Asymptotically robust variance estimation for person‐time incidence rates

Person‐time incidence rates are frequently used in medical research. However, standard estimation theory for this measure of event occurrence is based on the assumption of independent and identically distributed (iid) exponential event times, which implies that the hazard function remains constant over time. Under this assumption and assuming independent censoring, observed person‐time incidence rate is the maximum‐likelihood estimator of the constant hazard, and asymptotic variance of the log rate can be estimated consistently by the inverse of the number of events. However, in many practical applications, the assumption of constant hazard is not very plausible. In the present paper, an average rate parameter is defined as the ratio of expected event count to the expected total time at risk. This rate parameter is equal to the hazard function under constant hazard. For inference about the average rate parameter, an asymptotically robust variance estimator of the log rate is proposed. Given some very general conditions, the robust variance estimator is consistent under arbitrary iid event times, and is also consistent or asymptotically conservative when event times are independent but nonidentically distributed. In contrast, the standard maximum‐likelihood estimator may become anticonservative under nonconstant hazard, producing confidence intervals with less‐than‐nominal asymptotic coverage. These results are derived analytically and illustrated with simulations. The two estimators are also compared in five datasets from oncology studies.     

Estimation of log‐odds ratio from group testing data using Firth correction

We consider the estimation of the prevalence of a rare disease, and the log‐odds ratio for two specified groups of individuals from group testing data. For a low‐prevalence disease, the maximum likelihood estimate of the log‐odds ratio is severely biased. However, Firth correction to the score function leads to a considerable improvement of the estimator. Also, for a low‐prevalence disease, if the diagnostic test is imperfect, the group testing is found to yield more precise estimate of the log‐odds ratio than the individual testing.     

Some interrelationships among the regression coefficient estimates arising in a class of models appropriate to response-time data.

Interrelationships among three response-time models which incorporate covariate information are explored. The most general of these models is the logistic-exponential in which the log odds of the probability of responding in a fixed interval is assumed to be a linear function of the covariates; this model includes a parameter W for the width of discrete time intervals in which responses occur. As W leads to O this model is equivalent to a continuous time exponential model in which the log hazard is linear in the covariates. As W leads to infininity it is equivalent to a continuous time exponential model in which the hazard itself is a linear function of the covariates. This second model was fitted to the data used in an earlier publication describing the logistic exponential model, and very close agreement of the estimates of the regression coefficients is demonstrated.     

Smoothing spline ANOVA frailty model for recurrent event data

Gap time hazard estimation is of particular interest in recurrent event data. This article proposes a fully nonparametric approach for estimating the gap time hazard. Smoothing spline analysis of variance (ANOVA) decompositions are used to model the log gap time hazard as a joint function of gap time and covariates, and general frailty is introduced to account for between-subject heterogeneity and within-subject correlation. We estimate the nonparametric gap time hazard function and parameters in the frailty distribution using a combination of the Newton-Raphson procedure, the stochastic approximation algorithm (SAA), and the Markov chain Monte Carlo (MCMC) method. The convergence of the algorithm is guaranteed by decreasing the step size of parameter update and/or increasing the MCMC sample size along iterations. Model selection procedure is also developed to identify negligible components in a functional ANOVA decomposition of the log gap time hazard. We evaluate the proposed methods with simulation studies and illustrate its use through the analysis of bladder tumor data.     

Non-inferiority testing with a variable margin

There has been growing interest, when comparing an experimental treatment with an active control with respect to a binary outcome, in allowing the non-inferiority margin to depend on the unknown success rate in the control group. It does not seem universally recognized, however, that the statistical test should appropriately adjust for the uncertainty surrounding the non-inferiority margin. In this paper, we inspect a naive procedure that treats an "observed margin" as if it were fixed a priori, and explain why it might not be valid. We then derive a class of tests based on the delta method, including the Wald test and the score test, for a smooth margin. An alternative derivation is given for the asymptotic distribution of the likelihood ratio statistic, again for a smooth margin. We discuss the asymptotic behavior of these tests when applied to a piecewise smooth margin. A simple condition on the margin function is given which allows the likelihood ratio test to carry over to a piecewise smooth margin using the same critical value as for a smooth margin. Simulation experiments are conducted, under a smooth margin and a piecewise linear margin, to evaluate the finite-sample performance of the asymptotic tests studied.     

The exponential‐Poisson model for recurrent event data: An application to a set of data on malaria in Brazil

In this paper, we introduce a new model for recurrent event data characterized by a baseline rate function fully parametric, which is based on the exponential‐Poisson distribution. The model arises from a latent competing risk scenario, in the sense that there is no information about which cause was responsible for the event occurrence. Then, the time of each recurrence is given by the minimum lifetime value among all latent causes. The new model has a particular case, which is the classical homogeneous Poisson process. The properties of the proposed model are discussed, including its hazard rate function, survival function, and ordinary moments. The inferential procedure is based on the maximum likelihood approach. We consider an important issue of model selection between the proposed model and its particular case by the likelihood ratio test and score test. Goodness of fit of the recurrent event models is assessed using Cox‐Snell residuals. A simulation study evaluates the performance of the estimation procedure in the presence of a small and moderate sample sizes. Applications on two real data sets are provided to illustrate the proposed methodology. One of them, first analyzed by our team of researchers, considers the data concerning the recurrence of malaria, which is an infectious disease caused by a protozoan parasite that infects red blood cells.     

Nonparametric Tests for Trend: Jonckheere's Test,a Modification and a Maximum Test

Jonckheere's test is a frequently used nonparametric trend test for the evaluation of preclinical studies and clinical dose-finding trials. In this paper, a modification of Jonckheere's test is proposed. If the exact permutation distribution is used for inference, the modified test can fill out the level of the type I error in a much more complete way and is substantially more powerful than the common Jonckheere test. If the asymptotic normality is used for inference, the modified test is slightly more powerful. In addition, a maximum test is investigated which is more robust concerning an a priori unknown dose-response shape. The robustness is advantageous, especially in a closed testing procedure. The different tests are applied to two example data sets.     

Principles of formalizing a quantitative evaluation of the genetic danger of chemical compounds for man

Specific characteristics of the mutagenic effect of chemicals, which must be taken into account in developing the test system to assess the potential genetic risk caused by chemical substances, are considered. The organizational principles of the procedures currently available for testing and ranking chemicals by their mutagenic and carcinogenic hazard to humans are discussed. The use of selective information suggested by Wiener and Shannon as an efficiency measure of testing and estimating the potential genetic hazard of chemical substances is substantiated. The feasibility of this approach was demonstrated by testing the efficiency of the battery of two short-term in vitro tests as an example. It was shown that selective information is able to serve as an integral universal criterion of the efficiency of testing, if either one test or the test battery were used.     

Exact percentage points of the likelihood-ratio test for a change-point hazard-rate model

We wish to test that the hazard rate of survival or failure-time data is constant against the alternative of a change in hazard after an unspecified time. The likelihood ratio is unbounded but the exact null distribution of a restricted likelihood-ratio test statistic is found. This distribution is not affected by Type II censoring but it does depend very strongly on the interval in which the unknown change-point is assumed to lie. Some exact percentage points are given which are much larger than simulated points that have been reported in the literature.     

Combining multivariate bioassays

Linear multivariate theory is applied to the problem of combining several multivariate bioassays. Results are an asymptotic test of the hypothesis of a common log relative potency; the maximum likelihood estimator of the common log relative potency; and an exact and asymptotic confidence interval estimator for log relative potency.     

Efficiency of the prediction of carcinogenic activities of chemical substances based on scoring somatic mutations in the soybean Glycine max (L.) Merrill

The efficiency of scoring somatic mutations in soybean (Glycine max (L.) Merrill) leaves as a test for carcinogenic activity of chemical substances in rodents has been evaluated. The efficiency of the test used alone or as part of a battery of tests has been estimated. The mutagenic activities of some chemical substances estimated using the soybean test are presented. Selective information on the carcinogenic activities of substances obtained in special carcinogenicity tests has been used as a quantitative measure of the efficiency of the tests with soybean leaves. To estimate the weight of evidence for the presence of this activity in the tested substances, a special function has been used whose values are uniquely related to the complete information, which is the sum of a priori information and the information obtained after testing. In general, the results have shown that the somatic mutation score test using soybean leaves is at least as efficient as the well-known tests that are generally used now, such as the Ames test and the chromosome aberration score test using mammalian cells in vitro. This test may be promising for the formation of efficient short-term test batteries.     

Handedness and longevity: archival study of cricketers.

OBJECTIVE--To test whether handedness is associated with a change in longevity. DESIGN--Archival survey. SETTING--British Isles. SUBJECTS--All first class cricketers born before 1961 whose bowling hand was specified (right, n = 5041; left, n = 1132) in a comprehensive encyclopaedia. MAIN OUTCOME MEASURES--Bowling hand and life span. RESULTS--Regression analysis of the 5960 players born between 1840 and 1960 (3387 dead, 2573 alive) showed no significant relation between mortality and handedness (P = 0.3). Left handedness was, however, associated with an increased likelihood of death from unnatural causes (P = 0.03, log hazard 0.37, 95% confidence interval 0.04 to 0.70). This effect was especially related to deaths during warfare (P = 0.009, log hazard 0.53, 0.13 to 0.92). CONCLUSION--Left handedness is not, in general, associated with an increase in mortality.     

Efficiency of the prediction of carcinogenic activities of chemical substances based on scoring somatic mutations in the soybean <Emphasis Type="Italic">Glycine max</Emphasis> (L.) Merrill

The efficiency of scoring somatic mutations in soybean (Glycine max (L.) Merrill) leaves as a test for carcinogenic activity of chemical substances in rodents has been evaluated. The efficiency of the test used alone or as part of a battery of tests has been estimated. The mutagenic activities of some chemical substances estimated using the soybean test are presented. Selective information on the carcinogenic activities of substances obtained in special carcinogenicity tests has been used as a quantitative measure of the efficiency of the tests with soybean leaves. To estimate the weight of evidence for the presence of this activity in the tested substances, a special function has been used whose values are uniquely related to the complete information, which is the sum of a priori information and the information obtained after testing. In general, the results have shown that the somatic mutation score test using soybean leaves is at least as efficient as the well-known tests that are generally used now, such as the Ames test and the chromosome aberration score test using mammalian cells in vitro. This test may be promising for the formation of efficient short-term test batteries.     

Multilevel Mixture Cure Models with Random Effects

This paper extends the multilevel survival model by allowing the existence of cured fraction in the model. Random effects induced by the multilevel clustering structure are specified in the linear predictors in both hazard function and cured probability parts. Adopting the generalized linear mixed model (GLMM) approach to formulate the problem, parameter estimation is achieved by maximizing a best linear unbiased prediction (BLUP) type log‐likelihood at the initial step of estimation, and is then extended to obtain residual maximum likelihood (REML) estimators of the variance component. The proposed multilevel mixture cure model is applied to analyze the (i) child survival study data with multilevel clustering and (ii) chronic granulomatous disease (CGD) data on recurrent infections as illustrations. A simulation study is carried out to evaluate the performance of the REML estimators and assess the accuracy of the standard error estimates.     

Maximum Likelihood Methods for Fitting the Burr Type XII Distribution to Life Test Data

This paper describes mathematical and computational methodology for estimating the parameters of the Burr Type XII distribution by the method of maximum likelihood. Expressions for the asymptotic variances and covariances of the parameter estimates are given, and the modality of the log-likelihood and conditional log-likelihood functions is analyzed. As a result of this analysis for various a priori known and unknown parameter combinations, conditions are given which guarantee that the parameter estimates obtained will, indeed, be maximum likelihood estimates. An efficient numerical method for maximizing the conditional log-likelihood function is described, and mathematical expressions are given for the various numerical approximations needed to evaluate the expressions given for the asymptotic variances and covariances of the parameter estimates. The methodology discussed is applied in a numerical example to life test data arising in a clinical setting.     

Power of the scan statistic in detecting a changed segment in a Bernoulli sequence

Fu & Curnow (1990) derive recursive equations to find thelevel of significance and power of a likelihood ratio test fora changed segment of specified length, based on the scan statistic,the maximum number of successes within the specified length.Their method is computationally feasible for segment lengthsof 20 or less. We present and evaluate highly accurate approximationsas well as bounds for the power function of this test that arecomputationally feasible even for very large segment lengths.We also evaluate power when the duration of the increased lengthused in the test statistic does not correspond to the actuallength.     

A statistical distribution with an unbounded hazard function and its application to a theory from demography.

We introduce a statistical distribution with hazard function mu(t) proportional to (psi-t)beta-1 for 0 less than or equal to t less than psi and shape parameter beta satisfying 0 less than beta less than 1. This hazard function is suggested by a theory of aging in demography. We discuss properties of this distribution and maximum likelihood estimates of its parameters. Mixture distributions are considered to account for pooling across dissimilar populations. This model is compared with the Gompertz and generalized Pareto distributions and used to estimate a finite limit on human lifespan based on the survival of a group of female centenarians.     

Flow DNA-cytometric findings of paraffin-embedded primary cutaneous melanomas related to prognosis

In a retrospective study flow cytometry was performed in 82 paraffin-embedded clinical stage I cutaneous melanomas. A significant correlation of ploidy to melanoma thickness was found. Using a log rank test ploidy was found significant for prognosis. However, using the Cox's proportional hazard test the influence of ploidy vanished and melanoma thickness or Clark level of invasion together with sex and location were the factors most important for prognosis.