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Wachsmuth et. al (2003) noticed that a loess smooth through Galton's data on heights of mid-parents and their offspring exhibited a slightly non-linear trend, and asked whether this might be due to Galton having pooled the heights of fathers and mothers and sons and daughters in constructing his tables and graphs.

To answer this question, they used analogous data from English families at about the same time, tabulated by Karl Pearson and Alice Lee (1896, 1903), but where the heights of parents and children were each classified by gender of the parent.

1 |

A frequency data frame with 746 observations on the following 6 variables.

`child`

child height in inches, a numeric vector

`parent`

parent height in inches, a numeric vector

`frequency`

a numeric vector

`gp`

a factor with levels

`fd`

`fs`

`md`

`ms`

`par`

a factor with levels

`Father`

`Mother`

`chl`

a factor with levels

`Daughter`

`Son`

The variables `gp`

, `par`

and `chl`

are provided to allow stratifying
the data according to the gender of the father/mother and son/daughter.

Pearson, K. and Lee, A. (1896). Mathematical contributions to the theory
of evolution. On telegony in man, etc. *Proceedings of the Royal Society of
London*, 60 , 273-283.

Pearson, K. and Lee, A. (1903).
On the laws of inheritance in man: I. Inheritance of physical characters. *Biometika*, 2(4), 357-462.
(Tables XXII, p. 415; XXV, p. 417; XXVIII, p. 419 and XXXI, p. 421.)

Wachsmuth, A.W., Wilkinson L., Dallal G.E. (2003).
Galton's bend: A previously undiscovered nonlinearity in Galton's family stature regression data.
*The American Statistician*, 57, 190-192.
http://www.cs.uic.edu/~wilkinson/Publications/galton.pdf

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 | ```
data(PearsonLee)
str(PearsonLee)
with(PearsonLee,
{
lim <- c(55,80)
xv <- seq(55,80, .5)
sunflowerplot(parent,child, number=frequency, xlim=lim, ylim=lim, seg.col="gray", size=.1)
abline(lm(child ~ parent, weights=frequency), col="blue", lwd=2)
lines(xv, predict(loess(child ~ parent, weights=frequency), data.frame(parent=xv)),
col="blue", lwd=2)
# NB: dataEllipse doesn't take frequency into account
if(require(car)) {
dataEllipse(parent,child, xlim=lim, ylim=lim, plot.points=FALSE)
}
})
## separate plots for combinations of (chl, par)
# this doesn't quite work, because xyplot can't handle weights
require(lattice)
xyplot(child ~ parent|par+chl, data=PearsonLee, type=c("p", "r", "smooth"), col.line="red")
# Using ggplot [thx: Dennis Murphy]
require(ggplot2)
ggplot(PearsonLee, aes(x = parent, y = child, weight=frequency)) +
geom_point(size = 1.5, position = position_jitter(width = 0.2)) +
geom_smooth(method = lm, aes(weight = PearsonLee$frequency,
colour = 'Linear'), se = FALSE, size = 1.5) +
geom_smooth(aes(weight = PearsonLee$frequency,
colour = 'Loess'), se = FALSE, size = 1.5) +
facet_grid(chl ~ par) +
scale_colour_manual(breaks = c('Linear', 'Loess'),
values = c('green', 'red')) +
theme(legend.position = c(0.14, 0.885),
legend.background = element_rect(fill = 'white'))
# inverse regression, as in Wachmuth et al. (2003)
ggplot(PearsonLee, aes(x = child, y = parent, weight=frequency)) +
geom_point(size = 1.5, position = position_jitter(width = 0.2)) +
geom_smooth(method = lm, aes(weight = PearsonLee$frequency,
colour = 'Linear'), se = FALSE, size = 1.5) +
geom_smooth(aes(weight = PearsonLee$frequency,
colour = 'Loess'), se = FALSE, size = 1.5) +
facet_grid(chl ~ par) +
scale_colour_manual(breaks = c('Linear', 'Loess'),
values = c('green', 'red')) +
theme(legend.position = c(0.14, 0.885),
legend.background = element_rect(fill = 'white'))
``` |

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