B. Raja Rao

Equivalence of the tampered random variable and the tampered failure rate models in accelerated lifetesting for a class of life distributions having the ‘setting the clock back to zero property'

It is well-known that Nelsons (1980) Cumulative Exposure (CE) model in a step-stress accelerated life test is equivalent to the Tampered Random Variable (TRV) model of DeGroot and Goel (1979) if the life distributions under the two stresses belong to the same scale-parameter family. In the present paper, it is proved that if a step-stress accelerated life test is performed on a population of units whose life distribution belongs to a class of life distributions having the ‘setting the clock back to zero’ property, (see B.R. Rao and Talwaker 1989, 1990 and B.R. Rao 1990a), then the TRV is equivalent to the Tampered Failure Rate (TFR) model of Bhattacharyya and Soejoeti (1989). The proof of the necessity is simple. Simple examples of the family of life distributions with this property include the exponential, the linear hazard exponential and the Gompertz distributions. The Weibull distribution does not have this property and it is demonstrated that the TRV and the TFR models lead to different surival dist...

A simple time to tumor distribution and the ‘setting the clock back to zero’ property

The present paper introduces the ‘setting the clock back to zero’ property, of a family of life time distributions and proves some closure theorems for competing risks acting on an organism. It is proved that the family of simple time to tumor distributions proposed by Chiang and Conforti (1989) has this property. As a simple application of this result, the life expectancy at age X0 of an organism is derived for such a family. In our time to tumor application, if no tumor has been found in an organism of age x0, the life expectancy denotes that the organism can have, on the average, an additional time units of tumor-free life. If x0=0, the expression reduces to the ordinary expectation of life, see Chiang and Conforti (1989). Some further results and generalizations are discussed, with applications in competing risk theory.

Characterization of generalized markov-polya and generalized polya-eggenberger distributions

A discrete model is considered where the original observation is subjected to partial destruction according to the Generalized Markov-Polya (GMP) damage model. A characterization of the Generalized Polya-Eggenberger distribution (GPED) is given in the context of the Rao-Rubin condition. More specifically, if the probability that an observation n of a non-negative integer valued r.v.X is reduced to an integer k during a damage, process is given by the GMPD, and if the resulting r.v.Y is such thatrit satisfies the RR-conditlon, then X has a GPED. Secondly, if N = A + B, where B is the missing part and A is the recorded part such that the conditional distribution P(A= x|N=n) is the GMPD, then the r.v.s A and B are independent if, and only if, N has a GPED. Several other characterizations are also given for these two distributions. The results of Rao-Rubin ‘1964’, Patil-Ratnaparkhi (1977) and Consul (1975) follow as special cases.

On the Moments of Multivariate Discrete Distributions Using Finite Difference Operators

Abstract This article discusses a general approach to finding the moments of two classes of multivariate discrete distributions, which include those widely used in applied and theoretical statistics. The two classes of multivariate discrete distributions are the multivariate generalized power series distributions (GPSD) and the unified multivariate hypergeometric (UMH) Distributions. The results of Link (1981) follow as special cases.

Sidak-type simultaneous prediction intervals for the mortality measures RSRRi about the corresponding SePMRi for several competing risks of death in an epidemiologic study

Abstract Simultaneous prediction intervals are obtained involving the externally standardized proportional mortality measures S e PMR i which include the corresponding RSRR i with probability 0.95, or better, for several competing risks of death, under a reasonable assumption, namely, that the total number of deaths d .. due to all risks in the mortality study be large. The results are obtained under a very general situation when several risks compete for an individuals life in the study, irrespective of any covariance structure among the risks of death, known or unknown, positive or negative, or zero. As a particular case, if the risks of death may be assumed as independent, then the simultaneous prediction intervals reduce to those of the present authors (1985a). It is demonstrated that asymptotically (as d .. → ∞) the joint c.d.f. of the statistics Z 1 , Z 2 , ·, Z M , where Z i = RSRR i /S e PMR i , may be approximated by means of the joint c.d.f. of the multivariate ( M -variate) normal distribution. This is the basic result from which the prediction intervals are derived, using Sidaks (1967, 1968) multivariate normal probability inequalities. Our results are empirically evaluated using real data from a recent retrospective cohort mortality study of Enterline and Marsh (1983), concerning respiratory cancer among man-made mineral fiber workers. These prediction intervals do indeed quantify the observation that for each cause i of death, the measure S e PMR i is close in value to the corresponding RSRR i .

Partial Correlation in Terms of Path Coefficients

where r12.34, etc. may, in turn, be expressed in terms of r12.3, etc., until simple correlations (of zero order) are reached. Moreover, the expression for r12.345 is not unique, as any two of the subscripts after the dot may be chosen to remain after the dot in the correlations of the lower order. In example (1), the subscripts 34 remain after the dot throughout the expression. The purpose of this note is to present a new expression by the use of path coefficients. The new expression remains the same for partial correlations of any order. The new expression is based on multiple correlation coefficients rather than partial correlation coefficients. Hence a few words about multiple regression are in order. The linear regression of Xi on X3, . . ., Xk may be written, assuming zero means for all variables, as

Sidak-type simultaneous confidence intervals for the risk measures rsmr, based on proportional mortality measures in epidemiologic 1 studies involving several competing risks of death

In the present paper, simultaneous confidence interval estimates are constructed for the mortality measures RSMR. based on propor¬tional mortality measures SPMR. in epidemiologic studies for several competing risks of death to which the individuals in the study are exposed. It is demonstrated that, under a reasonable assumption, the joint sampling distribution of the statistics X. = RSMR./SPMR. for M competing risks9 may be approximated by means of a multi-variafe normal distribution, Sidaks (1967, 1968) mulfivariate normal probability inequalities are applied to construct the simultaneous confidence intervals for the measures RSMR., i=l3 2, ..., M. These are valid regardless of the covariance structure among the risks, As a particular case if the risks may be assumed as independent, our confidence intervals reduce to those for a single measure RSMR., which are narrower than those of Kupper et al., (1978), In this sense, our paper generalizes the results presently available in the literature in two direc...

The sufficient-component discrete-cause model and its extension to several risk factors

Abstract This paper is concerned with Koopmans (1981) discussion of using the ICDR as a measure of deviation from an additive model of no interaction. This measure has been shown to be zero or slightly negative when the assumptions of no interaction hold in the sufficient-component discrete-cause model. The discussion centers around a theoretical structure of sufficient causes for juvenile-onset diabetes where there is a component cause common to all sufficient causes and two binary factors or risks X and Y are measured ( X : Coxsackie B 4 infection; Y : HLA4). In this paper, we have extended Koopmans discussion to the case where three or more binary factors X , Y , Z ,… are measured. We have considered interaction between factors in every subset and have proved that the ICDR is again either zero or negative when the assumptions of no interaction hold in the sufficient-component cause model. It is supposed that these factors in conjunction will inevitably result in beta-cell destruction together with a set of immunologic conditions that permit dissemination of the enterovirus infection and permit an autoimmune response.

Approximate simultaneous inferential procedures for overall risk assessment of several competing causes in biomedical and epidemiologic studies

Abstract The object of the present paper is to develop approximate simultaneous inferential procedures which facilitate overall risk assessment of several competing causes in biomedical and epidemiologic studies when the available data consist only of number of cases or deaths but not the number of individuals at risk. The paper describes how, under some reasonable assumptions, asymptotically precise simultaneous inferences may be based on proportional mortality of morbidity analyses to estimate the indirectly standardized cause-specific risk measures RSMR i involving the corresponding SPMR i and to estimate the externally standardized measures RSRR i involving the corresponding measures S e PMR i . Our paper may be divided into two parts. In the first part, approximate simultaneous Scheffe-type confidence intervals (as d ..→∞) are presented for the measures RSMR i about the corresponding SPMR i when the individuals in the study are exposed to several (say M ) competing risks of death. The second part of our paper presents approximate simultaneous Scheffe-type prediction intervals (as d ..→∞) for the measures RSSR i involving the easily-computable externally standardized measures S e PMR i in the presence of M competing risks of death. The approximate simultaneous inferential techniques developed in the paper are valid regardless of any dependence structure among the competing risk factors. As a special case, if the risks may be assumed as independent, our simultaneous confidence and prediction intervals reduce to those for a single measure, obtained previously by the present authors (1987). The utility of the proportional mortality or morbidity measures SPMR i and S e PMR i is empirically evaluated as predictors of the corresponding cause-specific risk measures RSMR i and RSRR i using the tuberculosis mortality data of Kupper (1978) and the respiratory cancer mortality data of Enterline and Marsh (1983).

Approximate variance formulas and asymptotic joint sampling distributions of standardized risk ratios in the presence of competing risks in cohort studies

The present paper is concerned with some results in cohort studies, in which the individuals in two study population are exposed simultaneously to several risks of death, which compete for their lives. The morality experience of individuals in the two study populations is compared with respect to the morality experience of individuals in a well-defined and fixed population called the standard population. Under some reasonable assumptions, not only simple variance formulas are-developed for the standardized risk ratio statistics (S[Rcirc]Ri) but also their joint asymptotic sampling distribution. It is demonstrated that these SRcirc;Ris have asymptotically a multivariate normal distribtion corresponding to any given number of competing risks of death, These results are utilized to construct Scheffe-type and Sidak-type simultaneous confidence intervals for the SRRi parameters which hold regardless of any covariance structure among the competing risks of death. The corresponding results for the cause-specifi...