«Abstract. Two measures of association for dichotomous variables, the phi-coeﬃcient and the tetrachoric correlation coeﬃcient, are reviewed and ...»
THE PHI-COEFFICIENT, THE TETRACHORIC CORRELATION
COEFFICIENT, AND THE PEARSON-YULE DEBATE
Abstract. Two measures of association for dichotomous variables, the phi-coeﬃcient
and the tetrachoric correlation coeﬃcient, are reviewed and diﬀerences between the two
are discussed in the context of the famous so-called Pearson-Yule debate, that took place in the early 20th century. The two measures of association are given mathemat- ically rigorous deﬁnitions, their underlying assumptions are formalized, and some key properties are derived. Furthermore, existence of a continuous bijection between the phi-coeﬃcient and the tetrachoric correlation coeﬃcient under given marginal proba- bilities is shown. As a consequence, the tetrachoric correlation coeﬃcient can be com- puted using the assumptions of the phi-coeﬃcient construction, and the phi-coeﬃcient can be computed using the assumptions of the tetrachoric correlation construction.
The eﬀorts lead to an attempt to reconcile the Pearson-Yule debate, showing that the two measures of association are in fact more similar than diﬀerent and that between the two, the choice of measure of association does not carry a substantial impact on the conclusions of the association analysis.
Key words and phrases. Phi-coeﬃcient, Tetrachoric Correlation Coeﬃcient, 2×2 Contingency Tables, Measures of Association, Dichotomous Variables.
Financial support from the Jan Wallander and Tom Hedelius Research Foundation, project P2008- 0102:1, is gratefully acknowledged.
2 JOAKIM EKSTROM(a) Karl Pearson (1857-1936) (b) George Udny Yule (1871-1951) Figure 1. Pearson portrait is from Pearson (1938), and is in the public domain. Yule portrait is from Yule et al. (1971), reproduced with the kind permission of Hodder & Stoughton.
1. Introduction The phi -coeﬃcient and the tetrachoric correlation coeﬃcient are two measures of as- sociation for dichotomous variables. The association between variables is of fundamental interest in most scientiﬁc disciplines, and dichotomous variables occur in a wide range of applications. Consequently, measures of association for dichotomous variables are useful in many situations. For example in medicine, many phenomena can only be reliably measured in terms of dichotomous variables. Another example is psychology, where many conditions only can be reliably measured in terms of, for instance, diagnosed or not diagnosed. Data is often presented in the form of 2 × 2 contingency tables. A his- torically prominent example is Pearson’s smallpox recovery data, see Table 1, studying possible association between vaccination against, and recovery from, smallpox infection.
Another interesting data set is Pearson’s diphtheria recovery data, Table 2, studying possible association between antitoxin serum treatment and recovery from diphtheria.
Measures of association for dichotomous variables is an area that has been studied from the very infancy of modern statistics. One of the ﬁrst scholars to treat the subject was Karl Pearson, one of the fathers of modern statistics. In the 7th article in the seminal series Mathematical contributions to the theory of evolution, Pearson (1900) proposed what later became known as the tetrachoric correlation coeﬃcient, as well as, Pearson would later argue, the phi -coeﬃcient. The fundamental idea of the tetrachoric correlation coeﬃcient is to consider the 2 × 2 contingency table as a double dichotomization of a bivariate standard normal distribution, and then to solve for the parameter such that the volumes of the dichotomized bivariate standard normal distribution equal the joint
THE PHI -COEFFICIENT, THE TETRACHORIC CORRELATION COEF... 3Figure 2. Care at the Hampstead fever hospital, London 1872. One of many hospitals opened for the sick poor by the Metropolitan Asylums Board in the late 19th century. With the kind permission of workhouses.org.uk.
probabilities of the contingency table. The tetrachoric correlation coeﬃcient is then deﬁned as that parameter, which, of course, corresponds to the linear correlation of the bivariate normal distribution.
According to Pearson’s colleague Burton H. Camp (1933), Pearson considered the tetrachoric correlation coeﬃcient as being one of his most important contributions to the theory of statistics, right besides his system of continuous curves, the chi-square test and his contributions to small sample statistics. However, the tetrachoric correlation coeﬃcient suﬀered in popularity because of the diﬃculty in its computation. Throughout his career, Pearson published statistical tables aimed at reducing that diﬃculty (Camp,
1933), reﬂecting an interest in promoting a wider adoption of the tetrachoric correlation coeﬃcient among practitioners.
While the tetrachoric correlation coeﬃcient is the linear correlation of a so-called underlying bivariate normal distribution, the phi -coeﬃcient is the linear correlation of an underlying bivariate discrete distribution. This measure of association was independently proposed by Boas (1909), Pearson (1900), Yule (1912), and possibly others.
The question of whether the underlying bivariate distribution should be considered continuous or discrete is at the core of the so-called Pearson-Yule debate. In the historical context of the Pearson-Yule debate, though, it is important to understand that no one at the time looked upon these two measures of association as the linear correlations of diﬀerent underlying distributions, the framework in which both were presented in the preceding paragraph. On the contrary, according to Yule (1912) the tetrachoric correlation coeﬃcient is founded upon ideas entirely diﬀerent from those of which the phi -coeﬃcient is founded upon. The sentiment is echoed by Pearson & Heron (1913), which even claims that the phi -coeﬃcient is not based on a reasoned theory, while at the same time arguing for the soundness of the tetrachoric correlation coeﬃcient. In fact, the point of view that both measures of association are the linear correlations of underlying distributions is one of the contributions of the present article.
1.1. The Pearson-Yule debate. George Udny Yule, a former student of Pearson, favored the approach of an inherently discrete underlying distribution. Yule (1912) is a comprehensive review of the area of measures of association for dichotomous variables, as well as a response to Heron (1911), and contains blunt criticism of Pearson’s tetrachoric correlation coeﬃcient. Regarding the tetrachoric correlation coeﬃcient’s assumptions of
underlying continuous variables, Yule (1912) reads:
Here, I am concerned rather with the assumptions and their applicability.
[...] Those who are unvaccinated are all equally non-vaccinated, and similarly, all those who have died of small-pox are all equally dead. [...] From
THE PHI -COEFFICIENT, THE TETRACHORIC CORRELATION COEF... 5this standpoint Professor Pearson’s assumptions are quite inapplicable, and do not lead to the true correlation between the attributes. But this is not, apparently, the standpoint taken by Professor Pearson himself.
The example that Yule (1912) referes to is the smallpox recovery data which was prominently featured in Pearson (1900), see Table 1.
Yule (1912) also contains a bibliographical discussion which could be interpreted as a questioning of whether Pearson really is the originator of some of the ideas that Pearson claimed credit for. In all, Pearson quite evidently felt oﬀended by some of Yule’s wordings and was upset by his former student’s publicly expressed, and in Pearson’s opinion uninformed, misgivings about the tetrachoric correlation coeﬃcient. And from there on, it is by most accounts fair to say that the debate lost all proportions.
Pearson & Heron (1913) is a scathing, almost 200 pages long reply. The introduction
The recent paper by Mr Yule calls for an early reply on two grounds, ﬁrst because of its singularly acrimonious tone [...], and secondly because we believe that if Mr Yule’s views are accepted, irreparable damage will be done to the growth of modern statistical theory. Mr Yule has invented a series of methods which are in no case based on a reasoned theory, but which possess the dangerous fascination of easy application [...], and therefore are seized upon by those who are without adequate training in statistical theory.
With regards to the smallpox recovery example, Pearson & Heron (1913) replies:
Recovery and death in cases of small-pox were used to measure a continuous variable - the severity of the attack. [ Moreover, ] vaccination regarded as conferring immunity is an essentially continuous variable.
With respect to Yule’s contrasting view of the dichotomous variables as inherently discrete, while still unidimensional, Pearson & Heron (1913) rhetorically counter-asks:
Does Mr Yule look upon death as the addition of one unit to recovery?
Pearson may also have taken oﬀense at the fact that Yule wrote a review on one of the regarded Professor’s favorite topics. Pearson & Heron (1913) mentions Yule’s statistical textbook on several occasions.
It may be said that a vigorous protest against Mr Yule’s coeﬃcient is unnecessary. We believe on the contrary that, if not made now and made strongly, there will be great set-back to both modern statistical theory and practice. The publication of Mr Yule’s text-book has resuscitated the use of his coeﬃcient of association; it is now being used in all sorts of quarters on all sorts of unsuitable data. The coeﬃcient of association is in our opinion wholly fallacious, it represents no true properties of the actual distribution, and it has no adequate physical interpretation.
¨ 6 JOAKIM EKSTROM The exchange became known as the Pearson-Yule debate. The tone was indeed caustic, many readers likely felt intimidated by the gravity of the accusations, and Camp (1933) acknowledges that it may have contributed to Pearson’s reputation of being unkind.
Though in the end, it is important to point out, Yule wrote Pearson’s obituary for the Royal Society (Yule & Filon, 1936) and according to Kendall (1952), Yule was deeply aﬀected by Pearson’s death.
The unresolved nature of the debate must also have had the negative eﬀect that practitioners and fellow statisticians alike were left in doubt about what measure of association to use in diﬀerent situations. The tone of the debate leaves the reader with the impression that the choice of measure of association almost is a matter of life and death. And that is, of course, not quite the case. In fact, one of the conclusions of the present article is that between the two, the choice does not carry a substantial impact on the conclusions of the association analysis. So quite on the contrary, as it will be seen, practitioners have no reason to be anxious. And neither Pearson nor Yule, as will also be seen, had really any reason to fear for the future of modern statistics.
1.2. Outline of the present article. The core of the Pearson-Yule debate is about the assumptions implied by the two measures of association. In this article, a close look at the two measures of association will be taken and the implied assumptions will be pinpointed and formalized. Pearson & Heron (1913) argued that dichotomous variables should be considered dichotomizations of continuous underlying variables, while Yule (1912) argued that they should be considered inherently discrete. In this article, however, it is shown that under given marginal probabilities there exists a continuous bijection between the two, which moreover has a ﬁxed point at zero for all marginal probabilities. Consequently, both measures of association can be computed equally well no matter whether the variables are considered dichotomizations of continuous variables or not. As long as one of the assumptions is deemed appropriate, it does not make a diﬀerence which one it is. As a consequence, it turns out, whether to use the tetrachoric correlation coeﬃcient or the phi -coeﬃcient is in principle a matter of preference only.
The main result of this article, that there exists a continuous bijection between the phi coeﬃcient and the tetrachoric correlation coeﬃcient under given marginal probabilities, has not been found in the literature. Guilford & Perry (1951) and Perry & Michael (1952) use series expansion of the integral equation of the tetrachoric correlation coeﬃcient to ﬁnd an approximate formula of the tetrachoric correlation coeﬃcient as a function of the phi -coeﬃcient whose errors, according to Perry & Michael, “are negligible for values of [the approximate tetrachoric correlation coeﬃcient] less than |0.35| and probably relatively small for values of [the approximate tetrachoric correlation coeﬃcient] between |0.35| and |0.6|”. Though Guilford & Perry and Perry & Michael consider the relationship phi -coeﬃcient - tetrachoric correlation coeﬃcient, their result does, however, not imply a continuous bijection.
In Section 2, the phi -coeﬃcient and the tetrachoric correlation coeﬃcient are introduced, necessary assumptions formalized, and a proof that the tetrachoric correlation
THE PHI -COEFFICIENT, THE TETRACHORIC CORRELATION COEF... 7coeﬃcient is well deﬁned is given. In Section 3, the main theorem of this article is stated and proved, and its implications are brieﬂy discussed. Thereafter, in Section 4, some numerical examples and graphs of the relation phi -coeﬃcient - tetrachoric correlation coeﬃcient are considered. And ﬁnally, the article is concluded with Section 5.
2. The two measures of association
2.1. Dichotomous variables. Let X and Y be two dichotomous variables. In the most general setting, the values of a dichotomous variable cannot be added, multiplied, ordered, or otherwise acted on by any binary operator, save projection. The algebraically most stringent way to model a dichotomous variable is to deﬁne it as a random element X : Ω → C, where the sample space C is an