The continuing interest in our textbook together with the ongoing development of statistical applications in veterinary and animal science has encouraged us to prepare this third edition of Statistics for Veterinary and Animal Science. We have introduced some new material but we want to reassure all readers that our original intention of this being an introductory text still stands. Again, you will find everything that you need to begin to understand statistics and its application to your scientific and clinical endeavours; it remains an introduction for the novice with emphasis on understanding the application, rather than exhibiting mathematical competence in the calculations. Readily available statistical software packages, which provide the mechanics of the calculations, have become more extensive in the range of procedures they offer. Accordingly, we have augmented our text, within the bounds of an introductory exposition, to match their development.

As in previous editions, we use two commonly employed statistical software packages, SPSS and Stata, to analyse the data in our examples. We believe that by presenting you with different forms of computer output, you will have the confidence and proficiency to interpret output from other statistical packages. The previous edition of the book had an accompanying CD which contained the data sets (in ASCII, Excel, SPSS and Stata) used as examples in the text. These data sets are now available at www.wiley.com/go/petrie/statisticsforvets, and will be helpful if you wish to get to grips with various statistical techniques by attempting the analyses yourselves. You will find a website icon next to the examples for which the data are available on the website. Please note that, although we have provided details of a considerable number of websites that you may find useful, we cannot guarantee that these website addresses will remain correct over the course of time because of the mutability of the internet.

Some sections of the book are, as in previous editions, in small print and are accompanied by a jumping horse symbol. These sections contain information that relates to more advanced or obscure topics, and you may skip (jump over) them without loss of continuity. Our teaching experience has demonstrated that one of hardest tasks for the novice when analysing his or her own data set is deciding which test or procedure is most appropriate. To overcome this difficulty, we provide two flow charts (Figure E.2 for binary data and Figure E.3 for numerical data) which lead you through the various questions that need to be asked to aid that decision. Another flow chart (Figure E.1) organizes the tests and procedures into relevant groups and indicates the particular section of the book where each is located: you can find these flow charts in the Appendix as well as on the inside back/front covers for easy reference.

Many of the chapters in this third edition are similar to those in the second edition, apart from some minor modifications and additional exercises. However, Chapter 5 has been extended to include techniques for recognizing and dealing with confounding, and this chapter now provides a description of the different types of missing data that might be encountered. We have added a section on checking the assumptions underlying a logistic regression model to Chapter 11, and have included modifications of the sample size estimation process to take account of different group sizes and losses to follow-up in Chapter 13. Chapter 14 has been expanded considerably by extending the sections on diagnostic tests, measuring agreement and survival analysis as well as Bayesian analysis. Chapter 15 is entirely new, bringing together a group of specialist topics – ethical issues of animal investigation (some of which was in Chapter 5 of the second edition), spatial statistics, surveillance and its importance, and statistics in molecular and quantitative genetics. While none of these is intended as more than an introduction, you will find references to help you explore the topics more fully should you so desire. The section on evidence-based veterinary medicine (EBVM) in Chapter 16 is unchanged from that in the second edition’s Chapter 15, but in the third edition this chapter no longer provides guidelines for reporting results. Instead, we have devoted the new Chapter 17 to this topic by presenting different published guidelines relevant to veterinary medicine (i.e. for reporting of livestock trials, research using laboratory animals, diagnostic accuracy studies, observational studies in epidemiology, and systematic reviews and meta-analyses) as a ready reference for those wanting to follow best practice both in planning and in writing up their research. Lastly, in Chapter 18, which is entirely new, we bring together the concepts of EBVM and the guidelines provided in Chapter 17 by proffering a template for the critical appraisal of randomized controlled trials and observational studies. We use this template to critically appraise two published papers, both of which are reproduced in full, and hope that by providing these examples, we will help you develop your own skills in what is an essential, but frequently overlooked, component of statistics.

We are indebted as always to those who, for earlier editions of this book, have offered their data to us to use for examples or exercises, have assisted with the presentation of the illustrations and tables, and have provided critical advice on the text. These colleagues are all identified in the prefaces to the first and second editions. As in earlier editions, we have occasionally taken summary data or abstracts from published papers and have used them to develop exercises or to illustrate techniques: we extend our thanks to the authors and the publishers for the use of this material. For this third edition, we are most grateful to Dr Geoff Pollott and Professor Dirk Pfeiffer (both of the Royal Veterinary College, University of London) for their critical reading and suggestions for sections of Chapter 15. We wish to record our particular thanks to Professor Garry Anderson (University of Melbourne) for his critique of much of the new text. His suggestions have drawn our attention to errors and have considerably improved the presentation. Nonetheless, we remain responsible for all contained herein, and offer it, with all its shortcomings, to our readership.

This preface would not be complete without acknowledging our marriage partners, Gerald and Rosie, and our children, Nina, Andrew and Karen, and Oliver and Anna, who have allowed us once again to engage with this task to their inevitable exclusion, and offer them our most grateful thanks.

It is six years since this book was first available, and we are glad to acknowledge the positive responses we have received to the first edition and the evident uptake of the text for a number of courses around the world. In the intervening period much has happened to encourage us to update and expand our initial text. However, many of the chapters which were in the first edition of the book are changed only slightly, if at all, in this second edition. To these chapters, we have added some exercises and further explanations (for example, on equivalence studies, confounding, interactions and bias, Bayesian analysis and Cox survival analysis) to make the book more comprehensive. We have nevertheless retained our original intent of this being an introductory text starting with very basic concepts for the complete novice in statistics. You will still find sections marked for skipping unless you have a particular need to explore them, and these include the newer more complex analysis methods. This edition also contains the glossaries of notation and of terms, but we have expanded them to reflect the enhanced content of the text. For easy reference, the flow charts for choosing the correct statistical analyses in different situations are now found immediately before the index, and we hope these will serve to guide you to the appropriate procedures and text relating to their use.

Computer software to deal with increasingly sophisticated analytical tools has been developed in recent years in such a way that the associated methodology is more readily accessible to those who previously believed such techniques were out of their reach. As a consequence, we have substantially enhanced the material relating to regression analysis and created a new chapter (Chapter 11) to describe some advanced regression techniques. The latter incorporates the sections on multiple regression and an expanded section on logistic regression from Chapter 10 of the first edition, and introduces Poisson regression, different regression methods which can be used to analyse clustered data, maximum likelihood estimation and the concept of the generalized linear model. Because we have inserted this new Chapter 11, the numbering of the chapters which follow does not accord with that of the corresponding chapters in the first edition.

Chapter 15 is an entirely new chapter which is devoted in large part to introducing the concepts of evidence-based veterinary medicine (EBVM), stressing the role of statistical knowledge as a basis for its practice. The methodology of EBVM describes the processes for integrating, in a systematic way, the results of scientifically conducted studies into day-to-day clinical practice with the aim of improving clinical outcome. This requires the practitioner to develop the skills to evaluate critically the efforts of others in respect of the design of studies, and of the presenta­tion, analysis and interpretation of results. The recognition of the value of the evidence-based approach to veterinary medicine has followed a similar emphasis in human clinical medicine, and is influencing the whole veterinary profession. Accordingly, it is also very much a part of the mainstream veterinary curriculum. Whether you are a practitioner of veterinary medicine or of one of the allied sciences, you will now more than ever need to be conversant with modern biostatistical analysis. Knowing how best to report your own results is also vital if you are to impart knowledge correctly, and so, to this end, we include in Chapter 15 a section on the CONSORT Statement, designed to standardize clinical trial reporting.

Although we refer only to two common statistical packages in the text, SPSS and Stata, sufficient information is given to interpret output from other packages, even though the layout and content may differ to some degree. We have also mentioned a number of websites containing useful information, and which were correct at the time of printing. Given the mutability of the internet, we cannot guarantee that such sites will stay available.

Also included with this edition is a CD containing the data sets used as examples in the text. You can use these data sets to consolidate the learning process. It is only when you attempt the analyses yourself that you are fully able to get to grips with the techniques. Each data set is presented in four different formats (ASCII, Excel, SPSS and Stata), so you should be able to access the data and use the software that is available to you.

We would like to acknowledge the generosity of the late Dr Penny Barber, Mark Corbett, Dr J. E. Edwards, Professor Jonathan Elliott, Professor Gary England, Dr Oliver Garden, Dr Ilke Klaas, Dr Teresa Martinez, Dr Anne Pearson, Dr P. D. Warriss, Professor Avril Waterman-Pearson and Dr Susannah Williams who shared their original data with us, and to others who have allowed us to use their published data. In places, we have taken published summary data and constructed a primary data set to suit our own purposes; if we have misrepresented our colleagues’ data, we accept full responsibility. We are particularly grateful to Alex Hunte who lent us his skills in refining the illustrations in the first edition, and to Dr David Moles who assisted with the preparation of the statistical tables. We especially thank Dr Ben Armstrong, Professor Caroline Sabin and Dr Ian Martin who kindly gave us their critical advice as the text of the first edition was developed, and Professor John Smith who was instrumental in getting us to consider writing the book in the first place. In addition, we acknowledge our debt to a host of other colleagues who have helped with discussions over the telephone, with their expertise in areas we are lacking, and in their encouragement to complete what we hope will be a useful contribution to the field of veterinary and animal science. We are particularly indebted to those of our colleagues who have graciously pointed us to our errors, which we hope are now corrected.

Lastly, we again acknowledge with gratitude the patience and encouragement of our marriage partners, Gerald and Rosie, and our children, Nina, Andrew and Karen, and Oliver and Anna, who have once more graciously allowed us to become absorbed in the book and have had to suffer neglect in the process. We trust that they still appreciate the worthiness of the cause!

Although statistics is anathema to many, it is, unquestionably, an essential tool for those involved in animal health and veterinary science. It is imperative that practitioners and research workers alike keep abreast with reports on animal production, new and emerging diseases, risk factors for disease and the efficacy of the ever-increasing number of innovations in veterinary care and of developments in training methods and performance. The most cogent information is usually contained in the appropriate journals; however, the usefulness of these journals relies on the reader having a proper understanding of the statistical methodology underlying study design and data analysis. The modern animal scientist and veterinary surgeon therefore need to be able to handle numerical data confidently and properly. Often, for us, as teachers, there is little time in busy curricula to introduce the subject slowly and systematically; students find they are left bewildered and dejected because the concepts seem too difficult to grasp. While there are many excellent introductory books on medical statistics and on statistics in other disciplines such as economics, business studies and engineering, these books are unrelated to the world of animal science and health, and students soon lose heart. It is our intention to provide a guide to statistics relevant to the study of animal health and disease. In order to illustrate the principles and methods, the reader will find that the text is well endowed with real examples drawn from companion and agricultural animals. Although veterinary epidemiology is closely allied to statistics, we have concentrated only on statistical issues as we feel that this is an area which, until now, has been neglected in veterinary and animal health sciences.

Our book is an introductory text on statistics. We start from very simple concepts, assuming no previous knowledge of statistics, and endeavour to build up an understanding in such a way that progression on to advanced texts is possible. We intend the book to be useful for those with­out mathematical expertise but with the ability to utilize simple formulae. We recognize the influence of the computer and so we avoid the description of complex hand calculations. Instead, emphasis is placed on understanding of concepts and interpretation of results, often in the context of computer output. In addition to acquiring an ability to perform simple statistical techniques on original data, the reader will be able critically to evaluate the efforts of others in respect of the design of studies, and of the presentation, analysis and interpretation of results. The book can be used either as a self-instructional text or as a basis for courses in statistics. In addition, those who are further on in their studies will be able to use the text as a reference guide to the analysis of their data, whether they be postgraduate students, veterinary practitioners or animal scientists in various other settings. Every section contains sufficient cross referencing for the reader to find the relevant background to the topic.

We would like to acknowledge the generosity of Penny Barber, Mark Corbett, Dr J. E. Edwards, Dr Jonathan Elliott, Dr Gary England, Dr Oliver Garden, Dr Anne Pearson, Dr P. D. Warriss, Professor Avril Waterman-Pearson and Susannah Williams, who shared their original data with us. In places, we have taken published summary data and constructed a primary data set to suit our own purposes; if we have misrepresented our colleagues’ data, we accept full responsibility. We are particularly grateful to Alex Hunte who lent us his skills in preparing the illustrations, and to Dr David Moles who assisted with the preparation of the statistical tables. We especially thank Dr Ben Armstrong, Dr Caroline Sabin and Dr Ian Martin who kindly gave us their critical advice as the text was developed. Professor John Smith was instrumental in getting us to consider writing the text in the first place, and we thank him for his continual encouragement. In addition, we acknowledge our debt to a host of other colleagues who have helped with discussions over the telephone, with their expertise in areas we are lacking, and in general encouragement to complete what we hope will be a useful contribution to the field of veterinary and animal science.

Lastly, we acknowledge with gratitude the patience and encouragement of our families. Our marriage partners, Gerald and Rosie, have endured with fortitude our neglect of them while this work was in preparation. In particular, our children, Nina, Andrew and Karen, and Oliver and Anna, have had to cope with our absorption with the project and lack of involvement in their activities. We trust they will recognize that it was in a good cause.

This book is accompanied by a companion website:

www.wiley.com/go/petrie/statisticsforvets

The website includes:

• Data files which relate to some of the examples in the text. Each data file is provided for download in four different formats: ASCII, Excel, SPSS and Stata.
• Examples relating to the data files are indicated in the text using the following icon:

1.1 Learning objectives

By the end of this chapter, you should be able to:

• State what is meant by the term ‘statistics’.
• Explain the importance of a statistical understanding to the animal scientist.
• Distinguish between a qualitative/categorical and a quantitative/numerical variable.
• List the types of scales on which variables are measured.
• Explain what is meant by the term ‘biological variation’.
• Define the terms ‘systematic error’ and ‘random error’, and give examples of circumstances in which they may occur.
• Distinguish between precision and accuracy.
• Define the terms ‘population’ and ‘sample’, and provide examples of real (finite) and hypo­thetical (infinite) populations.
• Summarize the differences between descriptive and inferential statistics.

1.2 Aims of the book

1.2.1 What will you get from this book?

All the biological sciences have moved on from simple qualitative description to concepts founded on numerical measurements and counts. The proper handling of these values, leading to a correct understanding of the phenomena, is encompassed by statistics. This book will help you appreciate how the theory of statistics can be useful to you in veterinary and animal science. Statistical techniques are an essential part of communicating information about health and disease of animals, and their agricultural productivity, or value as pets, or in the sporting or working environment. We, the authors, aim to introduce you to the subject of statistics, giving you a sound basis for managing straightforward study design and analysis. Where necessary, we recommend that you extend your knowledge by reference to more specialized texts. Occasionally, we advocate that you seek expert statistical advice to guide you through particularly tricky aspects.

You can use this book in two ways:

1. The chapter sequence is designed to develop your understanding systematically and we therefore recommend that, initially, you work through the chapters in order. You will find certain sections marked in small type with a symbol, which indicates that you can skip these, at a first read through, without subsequent loss of continuity. These marked sections contain information you will find useful as your knowledge develops. Chapters 11, 14 and 15 deal with particular types of analyses which, depending on your areas of interest, you may rarely need.

2. When you are more familiar with the concepts, you can use the book as a reference manual; you will find sufficient cross-referenced information in any section to answer specific queries.

1.3 What is statistics?

The number of introductory or elementary texts on the subject of statistics indicates how important the subject has become for everyone in the biological sciences. However, the fact that there are many texts might also suggest that we have yet to discover a foolproof method of presenting what is required.

The problem confronted in biological statistics is as follows. When you make a set of numerical observations in biology, you will usually find that the values are scattered. You need to know whether the values differ because of factors you are interested in (e.g. treatments) or because they are part of a ‘background’ natural variation. You need to evaluate what the numbers actually mean, and to represent them in a way that readily communicates their meaning to others.

The subject of statistics embraces:

• The design of the study in order that it will reveal the most information efficiently.
• The collection of the data.
• The analysis of the data.
• The presentation of suitably summarized information, often in a graphical or tabular form.
• The interpretation of the analyses in a manner that communicates the findings accurately.

Strictly, this broad numerical approach to biology is correctly termed ‘biometry’ but we shall adopt the more generally used term ‘statistics’ to cover all aspects. Statistics (meaning this entire process) has become one of the essential tools in modern biology.

1.4 Statistics in veterinary and animal science

One of the common initial responses of both veterinary students and animal science students is: Why do I need to study statistics? The mathematical basis of the subject causes much uncertainty, and the analytical approach is alien. However, in professional life, there are many instances of the relevance of statistics:

• The published scientific literature is full of studies in which statistical procedures are employed. Look in any of the relevant scientific journals and notice the number of times reference is made to mean ± SEM (standard error of mean), to statistical significance, to P-values or to t-tests or Chi-squared analysis or analysis of variance or multiple regression analysis. The information is presented in the usual brief form and, without a working knowledge of statistics, you are left to accept the conclusions of the author, unable to examine the strength of the supporting data. Indeed, with the advent of computer-assisted data handling, many practitioners can now collect their own observations and summarize them for the advantage of their colleagues; to do this, they need the benefit of statistical insights.
• The subject of epidemiology (see Section 5.2) is gaining prominence in veterinary and animal science, and the concepts of evidence-based veterinary medicine (see Section 1.5 and Chapter 16) are being explicitly introduced into clinical practice. As never before, there is an essential need for you to understand the types of trials and investigations that are carried out and to know the meaning of the terms associated with them.
• In the animal health sciences, there are an increasing number of independent diagnostic services that will analyse samples for the benefit of health monitoring and maintenance. Those running such laboratory services must always be concerned about quality control and accuracy in measurements made for diagnostic purposes, and must be able to supply clear guidelines for the interpretation of results obtained in their laboratories.
• The pharmaceutical and agrochemical industries are required to demonstrate both the safety and the efficacy of their products in an indisputable manner. Such data invariably require a statistical approach to establish and illustrate the basis of the claim for both these aspects. Those involved in pharmaceutical product development need to understand the importance of study design and to ensure the adequacy of the numbers of animals used in treatment groups in order to perform meaningful experiments. Veterinary product licensing committees require a thorough understanding of statistical science so that they can appreciate the data presented to substantiate the claims for a novel therapeutic substance. Finally, practitioners and animal carers are faced with the blandishments of sales representatives with competing claims, and must evaluate the literature which is offered in support of specific agents, from licensed drugs to animal nutrition supplements.
• Increasingly, there is concern about the regulation of safety and quality of food for human consumption. Where products of animal origin are involved, the animal scientist and the veterinary profession are at the forefront. Examples are: pharmaceutical product withdrawal times before slaughter based on the pharmacokinetics and pharmacodynamics of the products, the withholding times for milk after therapeutic treatment of the animal, tissue residues of herbicides and insecticides, and the possible contamination of carcasses by antibiotic-resistant bacteria. In every case, advice and appropriate regulations are established by experimental studies and statistical evaluation. The experts need to be aware of the appropriate statistical procedures in order to play their proper roles.

In all these areas, a common basic vocabulary and understanding of biometrical concepts is assumed to enable scientists to communicate accurately with one another. It is important that you gain mastery of these concepts if you are to play a full part in your chosen profession.

1.5 Evidence-based veterinary medicine

The veterinary profession is following the medical profession in introducing a more objective basis to its practice. Under the term evidence-based veterinary medicine (EBVM) – by which we mean the conscientious, explicit and judicious use of current best evidence to inform clinical judgements and decision-making in veterinary care (see Cockcroft and Holmes, 2003) – we are now seeing a move towards dependence upon good scientific studies to underpin clinical decisions. In many ways, practice has implicitly been about using clinical experience to make the best decisions, but what has changed is the explicit use of the accessible information. No longer do clinicians have to depend on their own clinical experience and judgement alone; now they can benefit from other studies in a formalized manner to assist their work. The clinician has to know what information is relevant and how to access this evidence, and be able to use rigorous methods to assess it. Generally, this requires a familiarity with the terminology used and an understanding of the principles of statistical analysis. Moreover, the wider world of animal science is finding a need to understand these ideas as the evidence-based concepts are being applied not only in the treatment of clinical disease but also in aspects of production and performance.

One of the differences between the applica­tion of EBVM in veterinary science and in human medicine is that in the latter the body of literature is now very large, and this makes finding relevant information easier. In the vet­erinary field, EBVM is still hampered by the relatively small amount and variable quality of the evidence available. Nevertheless, EBVM is gaining momentum, and we have devoted Chapter 16 to its concepts. One of the key requirements of EBVM is reliably reported information and, as in the human medical field, the veterinary publishing field is in the process of consolidating a set of guide­lines for good reporting. We have addressed this in Chapter 17, outlining the information that is available at the time of writing. As critical appraisal of the published literature is invariably an essential component of evaluating evidence, we have devoted Chapter 18 to it. In this chapter, we provide templates for critically appraising randomized controlled trials and observational studies, and invite you to develop your skills by critically appraising two published articles.

1.6 Types of variable

A variable is a characteristic that can take values which vary from individual to individual or group to group, e.g. height, weight, litter size, blood count, enzyme activity, coat colour, percentage of the flock which are pregnant, etc. Clearly some of these are more readily quantifiable than others. For some variables, we can assign a number to a category and so create the appearance of a numerical scale, but others have a true numerical scale on which the values lie. We take readings of the variable which are measurements of a biological characteristic, and these become the values which we use for the statistical procedures. Both these terms are in general use, and both refer to the original measurements, the raw data.

Numerical data take various forms; a proper understanding of the nature of the data and the classification of variables is an important first step in choosing an appropriate statistical approach. The flow charts shown in Appendix E, and on the inside front and back covers, illustrate this train of thought, which culminates in a suitable choice of statistical procedure to analyse a particular data set.

We distinguish the main types of variable in a systematic manner by determining whether the variable can take ‘one of two distinct values’, ‘one of several distinct values’ or ‘any value’ within the given range. In particular, the variable may be one of the following: