Statistics and Scientific Method: An Introduction for Students and Researchers

189 Pages·2011·4.63 MB·English

by  Peter J. Diggle

Statistics and Scientific Method
Statistics and Scientific Method , First Edition, Peter J. Diggle, Amanda G. Chetwynd © Peter J. Diggle,
Amanda G. Chetwynd 2011. Published in 2011 by Oxford University Press
Statistics and Scientific Method
An Introduction for Students and Researchers
PETER J. DIGGLE
and
AMANDA G. CHETWYND
Lancaster University
1
3
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To Mike, especially for Chapter 8
Statistics and Scientific Method , First Edition, Peter J. Diggle, Amanda G. Chetwynd © Peter J. Diggle,
Amanda G. Chetwynd 2011. Published in 2011 by Oxford University Press
Acknowledgements
MostofthediagramsinthebookwereproducedusingR.Fortheremainder,
we thank the following people and organizations:
Figure 2.1. Institute of Astronomy library, University of Cambridge
Figure 4.1. Professor Andy Cossins, University of Liverpool
Figure 5.1. copyright Rothamsted Research Ltd.
Figure5.3.copyrightphotographbyAntonyBarringtonBrown,reproduced
with the permission of the Fisher Memorial Trust
Figure 10.1. Devra Davis (www.environmentalhealthtrust.org)
We have cited original sources of data in the text where possible, but
would like here to add our thanks to: Dr Bev Abram for the Arabadopsis
microarray data; Professor Nick Hewitt for the Bailrigg daily temperature
data;DrRaquelMenezesandDrJos´eAngelFernandezfortheGalicialead
pollutiondata;ProfessorStephenSennfortheasthmadata;CSIROCentre
forIrrigationResearch,Griffith,NewSouthWales,fortheglyphosatedata;
Dr Peter Drew for the survival data on dialysis patients; Professor Tanja
Pless-Mulloli for the Newcastle upon Tyne black smoke pollution data.
Statistics and Scientific Method , First Edition, Peter J. Diggle, Amanda G. Chetwynd © Peter J. Diggle,
Amanda G. Chetwynd 2011. Published in 2011 by Oxford University Press

Preface
Statistics is the science of collecting and interpreting data. This makes it
relevant to almost every kind of scientific investigation. In practice, most
scientific data involve some degree of imprecision or uncertainty, and one
consequence of this is that data from past experiments cannot exactly
predict the outcome of a future experiment. Dealing with uncertainty is
a cornerstone of the statistical method, and distinguishes it from math-
ematical method. The mathematical method is deductive: its concern is
the logical derivation of consequences from explicitly stated assumptions.
Statistical method is inferential: given empirical evidence in the form of
data, its goal is to ask what underlying natural laws could have generated
the data; and it is the imprecision or uncertainty in the data which
makes the inferential process fundamentally different from mathematical
deduction.
As a simple example of this distinction, consider a system with a single
variable input, x, and a consequential output, y. A scientific theory asserts
thattheoutputisalinearfunctionoftheinput,meaningthatexperimental
values of x and y will obey the mathematical relationship
y =a+b×x
forsuitablevaluesoftwoconstants,aandb.Toestablishthecorrectvalues
ofaandb,weneedonlyruntheexperimentwithtwodifferentvaluesofthe
input, x, measure the corresponding values of the output, y, plot the two
points (x,y), connect them with a straight line and read off the intercept,
a,andslope,b,oftheline.Ifthetruthoftheassumedmathematicalmodel
is in doubt, we need only run the experiment with a third value of the
input, measure the corresponding output and add a third point (x,y) to
our plot. If the three points lie on a straight line, the model is correct,
and conversely. However, if each experimental output is subject to any
amount, however small, of unpredictable fluctuation about the underlying
straight-line relationship, then logically we can neither determine a and b
exactly, nor establish the correctness of the model, however many times
we run the experiment. What we can do, and this is the essence of the
statistical method, is estimate a and b with a degree of uncertainty which
diminishesasthenumberofrunsoftheexperimentincreases,andestablish
Statistics and Scientific Method , First Edition, Peter J. Diggle, Amanda G. Chetwynd © Peter J. Diggle,
Amanda G. Chetwynd 2011. Published in 2011 by Oxford University Press
viii PREFACE
the extent to which the postulated model is reasonably consistent with the
experimental data.
In some areas of science, unpredictable fluctuations in experimental
results are a by-product of imperfect experimental technique. This is
presumably the thinking behind the physicist Ernest Rutherford’s much-
quoted claim that ‘If your experiment needs statistics, you ought to have
doneabetterexperiment.’ Perhapsforthisreason,inthephysicalsciences
unpredictable fluctuations are often described as ‘errors’. In other areas
of science, unpredictability is an inherent part of the underlying scientific
phenomenon, and need carry no pejorative associations. For example, in
medicinedifferentpatientsshowdifferentresponsestoagiventreatmentfor
reasonsthatcannotbewhollyexplainedbymeasurabledifferencesamongst
them, suchastheirage, weight orotherphysiological characteristics. More
fundamentally, in biology unpredictability is an inherent property of the
process of transmission and recombination of genetic material from parent
to offspring, and is essential to Darwinian evolution.
It follows that the key idea in the statistical method is to understand
variationindataandinparticulartounderstandthatsomeofthevariation
whichweseeinexperimentalresultsispredictable,orsystematic,andsome
unpredictable,orrandom.Mostformaltreatmentsofstatisticstend,inthe
authors’ opinion, to overemphasize the latter, with a consequential focus
on the mathematical theory of probability. This is not to deny that an
understanding of probability is of central importance to the statistics dis-
cipline, but from the perspective of a student attending a service course in
statisticsanemphasisonprobabilitycanmakethesubjectseemexcessively
technical,obscuringitsrelevancetosubstantivescience.Evenworse,many
service courses in statistics respond to this by omitting the theory and
presenting only a set of techniques and formulae, thereby reducing the
subject to the status of a recipe book.
Our aim in writing this book is to provide an antidote to technique-
oriented service courses in statistics. Instead, we have tried to emphasize
statistical concepts, to link statistical method to scientific method, and to
show how statistical thinking can benefit every stage of scientific inquiry,
from designing an experimental or observational study, through collecting
and processing the resulting data, to interpreting the results of the data-
processing in their proper scientific context.
Each chapter, except Chapters 1 and 3, begins with a non-technical
discussion of a motivating example, whose substantive content is indicated
inthechaptersubtitle.Ourexamplesaredrawnlargelyfromthebiological,
biomedical and health sciences, because these are the areas of application
with which we are most familiar in our own research. We do include
some examples from other areas of science, and we hope that students
whose specific scientific interests are not included in the subject matter
of our examples will be able to appreciate how the underlying statistical
PREFACE ix
concepts are nevertheless relevant, and adaptable, to their own areas of
interest.
Our book has its origins in a service course at Lancaster University
which we have delivered over a period of several years to an audience of
first-year postgraduate students in science and technology. The scientific
maturity of students at this level, by comparison with undergraduates,
undoubtedly helps our approach to succeed. However, we do not assume
anypriorknowledgeofstatistics,nordowemakemathematicaldemandson
ourreadersbeyondawillingnesstogettogripswithmathematicalnotation
(itself a way of encouraging precision of thought) and an understanding of
basic algebra.
Even the simplest of statistical calculations requires the use of a com-
puter;andiftediumistobeavoided,thesameappliestographicalpresen-
tationofdata.Ourbookdoesnotattempttoteachstatisticalcomputation
inasystematicway.Manyoftheexercisescouldbedoneusingpencil,paper
and pocket calculator, although we hope and expect that most readers of
the book will use a computer.
We have, however, chosen to present our material in a way that will
encourage readers to use the R software environment (see the website
www.r-project.org). From our perspectives as teachers and as professional
statisticians,Rhasanumberofadvantages:itsopen-sourcestatus;thefact
that it runs on most platforms, including Windows, Macintosh and Linux
operating systems; its power in terms of the range of statistical methods
that it offers. Most importantly from a pedagogical perspective, using R
encourages the open approach to problems that the book is intended to
promote, and discourages the ‘which test should I use on these data’ kind
ofclosedthinkingthatweverymuchwanttoavoid.Ofcourse, Risnotthe
only software environment which meets these criteria, but it is very widely
used in the statistical community and does seem to be here to stay.
WehaveprovideddatasetsandRscripts(sequencesofRcommands)that
will enable any reader to reproduce every analysis reported in the book.
This material is freely available at: www.lancs.ac.uk/staff/diggle/intro-
stats-book.
We hope that readers unfamiliar with R will either be able to adapt
these datasets and programmes for their own use, or will be stimulated to
learn more about the R environment. But we emphasize that the book can
be read and used without any knowledge of, or reference to, R whatsoever.