Introduction to Categorical Data Analysis

11.2 R. A. FISHER’S CONTRIBUTIONS 327
introduced degrees of freedom to characterize the family of chi-squared distributions.
Fisher claimed that, for tests of independence in I × J tables, X 2 had df = (I − 1)
(J − 1). By contrast, in 1900 Pearson had argued that, for any application of his
statistic, df equalled the number of cells minus 1, or IJ − 1 for two-way tables.
Fisher pointed out, however, that estimating hypothesized cell probabilities using
estimated row and column probabilities resulted in an additional (I − 1) + (J − 1)
constraints on the fitted values, thus affecting the distribution of X 2 .
Not surprisingly, Pearson reacted critically to Fisher’s suggestion that his formula
for df was incorrect. He stated
I hold that such a view [Fisher’s] is entirely erroneous, and that the writer has done no
service to the science of statistics by giving it broad-cast circulation in the pages of the
Journal of the Royal Statistical Society....I trust my critic will pardon me for comparing
him with Don Quixote tilting at the windmill; he must either destroy himself, or the whole
theory of probable errors, for they are invariably based on using sample values for those of
the sampled population unknown to us.
Pearson claimed that using row and column sample proportions to estimate unknown
probabilities had negligible effect on large-sample distributions. Fisher was unable to
get his rebuttal published by the Royal Statistical Society, and he ultimately resigned
his membership.
Statisticians soon realized that Fisher was correct. For example, in an article in
1926, Fisher provided empirical evidence to support his claim. Using 11,688 2×2
tables randomly generated by Karl Pearson’s son, E. S. Pearson, he found a sample
mean of X 2 for these tables of 1.00001; this is much closer to the 1.0 predicted by
his formula for E(X 2 )ofdf = (I − 1)(J − 1) = 1 than Pearson’s IJ − 1 = 3. Fisher
maintained much bitterness over Pearson’s reaction to his work. In a later volume of
his collected works, writing about Pearson, he stated “If peevish intolerance of free
opinion in others is a sign of senility, it is one which he had developed at an early age.”
Fisher also made good use of CDA methods in his applied work. For example, he
was also a famed geneticist. In one article, Fisher used Pearson’s goodness-of-fit test
to test Mendel’s theories of natural inheritance. Calculating a summary P -value from
the results of several of Mendel’s experiments, he obtained an unusually large value
(P = 0.99996) for the right-tail probability of the reference chi-squared distribution.
In other words X 2 was so small that the fit seemed too good, leading Fisher in 1936
to comment “the general level of agreement between Mendel’s expectations and his
reported results shows that it is closer than would be expected in the best of several
thousand repetitions. ... I have no doubt that Mendel was deceived by a gardening
assistant, who knew only too well what his principal expected from each trial made.”
In a letter written at the time, he stated “Now, when data have been faked, I know very
well how generally people underestimate the frequency of wide chance deviations,
so that the tendency is always to make them agree too well with expectations.”
In 1934 the fifth edition of Fisher’s classic text Statistical Methods for Research
Workers introduced “Fisher’s exact test” for 2 × 2 contingency tables. In his 1935
book The Design of Experiments, Fisher described the tea-tasting experiment