Encyclopedia of Biostatistics || Statistics, Overview

Statistics, Overview

This article is an inevitably personal view of the gen-eral position of statistical science, as seen in the mid-1990s, a period of rapid development. Even withinmedical statistics the range of applications is great,and in statistics more broadly the variety is even moreextreme, making sweeping statements about the rela-tive importance of, for example, different techniquesand different approaches difficult to substantiate inany generality.

The term statistical science is sometimes used forstatistical theory and its applications to the naturaland social sciences and to science-based technologyand this roughly corresponds to the scope of thepresent article.

The interlinked pillars of statistics as a field ofstudy are

1. The mathematics of probability.2. The general principles for the design, analysis

and interpretation of investigations. Formal prin-ciples of statistical inference are a part, but onlya part, of this.

It may be tempting to add a third pillar, which gavethe subject its name, namely the collection and studyof economic and social statistics for government, so-called official statistics, and the closely related issuesconcerning large enterprises. While this aspect of thesubject has indeed developed rather separately overmany years, at some level the general principles seemunlikely to be different from those involved withapplications to science and science-based technology.For medical statisticians official statistics connectedwith health have always been important (see VitalStatistics, Overview), and if the term biostatisticsis interpreted more widely to include, for example,agricultural statistics, then there are other links withofficial statistics.

Probability

The first part of the article thus concerns the math-ematics of probability; that is, issues concernedwith the meaning and philosophy of probability areexcluded.

Historically and, often but not always, in introduc-tory teaching, probability starts from combinatorial

problems, i.e. from the counting of the proportionof “favorable” cases in the enumeration of a set ofpossibilities assumed equally likely a priori. Mod-ern probability theory has blossomed from that intoa rich chapter of modern mathematics with links toother areas of pure mathematics. Some of the moderndevelopments are, at the moment, fairly remote fromstatistical applications although study at a relativelyadvanced level is required:

1. To derive and underpin various statistical meth-ods. Instances where mathematically elaboratemethods have been deployed include the use ofmartingale theory in connection with survivaldata [1] (see Counting Process Methods inSurvival Analysis) and more generally the rigor-ous derivation of limiting results in semiparamet-ric inference (see Semiparametric Regression).

2. To derive special stochastic models for phe-nomena, usually systems developing in time(see Stochastic Processes; Epidemic Models,Stochastic).

We deal here with the second of these.Stochastic models supply an important route for

developing mathematical models of systems involv-ing a nontrivial random element, both for the insightthat the models themselves can supply and as a basisfor introducing a substantive base into the interpreta-tion of empirical data.

Among the fields in which such work has a solidhistory combined with much current activity are

1. Epidemic theory, again with a long history, withdevelopments up to 1970 summarized by Bai-ley [4], and with major recent developments [26]stemming largely but not entirely from the studyof AIDS [10, 2]. For an application to BSE(bovine spongiform encephalopathy), see [3].

2. The study of congestion and more broadly inoperational research, dating back to the workof A.K. Erlang at the Copenhagen TelephoneCompany and congestion theory being stimulatednowadays by the study of complex networks.

3. Finance theory.4. Genetics, for example, in particular, phylogenet-

ics and genetic epidemiology.5. A number of other areas of mathematical biol-

ogy, such as competition processes, includingpredator–prey models.

6. Geomorphology and hydrology.

Encyclopedia of Biostatistics, Online © 2005 John Wiley & Sons, Ltd.This article is © 2005 John Wiley & Sons, Ltd.This article was published in the Encyclopedia of Biostatistics in 2005 by John Wiley & Sons, Ltd.DOI: 10.1002/0470011815.b2a00006

2 Statistics, Overview

7. Statistical physics, with a very long historyof the use of probabilistic ideas, often in away that seems idiosyncratic from the viewpointof other developments in probability theory.More recently, however, there has been rathermore contact between statistical physics andthe mainstream of work in stochastic processes.Two fundamental problems in physics, i.e. thefoundations of quantum theory and the natureof the process generating turbulence, both seemlikely to have some stochastic element to them.

In the present context, it is helpful to draw a roughdistinction between four types of probability model:

1. Purely empirical models.2. “Toy” models.3. Intermediate models.4. Quasi-realistic models.

Models for rainfall provide a convenient illustra-tion. A purely empirical model for, say, daily rainfallat a single site [41] might specify the binary sequenceof (no rain, rain) as an m-dependent Markov chainwith seasonally varying transition matrix and theamount of rain, conditionally on its being nonzero,as having a lognormal distribution or gamma dis-tribution with seasonally varying parameters. Thismight provide a valuable and accurate representationof the frequency properties of the rainfall process,but there would be no direct link with the underly-ing physical process or corresponding interpretationof the parameters.

A “toy” model is one in which a highly idealizedrepresentation is used to explore the particular cir-cumstances under which a phenomenon of interestcould be generated from simple starting assumptions.Examples are the use of idealized cascade models(clusters of clusters of . . .) to show some conditionsunder which scaling, i.e. self-similarity, of the rain-fall spatial field can occur [21], and simple modelsshowing conditions for the explosion or extinctionof epidemics, or the extinction of species by com-petition. Elaborate fitting to empirical data is ofteninappropriate.

An intermediate model is one in which someaspects of a complex physical or other process arerepresented with the objective of obtaining a formthat can be fitted to empirical data in such a waythat the resulting parameter estimates do have a linkwith the underlying generating process. An example

with rainfall is the use of models in which there isa Poisson process of storms. Each storm consists ofa random number of rain cells, displaced from thestorm origin, each cell being of random duration anddepth, the total rainfall depth consisting of the sum ofall contributing cells. The notion of a rain cell has aphysical interpretation and the resulting process canproduce a reasonable fit to the rather complicatedtime series of say five-minute rainfalls, in whichwithin periods of rain there is a large highly non-Gaussian distribution of intensity interspersed withshort periods of zero rainfall. The models [36] alsohave the major advantage that they can be generalizedto spatial–temporal form [14].

A corresponding quasi-realistic model would bea global circulation model in which the nonlinearpartial differential equations representing the phys-ical processes involved are solved numerically [30].Similar models involving complex processes are usedin studying global warming [24] and many othertypes of system, physical, biological, or economic.The models are frequently, although not inevitably,deterministic rather than having an explicit stochasticelement.

A few general issues in this broad area of workare as follows.

1. When is the introduction of a stochastic elementinto a model likely to be crucial, i.e. when aredeterministic models broadly adequate?

2. The relationships between a deterministicmodel and a roughly corresponding stochasticmodel [44, 25] are important in settling the kindof formulation suitable. For models consistingof linearly superimposable components thedeterministic model gives the correspondingstochastic mean, but even then the mean may, forsmall systems, give a poor idea of the behaviorof sample paths. For nonlinear systems, suchas epidemic models, the deterministic modelgives an approximation to the stochastic meanvalid in large systems (see Epidemic Models,Deterministic).

3. “Toy” models can be highly enlightening. (Theterm “toy” should not be taken pejoratively!)How can they best be used in combinationwith more realistic and elaborate models? Oneroute is in the interpretation of results from acomplex simulation model by examining theratio of relevant response variables as simulated


Statistics, Overview 3

to those predicted by a “toy” model. This ratio, orcorrection factor, may be expected to vary muchmore slowly with relevant parameters than theresponse itself [12].

4. Issues arise over the fitting of intermediate mod-els in that likelihood functions may be hard tocompute and, not always relevant and estimatingfunctions constructed by equating observed andfitted features may have a strong element of arbi-trariness.

The study of deterministic models, of stochasticmodels and of the analysis of empirical data has oftenbeen undertaken by separate groups of investigators.While the reasons for this may be clear the separationis to be regretted.

Design, Analysis, and Interpretation

Preliminaries

The remainder of the article is concerned with thedesign, analysis, and interpretation of investigations.In a broad sense the same principles apply to exper-iments, observational studies, and the secondaryanalysis of data collected for some not very specificresearch purpose, for example a large family expen-diture survey or a cancer registry. We assume thatthe starting point is a question or issue or researchhypothesis of interest, although sometimes prelim-inary analysis may be needed to clarify the issueinvolved. In all types of study the key initial questionsof design are

1. What individuals should be studied?2. What properties should be measured and what do

the measurements really mean?3. What contrasts, including interactions, should

be examined?

The broad requirements are

1. The avoidance of systematic error.2. The control of random error.3. The exploitation of the factorial structure of

contrasts (see Factorial Experiments).4. The formulation of special objectives.

So far as the last point is concerned we can forthe most part regard the purpose to be the estimationof relevant parameters and corresponding standard

errors, but a specific decision or prediction objectivemay alter the whole focus of the study. For example,the design of a plant-breeding programme for varietalselection would involve quite different considerationsfrom that for the comparison of a small numberof specific varieties. In the former, emphasis is tobe placed on the properties of the small number ofvarieties ultimately chosen for intensive investigationrather than on specific internal comparisons among asmall number of varieties.

The differing relative emphasis on the aboverequirements, especially the first three, explains whythe literature on design appears so different in theclinical trial context from that in, say, the chemicalengineering field. In the former, but not the latter,avoidance of systematic error is of key importance.In the design and analysis of observational studiestoo, attempts to eliminate systematic error are oftenof central importance.

It is disappointing that awareness of some of thebasic principles of design has not percolated morewidely into the laboratory sciences. Even in physics,where investigations of great subtlety are common,the widely held view that refinement of laboratorytechnique is always to be preferred to statisticaltechnique as a base for error control is probably muchless valid than it used to be.

The reason for neglect of the statistical aspects ofthe design of investigations may partly be that thetheory of statistical design is quite widely identifiedwith the use of complex designs. These have theirplace, but they are often not appropriate, key issuesmore commonly being simple techniques for method-ical bias elimination and error control.

Measurement Issues

The techniques of analysis connected with variancecomponents were developed in the 1930s and 1940stied to balanced data and continuous roughly norma-lly distributed data. The restriction to balanced datawas removed in pioneering work by C.R. Hendersonin connection with animal breeding and synthesizedmost satisfactorily in the residual (or restricted)maximum likelihood (REML) method of Pattersonand Thompson [32]. Systematic extensions to Pois-son, ordinal, and binomial data are the focus ofcurrent work [29].

Such techniques provide the basis for the designand analysis of interpersonal and interlaboratory



studies of measuring techniques, common in somefields of study, especially connected with the physicalsciences, but regrettably less common in a medicalcontext.

They also are in principle appropriate in instru-ment development. The psychometric notions ofreproducibility, and of face, criterion, and conceptvalidity and of comparison with a gold standardare not normally presented in terms of compo-nents of variance but would probably be betterdone so (see Psychometrics, Overview). Systematictechniques for the more detailed analysis of instru-ments consisting of many relatively similar items areneeded. This is especially relevant in connection withquality of life, i.e. health status [13].

Classical work on error assessment in experimentsand surveys emphasized the often multilevel or timeseries structure of error. Treating error variables asindependent and identically distributed typically leadsto an underestimate of the standard error of contrastsof primary interest. A more empirical adjustmentfor such overdispersion may be via the direct esti-mation of correction factors to apply to standarderrors [6, 31, 19].

Methods of Analysis

Some methods of analysis do not depend on anexplicit probability model and recent developments,especially in computer graphics, are of interest bothfor exploratory work and also in presenting the con-clusions of more elaborate analyses; indeed, as invery elaborate analyses the connection between thedata and the conclusions may get rather remote,the need for insightful methods of presentationincreases.

Nevertheless the rest of this section concentrateson methods that depend at least in part on an explicitprobabilistic base.

Some requirements for a probabilistic model areas follows, although not all are relevant in everyapplication.

1. The model should establish a link with underly-ing substantive knowledge or theory.

2. The model should allow comparisons with pre-vious related studies of the topic.

3. The model should be consistent with or suggesta possible process that might have generated thedata.

4. Parameters defining primary features of the sys-tem should have individually clear substantiveinterpretations.

5. The error structure should be represented suffi-ciently realistically that meaningful measures ofprecision are obtained for the primary compar-isons.

6. The fit to data should be adequate.

We comment here on only some of these points.The first three items are related to the general issue

of preferring what we previously called intermediatemodels to purely empirical models. Such a prefer-ence was indicated in much of the applied work ofJ. Neyman; it can have the disadvantage of makingthe analysis of fairly simple sets of data overcompli-cated and there is a difficult broad strategical issue tobe faced in each application concerning the weight tobe placed on substantive vs. purely empirical models.

In fields with a quantitative theoretical base thiswill typically provide a key to a suitable model. Inthe social sciences and in some areas of biology thereis often the problem of incorporating backgroundknowledge that is essentially qualitative. Here theideas of chain graphs, expressing directional relation-ships between variables and of substantive researchhypotheses [43] expressing some conditional inde-pendencies and some strong dependencies provide aroute to insertion of such knowledge. The graph the-ory ideas are a development from Sewell Wright’spath analysis. For accounts with a strong statisticalfocus, see [18, 15] and [16], and for a more theoreti-cal account see [28]. Spiegelhalter et al. [39] discussapplications to probabilistic expert systems.

The need to connect to previous work is insuperficial conflict with the Fisherian notion thatinvestigations provide their own estimate of error.However, the need to relate the primary conclu-sions in different studies is clear; this includesthe examination of consistency of the conclusions.There is a broad connection with overviews, or so-called meta-analysis, of much current interest inmedical research. The statistical principles were setout by Yates and Cochran [45] and developed fur-ther by Cochran [11]. The most challenging issuesthere, however, concern the choice of material forsynthesis.

Models suggesting or consistent with a data-generating process provide some possible link witha causal interpretation; see the further discussion



below. While the term causal has a number of inter-pretations, a very cautious usage tends to be favoredin statistical discussions, in particular that strongevidence for causality can only come from the syn-thesis of different kinds of data. Analysis that pointstowards a potentially causal interpretation can, how-ever, be valuable; this is one reason for the impor-tance of chain graph representations (see Causation).

The preference for primary parameters that haveindividual interpretations links to the previous pointand to the Fisherian notion that in analysis ofvariance “treatment” sums of squares should ifpossible be split into single degrees of freedom.

Detailed Techniques

Specific developments in methods of analysis aredescribed throughout this Encyclopedia. Amongsome predominant themes in current research are thefollowing:

1. Nonlinearity is a widely occurring theme, asin so many areas of modern mathematical sci-ence. Nonlinear time series models provide oneimportant example;

2. Nonparametric regression and density estima-tion have been intensively studied in their theo-retical aspects. The main value in applications islikely to be in the preliminary stages of analysis.

3. Semiparametric methods in which the primaryaspects of the model are represented by param-eters, and such issues as distributional formare left nonparametric, raise very interestingtheoretical issues. It is, of course, for consid-eration in each case whether the greater com-plication and loss of transparency involved ascompared with fully parametric formulations isreally justified.

4. Markov chain Monte Carlo methods provide apowerful general tool for the fitting of relativelycomplex models.

5. Computer simulation methods, cross-validation,and the bootstrap, provide fairly generalmethods of assessing precision without elabo-rate theoretical analysis, although some under-lying assumed simple structure is needed(see Computer-intensive Methods).

6. Methods for addressing data imperfections, suchas missing data, including selective nonre-sponse, and for the analysis of nonstandard sam-pling schemes, are important in many fields.

7. Higher-order asymptotic theory [5] aims to pro-vide a basis for choosing between proceduresequivalent to the first order of asymptotic the-ory and of providing more refined distributionalapproximations (see Large-sample Theory).

Interpretation

Design, analysis, and interpretation might suggest asequence in which narrowly statistical considerationsstop at analysis, for example ending with the esti-mation of relevant parameters, whereas interpretationinvolves essentially subject-matter considerations notspecifically statistical. While, of course, there is sometruth to this distinction, the emphasis on model for-mulation, in particular some of the criteria listedunder Methods of Analysis, and the considerationsof the section on Probability, are aimed, in line withcurrent thinking, to break down that barrier.

For example, there has been increasing discussionin statistical circles of the conditions under whichconclusions can be said to be causal; see, for exam-ple, [23, 38] and [16], Section 8.7. The discussion isprompted in part by developments in the computerscience and philosophical literature in which a weakerdefinition of causality tends to be employed [33, 34,40], much less cautious than the traditional statisti-cal and epidemiological view summarized in Hill’scriteria [22] (see Causation).

Closely connected with this are the issues of gen-eralizability and specificity and the importance ofabsence of interaction. For example, in the light of awell-conducted randomized clinical trial showing evi-dence of the superiority of treatment A over treatmentB, what is the basis for hoping that the conclusionsgeneralize to a new population of patients and whatis the basis for thinking that A rather than B will bebeneficial for a specific new patient? (See Validityand Generalizability in Epidemiologic Studies.)

Decision Analysis

Wald [42], in effect continuing in the Neyman–Pear-son tradition, proposed that all statistical problemscould be formulated as a choice between possi-ble decisions (see Decision Theory). In Wald’s for-mulation a utility (or loss) function was assumedknown but prior distributions enter only as tech-nical devices to produce a complete class of decisionfunctions.



The qualitative ideas that one should consider theobjectives of the study, the possible actions that mightbe taken on the basis of the results and their potentialconsequences are clearly important. In fields such assampling inspection and control theory, or where spe-cific point forecasting is involved, a full specificationof utilities and prior distribution will yield the pre-ferred solution, but for most purposes summarizationof evidence via the estimation of relevant parametersseems a more suitable objective. Thus formulationsof clinical trials as decision making procedures, whilegiving valuable qualitative insights, have not beenwidely accepted as a realistic model of how trials areused.

Formal Theories of Inference

Much discussion in the more theoretical literaturehas for many years focused on the formal theory ofstatistical inference, in particular on the meaning ofprobability when used to assess the uncertainty inconclusions. The issues are important partly in settingthe broad approach to specific problems and partly indetail in developing particular methods of analysisand interpretation.

There are many different approaches but, leavingaside a pure decision-theoretical approach, they canbe broadly classified as

1. Pure likelihood [17].2. Fisherian, putting emphasis on likelihood, suffi-

ciency, conditionality, ancillarity.3. Neyman–Pearson, reaching many of the same

conclusions as 2 but emphasizing operationalcriteria such as power.

4. A Bayesian approach based on standardizedimpersonal priors [27], now often called refer-ence priors [8, 7] (see Bayesian Methods).

5. An emphasis on personalistic (or subjective)probability leading to a wholly Bayesian analy-sis and rejecting the above approaches as inco-herent, or at best as approximations to somethingelse [9].

This is a controversial area on which it seemsimprobable (in any sense!) that unanimity will bereached. There are, however, some signs of a fairlybroad agreement perhaps along the following eclecticlines:

1. Many of the issues addressed in this article, andof direct concern in applied statistical work, donot depend critically on the choice of approachto formal inference.

2. Likelihood, or some adaptation thereof, is ofkey importance but typically needs calibrationinto posterior intervals, confidence intervals orwhatever.

3. Probability, as representing idealized propertiesof the real world, has to be distinguishedfrom probability as measuring a state of anindividual’s knowledge.

4. Problems with many similar parameters areusually best formulated in empirical Bayesform, powerful numerical methods now beingavailable for their solution.

5. Reference priors in a small number ofdimensions usually produce answers with goodproperties also from the confidence intervalviewpoint.

6. It is necessary to have some notion thatone’s methods of analysis have good proper-ties, or at least are not systematically mislead-ing, when hypothetically they are used repeat-edly.

7. To the extent that the previous notion isformalized, some element of conditioning isneeded, although overconditioning must beavoided, for example to escape the C.R. Raoparadox of sampling theory [35].

8. The Bayesian formalism provides a valuablerepresentation of the merging of “prior” knowl-edge with new knowledge from data underanalysis, although it does not deal adequatelywith the possibility of conflict between the twosources.

9. The Bayesian axioms of coherent personalisticprobabilities are a valuable guide to opinionformation by individuals, but are not com-pelling as a basis for public discussion, partlybecause they put weight on internal consis-tency rather than on consistency with the realworld.

10. While the importance of formal statistical sig-nificance is commonly overstated, some suchnotion, with a null hypothesis and alter-natives (usually not formalized probabilisti-cally), is needed partly to formalize an escaperoute from an initial unsatisfactory formaliza-tion.



Statistics and Public Affairs

The primary emphasis in this article has been onstatistics in science and science-based technology.The organization of government statistics variesbetween countries with somewhat differing emphasisplaced on the one hand on the provision ofinformation and advice for direct use by government,somewhat akin to the provision of such informationin a business context, and on the other hand onthe provision of information to society at largeand to social science research workers in particular.For both purposes independence is crucial; in theprovision of advice one would hope for governmentstatisticians to be the firm voice of reason inthe face of political and economic dogmatism. Inthe second role, collaboration between governmentstatisticians and social science research workers,including statisticians, is important.

One important area of social policy concerns riskassessment and management, i.e. especially as con-cerns extremely small or maybe even nonexistentrisks. This has been discussed extensively by engi-neers, toxicologists, epidemiologists, psychologists,social anthropologists, sociologists, economists, andpolitical scientists [37], but surprisingly little hasappeared in the statistical literature. The role ofjudgmental probabilities in such situations is cen-tral.

The importance of appreciation by the generalpublic of central principles of the interpretation ofevidence shows itself in many aspects of materialappearing in newspapers and presented on radio ortelevision. For sample surveys, issues like the sam-ple size, the sampling scheme, the response rate, andlimits of error, are probably reported rather moreoften nowadays than in the past. Sensible interpreta-tion of so-called league tables of the performance ofschools, hospitals, and the like [20] depends cruciallyon a critical attitude to empirical data. The reporting,sometimes rather sensationally, of the results of oftenbadly designed small medical studies is of particularconcern.

Conclusion

The years 1925–1960 can be regarded as a golden eraof statistical thought. For example, in terms of issuesof formal inference, the period embraces most of the

work of R.A. Fisher, of Neyman and E.S. Pearson,and of Wald, the objectivist Bayesian contribu-tions of Jeffreys and the personalistic approach ofF.P. Ramsey, de Finetti, and Savage. Aspects ofthe design of experiments and sample surveys weredeveloped to a high pitch of elaboration; many of thekey ideas of time series analysis and multivariateanalysis were formulated. Statistical quality controland randomized clinical trials were firmly established.

While further important developments took placebetween 1960 and 1985, these years may best beseen as primarily a period of consolidation. At thebeginning of that time most statisticians had accessto an electronic computer but obtaining useful resultscould be a lengthy chore. By the end of the period allthe “standard” methods, and more, were fairly readilyavailable to a wide spectrum of users.

Encouraging features of the last 10 years or soare that while a massive educational job remains,the appreciation of statistical ideas is more widelyspread among research workers in many disciplines,as shown by the relative sophistication of statisticalideas in subject-matter journals, and by an increase inthe amount of substantial collaborative work involv-ing statisticians, in contrast to short-term “consulting”on very specific and often minor details.

Viewed over a rather longer period, there hasbeen a massive growth in the subject, as indicatedby the amount of work published per year, by theintroduction of new journals, by the number ofpeople employed and by the career prospects for newgraduates.

There is currently no shortage of interesting newideas and challenging problems, many stemmingfrom the relatively large sets of data now so common.For individual research workers freedom to followone’s own judgment of topics likely to be importantand to which one is equipped to contribute is neededand is under threat from the short-term policies ofmany of the sources of financial support. Neverthe-less, if statisticians as a group become increasinglyinvolved in important issues in science, technology,and public affairs, if imaginative new ideas can beencouraged, and if fragmentation of the subject canbe avoided, then the prospects for an important newperiod of major development are strong.

Acknowledgment

Support from the Leverhulme Foundation is gratefullyacknowledged.



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(See also Biostatistics, Overview; ExperimentalDesign)

D.R. COX


Documents

Encyclopedia of Biostatistics || Statistics, Overview