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In 1876 Charles Darwin reported the results of a series of experiments on “The Effects of Cross- and Self- in the Vegetable Kingdom.” The description of his experiment and the table of data for this problem are based on Fisher’s discussion of “A Historical Experiment on Growth Rate” (Fisher, 1935/1971, Chapter III). The experimental method adopted by Darwin was to pit each self-fertilized plant against a cross-fertilized one under conditions that were as similar as possible for the two plants. Darwin emphasized this similarity by indicating “my crossed and self-fertilised plants… were of exactly the same age, were subjected from first to last to the same conditions, and were descended from the same parents” (as quoted in Fisher, 1935/1971, p. 28). One of the ways Darwin used to equalize conditions for the two members of a pair was to plant them in the same pot. The dependent measure was the height of the plant. (Darwin did not specify when this was measured, other than to say that all plants were of the same age when their height was measured.) Although sample sizes were relatively small, Darwin indicated in his report that the experiment required 11 years to complete. To be certain that his analysis of these valuable data was correct, Darwin requested and obtained statistical consulting from his half-cousin, Francis Galton. Darwin’s data and Gallon’s rearrangements of the data are shown in Table 2.5. Darwin’s paired data are shown in columns II and III, where the reader sees that varying numbers of pairs of plants were put in each pot. For example, there were three pairs in Pot I, five pairs in Pot III, and so on. Galton complained that the data had no “prima facie appearance of regularity.” He attempted to rectify this problem by arranging the data by rank ordering according to heights, first within pots in columns IV and V, and then collapsing across pots in columns VI and VII. Gallon’s differences between the reordered lists are shown in column VIII. a. Criticize Darwin’s experimental design. b. Perform appropriate analyses of these data Perform appropriate analyses of these data. (i) Begin simply. Determine how many of the within-pair differences in heights in the original data of columns II and III favor cross-fertilization. If the cross-fertilization had no effect, how many differences would you expect on the average out of 15 to favor the cross-fertilized member of a pair? Is the observed number of differences favoring cross-fertilization significantly different from what you would expect by chance? (ii) Perform the simplest possible parametric statistical test appropriate for analyzing Darwin’s data. How does the p value for this test compare to that in part (i)? Why is the difference between the p values in this case in the direction it is? (iii) What assumptions are required for your analyses in parts (i) and (ii)? (iv) One could, and Fisher, in fact, did, carry out a randomization test on these data. What assumptions does that test require, and what hypothesis would it test here? c. Criticize Galton’s analysis. How differently would the strength of the evidence have appeared if the data in columns VI and VII had been used for analysis rather than that in columns II and III?
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