Galton's blinding glasses Modern statistics hiding causal structure in early theories of inheritance Bert Leuridan 14 June 2006 Centre for Logic and Philosophy of Science Ghent University Belgium

2 1. The Problem

3 Gregor Mendel ( ) Versuche über Pflanzenhybriden (sunk into oblivion) Rediscovery: Correns, de Vries (1900) In the meantime: Francis Galton –Hereditary Genius (1869) –“Typical Laws of Heredity” (1877) –Natural Inheritance (1889) –“The Average Contribution of each several Ancestor to the total Heritage of the Offspring” (1897) –‘‘A Diagram of Heredity’’ (1898) Early 20th century: strong rivalry between both their adherents Mendel Galton

4 Gregor Mendel: –Now considered to be the founding father of classical genetics –Pointed at the right causal structure –Not acquainted with modern statistics Francis Galton: –One of the major statisticians, founding father of linear regression –Unfruitful theory of inheritance (wrong causal structure) Claim: because of Galton’s modern statistics –His statistics generated two explananda (the normal distribution of characteristics and regression towards the mean) –The explananda generated constraints that biased Galton’s search for the causal mechanism of inheritance The Problem

5 Outline 1.The Problem 2.The contrast between Mendel and Galton 3.Galton’s statistics → two explananda –Normal distribution of characteristics –Regression towards the mean 4.Galton’s explanans: the theory of ancestral inheritance 5.Conclusion

6 2. The contrast between Mendel and Galton

7 Contrast Theory of ancestral inheritance (Galton) Theory of parental inheritance (Mendel) Mostly (but not exclusively!) studied continuous traits: e.g. Humans Stature (observational data) Size of sweat peas (experimental data) Studied pairs of opposing traits (no intermediate traits): e.g. stem length (tall vs dwarf) in P.sativum (experimental data) Indefinite number of elements/particles for each trait Pairs of unit factors for each trait The hereditary influence of the ancestors is not screened off by the parents or the parental gametes The hereditary influence of the ancestors is screened off by the parents or even the parental gametes Compatible with 19th century cell- biological knowledge

8 3. Galton’s modern statistics generating two explananda

9 Statistics steering Biology Galton’s biological theory served as an explanans for two explananda: 1.Normal distribution of characteristics 2.Regression towards the mean These phenomena he observed in different organisms: Human Stature: Record of Family Faculties (observational data gathered by Galton) Size of the seeds of sweat peas (experimental data gathered by Galton)

10 Normal Distribution Very dominant concept in the 19th century (cf. Adolphe Quetelet and Quetelismus) Median: P = 68,25 inch Dispersion: Q = half interquartile range = 1/2(Q 2 -Q 1 ) = 1,7 inch “In particular, the agreement of the Curve of Stature with the Normal Curve is very fair, and forms a mainstay of my inquiry into the laws of Natural Inheritance.” (Natural Inheritance, p. 57) P Q1Q1 Q2Q2 Q

11 Causes of Normal Distributions? Hypothesis of Elementary Errors : Laplace, 1810: The joint action of a multitude of independent ‘errors’ produces a normal distribution. “The Law of Error finds a footing wherever the individual peculiarities are wholly due to the combined influence of a multitude of “accidents,” (…).” (Natural Inheritance, p. 55) First Constraint: any theory of inheritance should introduce a "variety of petty influences" Note: this is prima facie impossible in the Mendelian picture There you have two gross influences for each trait: –unit factor of the pollen cell –unit factor of the germ cell Mendel’s traits were not normally distributed

12 Preliminaries: –Human stature = P + D P = population mean/median D = individual’s deviation from the mean –D follows the Law of Frequency of Error, –i.e. D has a normal distribution –Transmutation: On average, women are smaller than men, therefore … Stat. Transmuted Female = 1.08 x Stat. Female The distribution of the transmuted female statures almost exactly fits the distribution of the male statures –Mid-Parent: Stat. Mid-Parent = Stat.Father + Stat.Transm.Mother 2 Regression towards the Mean

13 Regression towards the Mean If Stat. Mid-Parent = P + D Then Stat. Son = P + 2/3 D “I call this ratio of 2 to 3 the ratio of ‘Filial Regression.’ It is the proportion in which the Son is, on the average, less exceptional than his Mid- Parent.” (Natural Inheritance, p. 97) Second Constraint: Every theory of inheritance should be able to explain or predict regression towards the mean Note: this was the birth of Linear Regression D 2/3*D P

14 Causes of Regression What are the causes responsible for regression towards the mean? see later

15 4. Galton’s explanans: the theory of ancestral inheritance

16 Galton’s theory of ancestral inheritance Galton introduced a causal mechanism based on an indefinite number of hereditary elements/particles transmitting deviations D from generation to generation –First Constraint is satisfied: we have a variety of petty influences The normal distribution of Human Stature is explained

17 Galton’s theory of ancestral inheritance Law of Ancestral Heredity: –Hereditary influence of all ancestors (no screening off) –Every ancestor may contribute its deviation D ancestor’s personal allowance ancestor’s ancestral allowance (= passing of influence of more remote ancestry) –Taxation: 1/2 in every generation Parents determine 0.5 of the total heritage Grandparents determine (0.5)² of the total heritage Great-Grandparents determine (0.5)³ of the total heritage etc (0.5)² + (0.5)³ +... = 1 –Note: the Law of Ancestral Heredity also applied to discrete traits such as Eye-colour

stature personal elements latent elements

19 Explaining regression Preliminaries: Mid-Parental Regression: –If the deviation of the Mid-Parent = D, –Then the deviation of the Mid-Grand-Parent = 1/3 D, –And the deviation of the Mid-Great-Grand-Parent = (1/9) D –… Taxation: Law of Ancestral Heredity –The transmission of particles is taxed by 50% per generation Effective heritage: (D)*(0.5) + (1/3D)*(0.5)² + (1/9D)*(0.5)³ + … ≈ 2/3 D If Stat.Mid-Parent = P + D, Then Stat. Son = P + 2/3 D (QED)

20 Explaining regression Dilution-theory The deviation D of the Mid-Parent is mixed with the smaller deviations of more remote ancestry, the result being a regression towards the mean. Metaphor: ‘[The] effect resembles that of pouring a measure of water into a vessel of wine. The wine is diluted to a constant fraction [2/3] of its alcoholic strength [D], whatever that strength may have been.’ (Natural Inheritance, p. 105) Second Constraint is satisfied: Taxation and Dilution together explain regression towards the mean

21 5. Conclusion

22 Conclusion Galton’s causal picture differed sharply from Mendel’s. This can be explained by –The absence of cytological constraints –The presence of statistical constraints, constraints that Mendel did not had to account for. Mendel’s theory proved a fruitful basis for further genetic research from 1900 onwards, even if during the development of classical genetics it was subject to a lot of changes, Galton’s causal theory (but not his empirical results) was abandoned in the beginning of the 20th century. Conclusion: modern statistics played a blinding role, it hided causal structure in Galton’s early theory of inheritance. In general: although probability and statistics are nice tools to capture causality, in practice they should be applied very carefully.

Galton's blinding glasses Modern statistics hiding causal structure in early theories of inheritance Bert Leuridan 14 June 2006 Centre for Logic and Philosophy of Science Ghent University Belgium