999 67.34 13.41 46.85 51.00 60.00 69.00 75.00 82.00 85.00
baseEF Baseline cardiac ejection fraction % 558 0 54 0.994 55.6 10.71 32 40 52 57 62 65 66
dobEF Ejection fraction on dobutamine % 558 0 60 0.992 65.24 12.38 40.0 49.7 62.0 67.0 73.0 76.0 80.0
d Descriptives
18 Categorical Variables of 31 Variables, 558 Observations
Variable Label Units n Missing Distinct Info Sum Mean Gini |Δ|
dose Dose of dobutamine given mg 558 0 7 0.840
33.75 8.334
dobdose Dobutamine dose at max double product mg 558 0 8 0.941
30.24 10.55
gender 558 0 2



chestpain Chest pain 558 0 2 0.640 172 0.3082 0.4272
restwma Resting wall motion abnormality on echocardiogram 558 0 2 0.745 257 0.4606 0.4978
posSE Positive stress echocardiogram 558 0 2 0.553 136 0.2437 0.3693
newMI New myocardial infarction 558 0 2 0.143 28 0.05018 0.09549
newPTCA Recent angioplasty 558 0 2 0.138 27 0.04839 0.09226
newCABG Recent bypass surgery 558 0 2 0.167 33 0.05914 0.1115
death 558 0 2 0.123 24 0.04301 0.08247
hxofHT History of hypertension 558 0 2 0.625 393 0.7043 0.4173
hxofDM History of diabetes 558 0 2 0.699 206 0.3692 0.4666
hxofCig History of smoking 558 0 3



hxofMI History of myocardial infarction 558 0 2 0.599 154 0.276 0.4004
hxofPTCA History of angioplasty 558 0 2 0.204 41 0.07348 0.1364
hxofCABG History of coronary artery bypass surgery 558 0 2 0.399 88 0.1577 0.2661
any.event Death, newMI, newPTCA, or newCABG 558 0 2 0.402 89 0.1595 0.2686
ecg Baseline electrocardiogram diagnosis 558 0 3



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)
Figure 9.1: Regular plot(describe) output

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)

plotly plot(describe) output

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.
maketabs(
  ` `                         ~ ` `,   # empty tab
  `Table 1`                   ~ print(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)
Descriptive Statistics (N=558).
No history of MI
N=404
History of MI
N=154
Test Statistic
Basal heart rate
bpm
65 74 85 63 72 84 F1 556=1.41, P=0.2351
Basal blood pressure
mmHg
120 134 150 120 130 150 F1 556=1.39, P=0.2381
Basal Double Product bhr*basebp
bpm*mmHg
8514 9874 11766 8026 9548 11297 F1 556=3.32, P=0.0691
Peak heart rate
mmHg
107 123 136 104 120 132 F1 556=2.35, P=0.1261
Systolic blood pressure
mmHg
124 146 174 115 134 158 F1 556=12.1, P<0.0011
Double product pkhr*sbp
bpm*mmHg
14520 17783 21116 13198 15539 18885 F1 556=15, P<0.0011
Dose of dobutamine given
mg
: 10
0.00 2404 0.00 0154 χ26=8.77, P=0.1872
  15 0.05 21404 0.05 7154
  20 0.10 40404 0.05 7154
  25 0.11 45404 0.07 11154
  30 0.11 43404 0.14 21154
  35 0.11 45404 0.10 16154
  40 0.51 208404 0.60 92154
Maximum heart rate
bpm
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 pressure
mmHg
138 154 180 130 142 162 F1 556=13, P<0.0011
Double product on max dobutamine dose
bpm*mmHg
15654 18666 21664 14489 16785 19680 F1 556=16.1, P<0.0011
Dobutamine dose at max double product
mg
: 5
0.01 4404 0.02 3154 χ27=8.5, P=0.292
  10 0.01 6404 0.01 1154
  15 0.11 43404 0.08 12154
  20 0.14 58404 0.10 15154
  25 0.13 54404 0.11 17154
  30 0.13 51404 0.18 27154
  35 0.10 40404 0.14 22154
  40 0.37 148404 0.37 57154
Age
years
59.0 68.0 75.0 63.2 71.0 76.8 F1 556=9.75, P=0.0021
gender : female 0.68 273404 0.42 65154 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 119404 0.34 53154 χ21=1.29, P=0.2572
Resting wall motion abnormality on echocardiogram 0.57 230404 0.18 27154 χ21=69.7, P<0.0012
Positive stress echocardiogram 0.21 86404 0.32 50154 χ21=7.56, P=0.0062
New myocardial infarction 0.03 14404 0.09 14154 χ21=7.4, P=0.0072
Recent angioplasty 0.02 10404 0.11 17154 χ21=17.8, P<0.0012
Recent bypass surgery 0.05 21404 0.08 12154 χ21=1.35, P=0.2462
death 0.04 15404 0.06 9154 χ21=1.23, P=0.2672
History of hypertension 0.69 280404 0.73 113154 χ21=0.89, P=0.3462
History of diabetes 0.36 147404 0.38 59154 χ21=0.18, P=0.6742
History of smoking : heavy 0.21 83404 0.25 39154 χ22=3.16, P=0.2062
  moderate 0.24 96404 0.27 42154
  non-smoker 0.56 225404 0.47 73154
History of angioplasty 0.04 15404 0.17 26154 χ21=28.4, P<0.0012
History of coronary artery bypass surgery 0.08 34404 0.35 54154 χ21=59.6, P<0.0012
Death, newMI, newPTCA, or newCABG 0.12 48404 0.27 41154 χ21=18.1, P<0.0012
Baseline electrocardiogram diagnosis : normal 0.59 240404 0.46 71154 χ22=8, P=0.0182
  equivocal 0.29 117404 0.38 59154
  MI 0.12 47404 0.16 24154
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 .

summaryM plots

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)]
Figure 9.2: Spike histograms of heart rate stratified by ECG category

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)
Figure 9.3: Representative curves determined by curveRep stratified by per-subject sample size ranges

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)
pt day status
1 0 well
1 1 well
1 2 sick
1 3 very sick
pt day status
2 1 sick
2 2 very sick
2 4 coma
2 6 death
pt day status
3 1 sick
3 2 very sick
3 3 sick
3 4 very sick
3 5 discharged
pt day status
4 1 well
4 2 sick
4 3 very sick
4 4 well
4 10 discharged
Figure 9.4: Multi-event chart for 4 patients
multEventChart(status ~ day + pt, data=d,
               absorb=.q(death, discharged),
               colorTitle='Status', sortbylast=TRUE) +
  theme_classic() + theme(legend.position='bottom')
Figure 9.5: Multi-event chart for 4 patients

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))
Figure 9.6: State transition proportions

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))
Figure 9.7: State occupancy proportions by time and male/female

9.3 Multiple Longitudinal Continuous Variables

When there are multiple longitudinal measurements, exploratory or descriptive analysis is more challenging. One can learn how the different longitudinal processes relate to each other by computing pairwise cross-correlations, time-stratified variable clustering, a matrix of pairwise correlation time-trends, and, in what may be called a third-order analysis, a thermometer-plot matrix of pairwise correlation coefficients where each correlation coefficient quantifies the correlation between time and the correlation coefficient between the indicated pair of variables.

Consider a longitudinal dataset of participants in a clinical trial safety study that was described here. The dataset is in the Vanderbilt Biostatistics data repository and is called safety. Fetch the data, keep only the needed variables, and show descriptive statistics.

getHdata(safety)   # Internal dataset name was All
d <- All[, .q(id, week,
              sbp, dbp, hr, axis, corr.qt, pr, qrs, rr, neutrophils,
              alat, albumin, alk.phos, asat, basophils, bilirubin, bun, 
              chloride, creatinine, eosinophils, ggt, glucose,
              hematocrit, hemoglobin, potassium, lymphocytes,
              monocytes, sodium, platelets, protein, rbc, uric.acid, wbc)]
setDT(d, key=.q(id, week))
sparkline::sparkline(0)
w <- describe(d)
maketabs(print(w, 'both'), wide=TRUE, initblank=TRUE)
d Descriptives
33 Continous Variables of 34 Variables, 16464 Observations
Variable Label Units n Missing Distinct Info Mean Gini |Δ| Quantiles
.05 .10 .25 .50 .75 .90 .95
id 16464 0 2058 1.000 1030 686 103.2 206.0 515.0 1029.5 1544.0 1853.0 1955.8
sbp systolic blood pressure mmHg 12460 4004 114 0.992 131.7 18.2 108 110 120 130 140 152 160
dbp diastolic blood pressure mmHg 12460 4004 79 0.978 78.07 10.51 60 68 70 80 84 90 94
hr heart rate bpm 12458 4006 87 0.996 76.15 12.12 60 63 68 76 83 90 96
axis degree 12356 4108 326 1.000 34.43 52.36 -61 -37 7 48 70 81 88
corr.qt corrected qt msec 12355 4109 1832 1.000 424.2 24.27 397.3 400.6 408.0 419.6 436.4 454.7 467.1
pr msec 12030 4434 121 0.997 158.7 27.08 124.0 132.0 144.0 156.0 172.0 188.0 201.5
qrs msec 12356 4108 70 0.990 92.91 17.09 72 76 84 88 100 112 128
rr msec 12356 4108 97 0.999 837.8 164.3 618.5 659.3 731.7 821.9 937.5 1034.4 1090.9
neutrophils neutrophils absolute 109/L 9079 7385 868 1.000 4.593 1.734 2.53 2.84 3.50 4.33 5.36 6.58 7.54
alat alanine aminotransferase IU/L 9129 7335 113 0.998 18.99 10.76 8 9 12 16 22 31 39
albumin G/L 9118 7346 26 0.982 42.15 2.6 38 39 41 42 44 45 46
alk.phos alkaline phosphatase IU/L 9110 7354 185 1.000 76.3 24.44 46 52 61 73 87 104 118
asat aspartate aminotransferase IU/L 9123 7341 99 0.996 19.78 7.614 12 13 15 18 22 27 33
basophils 109/L 2824 13640 596 0.999 0.03488 0.03131 0.00000 0.00440 0.01420 0.02760 0.04900 0.07500 0.09265
bilirubin total bilirubin UMOL/L 9112 7352 67 0.997 10.5 4.72 5.13 6.00 7.00 10.00 12.00 16.00 19.00
bun blood urea nitrogen MMOL/L 9126 7338 174 1.000 5.776 2.051 3.200 3.600 4.500 5.500 6.783 8.200 9.000
chloride MMOL/L 9167 7297 30 0.990 103.9 3.589 98 100 102 104 106 108 109
creatinine UMOL/L 9122 7342 174 0.999 76.73 21.08 51 54 63 74 87 99 110
eosinophils 109/L 2824 13640 1450 1.000 0.2289 0.1664 0.04805 0.07320 0.11800 0.19090 0.29677 0.43375 0.53868
ggt gamma glutamyl transferase IU/L 9114 7350 240 0.999 36.74 27.8 13 15 19 26 41 68 92
glucose glucose - random MMOL/L 8323 8141 349 1.000 6.055 1.797 4.200 4.500 5.000 5.600 6.384 7.900 9.688
hematocrit % 9129 7335 265 1.000 43.83 4.356 37.5 38.9 41.3 43.9 46.3 48.6 50.1
hemoglobin G/L 9129 7335 183 1.000 146.6 14.35 125.0 130.0 139.0 146.6 154.7 161.9 166.8
potassium MMOL/L 9138 7326 48 0.993 4.516 0.4576 3.9 4.0 4.3 4.5 4.7 5.0 5.2
lymphocytes 109/L 2824 13640 2228 1.000 1.902 0.7091 1.031 1.155 1.452 1.818 2.244 2.738 3.119
monocytes 109/L 2824 13640 1492 1.000 0.4611 0.2053 0.1947 0.2451 0.3355 0.4413 0.5619 0.6911 0.7923
sodium MMOL/L 9163 7301 33 0.985 140.8 3.022 136 137 139 141 143 144 145
platelets 109/L 9124 7340 419 1.000 235.9 69.36 147 164 193 229 270 315 349
protein total protein G/L 9124 7340 42 0.995 70.44 4.963 63 65 67 70 73 76 78
rbc red blood cell count 1012/L 9128 7336 143 0.995 4.673 0.5051 3.9 4.1 4.4 4.7 5.0 5.2 5.4
uric.acid uric acid UMOL/L 9142 7322 571 1.000 323.2 95.77 192.0 219.0 264.0 317.0 376.0 434.2 473.0
wbc white blood cell count 109/L 9129 7335 154 1.000 7.34 2.139 4.7 5.1 6.0 7.1 8.4 9.8 10.9
d Descriptives
1 Categorical Variables of 34 Variables, 16464 Observations
Variable Label n Missing Distinct Info Mean Gini |Δ|
week Week 16464 0 8 0.984 7.875 7.782

Show the frequency distribution of the set of distinct time points per subject.

w <- d[, .(uw=paste(sort(week), collapse=' ')), by=.(id)]
w[, table(uw)]
uw
0 1 2 4 8 12 16 20 
              2058 

Every subject has a record for every week. However there are many NAs for several of the variables.

plot(naclus(d))

Let’s remove the variables that are missing more than 0.8 of the time.

set(d, j=.q(monocytes, lymphocytes, basophils, eosinophils), value=NULL)

9.3.1 Cross-Correlation

Compute all possible cross-correlations. The cross-correlation of two signals quantifies how closely the two signals track over time. It uses time-paired data but does not otherwise use the absolute times or the ordering of times. The dataset is a tall and thin dataset with one row per subject per time and one column per lab measurement, so it is already set up for computing cross-correlations. We just need to compute the correlations separately by subject, then average them over subjects. Spearman rank correlations are used throughout.

# Get variables to correlate
v <- setdiff(names(d), .q(week, id))
# Function to create a matrix of Spearman correlations using
# Hmisc::rcorr
r <- function(u) rcorr(as.matrix(u), type='spearman')$r
# Make a list weith raw data, one element per subject
s <- split(d[, ..v], d$id)
# Make a list of correlation matrices, one per subject
R <- lapply(s, r)
# Combine all these into an array with an extra dimension for subjects
R <- array(unlist(R), dim=c(length(v), length(v), length(R)),
           dimnames=list(v, v, NULL))
# Compute mean (separately by row and col var. combinations) 
# over subjects, ignoring NAs
R <- apply(R, 1:2, mean, na.rm=TRUE)
# Plot the matrix using Hmisc::plotCorrM
plotCorrM(R, xangle=45)[[1]]

The variables that move strongly together over time are WBC and neutrophils, RBC and hemotocrit, RBC and hemoglobin, hematocrit and hemoglobin, no surprises. Let’s see how this compares to the ordinary correlation matrix obtained by pooling all the subjects, and ignoring which measurements came from which subjects.

R <- r(d[, ..v])
plotCorrM(R, xangle=45)[[1]]

9.3.2 Time-Stratified Variable Clustering

Let’s describe how the relationships among variables appear to change over time, if they do. We do this by running variable clustering separately by follow-up time. Ignore week 1 when little data were collected.

par(mfrow=c(4,2))
g <- function(x, wk) {
  plot(varclus(as.matrix(x)))
  title(sub=paste('Week', wk), adj=0)
  invisible()
}
d[week != 1, g(.SD, week), by=.(week), .SDcols=v]
Empty data.table (0 rows and 1 cols): week

9.3.4 Correlation Time Trend Summaries

Take the similarity array just computed, and reduce it by one dimension by computing the linear correlation between time and Spearman’s \(\rho^2\).

k <- dim(s)
r <- matrix(0., nrow=k[1], ncol=k[2],
            dimnames=dimnames(s)[1:2])
for(i in 1 : k[1])
  for(j in 1 : k[2])
    if(i != j) r[i, j] <- cor(wks, s[i, j, ])
plotCorrM(r, xangle=45)[[1]]

The correlation that is changing the most over time is correlation between the following two variables.

v[row(r)[abs(r) == max(abs(r))]]
[1] "wbc"       "platelets"

9.4 Adverse Event Chart

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 qreport. 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).

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')
image

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

Proportion of adverse events by Treatment sorted by descending risk difference. 3 events with less than 0.05 incidence in all groups are not shown (Hyperkalemia, Rash, Weight Decrease).

9.5 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))) 
Figure 9.8: Event chart

9.6 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 qreport 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,
               d.time, death,   hospdead, sfdm2)
vClus(support, exclude=outcomes, corrmatrix=TRUE,
      fracmiss=0.4, maxlevels=10, minprev=0.05,
      label='fig-varclus')
Variables removed or modified
Variable Reason
dzgroup grouped categories
race grouped categories
glucose fraction missing>0.4
bun fraction missing>0.4
urine fraction missing>0.4
adlp fraction missing>0.4
Figure 9.9: Spearman rank correlation matrix. Positive correlations are blue and negative are red.

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.