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5 - Measurement error and latent variables
- Bill Shipley, Université de Sherbrooke, Canada
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Summary
Ambient temperature affects the metabolic rate of animals. When it is cold a homeothermic animal has to burn stored energy reserves – first glycogen and fat and then, when these are exhausted, protein – in order to generate heat and maintain its body temperature. The scaling of surface area (the site of heat loss to the atmosphere) to body volume (where the heat is generated) means that small homeothermic animals, such as songbirds, can lose up to 15 per cent of their body fat in one cold night. To burn this fat the bird must increase its metabolic rate, which increases its oxygen consumption. Imagine that we conduct an experiment in which we place small birds inside metabolic chambers overnight and vary the air temperature. The hypothesised causal process is shown in Figure 5.1.
Unfortunately, we can't directly measure any of these three variables; they are unmeasured, or latent, and so I have enclosed them in circles following the conventions of path diagrams. If we measure the air temperature using a thermometer then we aren't directly measuring temperature – the average kinetic energy of the molecules in the air. Instead, we are measuring the height of a column of mercury in a vacuum and enclosed in a hollow glass tube. In fact, we can't even measure the actual height of the mercury exactly, since our observed height will include some measurement error. Nor can we directly measure metabolic rate. Typically, one measures the rate of gas exchange (oxygen decrease or carbon dioxide increase) between the air entering and leaving the metabolic chamber. If we measure oxygen consumption using an infrared gas analyser then we aren't even directly measuring oxygen consumption. Instead, we are measuring differences in the amount of light of particular wavelengths that is absorbed as the light passes through the air. Again, even this variable is not perfectly measured, since the observed values will also contain measurement error. When we measure the fat reserves that are burned by the birds we might actually be measuring the difference in body weight over the course of the experiment, and this too will include measurement error. One simplified representation of the actual causal process is depicted in Figure 5.2.
Frontmatter
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Index
- Bill Shipley, Université de Sherbrooke, Canada
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8 - Exploration, discovery and equivalence
- Bill Shipley, Université de Sherbrooke, Canada
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Summary
Hypothesis generation
If this were a textbook of statistics then this chapter would not exist. Modern statistics is almost entirely concerned with testing hypotheses, not developing them. Such a bureaucratic approach views science as a compartmentalised activity in which hypotheses are constructed by one group, data are collected by another group and then the statistician confronts the hypothesis with the data. Since this book is a user's guide to causal modelling, such a compartmentalised approach will not do. One of the main challenges faced by the practising biologist is not in testing causal hypotheses but in developing causal hypotheses worth testing.
If this were a book about the philosophy of science then this chapter might not exist either. The philosophy of science mostly deals with questions such as: how can we know if a scientific hypothesis is true or not? What demarcates a scientific hypothesis from a non-scientific hypothesis? For most philosophers of science, the question of how one looks for a useful scientific hypothesis in the first place is someone else's problem. For instance, Karl Popper, in his influential Logic of Scientific Discovery (Popper 1980: 32), says that ‘there is no such thing as a logical method of having new ideas, or a logical reconstruction of this process. My view may be expressed by saying that every discovery contains “an irrational element”, or “a creative intuition”…’ Later, he says that ‘[scientific laws] can only be reached by intuition, based on something like an intellectual love of the objects of experience’. Again, one gets the impression that science consists to two hermetically sealed compartments. One compartment, labelled hypothesis generation, consists of an irrational fog of thoughts and ideas, devoid of method, out of which a few gifted people are able to extract brilliant insights. The other compartment, labelled hypothesis testing, is the public face of science. Here, one finds method and logic, in which established rules govern how observations are to be taken, statistically manipulated and interpreted.
At a purely analytic level there is much to be gained by taking this schizophrenic view of the scientific process. After all, how a scientific idea is developed is irrelevant to its truth. For instance, the history of science documents many important ideas whose genesis was bizarre.
2 - From cause to correlation and back
- Bill Shipley, Université de Sherbrooke, Canada
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Translating from causal to statistical models
The official language of statistics is the probability calculus, based on the notion of a probability distribution. For instance, if you conduct an analysis of variance (ANOVA) then the key piece of information is the probability of observing a particular value of Fisher's F statistic in a random sample of data, given a particular hypothesis or model. To obtain this crucial piece of information, you (or your computer) must know the probability density function of the F statistic. Certain other (mathematical) languages are tolerated within statistics but, in the end, one must link one's ideas to a probability distribution in order to be understood. If we wish to study causal relationships using statistics, it is necessary that we translate, without error, from the language of causality to the only language that statistics can understand: probability theory.
Such a rigorous translation device did not exist until very recently (Pearl 1988). It is no wonder that statisticians have virtually banished the word ‘cause’ from statistics; such a word has no equivalent in their language. Within the world of statistics the scientific notion of causality has, until recently, been a stranger in a strange land. Posing causal questions in the language of the probability calculus is like a unilingual Englishman asking for directions to the Louvre from a Frenchman who can't speak English. The Frenchman might understand that directions are being requested, and the Englishman might see fingers pointing in particular directions, but it is not at all certain that works of art will be found. Imperfect translations between the language of causality and the language of probability theory are equally disorienting.
Mistakes in translation come in all kinds. The most dangerous ones are the subtle errors in which a slight change in inflection or context of a word can change the meaning in disastrous ways. Because the French word demande both sounds like the English word ‘demand’ and has roughly the same meaning (it simply means ‘to ask for’, without any connotation of obligation), I have seen French-speaking people come up to a shop assistant and, while speaking English, ‘demand service’. They think that they are politely asking for help, while the assistant thinks they are issuing an ultimatum.
References
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3 - Sewall Wright, path analysis and d-separation
- Bill Shipley, Université de Sherbrooke, Canada
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Summary
A bit of history
The ideal method of science is the study of the direct influence of one condition on another in experiments in which all other possible causes of variation are eliminated. Unfortunately, causes of variation often seem to be beyond control. In the biological sciences, especially, one often has to deal with a group of characteristics or conditions which are correlated because of a complex of interacting, uncontrollable, and often obscure causes. The degree of correlation between two variables can be calculated with well-known methods, but when it is found it gives merely the resultant of all connecting paths of influence.
The present paper is an attempt to present a method of measuring the direct influence along each separate path in such a system and thus of finding the degree to which variation of a given effect is determined by each particular cause. The method depends on the combination of knowledge of the degrees of correlation among the variables in a system with such knowledge as may be possessed of the causal relations. In cases in which the causal relations are uncertain the method can be used to find the logical consequences of any particular hypothesis in regard to them.
So begins Sewall Wright's 1921 paper (Wright 1921), in which he describes his ‘method of path coefficients’. In fact, he invented this method while still in graduate school (Provine 1986) and had even used it without presenting its formal description in a paper published the previous year (Wright 1920). The 1920 paper used his new method to describe and measure the direct and indirect causal relationships that he had proposed to explain the patterns of inheritance of different colour patterns in guinea pigs. The paper came complete with a path diagram – i.e. a causal graph – in which actual drawings of the colour patterns of guinea pig coats were used instead of variable names.
Wright was one of the most influential evolutionary biologists of the twentieth century, being one of the founders of population genetics and intimately involved in the modern synthesis of evolutionary theory and genetics.
Appendix - A cheat-sheet of useful R functions
- Bill Shipley, Université de Sherbrooke, Canada
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The purpose of the Appendix is to summarise the usage of the two main R libraries that are discussed in this book: ggm and lavaan.
The ggm (graphical Gaussian model) library
There is no R library that is dedicated to the d-sep test and its generalisations. However, the ggm library of R does have a number of useful functions that can help you. First, the basiSet() function can always be used to obtain the union basis set of d-separation claims that lies at the heart of the d-sep test. This function requires that you specify the DAG, and this is done using the DAG() function. The dSep() function is not required but is a very useful little function for helping you test your understanding of this important concept. The shipley.test() function implements the d-sep test for the special case in which your DAG involves only normally distributed variables and linear relationships.
DAG(…, order = FALSE)
The R function DAG() is used for imputing a directed acyclic graph. The output of this function is the adjacency matrix of the DAG – i.e. a square Boolean matrix of order equal to the number of nodes of the graph and a 1 in position (i,j) if there is an arrow from i to j and a zero otherwise. The row names of the adjacency matrix are the nodes of the DAG.
Arguments
…= a sequence of model formulae, using the regression (∼) operator. For each formula, the right-hand response defines an effect node (a child in the DAG) and the left-hand explanatory variables the parents of that node. If the regressions are not recursive (i.e. if there is a feedback loop in the DAG) then the function returns an error message.
order = logical, defaulting to FALSE. If TRUE then nodes of the DAG are permuted according to the topological order. If FALSE then nodes are in the order they first appear in the model formulae (from left to right). This argument is used for purely aesthetic reasons.
Consider this simple causal chain model: X→Y→Z. To input this model as a DAG you would specify My.DAG<-DAG(Y∼X, Z∼Y). Note that this uses the same syntax as when specifying linear models in R. For each child node (dependent variable, endogenous variable) you specify dependent variable ∼ parent variable1 + parent variable2, etc.
6 - The structural equation model
- Bill Shipley, Université de Sherbrooke, Canada
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The structural equation model is commonly described as the combination of a measurement model and a structural model. These terms derive from the history of SEM as being a union of the factor analytic, or measurement, models of psychology and sociology and the simultaneous structural equations of the econometricians. In its pure form it therefore explicitly assumes that every variable that we can observe is an imperfect measure of some underlying latent causal variable and that the causal relationships of interest are always between these latent variables. As in many other things, purity is more a goal than a requirement. Using the example in Chapter 5 of the effect of air temperature on metabolic rate (Figure 6.1), the things that we can measure (the height of the mercury in the thermometer or the change in CO2 in the metabolic chamber) always contain measurement error (εi). The measurement model, shown by the dashed squares in Figure 6.1, describes the relationship between the observed measures and the underlying latent variables (the average kinetic energy of the molecules in the air and the metabolic rate of the animal). The structural model, shown by the dashed circle in Figure 6.1, describes the relationship between the ‘true’ underlying causal variables. If we have only one measured variable per latent variable and we assume that the measured variable contains no measurement error (i.e. the correlation between the measured variable and the underlying latent variable is perfect) then we end up with a path model. If we have a set of measured variables for each latent variable and we do not assume any causal relationships between the latent variables then we have a series of measurement models. If we have more complicated combinations, in which we assume causal relationships between the latent variables, then we have a full structural equation model. Therefore, if you have understood Chapters 1 to 5, you already know how to construct and test a structural equation model; you simply have to put the pieces together.
The goal of this chapter is therefore to deal with some technical details that I have ignored up to now. The first detail is the problem of identification.
1 - Preliminaries
- Bill Shipley, Université de Sherbrooke, Canada
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The shadow's cause
The Wayang Kulit is an ancient theatrical art, practised in Malaysia and throughout much of the Orient. The stories are often about battles between good and evil, as told in the great Hindu epics. What the audience actually see are not actors, nor even puppets, but, instead, the shadows of puppets projected onto a canvas screen. Behind the screen is a light. The puppet master creates the action by manipulating the puppets and props so that they will intercept the light and cast shadows. As these shadows dance across the screen the audience must deduce the story from these two-dimensional projections of the hidden three-dimensional objects. However, shadows can be ambiguous. In order to imply the three-dimensional action, the shadows must be detailed, with sharp contours, and they must be placed in context.
Biologists are unwitting participants in nature's shadow play. These shadows are cast when the causal processes in nature are intercepted by our measurements. Like the audience at the Wayang Kulit, the biologist cannot simply peek behind the screen and directly observe the actual causal processes. All that can be directly observed are the consequences of these processes in the form of complicated patterns of association and independence in the data. As with shadows, these correlational patterns are incomplete – and potentially ambiguous – projections of the original causal processes. As with shadows, we can infer much about the underlying causal processes if we can learn to study their details and sharpen their contours, especially if we can study them in context.
Unfortunately, unlike the puppet master in a Wayang Kulit, who takes care to cast informative shadows, nature is indifferent to the correlational shadows that it casts. This is the main reason why researchers go to such extraordinary lengths to randomise treatment allocations and to control variables. These methods, when they can be properly done, simplify the correlational shadows to manageable patterns that can be more easily mapped onto the underlying causal processes.
It is uncomfortably true, though rarely admitted in statistics texts, that many important areas of science are stubbornly impervious to experimental designs based on the randomisation of treatments to experimental units.
Preface
- Bill Shipley, Université de Sherbrooke, Canada
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This book describes a series of statistical methods for testing causal hypotheses using observational data – but it is not a statistics book. It describes a series of algorithms, derived from research in artificial intelligence (AI), that can discover causal relationships from observational data – but it is not a book about artificial intelligence. It describes the logical and philosophical relationships between causality and probability distributions – but it is certainly not a book about the philosophy of statistics. Rather, it is a user's guide, written for biologists, whose purpose is to allow the practising biologist to make use of these important new developments when causal questions cannot be answered with randomised experiments.
I have written the book assuming that you have no previous training in these methods. If you have taken an introductory statistics course – even if it was longer ago than you want to acknowledge – and have managed to hold on to some of the basic notions of sampling and hypothesis testing using statistics then you should be able to understand the material in this book. I recommend that you read each chapter through in its entirety even if you do not feel that you have mastered all the notions. This will at least give you a general feeling for the goals and vocabulary of each chapter. You can then go back and pay closer attention to the details.
The book is addressed to biologists, mostly because I am a practising biologist myself, but I hope that it will also be of interest to statisticians, scientists in other fields and even philosophers of science. I have not written the book as a textbook simply because the discipline to which the material in this book naturally belongs does not yet exist. Whatever the name eventually given to this new discipline, I firmly believe that it will exist, and be generally recognised as a distinct discipline, in the future. The questions that this new discipline addresses, and the elegance of its results, are too important for this not to be the case. Nonetheless, the chapters follow a logical progression that would be well suited to an upper-level undergraduate, or graduate, course. I have used the manuscript of this book for such a purpose, and every one of my students is still alive.
Dedication
- Bill Shipley, Université de Sherbrooke, Canada
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Contents
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7 - Multigroup models, multilevel models and corrections for the non-independence of observations
- Bill Shipley, Université de Sherbrooke, Canada
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Like successful politicians, good statistical models must be able to lie without getting caught. For instance, no series of observations from nature are really normally distributed. The normal distribution is just a useful abstraction – a myth – that makes life bearable. In constructing statistical models we pretend that the normal distribution is real and then check to ensure that our data do not deviate from it so much that the myth becomes a fairy tale. In the last chapter we saw how far we could stretch the truth about the distributional properties of our data before our data called us a liar. The goal of this chapter is to describe how SEM can deal with two other statistical myths that people often tell with respect to their data. These two myths are (a) that the observations in our data sets are generated by the same causal process (causal homogeneity) and (b) that these observations are independent draws from this single causal process.
Consider first the myth of causal homogeneity. It is easy to imagine cases in which different groups of observations might be generated by partially different causal processes. For instance, a behavioural ecologist studying a series of variables related to aggression and social dominance in primates would not necessarily want to combine together the observations from males and females, since it is possible that the behavioural responses of males and females are generated by different causal stimuli. When we sample from populations with different causal processes, either in terms of the causal structure or of the quantitative strengths between the variables, and we wish to compare the causal relationships across the different groups, we require a model that can explicitly take into account these differences between groups. Such modelling is called multigroup SEM, and this, in turn, requires the notion of nested models.
The assumption of the independence of observations can often be violated as well, because the observations are nested in space or time. The process of speciation itself suggests one way in which we can get non-independence of observations (Felsenstein 1985; Harvey and Pagel 1991). The attributes of organisms, if they have a genetic component, will often tend to be more similar to those of close relatives than to genetic strangers.
4 - Path analysis and maximum likelihood
- Bill Shipley, Université de Sherbrooke, Canada
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James Burke (1996), in his fascinating book The Pinball Effect, demonstrates the curious and unexpected paths of influence leading to most scientific discoveries. People often speak of the ‘marriage of ideas’. If this is true then the most prolific intellectual offspring come not from the arranged marriages preferred by research administrators but from chance meetings, and even illicit unions. The popular view of scientific discoveries as being linear causal chains from idea to solution is profoundly wrong; a better image would be a tangled web with many dead ends and broken strands. If most present knowledge depends on unlikely chains of events and personalities, what paths of discovery have been deflected because the right people did not come together at the right time? Which historical developments in science have been changed because two people, each with one half of the solution, were prevented from communicating due to linguistic or disciplinary boundaries? The second stage in the development of modern structural equation modelling is a case study in such historical contingencies and interdisciplinary incomprehension.
During the First World War, and in connection with the American war effort, Sewall Wright was on a committee allocating pork production to various states based on the availability of corn. He was confronted with a problem that had a familiar feel. Given a whole series of variables related to corn availability and pork production, how do all these variables interact to determine the relationship between supply and demand, and the fluctuations between these two? It occurred to him that his new method of path analysis might help. He calculated the correlation coefficients between each pair of variables for five years, giving 510 separate correlations. After much trial and error he developed a model involving only four variables (corn price, summer hog price, winter hog price and hog breeding) and only fourteen paths that still gave a ‘good match’ between observed and predicted correlations. He described his results in a manuscript that was submitted as a bulletin of the US Bureau of Animal Industry. It was promptly rejected, perhaps because officials at the Bureau of Agricultural Economics considered it as an intrusion onto their turf. Happily for Wright, he had also shown it to the son of the secretary of agriculture (Henry A. Wallace), who was interested in animal breeding and quantitative modelling.
Preface to the second edition
- Bill Shipley, Université de Sherbrooke, Canada
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I had two motives, one positive and one more selfish, in writing the first edition of this book. The positive motive was to provide a detailed introduction of these methods to practising biologists, since they were largely unknown to students and researchers in this discipline. The more selfish motive was to provide a detailed justification of these methods to practising biologists. You see, I was frustrated. My research manuscripts that included these methods were being rejected by reviewers, who viewed the analyses as the statistical equivalents of conjurer's tricks. I concluded that a book-length explanation that was written specifically for biologists would provide such a justification. Now, writing fifteen years later, the situation is quite different. These methods have been increasingly adopted by biologists working in ecology, evolution, genetics and molecular biology. I hope that the first edition of this book, as well as Jim Grace's (2006) very fine book, have contributed to this change. Virtually every chapter has been updated in this second edition. These changes include, inter alia, new additions to the d-sep test, the inclusion of phylogenetic information and an expanded treatment of latent variables. The most extensive change is the detailed explanation for implementing these methods using the R programming language. The only computer programs for structural equation modelling that were available when I wrote the first edition were commercial ones. Since I didn't want to become a salesman for any particular commercial package, I didn't include the actual code and steps for carrying out the analyses. However, a ‘user's guide’ that omits such vital information is clearly lacking. Now that the freely available R program has become so ubiquitous for statistical analysis by biologists, and now that the methods in this book have been included in several R libraries, I have included detailed instructions in this second edition for carrying out the analyses.
Cause and Correlation in Biology
- A User's Guide to Path Analysis, Structural Equations and Causal Inference with R
- 2nd edition
- Bill Shipley
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Many problems in biology require an understanding of the relationships among variables in a multivariate causal context. Exploring such cause-effect relationships through a series of statistical methods, this book explains how to test causal hypotheses when randomised experiments cannot be performed. This completely revised and updated edition features detailed explanations for carrying out statistical methods using the popular and freely available R statistical language. Sections on d-sep tests, latent constructs that are common in biology, missing values, phylogenetic constraints, and multilevel models are also an important feature of this new edition. Written for biologists and using a minimum of statistical jargon, the concept of testing multivariate causal hypotheses using structural equations and path analysis is demystified. Assuming only a basic understanding of statistical analysis, this new edition is a valuable resource for both students and practising biologists.
7 - The statistical mechanics of species abundance distributions
- Bill Shipley, Université de Sherbrooke, Canada
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So far in this book we have been looking for an answer to the following general question: If we know which species exist in the species pool, and we know their functional traits, can we predict the relative abundance of each species in different environmental contexts? What happens if we don't know which species – or even how many species – are in the species pool? Clearly, if we don't know which species are in the pool then we are in no position to predict their relative abundances. Even if we do know that a particular species is in the pool we still can't predict its relative abundance if we don't know how many other species are in the pool; after all, relative abundance is a proportion from a total. Therefore, if we don't know the composition of the species pool then we can't possibly answer the central question posed in this book.
However, even when missing this vital information about the composition of the species pool, we can ask a different, but related, question. Since we can't inquire about the abundance of species i, we might want to know how many species will have a given abundance. We can ask, for example: how many species will have only one individual or unit of biomass? How many species will be a little less rare and have two individuals, or units of biomass, and so on?
References
- Bill Shipley, Université de Sherbrooke, Canada
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6 - Community assembly during a Mediterranean succession
- Bill Shipley, Université de Sherbrooke, Canada
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Theory can be dangerously seductive. Once one has built up an argument that is internally consistent, and once conclusions appear to follow inexorably from premises through clean lines of logic, it is sometimes enticing to conflate logical argument with reality. A good defense against such logical seduction is to let Nature into the conversation. A proper empirical evaluation of the method presented in Chapter 4 would involve an accurately measured environmental gradient involving all of the relevant environmental variables driving natural selection plus measured values of the key functional traits that respond to this selection of all species in the regional pool. This would be replicated in different localities along with evidence of quantitative generality of the community-aggregated traits. Hopefully, this book will have sufficiently convinced you of the potential of the approach that you will contribute to the hard work of assembling such empirical information. When this is done then we will know if the model actually works.
I'm easy to seduce. I think that it will work. However, I'm old enough to know the difference between seduction and commitment and I have had enough experience with field ecology to know that it might not work after all. I certainly won't hang myself in the barn if the model fails. As Thomas Henry Huxley famously pointed out, many beautiful theories have been killed by ugly facts.