An age dependent stochastic model of periodic screening: Length bias at a prevalence screen

An age dependent stochastic model for the periodic screening of a progressive chronic disease is used to investigate the length bias phenomenon for the case of a single screen. The preclinical state sojourn time distribution is obtained for control group cases, cases detected at the screen, and interval cases which surface after the screen in the screened group within an evaluation trial setting. Properties of these distributions are compared among themselves and with those of the underlying population to investigate the magnitude and direction of the length bias. The effect upon length bias of the magnitude of the false negative probability, the length of followup, the age at screening, the variance of the preclinical state sojourn time, and the correlation between the sojourn times in the disease free and preclinical disease states are investigated. Numerical results indicate that certain combinations of correlation and age at screening can result in substantial length bias in either the positive or the negative direction. It is also apparent that use of the randomized trial design and a suitable age range for screening can help to eliminate most of the extreme or negative length bias effects.     

Quantification of length‐bias in screening trials with covariate‐dependent test sensitivity

Length‐biased sampling exists in screening programs where longer duration disease is detected during the preclinical stage because a longer sojourn time (preclinical duration) has a higher probability of being screen detected. By modeling the course of disease, we quantify the effect of length‐biased sampling on clinical duration when cases are subject to periodic screening with variable test sensitivity. We use the highly flexible bivariate lognormal density to jointly model preclinical and clinical durations, and we model screening test sensitivity as a function of the sojourn time and number of previous false negative screens. We show that the mean clinical duration among screen‐detected cases can be up to 40% higher, with shrinking standard deviation, than those among nonscreen‐detected cases, due to biased sampling alone, irrespective of any possible benefit (increased survival time arising from earlier detection or reduction in mortality). These findings will aid in the design and interpretation of screening trials.     

Robust modeling in screening studies: estimation of sensitivity and preclinical sojourn time distribution

In early-detection clinical trials, quantities such as the sensitivity of the screening modality and the preclinical duration of the disease are important to describe the natural history of the disease and its interaction with a screening program. Assume that the schedule of a screening program is periodic and that the sojourn time in the preclinical state has a piecewise density function. Modeling the preclinical sojourn time distribution as a piecewise density function results in robust estimation of the distribution function. Our aim is to estimate the piecewise density function and the examination sensitivity using both generalized least squares and maximum likelihood methods. We carried out extensive simulations to evaluate the performance of the methods of estimation. The different estimation methods provide complimentary tools to obtain the unknown parameters. The methods are applied to three breast cancer early-detection trials.     

Estimation of sojourn time in chronic disease screening without data on interval cases

Estimation of the sojourn time on the preclinical detectable period in disease screening or transition rates for the natural history of chronic disease usually rely on interval cases (diagnosed between screens). However, to ascertain such cases might be difficult in developing countries due to incomplete registration systems and difficulties in follow-up. To overcome this problem, we propose three Markov models to estimate parameters without using interval cases. A three-state Markov model, a five-state Markov model related to regional lymph node spread, and a five-state Markov model pertaining to tumor size are applied to data on breast cancer screening in female relatives of breast cancer cases in Taiwan. Results based on a three-state Markov model give mean sojourn time (MST) 1.90 (95% CI: 1.18-4.86) years for this high-risk group. Validation of these models on the basis of data on breast cancer screening in the age groups 50-59 and 60-69 years from the Swedish Two-County Trial shows the estimates from a three-state Markov model that does not use interval cases are very close to those from previous Markov models taking interval cancers into account. For the five-state Markov model, a reparameterized procedure using auxiliary information on clinically detected cancers is performed to estimate relevant parameters. A good fit of internal and external validation demonstrates the feasibility of using these models to estimate parameters that have previously required interval cancers. This method can be applied to other screening data in which there are no data on interval cases.     

Mortality modeling of early detection programs

Summary .   Consider a group of subjects who are offered an opportunity to receive a sequence of periodic special examinations for the purpose of diagnosing a chronic disease earlier relative to usual care. The mortality for the early detection group is to be compared with a group receiving usual care. Benefit is reflected in a potential reduction in mortality. This article develops a general probability model that can be used to predict cumulative mortality for each of these groups. The elements of the model assume (i) a four-state progressive disease model in which a subject may be in a disease-free state (or a disease state that cannot be detected), preclinical disease state (capable of being diagnosed by a special exam), clinical state (diagnosis by usual care), and a death state; (ii) age-dependent transitions into the states; (iii) age-dependent examination sensitivity; (iv) age-dependent sojourn time in each state; and (v) the distribution of disease stages on diagnosis conditional on modality of detection. The model may be used to (i) compare mortality rates for different screening schedules; (ii) explore potential benefit of subpopulations; and (iii) compare relative reductions in disease-specific mortality due to advances and dissemination of both treatment and early detection screening programs.     

An early- and late-stage convolution model for disease natural history

The standard convolution model of disease natural history posits an asymptomatic (preclinical) and a symptomatic (clinical) state. An augmented model includes, in both the preclinical and clinical states, an early and late stage of disease. In the case of cancer, the early stage would generally correspond to the organ-confined stages before there is evidence of cancer spread. We compute the number of screen-detected (preclinical) and clinical cases in the early and late stages expected under a given screening program and show how the model can be fit to data from a screening trial using maximum likelihood. We also develop expressions for sojourn time, lead time, and overdiagnosis in the context of the model, where each of the above concepts incorporates disease stage. As an example, we fit the model to data from the Mayo Lung Cancer Screening trial.     

Neuron firing in driven nonlinear integrate-and-fire models

Statistical properties of neuron firing are studied in the framework of a nonlinear leaky integrate-and-fire model that is driven by a slow periodic subthreshold signal. The firing events are characterized by first passage time densities. The experimentally better accessible interspike interval density generally depends on the sojourn times in a refractory state of the neuron. This aspect is not part of the integrate-and-fire model and must be modelled additionally. For a large class of refractory dynamics, a general expression for the interspike interval density is given and further evaluated for the two cases with an instantaneous resetting (i.e. no refractory state) and a refractory state possessing a deterministic lifetime. First passage time densities and interspike interval densities following from the proposed theory compare favorably with precise numerical simulations.     

Two-stage models for the analysis of cancer screening data

Methods are proposed for the analysis of the natural history of disease from screening data when it cannot be assumed that untreated preclinical disease always progresses to clinical disease. The methodology is based on a two-stage model for preclinical disease in which stage 1 lesions may or may not progress to stage 2, but all stage 2 lesions progress to clinical disease. The focus is on joint estimation of the total preclinical duration and the sensitivity of the screening test. A partial likelihood is proposed for the analysis of prospectively collected screening data, and an analogous conditional likelihood is proposed for retrospective data. Some special cases for the joint sojourn distribution of the two stages are considered, including the independent model and limiting models where the duration of stage 2 is short relative to stage 1. The methods are applied to a case-control study of cervical cancer screening in Northeast Scotland.     

Nonparametric estimation of asymptomatic duration from a randomized prospective cancer screening trial

We propose a nonparametric estimation of preclinical duration distribution in cancer based on data from a randomized early detection trial. In cancer screening studies, the preclinical duration of a disease is of great interest for better understanding the natural history of the disease, and for developing optimal screening strategies. To estimate the sojourn time distribution nonparametrically, we first estimate the distribution of the age at onset of preclinical disease nonparametrically using data from the screening arm in a randomized screening trial, and the distribution for the age at onset of clinical disease from the control arm of the randomized screening trial. Finally, by using deconvolution the two estimated distributions lead to a nonparametric estimate of the distribution for the gap time between the onset of preclinical disease and the onset of clinical disease. We illustrate the methodology using data from a randomized breast cancer screening trial.     

Estimation and Prediction for Cancer Screening Models Using Deconvolution and Smoothing

The model that specifies that cancer incidence, I, is the convolution of the preclinical incidence, g, and the density of time in the preclinical phase, f, has frequently been utilized to model data from cancer screening trials and to estimate such quantities as sojourn time, lead time, and sensitivity. When this model is fit to the above data, the parameters of f as well as the parameter(s) governing screening sensitivity must be estimated. Previously, g was either assumed to be equal to clinical incidence or assumed to be a constant or exponential function that also had to be estimated. Here we assume that the underlying incidence, I, in the study population (in the absence of screening) is known. With I known, g then becomes a function of f, which can be solved for using (numerical) deconvolution, thus eliminating the need to estimate g or make assumptions about it. Since numerical deconvolution procedures may be highly unstable, however, we incorporate a smoothing procedure that produces a realistic g function while still closely reproducing the original incidence function I upon convolution with f. We have also added the concept of competing mortality to the convolution model. This, along with the realistic preclinical incidence function described above, results in more accurate estimates of sojourn time and lead time and allows for estimation of quantities related to overdiagnosis, which we define here.     

MLE and Bayesian inference of age-dependent sensitivity and transition probability in periodic screening

This article extends previous probability models for periodic breast cancer screening examinations. The specific aim is to provide statistical inference for age dependence of sensitivity and the transition probability from the disease free to the preclinical state. The setting is a periodic screening program in which a cohort of initially asymptomatic women undergo a sequence of breast cancer screening exams. We use age as a covariate in the estimation of screening sensitivity and the transition probability simultaneously, both from a frequentist point of view and within a Bayesian framework. We apply our method to the Health Insurance Plan of Greater New York study of female breast cancer and give age-dependent sensitivity and transition probability density estimates. The inferential methodology we develop is also applicable when analyzing studies of modalities for early detection of other types of progressive chronic diseases.     

A two-sample comparison for multiple ordered event data

A longitudinal study is conducted to compare the process of particular disease between two groups. The process of the disease is monitored according to which of several ordered events occur. In the paper, the sojourn time between two successive events is considered as the outcome of interest. The group effects on the sojourn times of the multiple events are parameterized by scale changes in a semiparametric accelerated failure time model where the dependence structure among the multivariate sojourn times is unspecified. Suppose that the sojourn times are subject to dependent censoring and the censoring times are observed for all subjects. A log-rank-type estimating approach by rescaling the sojourn times and the dependent censoring times into the same distribution is constructed to estimate the group effects and the corresponding estimators are consistent and asymptotically normal. Without the dependent censoring, the independent censoring times in general are not available for the uncensored data. In order to complete the censoring information, pseudo-censoring times are generated from the corresponding nonparametrically estimated survival function in each group, and we can still obtained unbiased estimating functions for the group effects. A real application and a simulation study are conducted to illustrate the proposed methods.     

On the use of time constants as estimates of mean lifetimes in single channel data analysis

For Markov models of single channel kinetics, a sojourn time in a class of states has a density function which is usually a linear combination of exponential densities. There are many instances in the single channel literature where the time constants of exponentials fitted to sojourn time data have been used as estimated mean sojourn times in individual states, though the two may be very different. In the present study the nature and magnitude of this difference in the case of a two state class is illustrated analytically and numerically. The time constants should be viewed at best as approximations, possibly poor, to the estimated mean sojourn times. Estimates of kinetic parameters cannot in general be obtained explicitly from the fitted parameters of the density alone. However, this is shown to be possible in some special cases and enables direct estimation of, for example, the channel opening rate constant (or an upper limit to the estimate of in the case of multiple channels) in standard sequential three or four state models of nicotinic receptor kinetics, using only the fitted parameters of the closed-time density. Offprint requests to: R. O. Edeson     

Estimating key parameters in periodic breast cancer screening—Application to the Canadian National Breast Screening Study data

Problem statement: Breast cancer screening in women of younger age has been controversial. The screening sensitivities, transition probabilities and sojourn time distributions are estimated for females aged 40–49 years and 50–59 years separately, using the Canadian National Breast Screening Study (CNBSS) data. The purpose is to estimate the lead time distribution and the probability of not detecting the cancer early. Approach: Within the 40–49-year-old and 50–59-year-old cohorts separately, the age-independent statistical model was applied. Bayesian estimators along with 95% highest probability density (HPD) credible intervals (CI) were calculated. Bayesian hypothesis testing was used to compare the parameter estimates of the two cohorts. The lead time density was also estimated for both the 40–49 and 50–59-year-old cohorts. Results: The screening sensitivity, transition probability of the disease, and mean sojourn time were all found to increase with age. For the 40–49-year-old and 50–59-year-old cohorts, the posterior mean sensitivities were 0.70 (95% HPD-CI: 0.46, 0.93) and 0.77 (0.61, 0.93), respectively. The posterior mean transition probabilities were 0.0023 (0.0018, 0.0027) and 0.0031 (0.0024, 0.0038), while the posterior mean sojourn times were 2.55 (1.56, 4.26) years and 3.15 (2.12, 4.96) years. Bayes factors for the ratio of posterior probabilities that the respective parameter was larger vs. smaller in the 50–59-year-old cohort were estimated to be 2.09, 40.8 and 3.0 for the sensitivity, transition probability, and mean sojourn time, respectively. All three Bayes factors were larger than two, indicating greater than 2:1 odds in favor of the hypothesis that each of these parameters was greater in the 50–59-year-old cohort. The estimated mean lead times were 0.83 years and 0.96 years if the two cohorts were screened annually. Conclusions: The increase in sensitivity corresponds to an increase in the mean sojourn time. Breast cancer in younger women is more difficult to detect by screening tests and is more aggressive than breast cancer in older women. Women aged 50–59 tend to benefit more from screening compared with women aged 40–49.     

Modeling and optimization in early detection programs with a single exam

The choice of timing of screening examinations is an important element in determining the efficacy of strategies for the early detection of occult disease. In this article, we describe a flexible decision-making framework for the design of early detection programs, and we investigate the choice of timing when each individual in the screening program is examined only once. We focus on the theoretical relation between the optimal examination time and the distributions of sojourn times in health-related states. Specifically, we derive closed-form solutions of the optimal age using two specifications of utility functions, discuss the effects of natural history and utility specifications on the optimal solution, and present an application to early detection of colorectal cancer by once-only sigmoidoscopy or colonoscopy.     

Graphs, random sums, and sojourn time distributions, with application to ion-channel modeling.

This paper considers the distribution of a sojourn time in a class of states of a stochastic process having finite discrete state space where sojourn times in any individual state are independent and identically distributed, and transitions between states follow a Markov chain. The state space and possible transitions of the process are represented by a graph. Class sojourn time distributions are derived by modifying this graph using 'composition' of states, defining a new Markov chain on the modified graph, and expressing the sojourn time in a composition state as a random sum. Appropriate compositions are chosen according to the possible "cores" of sojourns in the particular class, where a core describes the structure of a sojourn in terms of a single state or a chain in the original graph. Graph methods provide an algorithmic basis for the derivation, which can be simplified by using symmetry results. Models of ion-channel kinetics are used throughout for illustration; class sojourn time distributions are important in such models because individual states are often indistinguishable experimentally. Markov processes are the special case where sojourn times in individual states are exponentially distributed. In this case kinetic parameter estimation based on the observed class sojourn time distribution is briefly discussed; explicit estimating equations applicable to sequential models of nicotinic receptor kinetics are given.     

Screening for squamous cervical cancer: duration of low risk after negative results of cervical cytology and its implication for screening policies. IARC Working Group on evaluation of cervical cancer screening programmes.

A collaborative study of screening programmes in eight countries was performed to estimate the risks of cervical cancer associated with different screening policies. Most of the data came from centrally organised screening programmes. Relative protection was higher in women who had had two or more negative results of screening tests than in those who had had only one negative smear, particularly in the first five years after the last test. There was little difference in the protection afforded by screening every year compared with every three years, but screening only once every five or 10 years offered appreciably less protection. The age of the women did not affect the sensitivity of the test or the sojourn time of the disease (the length of the detectable preclinical phase during which abnormal cytology could be picked up if a smear were taken); invasive cancer in women under 25 was rare. Centrally organised screening programmes were more effective than uncoordinated screening. Screening programmes should be aimed principally at women aged 35-60 but should start some years before the age of 35, and the intervals between screening should be three years or less.     

Bayesian inference for the lead time in periodic cancer screening

This article develops a probability distribution for the lead time in periodic cancer screening examinations. The general aim is to allow statistical inference for a screening program's lead time, the length of time the diagnosis is advanced by screening. The program's lead time is distributed as a mixture of a point mass and a piecewise continuous distribution. Simulation studies using the HIP (Health Insurance Plan for Greater New York) study's data provide estimates of different characteristics of a screening program under different screening frequencies. The components of this mixture represent two aspects of screening's benefit, namely, a reduction in the number of interval cases and the extent by which screening advanced the age of diagnosis. We present estimates of these two measures for participants in a breast cancer screening program. We also provide the mean, mode, variance, and density curve of the program's lead time. The model can provide policy makers with important information regarding the screening period, frequency, and the endpoints that may serve as surrogates for the benefit to women who take part in a periodic screening program. Though the study focuses on breast cancer screening, it is also applicable to other kinds of chronic disease.     

Single ion channel models incorporating aggregation and time interval omission.

We present a general theoretical framework, incorporating both aggregation of states into classes and time interval omission, for stochastic modeling of the dynamic aspects of single channel behavior. Our semi-Markov models subsume the standard continuous-time Markov models, diffusion models and fractal models. In particular our models allow for quite general distributions of state sojourn times and arbitrary correlations between successive sojourn times. Another key feature is the invariance of our framework with respect to time interval omission: that is, properties of the aggregated process incorporating time interval omission can be derived directly from corresponding properties of the process without it. Even in the special case when the underlying process is Markov, this leads to considerable clarification of the effects of time interval omission. Among the properties considered are equilibrium behavior, sojourn time distributions and their moments, and auto-correlation and cross-correlation functions. The theory is motivated by ion channel mechanisms drawn from the literature, and illustrated by numerical examples based on these.     

Contributions to the mathematical theory of epidemics—II. The problem of endemicity

(1) A mathematical investigation has been made of the prevalence of a disease in a population from which certain individuals are being removed as the result of the disease, whilst fresh individuals are being introduced as the result of birth or immigration. Allowance is made for the effects of the immunity produced as the result of an attack of the disease, but the effect of deaths from other causes is not taken into account, and the action of the disease is supposed to be independent of the age of the individual. (2) As a special case of the above, results have been obtained for a closed population in which no deaths occur and to which no fresh individuals are added, but in which the individuals after being infected acquire immunity, and then may be again infected. A threshold density of population exists analogous to that described in the previous paper, which is such that no disease can exist in a population, the density of which is below the threshold. (3) In other special cases investigated when either immigration or birth is operative in the supply of fresh individuals, as well as in the general case, only one steady state of disease is possible. To reach this state the population must be of a certain density which will be determined by the functions characterizing the infectivity, morbidity, etc., of the disease. (4) Increase of the immigration rate or of the birth-rate results in an increase in the rate of infection of the healthy individuals and also in the percentage rate of infection, the percentage of sick, and in the percentage of mortality from the disease. This result is, of course, a necessary consequence of our assumption that the disease is the only cause of death. (5) More particular results have been obtained by substituting constants in the place of the undetermined functions assumed in the general theory. Further, under these conditions the nature of the steady states has been more fully investigated and it has been shown that in all cases, except one, the steady states are stable ones. In the exception, a disturbance would result in purely periodic oscillations about the steady state.