The number and timing of time -dependent covariate measurements for the Cox regression model

The problem of analyzing data with updated measurements in the time-dependent proportional hazards model arises frequently in practice. One available option is to reduce the number of intervals (or updated measurements) to be included in the Cox regression model. We empirically investigated the bias of the estimator of the time-dependent covariate while varying the effect of failure rate, sample size, true values of the parameters and the number of intervals. We also evaluated how often a time-dependent covariate needs to be collected and assessed the effect of sample size and failure rate on the power of testing a time-dependent effect.^ A time-dependent proportional hazards model with two binary covariates was considered. The time axis was partitioned into k intervals. The baseline hazard was assumed to be 1 so that the failure times were exponentially distributed in the ith interval. A type II censoring model was adopted to characterize the failure rate. The factors of interest were sample size (500, 1000), type II censoring with failure rates of 0.05, 0.10, and 0.20, and three values for each of the non-time-dependent and time-dependent covariates (1/4,1/2,3/4).^ The mean of the bias of the estimator of the coefficient of the time-dependent covariate decreased as sample size and number of intervals increased whereas the mean of the bias increased as failure rate and true values of the covariates increased. The mean of the bias of the estimator of the coefficient was smallest when all of the updated measurements were used in the model compared with two models that used selected measurements of the time-dependent covariate. For the model that included all the measurements, the coverage rates of the estimator of the coefficient of the time-dependent covariate was in most cases 90% or more except when the failure rate was high (0.20). The power associated with testing a time-dependent effect was highest when all of the measurements of the time-dependent covariate were used. An example from the Systolic Hypertension in the Elderly Program Cooperative Research Group is presented. ^

AN ILLNESS-DEATH PROCESS WITH TIME-DEPENDENT COVARIATES

A general model for the illness-death stochastic process with covariates has been developed for the analysis of survival data. This model incorporates important baseline and time-dependent covariates to make proper adjustment for the transition probabilities and survival probabilities. The follow-up period is subdivided into small intervals and a constant hazard is assumed for each interval. An approximation formula is derived to estimate the transition parameters when the exact transition time is unknown.^ The method developed is illustrated by using data from a study on the prevention of the recurrence of a myocardial infarction and subsequent mortality, the Beta-Blocker Heart Attack Trial (BHAT). This method provides an analytical approach which simultaneously includes provision for both fatal and nonfatal events in the model. According to this analysis, the effectiveness of the treatment can be compared between the Placebo and Propranolol treatment groups with respect to fatal and nonfatal events. ^

Analysis of recurrent failure times: A time-dependent Yule process approach

Analysis of recurrent events has been widely discussed in medical, health services, insurance, and engineering areas in recent years. This research proposes to use a nonhomogeneous Yule process with the proportional intensity assumption to model the hazard function on recurrent events data and the associated risk factors. This method assumes that repeated events occur for each individual, with given covariates, according to a nonhomogeneous Yule process with intensity function λx(t) = λ 0(t) · exp( x′β). One of the advantages of using a non-homogeneous Yule process for recurrent events is that it assumes that the recurrent rate is proportional to the number of events that occur up to time t. Maximum likelihood estimation is used to provide estimates of the parameters in the model, and a generalized scoring iterative procedure is applied in numerical computation. ^ Model comparisons between the proposed method and other existing recurrent models are addressed by simulation. One example concerning recurrent myocardial infarction events compared between two distinct populations, Mexican-American and Non-Hispanic Whites in the Corpus Christi Heart Project is examined. ^

Is sensitization to stimulants time dependent and mediated via NMDA receptors?

Chronic administration of psychomotor stimulants has been reported to produce behavioral sensitization to its effects on motor activity. This adaptation may be related to the pathophysiology of recurrent psychiatric disorders. Since disturbances in circadian rhythms are also found in many of these disorders, the relationship between sensitization and chronobiological factors became of interest. Therefore, a computerized monitoring system investigated the following: whether repeated exposure to methylphenidate (MPD) and amphetamine (AMP) could produce sensitization to its locomotor effects in the rat; whether sensitization to MPD and AMP was dependent on the circadian time of drug administration; whether the baseline levels of locomotor activity would be effected by repeated exposure to MPD and AMP; whether the expression of a sensitized response could be affected by the photoperiod; and whether MK-801, a non-competitive NMDA antagonist, could disrupt the development of sensitization to MPD. Dawley rats were housed in test cages and motor activity was recorded continuously for 16 days. The first 2 days served as baseline for each rat, and on day 3 each rat received a saline injection. The locomotor response to 0.6, 2.5, or 10 mg/kg of MPD was tested on day 4, followed by five days of single injections of 2.5 mg/kg MPD (days 5–9). After five days without injection (days 10–14) rats were re-challenged (day 15) with the same doses they received on day 4. There were three separate dose groups ran at four different times of administration, 08:00, 14:00, 20:00, or 02:00 (i.e. 12 groups). The same protocol was conducted with AMP with the doses of 0.3, 0.6, and 1.2 mg/kg given on day 4 and 15, and 0.6 mg/kg AMP as the repeated dose on days 5 to 9. In the second set of experiments only sensitization to MPD was investigated. The expression of the sensitized response was dose-dependent and mainly observed with challenge of the lower dose groups. The development of sensitization to MPD and ANT was differentially time-dependent. For MPD, the most robust sensitization occurred during the light phase, with no sensitization during the middle of the dark phase. (Abstract shortened by UMI.) ^

Age -period -cohort analysis: An extension of Cox's lifetable regression with time -dependent covariates

It is well known that an identification problem exists in the analysis of age-period-cohort data because of the relationship among the three factors (date of birth + age at death = date of death). There are numerous suggestions about how to analyze the data. No one solution has been satisfactory. The purpose of this study is to provide another analytic method by extending the Cox's lifetable regression model with time-dependent covariates. The new approach contains the following features: (1) It is based on the conditional maximum likelihood procedure using a proportional hazard function described by Cox (1972), treating the age factor as the underlying hazard to estimate the parameters for the cohort and period factors. (2) The model is flexible so that both the cohort and period factors can be treated as dummy or continuous variables, and the parameter estimations can be obtained for numerous combinations of variables as in a regression analysis. (3) The model is applicable even when the time period is unequally spaced.^ Two specific models are considered to illustrate the new approach and applied to the U.S. prostate cancer data. We find that there are significant differences between all cohorts and there is a significant period effect for both whites and nonwhites. The underlying hazard increases exponentially with age indicating that old people have much higher risk than young people. A log transformation of relative risk shows that the prostate cancer risk declined in recent cohorts for both models. However, prostate cancer risk declined 5 cohorts (25 years) earlier for whites than for nonwhites under the period factor model (0 0 0 1 1 1 1). These latter results are similar to the previous study by Holford (1983).^ The new approach offers a general method to analyze the age-period-cohort data without using any arbitrary constraint in the model. ^

A reduced model clarifies the role of feedback loops and time delays in the Drosophila circadian oscillator.

Although several detailed models of molecular processes essential for circadian oscillations have been developed, their complexity makes intuitive understanding of the oscillation mechanism difficult. The goal of the present study was to reduce a previously developed, detailed model to a minimal representation of the transcriptional regulation essential for circadian rhythmicity in Drosophila. The reduced model contains only two differential equations, each with time delays. A negative feedback loop is included, in which PER protein represses per transcription by binding the dCLOCK transcription factor. A positive feedback loop is also included, in which dCLOCK indirectly enhances its own formation. The model simulated circadian oscillations, light entrainment, and a phase-response curve with qualitative similarities to experiment. Time delays were found to be essential for simulation of circadian oscillations with this model. To examine the robustness of the simplified model to fluctuations in molecule numbers, a stochastic variant was constructed. Robust circadian oscillations and entrainment to light pulses were simulated with fewer than 80 molecules of each gene product present on average. Circadian oscillations persisted when the positive feedback loop was removed. Moreover, elimination of positive feedback did not decrease the robustness of oscillations to stochastic fluctuations or to variations in parameter values. Such reduced models can aid understanding of the oscillation mechanisms in Drosophila and in other organisms in which feedback regulation of transcription may play an important role.

TIME DEPENDENT ALTERATIONS IN CENTRAL NERVOUS SYSTEM DOPAMINE RECEPTORS INDUCED BY DOPAMINE AGONIST TREATMENT

Levodopa, the precursor of dopamine, is currently the drug of choice in the treatment of Parkinson's disease. Recently, two direct dopamine agonists, bromocriptine and pergolide, have been tested for the treatment of Parkinson's disease because of reduced side effects compared to levodopa. Few studies have evaluated the effects of long-term treatment of dopamine agonists on dopamine receptor regulation in the central nervous system. Thus, the purpose of this study was to determine whether chronic dopamine agonist treatment produces a down-regulation of striatal dopamine receptor function and to compare the results of the two classes of dopaminergic drugs.^ Levodopa with carbidopa, a peripheral decarboxylase inhibitor, was administered orally to rats whereas bromocriptine and pergolide were injected intraperitoneally once daily. Several neurochemical parameters were examined from 1 to 28 days.^ Levodopa minimally decreased striatal D-1 receptor activity but increased the number of striatal D-2 binding sites. Levodopa increased the V(,max) of tyrosine hydroxylase (TH) in all brain regions tested. Protein blot analysis of striatal TH indicated a significant increase in the amount of TH present. Dopamine-beta-hydroxylase (DBH) activity was markedly decreased in all brain regions studied and mixing experiments of control and drug-treated cortices did not show the presence of an increased level of endogenous inhibitors.^ Bromocriptine treatment decreased the number of D-2 binding sites. Striatal TH activity was decreased and protein blot analysis indicated no change in TH quantity. The specificity of bromocriptine for striatal TH suggested that bromocriptine preferentially interacts with dopamine autoreceptors.^ Combination levodopa-bromocriptine was administered for 12 days. There was a decrease in both D-1 receptor activity and D-2 binding sites, and a decrease in brain HVA levels suggesting a postsynaptic receptor action. Pergolide produced identical results to the combination levodopa-bromocriptine studies.^ In conclusion, combination levodopa-bromocriptine and pergolide treatments exhibited the expected down-regulation of dopamine receptor activity. In contrast, levodopa appeared to up-regulate dopamine receptor activity. Thus, these data may help to explain, on a biochemical basis, the decrease in the levodopa-induced side effects noted with combination levodopa-bromocriptine or pergolide therapies in the treatment of Parkinson's disease. ^

Effect of patient accrual patterns on power and size of log-rank test for time-to-event outcome in clinical trials

The determination of size as well as power of a test is a vital part of a Clinical Trial Design. This research focuses on the simulation of clinical trial data with time-to-event as the primary outcome. It investigates the impact of different recruitment patterns, and time dependent hazard structures on size and power of the log-rank test. A non-homogeneous Poisson process is used to simulate entry times according to the different accrual patterns. A Weibull distribution is employed to simulate survival times according to the different hazard structures. The current study utilizes simulation methods to evaluate the effect of different recruitment patterns on size and power estimates of the log-rank test. The size of the log-rank test is estimated by simulating survival times with identical hazard rates between the treatment and the control arm of the study resulting in a hazard ratio of one. Powers of the log-rank test at specific values of hazard ratio (≠1) are estimated by simulating survival times with different, but proportional hazard rates for the two arms of the study. Different shapes (constant, decreasing, or increasing) of the hazard function of the Weibull distribution are also considered to assess the effect of hazard structure on the size and power of the log-rank test. ^

Signalling ballet in space and time.

Although we have amassed extensive catalogues of signalling network components, our understanding of the spatiotemporal control of emergent network structures has lagged behind. Dynamic behaviour is starting to be explored throughout the genome, but analysis of spatial behaviours is still confined to individual proteins. The challenge is to reveal how cells integrate temporal and spatial information to determine specific biological functions. Key findings are the discovery of molecular signalling machines such as Ras nanoclusters, spatial activity gradients and flexible network circuitries that involve transcriptional feedback. They reveal design principles of spatiotemporal organization that are crucial for network function and cell fate decisions.

A comparison of Markov and generalized linear models of hospital mortality

This paper reports a comparison of three modeling strategies for the analysis of hospital mortality in a sample of general medicine inpatients in a Department of Veterans Affairs medical center. Logistic regression, a Markov chain model, and longitudinal logistic regression were evaluated on predictive performance as measured by the c-index and on accuracy of expected numbers of deaths compared to observed. The logistic regression used patient information collected at admission; the Markov model was comprised of two absorbing states for discharge and death and three transient states reflecting increasing severity of illness as measured by laboratory data collected during the hospital stay; longitudinal regression employed Generalized Estimating Equations (GEE) to model covariance structure for the repeated binary outcome. Results showed that the logistic regression predicted hospital mortality as well as the alternative methods but was limited in scope of application. The Markov chain provides insights into how day to day changes of illness severity lead to discharge or death. The longitudinal logistic regression showed that increasing illness trajectory is associated with hospital mortality. The conclusion is reached that for standard applications in modeling hospital mortality, logistic regression is adequate, but for new challenges facing health services research today, alternative methods are equally predictive, practical, and can provide new insights. ^

THE RELATIONSHIP BETWEEN DEGREE OF BLOOD PRESSURE REDUCTION AND MORTALITY AMONG HYPERTENSIVES IN THE HYPERTENSION DETECTION AND FOLLOW-UP PROGRAM

The relationship between degree of diastolic blood pressure (DBP) reduction and mortality was examined among hypertensives, ages 30-69, in the Hypertension Detection and Follow-up Program (HDFP). The HDFP was a multi-center community-based trial, which followed 10,940 hypertensive participants for five years. One-year survival was required for inclusion in this investigation since the one-year annual visit was the first occasion where change in blood pressure could be measured on all participants. During the subsequent four years of follow-up on 10,052 participants, 568 deaths occurred. For levels of change in DBP and for categories of variables related to mortality, the crude mortality rate was calculated. Time-dependent life tables were also calculated so as to utilize available blood pressure data over time. In addition, the Cox life table regression model, extended to take into account both time-constant and time-dependent covariates, was used to examine the relationship change in blood pressure over time and mortality.^ The results of the time-dependent life table and time-dependent Cox life table regression analyses supported the existence of a quadratic function which modeled the relationship between DBP reduction and mortality, even after adjusting for other risk factors. The minimum mortality hazard ratio, based on a particular model, occurred at a DBP reduction of 22.6 mm Hg (standard error = 10.6) in the whole population and 8.5 mm Hg (standard error = 4.6) in the baseline DBP stratum 90-104. After this reduction, there was a small increase in the risk of death. There was not evidence of the quadratic function after fitting the same model using systolic blood pressure. Methodologic issues involved in studying a particular degree of blood pressure reduction were considered. The confidence interval around the change corresponding to the minimum hazard ratio was wide and the obtained blood pressure level should not be interpreted as a goal for treatment. Blood pressure reduction was attributed, not only to pharmacologic therapy, but also to regression to the mean, and to other unknown factors unrelated to treatment. Therefore, the surprising results of this study do not provide direct implications for treatment, but strongly suggest replication in other populations. ^