3140 BOOK REVIEWS

tables for many speci�c designs, multiple compar-isons procedures, and ANCOVA models. The mate-rial on more general regression modelling includesmany diagnostics and a brief mention of other mod-elling techniques such as ridge regression and thegeneralized linear model. There are short but well-written sections on causation, confounding and pathanalysis. The discussion of confounding is aided bythe welcome use of notation that distinguishes theregression coe�cients of the same variable depend-ing on what other variables are in the model.Despite the e�ort that has clearly gone into updat-

ing and expanding this book it is unavoidably lesscomprehensive than in its earlier days. The ‘classi-cal inference’ of the title is now classical both incontrast to Bayesian inference and in the choice oftopics. While many areas of modern statistical the-ory would be better reserved for specialized booksthere are some areas where a more expanded treat-ment or added references would have �t in wellwith the ow of the text.For example, the modern extension of asymp-

totic e�ciency theory to models with in�nite-dimensional parameters is not mentioned. Estimat-ing functions and the generalized method of mo-ments are covered in just over two pages, despitetheir practical importance and the interesting exten-

sions of ancillarity, su�ciency and e�ciency thatthey inspire. Empirical likelihood, minimum mes-sage length and maximum entropy methods are notmentioned in the list of alternatives to parametricmaximum likelihood, and while pro�le likelihood isdiscussed brie y, the various adjusted pro�le likeli-hoods are not referenced. Similar comments couldbe made about the linear modelling section, whereheteroscedasticity, robustness and the generalizedlinear model each receive about the same space asthe tetrachoric correlation.Omissions of this sort are inevitable as statistical

theory expands beyond the physical limits of anysingle volume, and other books are available to �llthe gaps. Kendall’s Advanced Theory of Statisticsis still a valuable reference work for the statisticalteacher and researcher, even if it is not the compre-hensive authority it once was.

THOMAS LUMLEYSchool of Public Health and Community Medicine

Department of BiostatisticsUniversity of Washington

Box 357232Seattle, WA, 98195-7232

U.S.A.

3. A SURVEY OF MODELS FOR TUMOR-IMMUNE SYS-TEM DYNAMICS. J. A. Adam and N. Bellomo (eds),Birkh�auser, Boston, 1997. No. of pages: x+344.Price: 138DM. ISBN 0-8176-3901-2

This volume consists of seven self-contained chap-ters on tumour modelling, with a substantial eighthchapter functioning as a bibliography aiming to help‘the mathematically oriented, but biologically dis-oriented’ reader. This last phrase adequately servesas a description of the probable readership of thisbook. The biologically-motivated models of tumourgrowth discussed in each of these chapters relateto the �nal (and relatively short) stage in tumourdevelopment, from the appearance of the �rst ma-lignant cell up to the clinically overt tumour, andshould be distinguished from the general class ofmodels dealing with the generally more prolongedmulti-stage process whereby a cell and its o�springsuccessively accumulate mutations which result inthe production of a cell with a malignant pheno-type, exempli�ed by the models of Armitage andDoll [1] and others [2; 3].Chapter 1 is a philosophical introduction to the

subject. While not without merit, it is at odds with

the mathematical and biological development else-where in the book and I suspect will not be of inter-est to the mathematically inclined readers at whomthe book is mainly aimed. Something of the tone ofthis �rst chapter may be deduced from the fact thatthere are no references, although Aristotle, FrancisBacon, Conan Doyle and A.A. Milne are discussedat some length!Chapter 2 is a wide ranging and valuable

overview of the mathematical models of solid tu-mour growth. However, it has not been written orproof-read with very much care, so that unde�nedmathematical symbols and other inconsistenciesabound. At times these are so serious, for example,in Section 2.4, that the development is rendered al-most meaningless. Another failing of this chapter,as of others in this book, is any substantive compar-ison of model predictions with experimental data,or any attempt to compare di�erent sorts of mod-els. Also, while the author considers ‘solid tumours’as a class, there is no discussion of of the di�er-ent sorts of solid tumour; again this failing a�ictsvarious other chapters.Chapter 3 gives a thorough exposition of a

number of related models of tumour growth, in

Copyright ? 2000 John Wiley & Sons, Ltd. Statist. Med. 2000; 19:3136–3145

BOOK REVIEWS 3141

particular the Gompertz, logistic and von Bertalan�ymodels. This chapter might be criticized for the factthat there is not much in it relating to biological as-pects other than tumour size as a function of time,so that the tumour–immune system interactions arenot covered. The chapter discusses at some lengththe relative goodness-of-�t of the various models,and compelling evidence is presented that the Gom-pertz model provides the best description of solidtumour growth. The authors go on to discuss pos-sible biological rationales for the set of di�erentialequations de�ning the Gompertz model. Overall thischapter is perhaps the most valuable in the book.Chapter 4 is unusual in that it is the only part of

the book dealing with tumours in the freely mixingstate, such as leukaemias and lymphomas, ratherthan the solid tumours considered elsewhere. Thegeneral class of stochastic models that are describedare as a consequence quite di�erent from those con-sidered elsewhere, and at �rst glance would appearto be hugely overparameterized. The authors makea number of simplifying assumptions and demon-strate that even with these various quite distinctsorts of dynamical behaviour are possible, depend-ing on the choice of parameters. Conspicuous fail-ings of the chapter are any comparison of modelsimulations with experimental data, explanation ofwhat experimental support there is for the biolog-ical assumptions of the model (which admittedlyappear plausible), or justi�cation of the particularparameter sets used.Chapter 5 returns to some of the models of solid

tumour growth considered in chapter 2, in particu-lar to models describing the process of angiogen-esis. A convincing case is made that this processis dominated by chemotactic growth of endothelialcells rather than by di�usion. The author goes onto consider models for growth of the tumour in itspre-vascular state, and shows how relatively simpledi�usion models can result in heterogeneous pat-terns of growth on the tumour surface, agreeingwell with experimental data. It is a weakness of thisotherwise excellent chapter that one must accept theauthors’ assertion on this last point, since details ofthe experimental data from which one could judgethe model �t are not given.Chapter 6 describes the models of interactions of

tumour cells with cytotoxic cells and ‘e�ector’ cells.These are described by sets of di�erential equationswhich for the most part have to be solved numer-ically. This chapter is well written, although theotherwise excellent glossary does not cover quitea few of the terms used here. In particular, there isgood discussion of the comparisons of the models

considered with experimental data, from which itappears that the description provided by these mod-els is excellent. A weakness of this chapter is thelack of any discussion of how the results from theapparently exotic experimental system the authordescribes relate to the interactions of cytotoxic cellsand tumour cells in vivo, in particular which sorts oftumours in vivo might be expected to be describedby these models.Chapter 7 addresses models of tumour hetero-

geneity, and considers for the most part systemsof two di�erent cell populations within a tumour,namely quiescent and dormant cells, correspondingroughly, in the case of a solid tumour, to cells onthe periphery and in the interior of the tumour. Thischapter is long on discussion and short on mathe-matical development, and I suspect that the moremathematically-inclined readers may �nd them-selves skipping certain portions. This would be apity, as the discussion is quite illuminating on bio-logical matters, in particular the discussion of howapparently dormant tumours can be persuaded to re-grow as a result of systemic insults.In summary this book is a bit of a curate’s egg,

which I suspect will be of most interest to biomath-ematicians rather than to statisticians. A generalcriticism of the book, from which only chapters 3and 6 are exempt, is the lack of comparison ofthe models with experimental data. Another generalfailing is the lack of consideration of possible dif-ferences between di�erent varieties of solid tumour.However, the mathematical development and anal-ysis are of a generally high standard, and the bestparts of the book (chapters 2, 3, 5 and 6) are gener-ally excellent. Given the book’s comparatively mod-est price I would recommend it to anyone workingin the area of modelling of tumour growth.

MARK LITTLEDepartment of Epidemiology and Public Health

Imperial College School of MedicineNorfolk Place

London W2 1PG, U.K.

REFERENCES

1. Armitage P, Doll R. The age distribution of cancer anda multi-stage theory of carcinogenesis. British Journalof Cancer 1954; 8:1–12.

2. Moolgavkar SH, Venzon DJ. Two-event models forcarcinogenesis: incidence curves for childhood andadult tumors. Mathematical Biosciences 1979; 47:55–77.

3. Tan W-Y. Stochastic Models of Carcinogenesis.Marcel Dekker: New York, 1991.

Copyright ? 2000 John Wiley & Sons, Ltd. Statist. Med. 2000; 19:3136–3145