Regina C. Elandt-Johnson

Conditional failure time distributions under competing risk theory with dependent failure times and proportional hazard rates

Abstract Suppose that death (or any non-repetitive event) can occur due to various causes, each having its own failure time. Assuming independence of failure times and proportional hazard rates over the whole range of time, some authors have shown that the single cause failure time distributions conditional on the cause of death, each in presence of the remaining causes, are the same as the distribution of observable failure time, regardless of the cause. It has been shown in the present article, that this result is also valid without the assumption of independence (section 3). It has also been suggested (Section 5), that in the case of dependent failure times, a conditional limiting distribution as T α→∞ could represent the failure time distribution when cause C α is eliminated. Three examples (trivariate exponential, bivariate Gompertz, and US Life Table 1959–61 data) are given as illustrations.

Various estimators of conditional probabilities of death in follow-up studies: summary of results.

In medical follow-up studies the data on mortality experience of a specified population at risk are usually ‘incomplete’ in the sense that the study terminates before all the patients die, so that their final survival duration is unknown. The data are often grouped in fixed intervals, [ti, ti + 1), say, including individuals who die, are withdrawn alive or survive the full interval. An important problem in the statistical analysis is to estimate qi—the conditional probability that an individual alive at the beginning of the interval, will die within the interval. Various estimators have been proposed in the literature. The present paper gives a summary of the results on six possible estimators of qi and discusses briefly assumptions and methods of derivation. It is hoped that Table 1 will be of use to medical investigators in the analysis of survivorship from chronic diseases. An example on mortality of 5982 women diagnosed with cancer of the cervix uteri is used to illustrate some of the computing techniques and to compare the various estimators.

Some properties of bivariate Gumbel Type A distributions with proportional hazard rates

We call a set of univariate distributions with the same mathematical form but different parameter values a family . Consider a bivariate Gumbel Type A survival distribution, S12(x1, x2), defined in (2.1), for which both marginal distributions, S1(x1), S2(x2), belong to the same family, of distributions. It is proved in this paper that subject to weak conditions, the crude hazard rates, h1(t) and h2(t), are proportional if and only if the marginal hazard rates, [lambda]1(t) and [lambda]2(t), are proportional (Theorem 1). It is also shown that the survival functions of W = min(X1, X2), and of the identified minimum, Wi = Xi, for Xi

Age-at-onset distribution in chronic diseases. A life table approach to analysis of family data☆

Abstract A method of studying age-at-onset distribution in age dependent chronic diseases is described. Some actuarial-type techniques are adapted to family data obtained from families ascertained by at least one affected child. Formulae for evaluation of personyears exposed to risk of the disease at different ages (last birthday) and the estimates of the probabilities of having the onset of the disease before or at a given age are derived. Some application of the method in fitting a model of a single recessive to family data on rheumatic fever, is demonstrated.

General purpose probability models in histocompatibility testing. I ‘Strong’ and ‘weak’ alleles at one locus.

1.1. The general model for allotransplantation of tissue is ‘immunogenetic ’, and its basic concepts of response are very much similar to those in blood transfusion. Where the donor possesses an antigen (or antigens), called ‘histocompatibility antigen(s) ’, which is lacking in the recipient, antibodies against the donor’s antigen(s) are manufactured (mostly in lymph nodes and spleen) and, after a certain time, the graft is rejected. But if the recipient has at least all the antigens possessed by the donor, the graft should be accepted. The tissue compatibility (histocompatibility) is under genetic control; that is, the histocompatibility antigens are the gene products. We do not attempt here to discuss the possible relations: gene-antigen-antibody, since the terminology is not clearlyestablished. There is someevidence that, for example, the relatively wellknown histocompatibility locus H-2 in mouse is really a H-2 ‘region ’ (or ‘complex locus’). Each ‘ site ’ (or pseudo-allele) controls one antigenic specificity (or determinant) which can be detected serologically. The set of ‘ closely linked pseudo-alleles’ corresponds to an allele which controls an antigen (i.e. a set of antigenic specificities). So far 18 alleles: H-2a, H-2b, . . . , etc., and 33 specificities have been discovered at locus H-2 (Snell et al. (1964)) Amos (1964)), and there are at least 14-17 histocompatibility loci in mice (Barnes & Krohn (1957), Lengerovh & Matouiek (1966)). 1.2. The histocompatibility problem in man is probably more complicated and also more difficult, because the experimentation in humans is not very easy. There is evidence that histocompatibility antigens are very much related to the leucocyte antigens (see Van Rood (1966) and his references), but it is not yet certain whether this relationship is unique. By analogy with blood group systems and/or H-loci in mice, one can make certain reasonable assumptions under which some general probability models can be developed. Fitting a model to the data obtained from experimental or clinical trials, one can attempt to estimate genetic parameters controlling histocompatibility. I n models to be suggested here the ‘strength ’ of antigenic substances and their role in organ transplantation will be taken into account. I n this paper we restrict ourselves to one locus with s multiple alleles, when k of them are ‘weak ’ and s k are ‘ strong ’. Extension will be made later to n loci. First, we make some assumptions and introduce a few definitions under which the models are constructed. 2. ASSUMPTIONS AND DEFINlTIONS

Analysis of distributional patterns of deaths from different causes

Abstract Piecewise exponential fitting in the construction of multiple decrement life table (MDLT) is briefly reviewed. Three methods of analysis of proportionate distributions of life table deaths from different causes and over different age ranges are presented. The following age intervals are considered: (a) [xi , x i+1) with x i+1-xi =5 years, for i=0, 1, ..., I;(b) [xi , ∞) for i=0, 1, ..., I; and (c) [x 0, xa ) for fixed x 0 and xa =x 1, x 2 ... Different items of information obtained from such analyses are interpreted jointly. Two examples are given for illustrative purposes.

Survival Models and Data Analysis.

Survival analysis deals with the distribution of life times, essentially the times from an initiating event such as birth or the start of a job to some terminal event such as death or pension. This book, originally published in 1980, surveys and analyzes methods that use survival measurements and concepts, and helps readers apply the appropriate method for a given situation. Four broad sections cover introductions to data, univariate survival function, multiple-failure data, and advanced topics.

Onset distributions of age dependent zero-one events. Estimation from population data

Abstract Distributions associated with ages at onset of the first episode of an all-or-none phenomenon are often referred as “age of onset distributions”. At least two kinds of such distributions have to be distinguished: (i) the incidence (or ‘crude’) onset distribution giving the probability that a newborn individual will actually experience onset before age x, and (ii) the waiting time onset distribution representing an individuals chance of having onset before age x, supposing he lives so long. Theoretical bases for derivation of these probability functions are similar to those in competing risks (Section 2). Using nonparametric approach, the onset distributions can be expressed in terms of multiple decrement life tables. The techniques for constructing these tables usually involve data on both deaths among those who have not, and those who have experienced onset. Our methods (Sections 3–5) do not require these data; they are based on a population life table together with incidence and prevalence dat...

General purpose probability models in histocompatibility testing

General probability models for finding the proportions of compatible and incompatible pairs donor‐recipient, in populations in equilibrium in respect of several histocompatibility loci, are derived. The cases when (i) all loci are only ‘strong’ or ‘weak’, (ii) some loci are ‘strong’, some ‘weak’, (iii) there are loci with ‘strong’ and ‘weak’ alleles at the same locus, (iv) combinations of (i), (ii) and (iii), are considered.