ZeaMays: Darwin's Heights of Cross- and Self-fertilized Zea May Pairs
In HistData: Data Sets from the History of Statistics and Data Visualization

Description Usage Format Details Source References See Also Examples

Darwin (1876) studied the growth of pairs of zea may (aka corn) seedlings, one produced by cross-fertilization and the other produced by self-fertilization, but otherwise grown under identical conditions. His goal was to demonstrate the greater vigour of the cross-fertilized plants. The data recorded are the final height (inches, to the nearest 1/8th) of the plants in each pair.

In the Design of Experiments, Fisher (1935) used these data to illustrate a paired t-test (well, a one-sample test on the mean difference, cross - self). Later in the book (section 21), he used this data to illustrate an early example of a non-parametric permutation test, treating each paired difference as having (randomly) either a positive or negative sign.

1	data(ZeaMays)

A data frame with 15 observations on the following 4 variables.

pair: pair number, a numeric vector
pot: pot, a factor with levels 1 2 3 4
cross: height of cross fertilized plant, a numeric vector
self: height of self fertilized plant, a numeric vector
diff: cross - self for each pair

In addition to the standard paired t-test, several types of non-parametric tests can be contemplated:

(a) Permutation test, where the values of, say self are permuted and diff=cross - self is calculated for each permutation. There are 15! permutations, but a reasonably large number of random permutations would suffice. But this doesn't take the paired samples into account.

(b) Permutation test based on assigning each abs(diff) a + or - sign, and calculating the mean(diff). There are 2^{15} such possible values. This is essentially what Fisher proposed. The p-value for the test is the proportion of absolute mean differences under such randomization which exceed the observed mean difference.

(c) Wilcoxon signed rank test: tests the hypothesis that the median signed rank of the diff is zero, or that the distribution of diff is symmetric about 0, vs. a location shifted alternative.

Darwin, C. (1876). The Effect of Cross- and Self-fertilization in the Vegetable Kingdom, 2nd Ed. London: John Murray.

Andrews, D. and Herzberg, A. (1985) Data: a collection of problems from many fields for the student and research worker. New York: Springer. Data retrieved from: https://www.stat.cmu.edu/StatDat/

Fisher, R. A. (1935). The Design of Experiments. London: Oliver & Boyd.

wilcox.test

independence_test in the coin package, a general framework for conditional inference procedures (permutation tests)

data(ZeaMays)

##################################
## Some preliminary exploration ##
##################################
boxplot(ZeaMays[,c("cross", "self")], ylab="Height (in)", xlab="Fertilization")

# examine large individual diff/ces
largediff <- subset(ZeaMays, abs(diff) > 2*sd(abs(diff)))
with(largediff, segments(1, cross, 2, self, col="red"))

# plot cross vs. self.  NB: unusual trend and some unusual points
with(ZeaMays, plot(self, cross, pch=16, cex=1.5))
abline(lm(cross ~ self, data=ZeaMays), col="red", lwd=2)

# pot effects ?
 anova(lm(diff ~ pot, data=ZeaMays))

##############################
## Tests of mean difference ##
##############################
# Wilcoxon signed rank test
# signed ranks:
with(ZeaMays, sign(diff) * rank(abs(diff)))
wilcox.test(ZeaMays$cross, ZeaMays$self, conf.int=TRUE, exact=FALSE)

# t-tests
with(ZeaMays, t.test(cross, self))
with(ZeaMays, t.test(diff))

mean(ZeaMays$diff)
# complete permutation distribution of diff, for all 2^15 ways of assigning
# one value to cross and the other to self (thx: Bert Gunter)
N <- nrow(ZeaMays)
allmeans <- as.matrix(expand.grid(as.data.frame(
                         matrix(rep(c(-1,1),N), nr =2))))  %*% abs(ZeaMays$diff) / N

# upper-tail p-value
sum(allmeans > mean(ZeaMays$diff)) / 2^N
# two-tailed p-value
sum(abs(allmeans) > mean(ZeaMays$diff)) / 2^N

hist(allmeans, breaks=64, xlab="Mean difference, cross-self",
	main="Histogram of all mean differences")
abline(v=c(1, -1)*mean(ZeaMays$diff), col="red", lwd=2, lty=1:2)

plot(density(allmeans), xlab="Mean difference, cross-self",
	main="Density plot of all mean differences")
abline(v=c(1, -1)*mean(ZeaMays$diff), col="red", lwd=2, lty=1:2)