Thursday, November 27, 2008

Wright, Fisher, Haldane, and odds and ends   posted by DavidB @ 11/27/2008 06:36:00 AM

From time to time I give links to those of my old posts that may still be worth reading. Previous guides are here: 1, 2, 3, 4.

It is over two years since the last update. In that time most of my posts have been on the history of population genetics, and especially on the 'founding fathers', R. A. Fisher, J. B. S. Haldane, and Sewall Wright. I recently finished a long series of Notes on Sewall Wright, so this is a convenient time to take stock.

Most of these posts are long, and aimed not so much at day-to-day readers as at people searching for specific topics.

Notes on Sewall Wright

On Reading Wright gave an overview of the planned series of notes, and includes some general reflections on Wright's reputation.

Before continuing with the series as planned, I realised that I needed to cover an additional topic, Wright's Method of Path Analysis This note is especially concerned to clarify the concept of a path coefficient, and the relationship between Wright's method and multiple regression.

In preparing the note on path analysis, I wanted to refer to some source containing the material on the statistical theory of correlation and regression that would be needed to understand Wright's work. I could not find a suitable source, so I decided to write it myself, using notes I have made on the subject over the years.

Notes on Correlation, Part 1 covers the general concepts of correlation and regression, and the justification for using them (which, like much in the foundations of statistics, is a moot point). Part 2 proves some key theorems on the correlation and regression of two variables, and discusses problems of interpretation. Part 3 outlines the theory of correlation and regression for more than two variables. This is particularly important for the understanding of Wright's path analysis.

After the note on Path Analysis I got back on the series as planned, with the following notes.

The measurement of kinship tries to explain Wright's approach to this, by contrasting it with the now more familiar methods of Gustave Malecot. The essential point is that Wright's kinship coefficients are in principle correlation coefficients rather than probabilities of identity (as in Malecot's system). A consequence of this is that kinship (or relatedness, or inbreeding) is relative to a specified population. The kinship between randomly selected individuals within such a population, relative to that population, is on average zero. This has implications for Hamiltonian inclusive fitness. Another implication is that Wright's kinship coefficients can be, and often are, negative (unlike Malecot's probabilities).

Wright's F-statistics. Wright devised a series of statistics known as F-statistics for measuring relationship and diversity within or between populations. The best known of these is FST, which is widely used as a measure of the genetic divergence between sub-populations of a species. My note traces the evolution of the F-statistics in Wright's work.

Genetic drift.. This note was originally going to be called 'Inbreeding and the decline of genetic variance', but that is not a very catchy title. I try to clarify the connection between genetic drift, inbreeding, and the decline of heterozygosis (a measure of genetic diversity). The note includes a detailed commentary on Wright's proof that heterozygosis tends to decline by 1/2N per generation.

Population size. I discuss the concept of effective population size and point out that Wright overlooked an important class of cases where effective population size is much larger than the current number of breeding adults.

Migration. Migration is important to Wright's theories because even very low rates of migration suffice to prevent subpopulations of a species diverging by genetic drift. The note traces Wright's work on the subject including his famous article on 'Isolation by distance'.

The adaptive landscape. Wright is closely associated with the concept of the adaptive landscape, though as far as I can find Wright himself never used this term. My note especially aims to explain the concept of a selective peak, and why Wright believed that there are a multitude of distinct selective peaks, usually of different fitness. In a related post on the Adaptive Landscape: Miscellaneous points, I discussed some issues not directly concerned with Wright, such as Stuart Kauffman's NK model, the relationship between selective peaks for genotypes and for gene frequencies, and the accessibility and stability of peaks.

The shifting balance theory of evolution.
This final note in the series is split into two parts. Part 1 examines the origins of Wright's famous shifting balance theory, and analyses the contents of the original version of the theory, as published in 1929-31. Part 2 explores subsequent developments in the theory, some of which are very important. Notably, as early as 1932 Wright abandoned his insistence that only genetic drift in small populations could take a population away from a suboptimal selective peak, as he now accepted that environmental fluctuations could have the same effect. In my view this removed much of the rationale for Wright's emphasis on population structure in evolution, though Wright himself never fully absorbed the implications of the change, which many biologists have overlooked.

Altogether, this series of posts would come to over 100 print pages. That's very nearly a book's worth! Alas, even if there were a market for such a boring book, I don't have the time, energy, or expertise to research and write it to the necessary standards, but I hope that anyone making a serious study of Wright will find something useful in my posts.

R. A. Fisher

My various notes on R. A. Fisher are mainly attempts to correct misunderstandings of his views which I have come across from time to time.

Fisher and Wright on population size (and here). These two notes were written shortly before I started my series of notes on Sewall Wright. Fisher is sometimes thought to have believed that entire species are randomly mating single populations. As this is palpably false, it is worth examining what Fisher really thought. In my first note I show, using Fisher's publications and letters, that he believed that migration between districts was usually frequent enough to offset their divergence by genetic drift. This does not imply that species are literally random mating (if they were, migration would be irrelevant), but only that for many purposes they can be treated as if they were. In the second note I examine what Fisher says about the actual population size of species. An Addendum is here.

Fisher on epistasis. It is sometimes claimed that Fisher ignored epistatic gene effects or considered them unimportant. My post shows that Fisher took account of epistasis in a variety of ways. Two further posts produce additional evidence: here and here.

Fisher on the adaptive landscape Following my note on Sewall Wright's adaptive landscape concept, I wrote this post on Fisher's views on the subject. Notably, he believed that environmental change, particularly in the biotic environment, made the idea of a constant landscape inapplicable.

Fisher on inclusive fitness

In this short post I draw attention to a passage by Fisher which contains a general anticipation of Hamilton's concept of inclusive fitness.

J. B. S. Haldane

I have written much less about Haldane than about Fisher and Wright. This is not because Haldane was less important or original. Haldane probably originated more of the basic results of population genetics than either of the others. But I tend to write posts mainly on issues that are obscure or controversial, whereas most of Haldane's results are clear and uncontroversial.

I have however devoted two posts to Haldane: one on Haldane's Dilemma, which examines Haldane's pioneering attempt to quantify the amount of genetic change possible by natural selection in a given period (see here for some corrections), and Haldane's Selection Theorem which comments on Haldane's proof that the probability that an individual favourable mutation will be successful is 2s, where s is the coefficient of selection.

Odds and ends

Finally, a few posts cover other issues.

Good Point? arises from a study by the economists Samuel Preston and Cameron Campbell. If intelligence is partly inherited, and less intelligent people on average have more children, it seems to follow that the average intelligence of the population will decline from one generation to the next. Preston and Campbell use an elaborate mathematical model to show that this is not necessarily the case. My post examines the argument, using a much simpler model due to the statistician I. J. Good. Briefly, I conclude that the argument is mathematically possible but biologically unrealistic. The case illustrates the danger of using sophisticated mathematics without properly considering the underlying assumptions.

Heterosis and the Flynn Effect looked sceptically at claims that heterosis (reduced inbreeding) might explain the long term increase in IQ scores.

Origins of the British is a piece examining the evidence on the ethnic origins of the people of the British Isles, following the recent book by Stephen Oppenheimer.

Group Selection and the Wrinkly Spreader takes a look at a recent defence of group selection by E. O. and D. S. Wilson, by examining in detail an example (the 'wrinkly spreader' variant of a certain bacterium) that they claim is a good case of group selection in action. It isn't.

Ethnic Genetic Interests Revisited looks at the new edition of Frank Salter's book Ethnic Genetic Interests, which includes comments on my own critique of the first edition.

Genophilia traces the origins of the term 'genophilia', which has been wrongly attributed to Francis Galton.

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