Practical Statistics for Environmental and Biological Scientists

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3 Practical Statistics for Environmental and Biological Scientists

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5 Practical Statistics for Environmental and Biological Scientists John Townend University of Aberdeen, UK JOHN WILEY & SONS, LTD

6 Copyright 2002 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex POl9 8SQ, England Telephone (+44) (for orders and customer service enquiries): Visit our Home Page on or Reprinted with corrections March Reprinted July 2004, April 2005, April 2006, April 2007, October 2008, April 2009 All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London WIT 4LP, UK, without the permission in writing of the Publisher. Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex POl9 8SQ, England, or ed to permreq@wiley.co.uk, or faxed to (+44) This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the Publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Other Wiley Editorial Offices John Wiley & Sons Inc., III River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA , USA Wiley-VCH Verlag GmbH, Boschstr. 12, D Weinheim, Germany John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02-01, Jin Xing Distripark, Singapore John Wiley & Sons Canada Ltd, 22 Worcester Road, Etobicoke, Ontario, Canada M9W I L I British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 13: (HB) ISBN 13: (PB) Typeset in 10.5 I 13pt Times by Vision Typesetting Manchester

7 Contents Preface ix PART I STATISTICS BASICS 1 1 Introduction Do you need statistics? What is statistics? Some important lessons I have learnt Statistics is getting easier I ntegrity in statistics About this book 8 2 A Brief Tutorial on Statistics Introduction Variability Samples and populations \0 2.4 Summary statistics The basis of statistical tests Limitations of statistical tests 24 3 Before You Start Introduction What statistical methods are available? Surveys and experiments Designing experiments and surveys - preliminaries Summary 43 4 Designing an Experiment or Survey Introduction Sample size Sampling Experimental design Further reading 60 5 Exploratory Data Analysis and Data Presentation Introduction Column graphs Line graphs Scatter graphs 69

8 VI Contents 5.5 General points about graphs Tables Standard errors and error bars 74 6 Common Assumptions or Requirements of Data for Statistical Tests Introduction Common assumptions Transforming data 84 PART II STA TISTICAL METHODS 91 7 I-tests and F-tests Introduction Limitations and assumptions t-tests F -test Further reading Analysis of Variance Introduction Limitations and assumptions One-way ANaYA III 8.4 Multiway ANaYA Further reading Correlation and Regression Introduction Limitations and assumptions Pearson's product moment correlation Simple linear regression Correlation or regression? Multiple linear regression Comparing two lines Fitting curves Further reading Multivariate ANOVA Introduction Limitations and assumptions Null hypothesis Description of the test Interpreting the results Further reading Repeated Measures Introduction Methods for analysing repeated measures data 166

9 Contents vii 11.3 Designing repeated measures experiments Further reading Chi-square Tests Introduction Limitations and assumptions Goodness of fit test Test for association between two factors Comparing proportions Further reading Non-parametric Tests Introduction Limitations and assumptions Mann-Whitney U-test Two-sample Kolmogorov-Smirnov test Two-sample sign test Kruskal-Wallis test Friedman's test Spearman's rank correlation Further reading Principal Component Analysis Introduction Limitations and assumptions Description of the method Interpreting the results Further reading Cluster Analysis Introduction Limitations and assumptions Clustering observations Clustering variables Further reading 228 APPENDICES 229 A Calculations for statistical tests 231 B Concentration data for Chapters 14 and C Using computer packages 249 D Choosing a test: decision table 261 E List of worked examples 265 Bibliography 271 Index 273

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11 Preface Statistics wasn't forced upon the environmental and biological sciences; it has been absorbed into their practice because it was realized that it had something to offer. Statistical methods provide us with ways of summarizing our data, objective methods to decide how much confidence we can place in experimental results, and ways of uncovering patterns that are initially masked by the complexity of a dataset. Also, if we carry out scientific investigations according to our instincts, there is a risk that we will bias the results by overlooking some important factor or through our desire to get a particular result. By carefully following accepted statistical procedures we can avoid these problems and, just as importantly, we will be seen to have avoided them, so our results will be more readily accepted by others. Statistics is also a useful means of communication. For example, a researcher might state that 'the molluscs had a mean shell length of 12.2 mm ± 1.6 mm (standard error)" or report that 'ANOVA showed significant differences between nitrogen contents in different groups of plants (P = 0.02)'. These are succinct ways of explaining a great deal of detail about how studies have been carried out and what can be concluded from them. Of course, they are only really a useful means of communication if you understand what the terms mean. Like it or not, though, they are widely used, so whether you intend to use statistics yourself or just read about others' research, it will still be a great help to know something about it. While teaching statistics in a university I found that, for the most part, the statistical methods required by both environmental and biological scientists were the same. Indeed this might be expected, because much of the science is common to both as well. I also found that requirements were very similar at all levels from undergraduate to experienced professional. Really there is seldom any necessity to use complex statistical methods to do world-class research in environmental and biological sciences. Those who are able to identify the key, simple questions to ask are likely to enjoy the greatest success. So it is that I ha ve tried to put together a book that addresses as many of the most common needs as possible. The choice of content is based on the questions I have most frequently been asked and the explanations that seemed to work best. Memorizing formulae will be of very little practical use to you, except perhaps to pass an exam; most calculations can be carried out by computer these days. However, computers do

12 x Preface not generally tell you whether you are carrying out the right calculations or exactly what you can conclude from the results. Here textbooks still have a part to play. In this book I try to unlock many of the codes commonly used to present scientific information and to provide you with the tools you need to be an effective user of statistics yourself. I wholeheartedly hope that it will provide you with the information you need.

13 PART I STATISTICS BASICS Chapters 1 to 6 introduce the ideas behind statistical methods and how practical studies should be set up to use them. They aim to give the required background for using the methods in Part II. Readers who are new to statistics or in need of a short refresher might find it useful to read this part in its entirety.

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15 1 Introduction UW lin Em _Mit If your first love was statistics, you probably wouldn't be studying or working in environmental or biological sciences. I am starting from this premise. 1.1 Do you need statistics? Somebody who is trying to sell you a statistics textbook is probably not the best person to ask whether you need statistics. Maybe you have opened this book because you have an immediate need for these techniques or because you have to study the subject as part of a course. In this case the answer for you is clearly yes, you need statistics. Otherwise, if you want to know whether statistics is really relevant to you, ask people who have been successful in your chosen area - academics, researchers or people doing the kind of job you want to do in the future. Some use it more than others, and certainly you will find some very successful people who are not confident with statistics and possibly dislike any involvement with it. I don't believe being a brilliant statistician is a necessary condition for being a brilliant biologist or environmental scientist. However, you will probably find that most of the people you ask would have found it useful to understand statistics at some stage in their career, perhaps very regularly. Even if you do not need it to present results yourself, you will need to understand some statistics in order to understand the real meaning of almost any scientific information given to you. The fact that most university degrees in environmental and biological sciences include a compulsory statistics course is simply a recognition of this. However, do not think that understanding statistics is all or nothing. Even a basic understanding of why and when it is used will be very valuable. If you can grasp the detail too, so much the better.