Piecewise Exponential Model Research Articles

Deliberate Birth Spacing in Nineteenth Century Northern Sweden

Fertility in nineteenth century Europe before the fertility transition has been described as high, unregulated, and stable; the extent of fertility control remains a controversial topic. The aim of this study is to determine whether there is evidence of deliberate birth spacing in northern Sweden prior to the onset of the fertility transition. This study analyses micro-level parish records of 9,636 women in nineteenth century northern Sweden—a remote but, at the time, economically dynamic frontier region of Sweden. Event history analysis reveals evidence of birth spacing that suggests some conscious birth control. Piecewise exponential models of the transition from second to third birth reveal circumstances in which parents increased or decreased the time to next birth. The results on the survival of previous children, geographic context, sex of previous children, and variations in grain prices all indicate that parents deliberately manipulated the spacing between births.

What is the true number needed to screen and treat to save a life with prostate-specific antigen testing?

The European Randomized Study of Screening for Prostate Cancer (ERSPC) reported a 20% mortality reduction with prostate-specific antigen (PSA) screening. However, they estimated a number needed to screen (NNS) of 1,410 and a number needed to treat (NNT) of 48 to prevent one prostate cancer death at 9 years. Although NNS and NNT are useful statistics to assess the benefits and harms of an intervention, in a survival study setting such as the ERSPC, NNS and NNT are time specific, and reporting values at one time point may lead to misinterpretation of results. Our objective was to re-examine the effect of varying follow-up times on NNS and NNT using data extrapolated from the ERSPC report. On the basis of published ERSPC data, we modeled the cumulative hazard function using a piecewise exponential model, assuming a constant hazard of 0.0002 for the screening and control groups for years 1 to 7 of the trial and different constant rates of 0.00062 and 0.00102 for the screening and control groups, respectively, for years 8 to 12. Annualized cancer detection and drop-out rates were also approximated based on the observed number of individuals at risk in published ERSPC data. According to our model, the NNS and NNT at 9 years were 1,254 and 43, respectively. Subsequently, NNS decreased from 837 at year 10 to 503 at year 12, and NNT decreased from 29 to 18. Despite the seemingly simplistic nature of estimating NNT, there is widespread misunderstanding of its pitfalls. With additional follow-up in the ERSPC, if the mortality difference continues to grow, the NNT to save a life with PSA screening will decrease.

Histology‐specific nomogram for primary retroperitoneal soft tissue sarcoma

This study was conducted to develop a histology-specific nomogram to predict postoperative overall survival (OS) at 5 and 10 years in primary retroperitoneal soft tissue sarcoma (STS). Data registered at a single institution (National Cancer Institute, Milan, Italy) prospective sarcoma database were used. In the present analysis, patients with primary localized retroperitoneal STS resected with curative intent between 1985 and 2007 were included. A parametric piecewise exponential survival multivariate model was used for nomogram development, and internal validation was performed with standard methodologies. Known prognostic variables, such as age, tumor burden, histologic variant (as reviewed by a sarcoma pathologist), grade, and surgical margins were considered as putative predictors. Among the 192 patients analyzed, within 10 years from surgery, 114 patients were alive, with a median follow-up time of 55 months (interquartile range, 25-104 months). Among the investigated factors, only histologic subtype did not reach significance at the 10% level. The relative hazard increased while increasing tumor size up to about 25 cm, and decreased thereafter. A histology-specific nomogram for retroperitoneal STS is now available. It can be used for better assessing the risk of the single patient and then making individualized decisions within the specific subset of retroperitoneal sarcomas. Cross-check external validation should be performed.

Risk Factors for Prostate Cancer Detection After a Negative Biopsy: A Novel Multivariable Longitudinal Approach

To introduce a novel approach for the time-dependent quantification of risk factors for prostate cancer (PCa) detection after an initial negative biopsy. Data for 1,871 men with initial negative biopsies and at least one follow-up biopsy were available. Piecewise exponential regression models were developed to quantify hazard ratios (HRs) and define cumulative incidence curves for PCa detection for subgroups with specific patterns of risk factors over time. Factors evaluated included age, race, serum prostate-specific antigen (PSA) concentration, PSA slope, digital rectal examination, dysplastic glands or prostatitis on biopsy, ultrasound gland volume, urinary symptoms, and number of negative biopsies. Four hundred sixty-five men had PCa detected, after a mean follow-up time of 2.8 years. All of the factors were independent predictors of PCa detection except for PSA slope, as a result of its correlation with time-dependent PSA level, and race. PSA (HR = 3.90 for > 10 v 2.5 to 3.9 ng/mL), high-grade prostatic intraepithelial neoplasia/atypical glands (HR = 2.97), gland volume (HR = 0.39 for > 50 v < 25 mL), and number of repeat biopsies (HR = 0.36 for two v zero repeat biopsies) were the strongest predictors. Men with high-risk versus low-risk event histories had a 20-fold difference in PCa detection over 5 years. Piecewise exponential models provide an approach to longitudinal analysis of PCa risk that allows clinicians to see the interplay of risk factors as they unfold over time for individual patients. With these models, it is possible to identify distinct subpopulations with dramatically different needs for monitoring and repeat biopsy.

Higher-risk periods for suicide among VA patients receiving depression treatment: Prioritizing suicide prevention efforts

BackgroundHealth systems with limited resources may have the greatest impact on suicide if their prevention efforts target the highest-risk treatment groups during the highest-risk periods. To date, few health systems have carefully segmented their depression treatment populations by level of risk and prioritized prevention efforts on this basis. MethodsWe conducted a retrospective cohort study of 887,859 VA patients receiving depression treatment between 4/1/1999 and 9/30/2004. We calculated suicide rates for five sequential 12-week periods following treatment events that health systems could readily identify: psychiatric hospitalizations, new antidepressant starts (>6 months without fills), “other” antidepressant starts, and dose changes. Using piecewise exponential models, we examined whether rates differed across time-periods. We also examined whether suicide rates differed by age-group in these periods. ResultsOver all time-periods, the suicide rate was 114/100,000 person-years (95% CI; 108, 120). In the first 12-week periods, suicide rates were: 568/100,000 p-y (95% CI; 493, 651) following psychiatric hospitalizations; 210/100,000 p-y (95% CI; 187, 236) following new antidepressant starts; 193/100,000 p-y (95% CI; 167, 222) following other starts; and 154/100,000 p-y (95% CI; 133, 177) following dose changes. Suicide rates remained above the base rate for 48 weeks following hospital discharge and 12 weeks following antidepressant events. Adults aged 61–80 years were at highest risk in the first 12-week periods. ConclusionsTo have the greatest impact on suicide, health systems should prioritize prevention efforts following psychiatric hospitalizations. If resources allow, closer monitoring may also be warranted in the first 12 weeks following antidepressant starts, across all age-groups.

Bayesian variable selection for the Cox regression model with missing covariates

In this paper, we develop Bayesian methodology and computational algorithms for variable subset selection in Cox proportional hazards models with missing covariate data. A new joint semi-conjugate prior for the piecewise exponential model is proposed in the presence of missing covariates and its properties are examined. The covariates are assumed to be missing at random (MAR). Under this new prior, a version of the Deviance Information Criterion (DIC) is proposed for Bayesian variable subset selection in the presence of missing covariates. Monte Carlo methods are developed for computing the DICs for all possible subset models in the model space. A Bone Marrow Transplant (BMT) dataset is used to illustrate the proposed methodology.

Breast Cancer Relative Hazard Estimates From Case–Control and Cohort Designs With Missing Data on Mammographic Density

We analyzed data from the Breast Cancer Detection Demonstration Project (BCDDP) to obtain multivariate relative hazard models for breast cancer that included mammographic density (MD) in addition to standard risk factors. Data from the BCDDP were collected from a stratified case–control study in the screening phase (1973–1980) and from follow-up of three subcohorts in the follow-up phase (1980–1995). For both phases, MD measurements were only available for about half the women who developed breast cancer (cases) and a small fraction of noncases. We used a logistic regression model for the stratified case–control study and developed a general pseudo-likelihood approach to accommodate missing covariate data (MD) by adapting the method of Scott and Wild and Breslow and Holubkov. We showed that this method was substantially more efficient than a previously proposed weighted-likelihood method. We assumed piecewise exponential models for the analysis of each subcohort, with the missing covariate (MD) distribution conditional on the observed information modeled with polytomous logistic regression. We developed an EM algorithm for estimation, which allowed for time-varying covariates, incomplete follow-up, and left truncation. We analyzed the three follow-up subcohorts separately and then combined the relative hazard models from the case–control and cohort data. The final model included main effects for MD, weight, age at first live birth, number of previous breast biopsies, and number of sisters or mother with breast cancer and was more discriminating (higher concordance) than the original model of Gail et al., which included standard risk factors but not MD. In a separate work, we combined this relative hazard model with other data to project absolute breast cancer risk.

Estimating the grid of time-points for the piecewise exponential model

One of the greatest challenges related to the use of piecewise exponential models (PEMs) is to find an adequate grid of time-points needed in its construction. In general, the number of intervals in such a grid and the position of their endpoints are ad-hoc choices. We extend previous works by introducing a full Bayesian approach for the piecewise exponential model in which the grid of time-points (and, consequently, the endpoints and the number of intervals) is random. We estimate the failure rates using the proposed procedure and compare the results with the non-parametric piecewise exponential estimates. Estimates for the survival function using the most probable partition are compared with the Kaplan-Meier estimators (KMEs). A sensitivity analysis for the proposed model is provided considering different prior specifications for the failure rates and for the grid. We also evaluate the effect of different percentage of censoring observations in the estimates. An application to a real data set is also provided. We notice that the posteriors are strongly influenced by prior specifications, mainly for the failure rates parameters. Thus, the priors must be fairly built, say, really disclosing the expert prior opinion.

A Multi-state Piecewise Exponential Model of Hospital Outcomes after Injury

To allow more accurate prediction of hospital length of stay (LOS) after serious injury or illness, a multi-state model is proposed, in which transitions from the hospitalized state to three possible outcome states (home, long-term care, or death) are assumed to follow constant rates for each of a limited number of time periods. This results in a piecewise exponential (PWE) model for each outcome. Transition rates may be affected by time-varying covariates, which can be estimated from a reference database using standard statistical software and Poisson regression. A PWE model combining the three outcomes allows prediction of LOS. Records of 259,941 injured patients from the US Nationwide Inpatient Sample were used to create such a multi-state PWE model with four time periods. Hospital mortality and LOS for patient subgroups were calculated from this model, and time-varying covariate effects were estimated. Early mortality was increased by anatomic injury severity or penetrating mechanism, but these effects diminished with time; age and male sex remained strong predictors of mortality in all time periods. Rates of discharge home decreased steadily with time, while rates of transfer to long-term care peaked at five days. Predicted and observed LOS and mortality were similar for multiple subgroups. Conceptual background and methods of calculation are discussed and demonstrated. Multi-state PWE models may be useful to describe hospital outcomes, especially when many patients are not discharged home.

Thermal fracture model for a functionally graded plate with a crack normal to the surfaces and arbitrary thermomechanical properties

The aim of this paper is to develop a general thermal fracture modelling method for a functionally graded plate (FGP) with a surface or embedded crack normal to the surfaces and arbitrary thermomechanical properties along the gradient direction. In the new model named as piecewise-exponential model (PE model), the FGP is modeled by dividing it into a multilayered medium with the mechanical properties varying exponentially in each layer and keeping continuous across the interfaces between layers. The thermal stress intensity factors (TSIFs) for some representative samples are presented. The results show that the variation profiles of the material property can significantly influence the fracture behavior of the FGP. Compared with the past piecewise homogeneous model, the advantages of the present PE model are discussed.

Modeling method for a crack problem of functionally graded materials with arbitrary properties—piecewise-exponential model

A new model, piecewise-exponential model (PE model), is developed to investigate the crack problem of the functionally graded materials (FGMs) with arbitrary properties. In the PE model, the functionally graded material is divided into some nonhomogeneous layers along the gradient direction of the properties, with each layer’s properties varying exponentially. By this way, the real material properties can be approached by a series of exponential functions. Since the real material properties are used on both surfaces of each nonhomogeneous layer, the nature of continuously varying properties of FGMs can be approached accurately. The influences of the local nonhomogeneity on the crack-tip fields can be fully considered. By using the new model, the fracture problem of a functionally graded strip with arbitrary properties and a crack vertical to the free surfaces is studied. The integral transform method, the theory of residues and the theory of singular integral equation are applied. Some representative samples with different kinds of nonhomogeneous properties are analyzed and the corresponding stress intensity factors (SIFs) are presented. It is shown that the PE mode is effective for investigating the crack problems of the FGMs with arbitrary properties.

Marginal regression models with a time to event outcome and discrete multiple source predictors

Information from multiple informants is frequently used to assess psychopathology. We consider marginal regression models with multiple informants as discrete predictors and a time to event outcome. We fit these models to data from the Stirling County Study; specifically, the models predict mortality from self report of psychiatric disorders and also predict mortality from physician report of psychiatric disorders. Previously, Horton et al. found little relationship between self and physician reports of psychopathology, but that the relationship of self report of psychopathology with mortality was similar to that of physician report of psychopathology with mortality. Generalized estimating equations (GEE) have been used to fit marginal models with multiple informant covariates; here we develop a maximum likelihood (ML) approach and show how it relates to the GEE approach. In a simple setting using a saturated model, the ML approach can be constructed to provide estimates that match those found using GEE. We extend the ML technique to consider multiple informant predictors with missingness and compare the method to using inverse probability weighted (IPW) GEE. Our simulation study illustrates that IPW GEE loses little efficiency compared with ML in the presence of monotone missingness. Our example data has non-monotone missingness; in this case, ML offers a modest decrease in variance compared with IPW GEE, particularly for estimating covariates in the marginal models. In more general settings, e.g., categorical predictors and piecewise exponential models, the likelihood parameters from the ML technique do not have the same interpretation as the GEE. Thus, the GEE is recommended to fit marginal models for its flexibility, ease of interpretation and comparable efficiency to ML in the presence of missing data.

Survival Analysis–Based Dynamic Travel Demand Models for Hurricane Evacuation

Two dynamic travel demand models for hurricane evacuation based on survival analysis are presented: a Cox proportional hazards model and a piecewise exponential model. These models were used to estimate the probability of a household's evacuating within discrete time intervals before hurricane landfall as a function of the household's socioeconomic characteristics, the characteristics of the hurricane, and management practice in terms of the issuance of an evacuation order. Data from southeast Louisiana collected following Hurricane Andrew were applied. Both models reproduced the evacuation behavior satisfactorily, although the piecewise exponential model was slightly more accurate. Both appear capable of modeling dynamic hurricane evacuation travel demand. However, the Cox model cannot accommodate certain time-dependent variables because of its structure. The piecewise exponential model does not have such a limitation. The piecewise exponential model also has the advantage of estimating the baseline hazards endogenously.

Survival Analysis-Based Dynamic Travel Demand Models for Hurricane Evacuation

Two dynamic travel demand models for hurricane evacuation based on survival analysis are presented: a Cox proportional hazards model and a piecewise exponential model. These models were used to estimate the probability of a household's evacuating within discrete time intervals before hurricane landfall as a function of the household's socioeconomic characteristics, the characteristics of the hurricane, and management practice in terms of the issuance of an evacuation order. Data from southeast Louisiana collected following Hurricane Andrew were applied. Both models reproduced the evacuation behavior satisfactorily, although the piecewise exponential model was slightly more accurate. Both appear capable of modeling dynamic hurricane evacuation travel demand. However, the Cox model cannot accommodate certain time-dependent variables because of its structure. The piecewise exponential model does not have such a limitation. The piecewise exponential model also has the advantage of estimating the baseline haza...

Genetic and environmental factors influencing mortality up to weaning of Bruna dels Pirineus beef calves in mountain areas. A survival analysis1

Survival analysis techniques were used to analyze survival up to weaning of beef calves in the Pyrenean mountains areas of Catalonia, Spain. The Kaplan-Meier curve showed that the survival experience was not constant throughout the lactation period, as the mortality rate was more pronounced during the first month of life. The proportional hazards model analysis showed that several factors influenced the instantaneous mortality rate, with the herd-year effect having the strongest influence. Calves born in the first part of the breeding season, from September to February, had the lowest mortality risk (P < 0.001), showing that mortality risk increases as births accumulate. Calves from cows younger than 1,300 d of productive life had a higher risk of mortality (P < 0.05). Unassisted calvings presented the smallest risk of mortality, and mortality risk increased up to five times as birth became more difficult (P < 0.001). This risk also tended to increase slightly when calf birth weight was small (P < 0.10); for bigger calves, no increase of risk was detected, probably because calving difficulty was included in the model. These results suggest the need for improving the environment in the second part of the breeding period and paying more attention to births from younger cows. The survival curve fitted a parametric piecewise exponential function very well, with cut points at 16 and 32 d. The lower risk corresponded to the period of 33 to 180 d, the risk for the periods 17 to 32 d and 1 to 16 d being multiplied by 7 and 26, respectively. Confirming the robustness of the Cox model, the relative risks estimated for the different factors under this piecewise exponential model or a Weibull time-dependent model were similar to those reported above, as well as to those estimated under a frailty model, including the sire as a random effect. The modal estimates of sire variance under different baseline functions were close to 0.3, although the standard errors were very large. At weaning, the heritability estimate in the binary scale reached a value of only 0.037 because the survival at weaning was very high (96.9%) in this population. Nevertheless, in populations with a higher mortality, the inclusion of survival to weaning in the breeding objective might be justified. Overall, these results show that survival analysis is a powerful tool to analyze the mortality curve until weaning of beef calves.

Joint models for efficient estimation in proportional hazards regression models.

In survival studies, information lost through censoring can be partially recaptured through repeated measures data which are predictive of survival. In addition, such data may be useful in removing bias in survival estimates, due to censoring which depends upon the repeated measures. Here we investigate joint models for survival T and repeated measurements Y, given a vector of covariates Z. Mixture models indexed as f (T/Z) f (Y/T,Z) are well suited for assessing covariate effects on survival time. Our objective is efficiency gains, using non-parametric models for Y in order to avoid introducing bias by misspecification of the distribution for Y. We model (T/Z) as a piecewise exponential distribution with proportional hazards covariate effect. The component (Y/T,Z) has a multinomial model. The joint likelihood for survival and longitudinal data is maximized, using the EM algorithm. The estimate of covariate effect is compared to the estimate based on the standard proportional hazards model and an alternative joint model based estimate. We demonstrate modest gains in efficiency when using the joint piecewise exponential joint model. In a simulation, the estimated efficiency gain over the standard proportional hazards model is 6.4 per cent. In clinical trial data, the estimated efficiency gain over the standard proportional hazards model is 10.2 per cent.

Assessing chronic disease progression using non-homogeneous exponential regression Markov models: an illustration using a selective breast cancer screening in Taiwan.

Previous research on estimation of the progression of chronic disease, from the normal preclinical screen-detectable phase (PCDP) to the final clinical phase, has usually assumed constant transition rates and has rarely addressed how relevant covariates affect multi-state transitions. The present study proposes two non-homogeneous models using the Weibull distribution and piecewise exponential model, together with covariate functions of the proportional hazard form, to tackle these problems. We illustrate the models by application to a selective breast cancer screening programme. The results of the Weibull model yield estimates of scale and shape parameters for annual preclinical incidence rate as 0.0000058 (SE=0.0000019) and 2.4755 (SE=0.1153), the latter being significantly higher than 1. Annual transition rate was estimated as 0.3153 (SE=0.1385). Relative risks for the effects of late age at first pregnancy (AP) and high body mass index (BMI) on preclinical incidence rate were 1.98 and 2.59, respectively. The corresponding figures on the transition from the PCDP to clinical phase were 1.56 and 1.99, respectively. Non-homogeneous Markov models proposed in this study can be easily applied to rates of progression of chronic disease with increasing or decreasing rates with time and to model the effect of relevant covariates on multi-state transition rates.

Joint modelling of cause-specific hazard functions with cubic splines: an application to a large series of breast cancer patients

The time of appearance of several kinds of relapses after a therapeutic intervention is of increasing interest in oncology. Typically, in breast cancer patients, events of clinical interest are intra-breast tumor recurrences and distant metastases, which act in a competitive way when considered as first failure. The evaluation of differential effects of clinical and biological variables on each event can improve the knowledge on the course of the disease and the targeting of future therapy. A simple tool for the joint smoothed estimation of cause-specific hazards functions and continuous covariate effects has been developed. Within the framework of generalized linear models with Poisson error, an extension of the piecewise exponential model is proposed, based on grouping follow-up times and continuous covariates. Interpolation of cause-specific hazards is obtained by resorting to cubic splines, which are piecewise polynomials of simple implementation with standard statistical software; their flexibility and smoothness are easily controlled by the number of knots and constraints on polynomial derivatives. The approach was applied to a data set of 2233 breast cancer patients treated with conservative surgery. It allowed modelling time-dependent and cause-specific effects of covariates on the hazard functions.

Bayesian Cure Rate Models for Malignant Melanoma: A Case-Study of Eastern Cooperative Oncology Group Trial E1690

Summary We propose several Bayesian models for modelling time-to-event data. We consider a piecewise exponential model, a fully parametric cure rate model and a semiparametric cure rate model. For each model, we derive the likelihood function and examine some of its properties for carrying out Bayesian inference with non-informative priors. We also examine model identifiability issues and give conditions which guarantee identifiability. Also, for each model, we construct a class of informative prior distributions based on historical data, i.e. data from similar previous studies. These priors, called power priors, prove to be quite useful in this context. We examine the properties of the power priors for Bayesian inference and, in particular, we study their effect on the current analysis. Tools for model comparison and model assessment are also proposed. A detailed case-study of a recently completed melanoma clinical trial conducted by the Eastern Cooperative Oncology Group is presented and the methodology proposed is demonstrated in detail.

Pseudo-likelihood estimates of the cumulative risk of an autosomal dominant disease from a kin-cohort study.

Wacholder et al. [1998: Am J Epidemiol 148:623-629] and Struewing et al. [1997: N Engl J Med 336:1401-1408] have recently proposed a design called the kin-cohort design to estimate the probability of developing disease (penetrance) associated with an autosomal dominant gene. In this design, volunteers (probands) agree to be genotyped and one also determines the disease history (phenotype) of first-degree relatives of the proband. They used this design to estimate that the chance of developing breast cancer by age 70 in Ashkenazi Jewish women who carried mutations of the genes BRCA1 or BRCA2 was 0.56, a figure that was lower than previously estimated from highly affected families. The method that they used to estimate the cumulative risk of breast cancer, while asymptotically correct, does not necessarily produce monotone estimates in small samples. To obtain monotone, weakly parametric estimates, we consider separate piecewise exponential models for carriers and non-carriers. As the number of intervals on which constant hazards are assumed increases, however, the maximum likelihood score equations become unstable and difficult to solve. We, therefore, developed alternative pseudo-likelihood procedures that are readily solvable for piecewise exponential models with many intervals. We study these techniques through simulations and a re-analysis of a portion of the data used by Struewing et al. [1997] and discuss possible extensions.