| Macdonell | R Documentation |
In the second issue of Biometrika, W. R. Macdonell (1902) published an extensive
paper, On Criminal Anthropometry and the Identification of Criminals
in which he included numerous tables of physical characteristics 3000 non-habitual
male criminals serving their sentences in England and Wales. His Table III (p. 216)
recorded a bivariate frequency distribution of height by finger length.
His main purpose was to show that Scotland Yard
could have indexed their material more efficiently, and find a given profile more quickly.
W. S. Gosset (aka "Student") used these data in two classic papers in 1908, in which he derived various characteristics of the sampling distributions of the mean, standard deviation and Pearson's r. He said, "Before I had succeeded in solving my problem analytically, I had endeavoured to do so empirically." Among his experiments, he randomly shuffled the 3000 observations from Macdonell's table, and then grouped them into samples of size 4, 8, ..., calculating the sample means, standard deviations and correlations for each sample.
data(Macdonell) data(MacdonellDF)
Macdonell: A frequency data frame with 924 observations on the following 3 variables giving
the bivariate frequency distribution of height and finger.
heightlower class boundaries of height, in decimal ft.
fingerlength of the left middle finger, in mm.
frequencyfrequency of this combination of height and finger
MacdonellDF: A data frame with 3000 observations on the following 2 variables.
heighta numeric vector
fingera numeric vector
Class intervals for height in Macdonell's table were given in 1 in.
ranges, from (4' 7" 9/16 - 4' 8" 9/16), to (6' 4" 9/16 - 6' 5" 9/16).
The values of height are taken as the lower class boundaries.
For convenience, the data frame MacdonellDF presents the same data, in expanded form, with
each combination of height and finger replicated frequency times.
Macdonell, W. R. (1902). On Criminal Anthropometry and the Identification of Criminals. Biometrika, 1(2), 177-227. doi:10.1093/biomet/1.2.177 http://www.jstor.org/stable/2331487
The data used here were obtained from:
Hanley, J. (2008). Macdonell data used by Student. http://www.medicine.mcgill.ca/epidemiology/hanley/Student/
Hanley, J. and Julien, M. and Moodie, E. (2008). Student's z, t, and s: What if Gosset had R? The American Statistican, 62(1), 64-69.
Gosett, W. S. [Student] (1908). Probable error of a mean. Biometrika, 6(1), 1-25. http://www.york.ac.uk/depts/maths/histstat/student.pdf
Gosett, W. S. [Student] (1908). Probable error of a correlation coefficient. Biometrika, 6, 302-310.
data(Macdonell)
# display the frequency table
xtabs(frequency ~ finger+round(height,3), data=Macdonell)
## Some examples by james.hanley@mcgill.ca October 16, 2011
## http://www.biostat.mcgill.ca/hanley/
## See: http://www.biostat.mcgill.ca/hanley/Student/
###############################################
## naive contour plots of height and finger ##
###############################################
# make a 22 x 42 table
attach(Macdonell)
ht <- unique(height)
fi <- unique(finger)
fr <- t(matrix(frequency, nrow=42))
detach(Macdonell)
dev.new(width=10, height=5) # make plot double wide
op <- par(mfrow=c(1,2),mar=c(0.5,0.5,0.5,0.5),oma=c(2,2,0,0))
dx <- 0.5/12
dy <- 0.5/12
plot(ht,ht,xlim=c(min(ht)-dx,max(ht)+dx),
ylim=c(min(fi)-dy,max(fi)+dy), xlab="", ylab="", type="n" )
# unpack 3000 heights while looping though the frequencies
heights <- c()
for(i in 1:22) {
for (j in 1:42) {
f <- fr[i,j]
if(f>0) heights <- c(heights,rep(ht[i],f))
if(f>0) text(ht[i], fi[j], toString(f), cex=0.4, col="grey40" )
}
}
text(4.65,13.5, "Finger length (cm)",adj=c(0,1), col="black") ;
text(5.75,9.5, "Height (feet)", adj=c(0,1), col="black") ;
text(6.1,11, "Observed bin\nfrequencies", adj=c(0.5,1), col="grey40",cex=0.85) ;
# crude countour plot
contour(ht, fi, fr, add=TRUE, drawlabels=FALSE, col="grey60")
# smoother contour plot (Galton smoothed 2-D frequencies this way)
# [Galton had experience with plotting isobars for meteorological data]
# it was the smoothed plot that made him remember his 'conic sections'
# and ask a mathematician to work out for him the iso-density
# contours of a bivariate Gaussian distribution...
dx <- 0.5/12; dy <- 0.05 ; # shifts caused by averaging
plot(ht,ht,xlim=c(min(ht),max(ht)),ylim=c(min(fi),max(fi)), xlab="", ylab="", type="n" )
sm.fr <- matrix(rep(0,21*41),nrow <- 21)
for(i in 1:21) {
for (j in 1:41) {
smooth.freq <- (1/4) * sum( fr[i:(i+1), j:(j+1)] )
sm.fr[i,j] <- smooth.freq
if(smooth.freq > 0 )
text(ht[i]+dx, fi[j]+dy, sub("^0.", ".",toString(smooth.freq)), cex=0.4, col="grey40" )
}
}
contour(ht[1:21]+dx, fi[1:41]+dy, sm.fr, add=TRUE, drawlabels=FALSE, col="grey60")
text(6.05,11, "Smoothed bin\nfrequencies", adj=c(0.5,1), col="grey40", cex=0.85) ;
par(op)
dev.new() # new default device
#######################################
## bivariate kernel density estimate
#######################################
if(require(KernSmooth)) {
MDest <- bkde2D(MacdonellDF, bandwidth=c(1/8, 1/8))
contour(x=MDest$x1, y=MDest$x2, z=MDest$fhat,
xlab="Height (feet)", ylab="Finger length (cm)", col="red", lwd=2)
with(MacdonellDF, points(jitter(height), jitter(finger), cex=0.5))
}
#############################################################
## sunflower plot of height and finger with data ellipses ##
#############################################################
with(MacdonellDF,
{
sunflowerplot(height, finger, size=1/12, seg.col="green3",
xlab="Height (feet)", ylab="Finger length (cm)")
reg <- lm(finger ~ height)
abline(reg, lwd=2)
if(require(car)) {
dataEllipse(height, finger, plot.points=FALSE, levels=c(.40, .68, .95))
}
})
############################################################
## Sampling distributions of sample sd (s) and z=(ybar-mu)/s
############################################################
# note that Gosset used a divisor of n (not n-1) to get the sd.
# He also used Sheppard's correction for the 'binning' or grouping.
# with concatenated height measurements...
mu <- mean(heights) ; sigma <- sqrt( 3000 * var(heights)/2999 )
c(mu,sigma)
# 750 samples of size n=4 (as Gosset did)
# see Student's z, t, and s: What if Gosset had R?
# [Hanley J, Julien M, and Moodie E. The American Statistician, February 2008]
# see also the photographs from Student's notebook ('Original small sample data and notes")
# under the link "Gosset' 750 samples of size n=4"
# on website http://www.biostat.mcgill.ca/hanley/Student/
# and while there, look at the cover of the Notebook containing his yeast-cell counts
# http://www.medicine.mcgill.ca/epidemiology/hanley/Student/750samplesOf4/Covers.JPG
# (Biometrika 1907) and decide for yourself why Gosset, when forced to write under a
# pen-name, might have taken the name he did!
# PS: Can you figure out what the 750 pairs of numbers signify?
# hint: look again at the numbers of rows and columns in Macdonell's (frequency) Table III.
n <- 4
Nsamples <- 750
y.bar.values <- s.over.sigma.values <- z.values <- c()
for (samp in 1:Nsamples) {
y <- sample(heights,n)
y.bar <- mean(y)
s <- sqrt( (n/(n-1))*var(y) )
z <- (y.bar-mu)/s
y.bar.values <- c(y.bar.values,y.bar)
s.over.sigma.values <- c(s.over.sigma.values,s/sigma)
z.values <- c(z.values,z)
}
op <- par(mfrow=c(2,2),mar=c(2.5,2.5,2.5,2.5),oma=c(2,2,0,0))
# sampling distributions
hist(heights,breaks=seq(4.5,6.5,1/12), main="Histogram of heights (N=3000)")
hist(y.bar.values, main=paste("Histogram of y.bar (n=",n,")",sep=""))
hist(s.over.sigma.values,breaks=seq(0,4,0.1),
main=paste("Histogram of s/sigma (n=",n,")",sep=""));
z=seq(-5,5,0.25)+0.125
hist(z.values,breaks=z-0.125, main="Histogram of z=(ybar-mu)/s")
# theoretical
lines(z, 750*0.25*sqrt(n-1)*dt(sqrt(n-1)*z,3), col="red", lwd=1)
par(op)
#####################################################
## Chisquare probability plot for bivariate normality
#####################################################
mu <- colMeans(MacdonellDF)
sigma <- var(MacdonellDF)
Dsq <- mahalanobis(MacdonellDF, mu, sigma)
Q <- qchisq(1:3000/3000, 2)
plot(Q, sort(Dsq), xlab="Chisquare (2) quantile", ylab="Squared distance")
abline(a=0, b=1, col="red", lwd=2)