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 |

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.

`independence_test`

in the `coin`

package, a general framework for conditional inference procedures
(permutation tests)

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 | ```
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)
``` |

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