QUESTIONS, ANSWERS AND STATISTICS T e r r y Speed CSIRO Division of Mathematics and Statistics Canberra, Australia
A major point, o n which I cannot y e t hope for universal agreement,
i s that o u r focus must b e 'on questions, not models. Models can and will g e t u s in deep troubles ifwe expect them t o t e l l us what the unique proper questions are.
J.W. Tukey (1977)
1 Introduction I n my view the value o f statistics, by which 1 mean both data and the techniques we use t o analyse data, stems from i t s use i n helping us t o give answers of a special t y p e t o more or less well defined questions. This is hardly a radical view, and not one with which many would disagree violently, y e t I believe t h a t much of the teaching of statistics and not a l i t t l e statistical practice goes on as if something quite different was the value of statistics. Just what the other t h i n g is I f i n d a l i t t l e hard t o say, b u t it seems t o be something like this: t o summarise, display and otherwise analyse data, o r t o construct, fit, test and evaluate models f o r data, presumably i n the belief t h a t if t h i s is done well, all (answerable) questions i n volving the data can then be answered. Whether t h i s is a fair statement o r not, it is certainly t r u e t h a t statistics and other graduates who f i n d themselves working with statistics i n government or semi-government agencies, business o r industry, i n areas such as health, education, welfare, economics, science and technology, are usually called upon t o answer questions, not t o analyse o r model data, although of course the latter will i n general The interplay bebe p a r t of their approach t o providing the answers. tween questions, answers and statistics seems t o me t o be something which should interest teachers of statistics, f o r if students have a good appreciation of this interplay, they will have learned some statistical thinking, not just some statistical methods. Furthermore, I believe t h a t a good understanding of this interplay can help resolve many of the difficulties commonly encountered i n making inferences from data. My primary aim i n this paper is quite simple. I would like t o encourage you t o seek o u t o r attempt t o discern the main question of interest associated with any given set of data, expressing this question i n the (usually nonstatistical) terminology of the subject area from whence the data came, before you even t h i n k of analysing or modelling the data. Having done this, I would also like t o encourage you t o view analyses, mode.ls etc. simply as means towards t h e end of providing an answer t o the question, where again the answer should be expressed in the terminology of the subject area, although there will always be the associated statement of uncertainty which characterises statistical answers. Finally, and regrettably t h i s last point is b y no means superfluous, I would then encourage you t o ask y o u r -
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self w h e t h e r t h e answer y o u gave r e a l l y did answer t h e question o r i g i n a l l y posed, a n d n o t some o t h e r question. A secondary aim, w h i c h I cannot hope t o achieve i n t h e time p e r m i t t e d t o me, w o u l d b e t o show y o u how many common d i f f i c u l t i e s experienced in a t t e m p t i n g t o d r a w inferences from data can b e resolved by c a r e f u l l y f r a m i n g t h e q u e s t i o n o f i n t e r e s t a n d t h e f o r m o f answer sought. A f e w remarks o n t h i s aspect a r e made i n Section 6 below.
2. W h y speak on this topic? O v e r t h e y e a r s I have h a d many experiences w h i c h have lead me t o t h i n k t h a t t h e i n t e r p l a y between questions, answers a n d s t a t i s t i c s i s w o r t h y o f consideration. L e t me b r i e f l y mention f o u r , each o f a d i f f e r e n t t y p e . T h e f i r s t experience i s a common one f o r me. Someone i s d e s c r i b i n g an application o f s t a t i s t i c s i n some area, say biology. T h e speaker u s u a l l y b e g i n s w i t h an o u t l i n e o f t h e b a c k g r o u n d science a n d goes o n t o g i v e an o f t e n detailed d e s c r i p t i o n o f t h e data a n d how t h e y were collected. T h i s p a r t i s new a n d i n t e r e s t i n g t o a n y statisticians listening, most o f whom w i l l b e unfamiliar w i t h t h a t p a r t i c u l a r p a r t o f biology. Sometimes t h e biologist who collected t h e data i s p r e s e n t a n d c o n t r i b u t e s t o t h e explanation, b u t a t a c e r t a i n stage t h e statistician s t a r t s t o e x p l a i n w h a t she/he did w i t h t h e data, how t h e y w e r e "analysed". B y now t h e b i o l o g i s t i s quiet, d e f e r r i n g t o t h e s t a t i s t i c i a n o n a l l matters statistical, a n d terms l i k e main e f fects, regression lines, homoscedacity, interactions, a n d covariates fly a r o u n d t h e room. Sooner o r l a t e r I find myself t h i n k i n g "Here a r e t h e answers, b u t w h a t was t h e question?" A l l too f r e q u e n t l y in such p r e s e n t a t i o n s n e i t h e r t h e statistician n o r t h e biologist has posed t h e main question o f biological i n t e r e s t in non-statistical terms, t h a t is, in terms w h i c h a r e i n d e p e n d e n t o f analyses o r models w h i c h may o r may n o t b e a p p r o p r i a t e f o r t h e data, a n d I can c e r t a i n l y remember occasions when t h e analysis p r e sented was seen t o b e i n a p p r o p r i a t e once t h e f o r g o t t e n question was f o r m u lated. O f course many s c i e n t i f i c questions b e t r a n s l a t e d i n t o statements a b o u t parameters i n a statistical model, so t h a t 1 am n o t condemning a l l instances o f t h e above practice.
A similar s o r t o f experience i s s u r e l y familiar t o a l l who have helped people w i t h t h e i r statistical problems. T h i s time a scientist, say a psychologist, comes t o me w i t h a set o f data and one o r more questions. She/he knows some statistics, o r a t least some o f t h e jargon. A f t e r b e i n g b r i e f e d o n t h e b a c k g r o u n d p s y c h o l o g y a n d t h e mode o f collection o f t h e data 1 u s u a l l y say something l i k e "What questions d o y o u w a n t t o answer w i t h these data?", ?" Not i n f r e q u e n t l y i m p l i c i t l y meaning "What psycholoqical questions . . t h e answer comes b a c k " I s t h e d i f f e r e n c e between such a n d such s i g n i f i cant?" meaning, o f course, s t a t i s t i c a l l y s i g n i f i c a n t . [I-n m y p e r v e r s i t y I o f t e n think t o myself: "Well, you should know; it's y o u r data a n d y o u a r e t h e psychologist! "1 A n o t h e r similar q u e r y m i g h t concern interactions, o r r e g r e s s i o n coefficients o f covariates etc. What t h i s has i n common w i t h t h e p r e v i o u s example i s t h e unwillingness o r i n a b i l i t y o f t h e psychologist t o state h e r / h i s questions o f i n t e r e s t in nonstatistical terms. We s h o u l d all b e familiar w i t h t h e idea t h a t s c i e n t i f i c (e.9. psychological) significance and
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statistical significance a r e n o t necessarily t h e same thing, b u t how many o f u s keep in m i n d t h e f a c t t h a t t h e l a t t e r involves an analysis o r a statistical model, a n d t h a t t h e r e may b e as many answers t o t h i s question as t h e r e a r e analyses o r models? S u r e l y much o f t h e blame f o r s u c h t h i n k i n g r e s t s w i t h us, t h e teachers o f statistics, who n e v e r fail t o popularize t h e rigid formalism o f Neyman- Pearson t e s t i n g t h e o r y . M y third t y p e o f experience concerns r e c e n t graduates i n statistics, s t u dents I a n d m y colleagues have t a u g h t and whom we believe should b e able t o operate i n d e p e n d e n t l y as statisticians. Many o f these graduates g o i n t o jobs i n big p u b l i c enterprises: railways, a g r i c u l t u r e bureaux, m i n i n g companies, government departments a n d so on, a n d a few f a r too many f o r comfort g e t in t o u c h w i t h u s when t h e y meet a d i f f i c u l t y i n t h e i r new job. It i s n o t t h e t h a t t h e y g e t i n t o u c h w h i c h i s discomfiting, b u t t h e questions t h e y ask! For we t h e n learn how l i t t l e t h e y have grasped. T h e y have questions in abundance, o f t e n important p o l i c y questions, access t o lots o f data, o r a t least t h e p o s s i b i l i t y o f collecting a n y data t h a t t h e y deem necessary, b u t t h e y a r e q u i t e u n s u r e how t o proceed, how t o answer t h e questions. O u t t h e r e i n t h e w o r l d t h e r e a r e "populations" o f real trains, f i e l d plots, c u b i c metres o f o r e o r people, a n d even t h e simplest question r e l a t i n g t o a mean o r a p r o p o r t i o n o r a sample size can b e f o r b i d d i n g . Perhaps t h e y should standardize something t o compare it w i t h something else, p e r h a p s i n c l u d e t h e v a r i a b i l i t y o f one f a c t o r when anal y s i n g another, o r something else again, all t h i n g s w h i c h we feel t h a t a g r a d u a t e o f o u r course should b e able t o cope w i t h unaided. B u t how well did we t r a i n them f o r t h i s experience?
and m y Finally, a n d b r i e f l y , l e t me castigate m y professional colleagues self, since I am n o exception f o r allowing ourselves t o f o r g e t t h e f u n d a mental importance o f t h e i n t e r p l a y o f questions, answers a n d statistics, f o r in so many o f o u r professional interactions we act as if it is i r r e l e v a n t . How many times have we p r e s e n t e d new statistical techniques t o one a n other, i l l u s t r a t e d o n sets o f "real" data, d r a w i n g conclusions a b o u t those data concerning questions no one e v e r asked, o r i s e v e r l i k e l y t o ask? A n d how o f t e n do we d e r i v e statistical models o r demonstrate p r o p e r t i e s o f models w h i c h a r e u n r e l a t e d t o a n y s e t o f data collected so far, a n d c e r t a i n ly n o t t o a n y questions f r o m a substantive f i e l d o f human endeavour. We are, so we t e l l ourselves, simply a d d i n g t o t h e stock o f statistical methods a n d models, f o r possible l a t e r use. I s it a n y wonder t h a t we o r o u r cow o r k e r s t h e n f i n d ourselves u s i n g these models a n d methods i n practice, regardless o f w h e t h e r o r n o t t h e y help u s t o answer t h e main questions o f i n t e r e s t . For a discussion o f some closely r e l a t e d issues o f g r e a t relevance t o teachers o f statistics, see t h e t w o excellent articles Preece (1982, 1986).
3. Why this audience? I don't t h i n k I w i l l b e v e r y wide o f f t h e m a r k if 1 assume t h a t most o f y o u a t least t h e a c t i v e teachers o f statistics amongst y o u - have come f r o m a b a c k g r o u n d o f mathematics r a t h e r t h a n statistics, a n d t h a t f e w o f y o u have a c t u a l l y been statisticians b e f o r e y o u s t a r t e d teaching t h e subject. I would f u r t h e r guess t h a t many o f y o u s t i l l teach mathematics, a n d perhaps a t t h e
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school level, statistics within a mathematics. curriculum. It is on t h i s assumption that I have chosen t o focus on non-mathematical aspects of our subject, ones with which I feel you will generally be less familiar. As I said i n the introduction, I hope t h a t my talk will encourage you t o give more attention t o the non-mathematical aspects of statistics i n y o u r teaching, in particular t o spend more time considering real questions of interest with real sets of data.
It is a curious t h i n g t h a t interest i n t h e teaching of statistics i n schools, colleges and universities has sprung u p worldwide as an extension of mathematics teaching, because I certainly feel t h a t t h e practice of statistics is no closer t o mathematics than cooking is t o chemistry. Both mathematics and chemistry are reasonably precise subjects i n t h e i r own ways, and i n general what goes on i n them both is repeatable; perhaps they are t r u e sciences. On the other hand, statistics and cooking are as much arts as t h e y are science, although both have strong links t o t h e i r corresponding science: mathematics i n the case of statistics, and chemistry i n t h e case of cooking. Who would recommend t h a t a chemistry teacher with no cooking experience be appointed as cooking teacher as well? If I can convey t o you some of the enjoyment and intellectual challenge t h a t lies i n my particular variety of cooking, and encourage you t o try it yourself, I will have succeeded i n my aims. 4. Two further examples I n this necessarily too b r i e f section I offer two more concrete illustrations of interplay of t h e questions, answer and statistics. The f i r s t bne is a v e r y simple paraphrase of Neyman's classic illustration of hypothesis testi n g involving X - r a y screening f o r tuberculosis, and I refer you t o Neyman (1950, Section 5.2.1) f o r a fuller background and f u r t h e r details. You have a single X - r a y examination and, after the photograph has been read b y the radiologist, you are given a clean b i l l of health, t h a t is, you are told t h a t there is no indication t h a t you are affected b y tuberculosis. With Neyman we will assume t h a t previous experience has led t o pr(clean b i l l ]no TB) = 0.99 pr(cleanbillpB)
You now ask the radiologist "What are the chances t h a t I have T B ? " She says "I can't answer t h a t question b u t I can say this: Of the people with T B who are examined i n t h i s way, 60% are correctly identified as having TB, and of . . . " You i n t e r r u p t her. "Doctor, I know the procedure is imperfect, b u t you have j u s t examined my X-ray . . . What are the chances t h a t I have TB?"
If your radiologist is sufficiently answer "Well, t h a t depends -on t h e t h a t is, on the proportion of people from which you may be regarded as application of Bayes' theorem yields:
flexible and well informed, she will prevalence of T B i n your population, affected b y T B i n the (a?) population a typical individual". Indeed a simple
ICOTS 2, 1986: Terry Speed
p r ( T B Iclean bill) = p r l c l e a n bill I T B ) ~ ~ ( T B ) pr(clean bill)
A t last you see how t o get an answer t o your question. It may not be easy t o obtain a value f o r p r ( T B ) : your smoking habits, the location of y o u r . may all play a p a r t i n defining home, your occupation, your ancestry . "your population", b u t this is what is needed t o answer the question and it is f a r better t o recognise this than t o fob you o f f with the answer t o another question not of interest t o you.
If this example smacks of Bayesian statistics it is not entirely accidental, f o r there are many occasions where the Bayesian view (which is certainly not necessary i n this example) helps answer t h e question of interest, whereas classical statistics refuses, frequently answering another, u n asked, question instead. For a more complex, explicitly Bayesian example, see the v e r y fine paper Smith and West (1983) concerning the monitoring o f renal transplants. My second example concerns the determination of the age of dingos, Australia's wild native dogs. A statistician was given a large body of data r e lating the age of a number of dingos t o a set of physical measurements i n cluding head length. The data concerned both males and females, a number of breeds and animals from a number of locations, b u t for t h i s discussion we will r e s t r i c t ourselves t o a single combination of sex, breed and location. The question, or at least the task, t o be addressed was the following: produce an age calibration curve f o r dingos based upon t h e most s u i t able physical measurement, t h a t is, produce a curve so t h a t the age of a dingo may be predicted by reading o f f the curve at the value of the p h y sical measurement. This curve was f o r use i n the field and it was taken f o r granted t h a t an estimate of the precision of any age so predicted would also be obtained.
It was found that a curve of the general form h = a + b [ l - exp(-ct)], where h and t are head length and age, respectively, and a, b and c are parameters of the curve, f i t t e d the data from each dingo extremely well over the range of ages used. This was an exercise i n non-linear regression with which the statistician took great care, special concern being given t o t h e different possible parametrizations of the curve, the convergence of t h e numerical algorithm used, the residuals about t h e fitted line and t o the validity of the resulting confidence intervals f o r a, b and c. The parameters estimated for different dingos naturally differed, although, not s u r prisingly, the values of a (head length at b i r t h ) showed less variation than those of b (ultimate head length -a) and c (a growth rate parameter). A l l this seems fine, and you might wonder why i am mentioning t h i s example a t all in the present context. My answer is as follows. The statistician in question knew, or knew where t o find, lots of information about t h e f i t t i n g of individual growth curves, and so he focussed on this aspect of the problem. To answer the original question, however, his attention
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should have been pointed i n quite a different direction, towards: t h e calculation of a population o r group growth c u r v e f o r t h e calibration procedure; features o f the sample of dingos measured t h a t may affect t h e use o f t h e i r measurements as a basis f o r the prediction of t h e age of a new dingo; properties of the parameters which are relevant t o t h i s question; and, finally, towards obtaining a realistic assessment o f the prediction e r r o r inherent i n t h e use of t h e curv e i n the field. I n summary, he was willing ual animals' curves; he was demanded b y t h e question, lation variability, problems issues including the use of fined populations.
and able t o spend a lot of time on t h e i n d i v i d less willing and less able t o focus on t h e issues those concerning population parameters, popuof selection, u n representativeness, and other normal theory, with real b u t not v e r y well de-
5. What is t h e problem? Let me oversimplify and p u t my message like this. I n the beginning we taught mathematics and called it statistics; much of t h i s was probability, a quite d i f f e r e n t subject. Then, with the help of computers, we started t o teach data analysis and statistical modelling; t h i s was fine apart from one feature: it was largely context-free. The real interest (for others and many statisticians), the important difficulties and t h e whole point o f statistics lies i n t h e interplay between the context and t h e statistics, t h a t is, i n the interplay between t h e items of my title. Let me offer a few similar views. A.T. James (1977, p. 157) said i n the discussion of a paper on statistical inference: The determination of what information in the data is relevant can only be made by a precise formulation of the question which the inference If one wants statistical methods to prove is designed to answer. reliable when important practical issues are at stage, the question which the inference is to answer should be formulated in relation to these issues.
Cox (1984, p. 309) makes the following characteristically b r i e f contribution t o o u r discussion :
It is t r i t e that specification of the purpose of a statistical analysis is important. Dawid (1986) is even more t o the point: Fitting models is one thing; interpreting and using them is another, If the model is correct and we know the parameters, how ought There is in fact no unique answer; it we to compare [schools]? all depends on our purpose. there remains a strong need for a careful prestatistical analysis of just what is required: following which i t may well be found that i t is conceptually impossible to estimate it!
Tukey and Mosteller (1977, p. 268) offer seven purposes of regression, or, as I would paraphrase it, seven types of questions which regression analysis may help answer. Summarized, these seven purposes are:
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1. t o get a summary;
2 . t o set aside the effect of a variable; 3. as a contribution t o an attempt a t causal analysis;
4. t o measure the size of an effect; 5. t o try t o discover a mathematical o r empirical law; 6. f o r prediction;
7. t o get a variable o u t of the way. Similarly, T u k e y (1980, series analysis;
pp. 10-11) gives the following six aims of time
1. Discovery of phenomena.
2 . "Modelling". 3. Preparation f o r f u r t h e r inquiry.
4. Reaching conclusions. 5. Assessment of predictability.. 6. Description of variability. Similar numbers of aims, purposes, o r types of questions could be given f o r t h e analysis of variance, the analysis of contingency tables, multiYet variate analysis, sampling and most other major areas of statistics. how often, do our students meet these techniques i n context with even one of these aims, much less the f u l l range? And how else are t h e y going t o learn t o cope with the special difficulties which arise when questions are asked of them i n context whose answers require statistics? This is the problem.
6. Some General Comments I n this section I will mention a few difficulties which I believe can be r e solved i n a given case when the relation between the questions asked, t h e form of t h e answers desired and the statistical analysis t o be conducted are carefully considered. A f u l l discussion of any one of the difficulties is o u t of the question, and even if t h a t had been given, there would probably remain an element of controversy, something which would be o u t of place i n a talk like this. The section closes with some f u r t h e r general comments about questions. Some elementary difficulties which I t h i n k arise include ? What is t h e population?
When are population characteristics (e. g . proportions) relevant?
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What is the "correct" variance t o attach t o a mean o r proportion? 0
When should we standardize (for comparison)?
I have found t h a t t h e relations between statistical models and analyses on the one hand, and populations and samples on t h e other, with parameters playing a role i n both, are something which puzzle many students of our subject. The former play a b i g role i n standard statistics courses whereas t h e latter are prominent i n applications. J u s t how t h e y connect is not a t r i v i a l matter. A few somewhat more advanced difficulties include 0
Which regression: y on x, x on y o r some other?
When should we use correlation and when regression analysis? When can/should we adjust y f o r x?
Which e r r o r terms do we compare (in anova)? Should we regard a given effect as fixed o r random?
Which classifications (of a multiway table) .correspond t o factors and which t o responses?
More subtle difficulties are associated with general questions such as 9
Should we do a joint, marginal or conditional analysis?
I believe t h a t i n all of t h e above cases the difficulties arise because insufficient attention has been given t o the nonstatistical context i n which the discussion is t a k i n g place, and t h a t when t h e question of interest is clarified and the form of answer sought understood, the d i f f i c u l t y either disappears completely o r is readily resolved. Of course doing so takes some experience. Note t h a t many of the difficulties listed involve, implicitly o r explicitly, the notion of conditioning, or i t s less probabilistic forms, standardizing o r adjusting. Just what we regard as being "held fixed" and what we "average over" i n any given context is crucial, and here our questions and answers determine everything. The simplest form of this issue is usually: "Are we interested i n j u s t these units (the ones we have seen), o r i n some population of units from which these may be regarded as a (random?) sample, o r both?" Models are no help here. A simple b u t easy t o forget aspect of the use of a statistical method is t h a t not all questions which could be asked and answered b y that method, are necessarily appropriate i n a particular context. Lord's paradox, see Cox & McCullagh (1982) and references therein, provides a good example here.
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7 . What can/should be done? It hardly needs saying t h a t t h e best way t o promote interest i n t h e i n t e r play between questions, answers and statistics is t o p u t trainee statisticians into situations where they are required t o provide answers t o clearly stated questions on t h e basis of real data sets. Note t h a t t h i s can be a v e r y different t h i n g from "illustrating" a statistical technique on a set of data. I n particular, much more background t o t h e data is usually r e quired, and this is r a r e l y available i n data sets presented i n statistics texts. Indeed technical journals are now so t i g h t with t h e i r space t h a t it is rare t o f i n d f u l l data sets published together with analyses and conclusions i n scientific articles. This means t h a t the best sources o f suitable material of the k i n d being discussed, t h a t is, of questions and data, are often one's colleagues o r clients: teachers and researchers i n other disciplines who make use of experimental o r observational data i n t h e i r work. Seeking out such material can be a way of f o r g i n g links with the users of statistics and of course sandwich courses are designed with this general aim i n mind. One practice which I believe is valuable is the conduct of regular practical statistics sessions where students are asked t o help answer specific questions on the basis of sets of data supplied together with background material. This is much more like the situation t h e y will meet after their t r a i n i n g is over. Two objections which are often expressed t o me when I recommend t h i s approach are (i) Surely it is unrealistic, except with the most advanced students, f o r unless t h e y have learned a wide range of techniques, they will not be able t o begin attacking "real" problems with any likelihood of success?; and (ii) Surely it is unrealistic, because real problems are so complex and real data sets so large, or even ill- defined, t h a t nothing like what happens in practice can be presented i n the classroom? Both these objections have some validity, b u t let me make a few observations concerning them. Firstly, it is not necessarily a bad t h i n g f o r a s t u dent (or anyone!) t o attempt t o answer a particular question (solve a p a r ticular problem) without knowing of the tools o r techniques t h a t may have been developed t o answer just t h a t t y p e of question (or problem). This goes on all the time i n the real world: parts of the wheel are rediscovered time and time again, and locomotion is even found t o be possible without the wheel! And of course there is v e r y seldom a single "correct" way t o answer a question; an approach using less knowledge of techniques may well be better than one which uses greater knowledge. I n the hands of a good teacher, such experiences can provide valuable object lessons, and, at the v e r y least, valuable motivation f o r techniques not y e t learned. Surely nothing could be more satisfying than hearing a student say: "What 1 need (to answer t h i s question) is a way of doing such and such, under the following circumstances (e.g. errors i n t h i s variable, t h a t factor misclassified, these observations missing o r censored, t h a t parameter chosen i n a particular way, etc.)? Group discussions, where ideas are shared and knowledge pooled, are also most appropriate f o r t h i s sort of work, and most enjoyable. The teacher can then play a subsidiary role, at times . focussing the discussion back on the questions, perhaps at other times supplying a sought-for technique.
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'It would seem t o me t h a t t h i s is j u s t t h e s o r t o f statistics which should be t a u g h t i n secondary schools, not t h e watered-down and frequently sterile mathematical material which is often found a t t h a t level. T h e second objection, t h a t real problems are often v e r y complex and r a r e l y amenable t o t h e s o r t o f trimming t h a t would b e necessary before t h e y could be used i n a classroom, is harder t o dismiss. It i s certainly t r u e t h a t many (most?) problems are like this, b u t s u r e l y t h i s highlights even more t h e difference between "illustrative" data sets, taken o u t o f context, w i t h no realistic questions o r idea what would b e satisfactory answers, and what we expect students t o be able t o cope w i t h upon graduation. There is cert a i n l y a b i g gap here - between "pseudo-applied" statistics involving context-free sets of numbers, t o illustrate arithmetic, and fully-fledged "warts and a l l " consulting problems - and I can o n l y state t h a t i n my e x perience it is possible t o f i n d problem data sets which can be presented i n t h e way I am suggesting. It certainly takes a l i t t l e e f f o r t t o f i n d such material, particularly if you are not in t h e habit o f meeting people w i t h data and statistical problems. B u t as teachers o f t h e subject, t h a t is not such an unreasonable t h i n g f o r me t o expect of you is i t ? A teaching strategy which could provide a means of p u t t i n g these ideas into practice might be t h e following: pair yourself (the statistics teacher) with a teacher i n an empirical field o f enquiry, e.g. biology, agriculture o r medicine, and also pair your statistics students w i t h students i n t h e corresponding class, r e q u i r i n g them t o w o r k together on a practical p r o ject which will enrich t h e i r understanding o f both disciplines, and how statistics helps t o answer questions. Many variants on t h i s suggesti,on could be devised; t h e important t h i n g is try something along these lines. Statistics students must meet more than mathematicszand sets of numbers i n t h e i r training, and it is t h e teachers o f statistics who must arrange f o r t h i s t o happen. Acknowledgement
I am v e r y grateful f o r t h e discussions and comments on t h i s topic offered t o me by my CSlRO colleagues Peter Diggle, Geoff Eagleson and Emlyn Williams. 8. References
Cox, D. R. (1984). Present position and potential developments: Some p e r sonal views. Design o f experiments and regression. J.R. Statist. Sac. Ser. A., 147,306-315. Cox D. R. & McCullagh P. (1982). ance. Biometries 38, 541-553. .
Some aspects o f t h e analysis o f covari-
Dawid, A.P. (1986). Contribution t o t h e Discussion of: "Statistical modelling issues i n school effectiveness studies" by M. A i t k i n and N. Longford. J. Roy. Statist. Soc. Ser. A, 149, 1-43.
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James, A.T. (1977). Contribution t o the Discussion of: "On resolving the controversy i n statistical inference" b y G. N. W i l kinson. J. Roy. Statist. Soc. Ser. B, 39, 157. Mosteller, Frederick & Tukey, John W. (1977). Data analysis and reqression. Sydney: Addison-Wesley Publishing Company. Neyman, J. (1950). F i r s t Course i n Probability and Statistics. New York: Henry Holt & Company. Preece, D.A. (1982). t is f o r trouble (and textbooks): a critique of some examples of the paired samples t-test. The Statistician, 91, 169-195. Preece, D. A. (1986). Illustrative examples: illustrative of what? The Statistician, 35, 33-44. Smith, A.F.M. & West, M. (1983). Monitoring renal transplants: An application of the multiprocess Kalman f i l t e r . Biometrics, 39, 867-878. Tukey, J.W. (1977). Contribution t o the Discussions of "A reformulation of linear models" b y J.A. Nelder. J. Roy. Statist. Soc. Ser. A, 140,72. Tukey, John W. (1980). Can we predict where "Time Series" should go next? Directions i n Time Series Eds D.K. Brillinger & G.C. Tiao. IMS. pp.1-31.
ICOTS 2, 1986: Terry Speed