Biostatistics

Concepts and Applications for Biologists

Brian Williams

• Publisher: CRC Press
• Year: 2017
• eBook ISBN: 9781351367905
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

This book is a first course in statistics for students of biology. Most of the examples have an ecological bias, but illustrate principles which have direct relevance for biologists doing laboratory work. The structured approach begins with basic concepts, and progresses towards an appreciation of the needs and use of analysis of variance and regression, and includes the use of computer statistical packages. The work is clearly explained with worked examples of real-life biological problems, and should be suitable for undergraduate students engaged in quantitative biological work. Biostatistics should give students a sound grasp of the key principles of biological statistics without overwhelming detail, and should allow students to quickly apply techniques to their own work and data.

1
Why mathematics?

It is interesting to contemplate a tangled bank, clothed with many plants of many kinds, with birds singing on the bushes, with various insects flitting about, and with worms crawling through the damp earth, and to reflect that these elaborately constructed forms, so different from each other, and dependent on each other in so complex a manner, have all been produced by laws acting around us. These laws, taken in the largest sense, being Growth and Reproduction; Inheritance which is almost implied by reproduction; Variability from the indirect and direct action of the conditions of life, and from use and disuse; a Ratio of Increase so high as to lead to a Struggle for Life, and as a consequence to Natural Selection, entailing Divergence of Character and the extinction of the less-improved forms.

C. Darwin (1906, p. 669).

Mathematics and biology are two of the oldest sciences, both dating back to the beginning of recorded history, but it is only in this century that they have come together in the creation of modern mathematical biology. Well before the development of our modern view of natural history, much effort had been devoted to the systematic classification of plants and animals; by the seventeenth century many species had been described and identified. In 1686 John Ray defined species as groups of similar individuals characterized by 'distinguishing features that perpetuate themselves in propagation from seed' and noted that 'One species never springs from the seed of another' (Burkhardt, 1981). In 1735 Linnaeus developed the two-name system of classification for plants and animals that is used to this day. Believing in the constancy of species, he said, 'There are as many species as the Creator produced different forms in the beginning' (Burkhardt 1981). By the nineteenth century this view of species as immutable entities was beginning to weaken; although naturalists accepted that species were 'genuine entities in nature, constant in their essential characters', they now added 'subject to non-essential, accidental variation' (Burkhardt, 1981). Although the classification of species cannot be called mathematical, we cannot do anything until we have defined the entities we wish to study.

Having defined and classified species, it was natural to ponder the reasons why these particular species were found in the world. The idea of change within the otherwise fixed species was unavoidable in a world in which selective breeding of plants and animals was widely practised, and this led to intense debates concerning the origin of the natural world: was the world created as we see it now or was it created in a simpler form from which our present world has evolved? Nearly all religions postulate a process of creation and the idea of a progressive, albeit rather rapid, creation of the world is found in all cultures. But if the world has evolved from a 'simpler' state, do we need to invoke a 'higher power' to guide and direct the process of evolution or can we find a theory that allows us to explain evolution without recourse to external powers? If we can find laws of evolution, what kind of laws should they be? Should the laws of evolution be causal and mechanistic as are the laws of classical physics? The only physical law that depends explicitly on the direction of time is the second law of thermodynamics and this law predicts increasing disorder as time passes. How can we reconcile this with the apparent increase in order as time passes that we seem to see in biological evolution?

Two thousand years ago Aristotle reflected on the same problem (Ross, 1952): Why should not nature work, not for the sake of something, nor because it is better so, but just as the sky rains, not in order to make com grow, but of necessity?... Why... should it not be... that our teeth come up of necessity—the front teeth sharp, fitted for tearing, the molars broad and useful for grinding down the food... [and] survived, being organized spontaneously in a fitting way; whereas those which grew otherwise perished and continue to perish?' Since Aristotle believed that chance would destroy rather than preserve the adaptations we find in members of a given population, he rejected his own proposal and continued: 'Yet it is impossible that this should be the true view. For teeth and all other natural things either invariably or normally come about in a given way; but of not one of the results of chance or spontaneity is this true'. In this discourse Aristotle is concerned with the problem of selection rather than evolution, but he touched on the idea that the best adapted organisms should survive while those less well adapted should perish. Two thousand years later we find Darwin wondering how undirected chance events could have produced the apparently directed change found in the fossil record.

Darwin's interest lay in the ways in which changes arose in natural populations and he knew that substantial changes can be brought about in animal and plant populations by cross-breeding and selection. But while the work of breeders, who systematically select desirable characteristics, hinted at a mechanism for the emergence of novelty, Darwin could not see how such apparently directed change could be brought about in nature, if we assume that she is neutral and does not regard any particular change as more desirable than any other.

The answer came to Darwin from a quite unexpected source, as he recorded in his autobiography: 'In October 1838, ...fifteen months after I had begun my systematic inquiry, I happened to read for amusement Malthus on Population, and being well prepared to appreciate the struggle for existence which everywhere goes on, from long-continued observation of the habits of animals and plants, it at once struck me that under these circumstances favourable variations would tend to be preserved, and unfavourable ones to be destroyed. The result of this would be the formation of a new species. Here, then, I had at last got a theory by which to work...' (Darwin, 1958). Wallace (1905) took his inspiration from the same source: 'One day something brought to my recollection Malthus' Principle of Population... I thought of his clear exposition of 'the positive checks' to increase... which kept down the population. ... It then occurred to me that these causes or their equivalents are continually acting in the case of animals also; and, as animals usually breed much more rapidly than does mankind, the destructions every year from these causes must be enormous in order to keep down the numbers of each species, since they evidently do not increase regularly from year to year, as otherwise the world would long ago have become densely crowded with those that breed most quickly .... Why do some die and some live? And the answer was clearly, that on the whole the best fitted live. From the effects of disease the most healthy escaped; from enemies the strongest, the swiftest, or the most cunning; from famine, the best hunters or those with the best digestion; and so on ..... The more I thought over it the more I became convinced that I had at length found the long-sought-for law of nature that solved the problem of the origin of species.' (Cronin, 1991, provides a fascinating account of the development of Darwinian theory from Darwin and Wallace to the present day.)

Malthus (1970) realized that if the growth of a natural population were unchecked it would increase exponentially, i.e. the numbers would double in a fixed time and then double again and again in each equivalent time period. He also argued that our ability to increase food production would increase only arithmetically, i.e. we can at best increase food production only by the same absolute amount in any fixed time. Therefore, population growth would always tend to outstrip increases in food production. Since human populations do not always increase exponentially, Malthus sought to identify the factors that act to limit population growth. After examining the evidence from a number of countries, Malthus concluded that 'vice' (including abortion and infanticide) and 'misery' (including hunger and disease) acted as checks, although in later works he added 'moral restraint' (abstinence), which he hoped might one day replace 'vice' and 'misery' as a check to population growth. In modem terms, Malthus understood that the growth of all populations must eventually be limited by an increase in mortality or a decrease in fecundity with increasing density. Darwin and Wallace now took Malthus's essentially mathema...

Table of contents

• Cover
• Half Title
• Dedication
• Title
• Copyright
• Contents
• Preface
• Acknowledgements
• 1 Why mathematics?
• 2 Probability
• 3 Representations
• 4 Measures
• 5 Distribution
• 6 Testing hypotheses
• 7 Comparisons
• 8 Analysis of variance
• 9 Regression
• 10 Tables
• References
• Index