The individual prognosis of adult IgA nephropathy patients was studied using the proportional hazards model for the time from biopsy until endstage renal disease. After selection of the most relevant prognostic factors, the 75 patients were stratified with respect to hypertension and its treatment. In these strata, individual prognosis was based on the initial age-adjusted glomerular filtration rate, the initial proteinuria, the presence/absence of gross hematuria, and the presence/absence of microscopic hematuria. Using the scores of a patient on these variables, the probability of surviving any given period of time can be estimated either graphically or by calculation. Prediction is feasible up to about 10 years. Attention has been given to supply all relevant estimates with confidence limits. For each patient the estimated 5-year survival probability as predicted by the model was compared with the actual outcome.

For counts it often occurs that the observed variance exceeds the nominal variance of the claimed binomial, multinomial or Poisson distributions. We study how models can be extended to cope with this phenomenon: a survey of literature is given. We focus on modelling, not on estimation or testing statistical hypotheses. The attention is restricted to independent observations.

The making of statistical inferences in distributional form is conceptionally complicated because the epistemic 'probabilities' assigned are mixtures of fact and fiction. In this respect they are essentially different from 'physical' or 'frequency-theoretic' probabilities. The distributional form is so attractive and useful, however, that it should be pursued. Our approach is In line with Walds theory of statistical decision functions and with Lehmann's books about hypothesis testing and point estimation: loss functions are defined, risk functions are studied, unbiasedness and equivariance restrictions are made, etc. A central theme is that the loss function should be 'proper'. This fundamental concept has been explored by meteorologists, psychometrists, Bayesian statisticians, and others. The paper should be regarded as an attempt to reconcile various schools of statisticians. By accepting what we regard 88 good and useful in the various approaches we are trying to develop a nondogmatic approach.Key Words & Phrases: decision theory, fiducial inference, foundations, invariance, unbiasedness. 1hmdUctm ' 0The making of statistical inferences is an uncertain affair. That is why different schools have emerged and the statistical controversy has appeared. We worry about this controversy, because the lack of consensus revealed by it impedes the respectabdity of our science. It is true that the debate between the various schools is less heated than it used to be. The statistical community has started to realize that every school has something useful to say and that the making of applications requires an eclectic attitude. Some situations are dealt with most appropriately by using a data-analytic 0 WS. 1995. PUblLbcd by BLd;nll P u b l i i IQ cow*y Rod. Oxford OX4 I f f , UK a d 238 M . * S U a r Canhidm MA 02142 USA.

SUMMARY The occurrence of more than one nucleolus within the cellular nucleus (polynucleolarity) is a well‐known phenomenon during the proliferative cell cycle, both under normal and pathological conditions (e.g. neoplasia). It can also be observed in neuronal nuclei at early stages of their maturation. Polynucleolarity merits investigation for cytological reasons. In an histological section, the observed number of nucleoli in a nucleus may be smaller than the actual number. In order to estimate the true distribution of the number of nucleoli per nucleus from the observed distribution, the mathematical relation between these distributions is derived (Section 4) on the basis of rather restrictive (Sections 3 and 11) stereological assumptions (Section 2). It is indicated how these distributions can be estimated from the data available and how the statistical uncertainties involved can be expressed (Sections 5, 6 and 7). This paper arose from making cell counts (Section 1). Two methods may be applied: (1) all visible nuclear profiles are counted, (2) nuclear profiles are only included if at least one nucleolus is visible in the section. We recommend a combination of these two methods (Section 8). An advantage of our theory for determining cell number (Section 9) is that one can often manage without the rather restrictive stereological assumptions needed hitherto. The advantage of expressing statistical uncertainties in estimated nucleolar multiplicity probabilities and cell numbers is indicated (Section 10).

The non-identifiability of relevant parameters is often used as a decisive argument to reject the mathematid model and (or) experimental design. This study shows by means of an example that the argument is not compelling. The experiment to be considered has been designed to determine the first few moments of the distribution of the number of gamma rays, emitted after a nuclear fusion reaction of a particular type. This task of determining moments is so complicated that certain systematic errors cannot be completely avoided. In consequence, the moments are not identifiable with respect to the usual mathematid model and the current experiment (section 2). There is no sense in rejecting the model or the experiment on the basis of this non-identiliability. It is easy to construct a simple estimator for the first moment, with properties that are not too bad (section 2). The current estimation procedures seem to be more accurate. They can even be used to estimate some higher moments. The systematic errors cannot be avoided but are negligible in many reasonable situations (section 3 and 4).The actual analysis of the data is more complicated than suggested in the first four sections. Nevertheless these sections are relevant because the underlying model gives an adequate description of the observations after a preliminary reduction of the data. This fact is intuitively clear but can also be proved (section 5).A test-case suggests that even the standard errors provided by the current technology are reliable (section 6). Key Words drphrpses: identijWility, mixture, systemuric errors, f i i o n reaction, gammtl rrry mulriplici'g 1 1 lm =p,,, + 1 { (1 -c)6* +€em} E 0 But @(lm)=y while xi ilm,pca as m+m.Remark. The non-identifiability of relevant parameters is sometimes used to