Weibull Proportional Hazards Model Research Articles

Towards Renewed Health Economic Simulation of Type 2 Diabetes: Risk Equations for First and Second Cardiovascular Events from Swedish Register Data

ObjectivePredicting the risk of future events is an essential part of health economic simulation models. In pursuit of this goal, the current study aims to predict the risk of developing first and second acute myocardial infarction, heart failure, non-acute ischaemic heart disease, and stroke after diagnosis in patients with type 2 diabetes, using data from the Swedish National Diabetes Register.Material and MethodsRegister data on 29,034 patients with type 2 diabetes were analysed over five years of follow up (baseline 2003). To develop and validate the risk equations, the sample was randomly divided into training (75%) and test (25%) subsamples. The Weibull proportional hazard model was used to estimate the coefficients of the risk equations, and these were validated in both the training and the test samples.ResultsIn total, 4,547 first and 2,418 second events were observed during the five years of follow up. Experiencing a first event substantially elevated the risk of subsequent events. There were heterogeneities in the effects of covariates within as well as between events; for example, while for females the hazard ratio of having a first acute myocardial infarction was 0.79 (0.70–0.90), the hazard ratio of a second was 1.21 (0.98–1.48). The hazards of second events decreased as the time since first events elapsed. The equations showed adequate calibration and discrimination (C statistics range: 0.70–0.84 in test samples).ConclusionThe accuracy of health economic simulation models of type 2 diabetes can be improved by ensuring that they account for the heterogeneous effects of covariates on the risk of first and second cardiovascular events. Thus it is important to extend such models by including risk equations for second cardiovascular events.

Bayesian analysis for step-stress accelerated life testing using weibull proportional hazard model

In this paper, we present a Bayesian analysis for the Weibull proportional hazard (PH) model used in step-stress accelerated life testings. The key mathematical and graphical difference between the Weibull cumulative exposure (CE) model and the PH model is illustrated. Compared with the CE model, the PH model provides more flexibility in fitting step-stress testing data and has the attractive mathematical properties of being desirable in the Bayesian framework. A Markov chain Monte Carlo algorithm with adaptive rejection sampling technique is used for posterior inference. We demonstrate the performance of this method on both simulated and real datasets.

Joint Modeling of Longitudinal and Survival Data

The joint modeling of longitudinal and survival data has received remarkable attention in the methodological literature over the past decade; however, the availability of software to implement the methods lags behind. The most common form of joint model assumes that the association between the survival and the longitudinal processes is underlined by shared random effects. As a result, computationally intensive numerical integration techniques such as adaptive Gauss–Hermite quadrature are required to evaluate the likelihood. We describe a new user-written command, stjm, that allows the user to jointly model a continuous longitudinal response and the time to an event of interest. We assume a linear mixed-effects model for the longitudinal submodel, allowing flexibility through the use of fixed or random fractional polynomials of time. Four choices are available for the survival submodel: the exponential, Weibull or Gompertz proportional hazard models, and the flexible parametric model (stpm2). Flexible parametric models are fit on the log cumulative-hazard scale, which has direct computational benefits because it avoids the use of numerical integration to evaluate the cumulative hazard. We describe the features of stjm through application to a dataset investigating the effect of serum bilirubin level on time to death from any cause in 312 patients with primary biliary cirrhosis.

Comparison of piecewise Weibull baseline survival models for estimation of true and functional longevity in Brown cattle raised in small herds

Piecewise Weibull proportional hazard models were used to investigate the effect of genetic and nongenetic factors on functional and true longevity traits of the Slovenian Brown cattle breed. Records of 37 908 Brown cows from 2401 Slovenian herds were used. As these herds were characterised by a relatively small average herd size starting from 6.7 in 1999 and increasing to 8.7 Brown cows per herd in 2008, milk yield classification was made within different herd size groups. The hazard rate was the lowest in the first part of each lactation and was increasing for later stages. Culling risk was lower for cows from herds increasing in size, for cows with higher milk production and for cows from a region with smaller herd sizes and tougher conditions for cattle breeding. The latter result is surprising and may be related to better attention to maintain the animals, despite their lower milk production. The introduction of the milk quota system and drought was found to have an important effect on culling policy between the last seasons of the years 2001 and 2003. Seasonal effects were not related to the milk quota year (from April to March), but to the effect of shortage in fodder during the winter time. The effect of age at first calving and the interaction between year and milk yield class were not found to be significant. Heritability for functional and for true longevity were similar at around 10% each. Inclusion of a correction for class of milk yield to approximate functional longevity increased the herd-year random effect variance by 53%, whereas the sire variance increased by only 14%. The correlation coefficient between ranks of breeding values for functional and true longevity was high (0.91), whereas genetic trends were not found to be significant. To assess their predictive ability, models were compared looking at the survival rate of 4212 second-crop daughters not included in the initial models. The average correlation between estimated breeding values and survival at different stages was 0.39 for true longevity and 0.43 for functional longevity. Results showed that ranking milk yield at population level is appropriate to correct for voluntary culling on low production in small herds.

Stgenreg: AStataPackage for General Parametric Survival Analysis

In this paper we present the Stata package stgenreg for the parametric analysis of survival data. Any user-defined hazard function can be specified, with the model estimated using maximum likelihood utilising numerical quadrature. Models that can be fitted range from the Weibull proportional hazards model to the generalized gamma model, mixture models, cure rate models, accelerated failure time models and relative survival models. We illustrate the features of stgenreg through application to a cohort of women diagnosed with breast cancer with outcome all-cause death.

Relationship between conformation traits and longevity in Polish Holstein Friesian cattle

A Weibull proportional hazard model was applied to examine the impact of conformation on true and functional longevity of Polish HF cattle. Lifetime records of 592,639 cows from 2369 herds calving from 1995 to 2008 were used. Conformation data available for 236,137 cows consisted of phenotypic scores for 14 linear traits. True longevity (LPL) was defined as the number of days from first calving to culling or censoring, and functional longevity (FLPL) as LPL adjusted for production. Linear type traits were added into the reference model, which included random effect of herd–year–season, and fixed effects of age at first calving, year–season, parity-stage of lactation, annual change in herd size, and (for FLPL) fat and protein yield. The strongest and positive influence on true longevity was found for udder traits. Capacity traits, rump traits, teat length and rear leg set displayed intermediate optima with respect to LPL. The highest impacts on functional longevity were from udder depth, body depth, chest width, and dairy character. Those traits can be considered early predictors. In respect to FLPL, udder depth had a positive impact, body depth showed an intermediate optimum, and the influences of chest width and dairy character were negative.

Comparison of linear, skewed-linear, and proportional hazard models for the analysis of lambing interval in Ripollesa ewes1

Lambing interval is a relevant reproductive indicator for sheep populations under continuous mating systems, although there is a shortage of selection programs accounting for this trait in the sheep industry. Both the historical assumption of small genetic background and its unorthodox distribution pattern have limited its implementation as a breeding objective. In this manuscript, statistical performances of 3 alternative parametrizations [i.e., symmetric Gaussian mixed linear (GML) model, skew-Gaussian mixed linear (SGML) model, and piecewise Weibull proportional hazard (PWPH) model] have been compared to elucidate the preferred methodology to handle lambing interval data. More specifically, flock-by-flock analyses were performed on 31,986 lambing interval records (257.3 ± 0.2 d) from 6 purebred Ripollesa flocks. Model performances were compared in terms of deviance information criterion (DIC) and Bayes factor (BF). For all flocks, PWPH models were clearly preferred; they generated a reduction of 1,900 or more DIC units and provided BF estimates larger than 100 (i.e., PWPH models against linear models). These differences were reduced when comparing PWPH models with different number of change points for the baseline hazard function. In 4 flocks, only 2 change points were required to minimize the DIC, whereas 4 and 6 change points were needed for the 2 remaining flocks. These differences demonstrated a remarkable degree of heterogeneity across sheep flocks that must be properly accounted for in genetic evaluation models to avoid statistical biases and suboptimal genetic trends. Within this context, all 6 Ripollesa flocks revealed substantial genetic background for lambing interval with heritabilities ranging between 0.13 and 0.19. This study provides the first evidence of the suitability of PWPH models for lambing interval analysis, clearly discarding previous parametrizations focused on mixed linear models.

Analysis of longevity in Slovenian holstein cattle

The longevity of Slovenian Holstein population was analysed using survival analysis with a Weibull proportional hazard model. Data spanned the period between January 1991 and January 2010 for 116,200 cows from 3,891 herds. Longevity was described as the length of productive life – from first calving till culling or censoring. Records above the sixth lactation were censored to partially avoid preferential treatment. Statistical model included the effect of age at first calving, stage of lactation within parity, yearly herd size deviation, season defined as year, herd, and sire-maternal grandsire (mgs). Some effects had time varying covariates, which lead to 1,839,307 or on average 16 elementary records per cow. Herd and sire-maternal grandsire effects were modelled hierarchically. Pedigree for sires and maternal grandsires included 2,284 entries. Estimated variance between herds was 0.12, while between sire variance was 0.04. Heritability was evaluated at 0.14. Genetic trend for sires was unfavourable, but not significant. A further research is needed to define the required number of daughters per sire and the dynamics of genetic evaluation for sires whose majority of daughters still have censored records.

Assessing the adequacy of Weibull survival models: a simulated envelope approach

The Weibull proportional hazards model is commonly used for analysing survival data. However, formal tests of model adequacy are still lacking. It is well known that residual-based goodness-of-fit measures are inappropriate for censored data. In this paper, a graphical diagnostic plot of Cox–Snell residuals with a simulated envelope added is proposed to assess the adequacy of Weibull survival models. Both single component and two-component mixture models with random effects are considered for recurrent failure time data. The effectiveness of the diagnostic method is illustrated using simulated data sets and data on recurrent urinary tract infections of elderly women.

Using cure models and multiple imputation to utilize recurrence as an auxiliary variable for overall survival

Intermediate outcome variables can often be used as auxiliary variables for the true outcome of interest in randomized clinical trials. For many cancers, time to recurrence is an informative marker in predicting a patient's overall survival outcome and could provide auxiliary information for the analysis of survival times. To investigate whether models linking recurrence and death combined with a multiple imputation procedure for censored observations can result in efficiency gains in the estimation of treatment effects and be used to shorten trial lengths. Recurrence and death times are modeled using data from 12 trials in colorectal cancer. Multiple imputation is used as a strategy for handling missing values arising from censoring. The imputation procedure uses a cure model for time to recurrence and a time-dependent Weibull proportional hazards model for time to death. Recurrence times are imputed, and then death times are imputed conditionally on recurrence times. To illustrate these methods, trials are artificially censored 2 years after the last accrual, the imputation procedure implemented, and a log-rank test and Cox model used to analyze and compare these new data with the original data. The results show modest, but consistent gains in efficiency in the analysis using the auxiliary information in recurrence times. Comparison of analyses show the treatment effect estimates and log-rank test results from the 2-year censored imputed data to be in between the estimates from the original data and the artificially censored data, indicating that the procedure was able to recover some of the lost information due to censoring. The models used are all fully parametric, requiring distributional assumptions of the data. The proposed models may be useful in improving the efficiency in estimation of treatment effects in cancer trials and shortening trial length.

Genetic evaluation of the longevity of the Holstein population in Japan using a Weibull proportional hazard model

The objective of this study was to confirm the stability of the genetic estimation of longevity of the Holstein population in Japan. Data on the first 10 lactation periods were obtained from the Livestock Improvement Association of Japan. Longevity was defined as the number of days from first calving until culling or censoring. DATA1 and DATA2 included the survival records for the periods 1991-2003 and 1991-2005, respectively. The proportional hazard model included the effects of the region-parity-lactation stage-milk yield class, age at first calving, the herd-year-season, and sire. The heritabilities on an original scale of DATA1 and DATA2 were 0.119 and 0.123, respectively. The estimated transmitting abilities (ETAs) of young sires in DATA1 may have been underestimated, but coefficient δ, which indicated the bias of genetic trend between DATA1 and DATA2, was not significant. The regression coefficient of ETAs between DATA1 and DATA2 was very close to 1. The proportional hazard model could steadily estimate the ETA for longevity of the sires in Japan.

Risk Factors for Tuberculosis After Highly Active Antiretroviral Therapy Initiation in the United States and Canada: Implications for Tuberculosis Screening

Screening for tuberculosis prior to highly active antiretroviral therapy (HAART) initiation is not routinely performed in low-incidence settings. Identifying factors associated with developing tuberculosis after HAART initiation could focus screening efforts. Sixteen cohorts in the United States and Canada contributed data on persons infected with human immunodeficiency virus (HIV) who initiated HAART December 1995-August 2009. Parametric survival models identified factors associated with tuberculosis occurrence. Of 37845 persons in the study, 145 were diagnosed with tuberculosis after HAART initiation. Tuberculosis risk was highest in the first 3 months of HAART (20 cases; 215 cases per 100000 person-years; 95% confidence interval [CI]: 131-333 per 100000 person-years). In a multivariate Weibull proportional hazards model, baseline CD4+ lymphocyte count <200, black race, other nonwhite race, Hispanic ethnicity, and history of injection drug use were independently associated with tuberculosis risk. In addition, in a piece-wise Weibull model, increased baseline HIV-1 RNA was associated with increased tuberculosis risk in the first 3 months; male sex tended to be associated with increased risk. Screening for active tuberculosis prior to HAART initiation should be targeted to persons with baseline CD4 <200 lymphocytes/mm³ or increased HIV-1 RNA, persons of nonwhite race or Hispanic ethnicity, history of injection drug use, and possibly male sex.

Estimation of genetic parameters for measures of calf survival in a population of Holstein heifer calves from a heifer-raising facility in New York State

The objectives of this study were to estimate the genetic parameters of survival to weaning and survival to exit for a population of Holstein calves from New York State, as well as to associate the estimated breeding values determined in the current study with traits from ongoing genetic evaluations used in Canada and the United States. Data were recorded for 7,372 heifer calves at a commercial rearing facility in New York State from arrival at 1 to 7 d of age for the duration of stay at the facility (on average, heifers departed the facility 1 mo before calving). Performance and disease up to weaning and mortality before and after weaning were recorded. Analyzed data were limited to daughters of sires with at least 10 calves originating from farms that had sent a minimum of 5 calves to be raised at the facility. As such, calves from 264 sires and 36 herds were studied using 2 statistical methods. The first method, calf survival, used a Weibull proportional hazards model, with survival defined as age at culling, death, or censoring. The second method, a 2-trait sire model, included survival from arrival to weaning (SUV1) and survival from weaning to exit (SUV2). Both models included fixed effects of arrival weight, serum total protein, weaning weight, season and year of birth, and calving ease score. Herd and sire were included as random effects. Significant associations among all fixed effects and calf survival were observed. In general, very light or heavy weight at arrival, low total protein, low weaning weight, and difficult birth increased risk of mortality for calves. The heritability of survival from the first method was 0.063. The heritabilities from the linear model were 0.001 for SUV1 and 0.036 for SUV2. The genetic correlation between SUV1 and SUV2 was 0.58. Genetic variance was close to zero for survival of calves to weaning, but was greater for survival after weaning through the growing period. Breeding values were estimated for SUV1 and SUV2 and correlated with routinely evaluated traits from Canadian and US genetic evaluations. Significant associations between genetic evaluations for survival traits and routinely evaluated traits in Canada and the United States were found, in particular with conformation body traits, somatic cell score, fertility, and longevity.

Analysis of the relationship between workability traits and functional longevity in Canadian dairy breeds

The aim of this study was to assess the effect of workability traits like milking speed and temperament on functional longevity of Canadian dairy cattle using a Weibull proportional hazards model. First-lactation data consisted of the following: 1,728,289 and 2,426,123 Holstein cows for milking temperament and milking speed, respectively, from 18,401 herds and sired by 8,248 sires; 39,618 and 60,121 Jersey cows for milking temperament and milking speed, respectively, from 1,845 herds and sired by 2,413 sires; and 54,391 and 94,847 Ayrshire cows for milking temperament and milking speed, respectively, from 1,316 herds and sired by 2,779 sires. Functional longevity was defined as the number of days from the first calving to culling, death, or censoring adjusted for production. Milking temperament and milking speed were recorded on a 1- to 5-point scale from very nervous to very calm and from very slow to very fast, respectively. The statistical model included the effects of stage of lactation; season of production; the annual change in herd size; type of milk recording supervision; age at first calving; effects of milk, fat, and protein yields calculated as within herd-year-parity deviations; herd-year-season of calving; sire; and milking temperament or milking speed class. The relative culling rate was calculated for animals in each milking temperament or milking speed class after accounting for the above-mentioned effects. The study showed that there was a statistically significant association between workability traits and functional longevity. Very nervous cows were 26, 23, and 46% more likely to be culled than very calm cows in Holstein, Ayrshire, and Jersey breeds, respectively. Similarly, very slow milkers were 36, 33, and 28% more likely to be culled than average milkers in Holstein, Ayrshire, and Jersey breeds, respectively. Additionally, very fast milkers were 11, 13, and 15% more likely to be culled than average milkers in Holstein, Ayrshire, and Jersey breeds, respectively. Producers might want to avoid consequences associated with the fast milkers such as udder health problems.

Age at which dairy cattle become Mycobacterium avium subsp. paratuberculosis faecal culture positive

Age at which cattle become faecal culture positive for Mycobacterium avium subsp. paratuberculosis ( Map) can be used as a proxy parameter for age at onset of faecal shedding, which is an important parameter in the control of Map in cattle herds. To investigate the age at becoming faecal culture positive, survival analysis methods were applied. The analyses were carried out on asynchronous interval censored data of faecal culture results of samples collected from 18,979 female Holstein-Frisian cattle in 353 Dutch herds between 1996 and 2002. The data were analysed with a Weibull proportional hazards model. The results indicate that the distribution of age at onset of faecal shedding in Holstein-Frisian dairy cattle in infected herds is associated with the within-herd prevalence. In higher classes of apparent prevalence, cattle started to shed Map at younger age on average. In herds with an apparent prevalence <0.05, 0.05–0.1, 0.1–0.2 and ≥0.2, the proportion (95% CI) of cattle with onset of faecal shedding before 2 years of age was estimated at 1% (0.5%; 2%), 4% (3%; 5%), 8% (5%; 10%) and 20% (11%; 32%), respectively. This study indicates that a considerable proportion of young stock is shedding Map, especially in high prevalence herds. Therefore, infectious young stock should be a major concern in the control of paratuberculosis.

Study on functional longevity of Tunisian Holstein dairy cattle using a Weibull proportional hazard model

Records of functional longevity (FL), from first calving to last recording (or culling), of 36,888 Tunisian Holstein cows in 169 herds were studied using survival analysis. Using a Weibull proportional hazard model, the influences of age at first calving, herd–year–season, lactation number–stage of lactation, milk production within the herd and year, annual change in herd size and random genetic effect of sire on FL were studied. Besides a random time-dependent effect of herd–year–season, the model included fixed effects of parity and stage of lactation, within herd and parity production level (all time-dependent), and age at first calving (time-independent). All factors had a significant effect on FL at a level of P < 0.001. Milk yield was by far the most important factor influencing FL and the hazard increased as the milk production decreased. Within the first three lactations, the relative culling risk increased stepwise with lactation stage. The highest risk of culling was found for cows at the beginning and at the end of the first lactation and at the end of any other lactation. The risk of culling decreased with parity.

Differences in Mortality among Heroin, Cocaine, and Methamphetamine Users: A Hierarchical Bayesian Approach.

Studies examining differences in mortality among long-term drug users have been limited. In this paper, we introduce a Bayesian framework that jointly models survival data using a Weibull proportional hazard model with frailty, and substance and alcohol data using mixed-effects models, to examine differences in mortality among heroin, cocaine, and methamphetamine users from five long-term follow-up studies. The traditional approach to analyzing combined survival data from numerous studies assumes that the studies are homogeneous, thus the estimates may be biased due to unobserved heterogeneity among studies. Our approach allows us to structurally combine the data from different studies while accounting for correlation among subjects within each study. Markov chain Monte Carlo facilitates the implementation of Bayesian analyses. Despite the complexity of the model, our approach is relatively straightforward to implement using WinBUGS. We demonstrate our joint modeling approach to the combined data and discuss the results from both approaches.

Outcome over seven years of healthy adults with and without subjective cognitive impairment

Background Subjective cognitive impairment (SCI) in older persons without manifest symptomatology is a common condition with a largely unclear prognosis. We hypothesized that (1) examining outcome for a sufficient period by using conversion to mild cognitive impairment (MCI) or dementia would clarify SCI prognosis, and (2) with the aforementioned procedures, the prognosis of SCI subjects would differ significantly from that of demographically matched healthy subjects, free of SCI, termed no cognitive impairment (NCI) subjects. Methods A consecutive series of healthy subjects, aged ≥40 years, presenting with NCI or SCI to a brain aging and dementia research center during a 14-year interval, were studied and followed up during an 18-year observation window. The study population (60 NCI, 200 SCI, 60% female) had a mean age of 67.2 ± 9.1 years, was well-educated (mean, 15.5 ± 2.7 years), and cognitively normal (Mini-Mental State Examination, 29.1 ± 1.2). Results A total of 213 subjects (81.9% of the study population) were followed up. Follow-up occurred during a mean period of 6.8 ± 3.4 years, and subjects had a mean of 2.9 ± 1.6 follow-up visits. Seven NCI (14.9%) and 90 SCI (54.2%) subjects declined ( P < .0001). Of NCI decliners, five declined to MCI and two to probable Alzheimer's disease. Of SCI decliners, 71 declined to MCI and 19 to dementia diagnoses. Controlling for baseline demographic variables and follow-up time, Weibull proportional hazards model revealed increased decline in SCI subjects (hazard ratio, 4.5; 95% confidence interval, 1.9–10.3), whereas the accelerated failure time model analysis with an underlying Weibull survival function showed that SCI subjects declined more rapidly, at 60% of the rate of NCI subjects (95% confidence interval, 0.45–0.80). Furthermore, mean time to decline was 3.5 years longer for NCI than for SCI subjects ( P = .0003). Conclusions These results indicate that SCI in subjects with normal cognition is a harbinger of further decline in most subjects during a 7-year mean follow-up interval. Relevance for community populations should be investigated, and prevention studies in this at-risk population should be explored.

Optimal maintenance strategy for Technology‐enhanced classrooms: a case study

AbstractThe purpose of the study was to collect data and evaluate an age‐based bulb replacement policy for projectors used in Technology‐enhanced classrooms in a university campus.The Office of Information Technology (OIT) had instituted a policy to replace bulbs in these projectors after either failure or 1800h of operation without failure. However, databases maintained by OIT included the time to bulb failure, the average number of times the bulbs were turned on and off per week, and the number of thermal events. Modeling the failure data with a Weibull proportional hazards model revealed that probability of bulb failure was a function of the number of hours a bulb was used and the frequency at which it was turned on and off. Combining this information with a cost model led to the recommendation for an improved condition‐based replacement policy. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Heifer rearing conditions affect length of productive life in Swedish dairy cows

Associations between rearing conditions and the risk of culling in dairy cows were studied by survival analysis. Data were collected from 1039 Swedish Red cows, 1029 Swedish Holsteins, and 56 cows of other milk or cross-breeds, representing all female animals born in 109 Swedish herds during 1998. Length of productive life was defined as the number of days from 1st calving to culling. The applied Weibull proportional hazards model included time-independent effects of breed, housing from 3 to 7 months of age, number of housing changes before calving, grazing before 1st calving, herd median age at 1st calving, age at 1st calving, cow housing, herd lactational incidence risk of veterinary-reported clinical mastitis, and the random effect of herd. Time-dependent effects were year, month, the interaction year by month, parity, number of breedings, pregnancy status, the interaction parity by pregnancy status, herd mean milk-production level, relative milk yield within breed-parity, and veterinary-reported clinical mastitis. The lactation was divided into six stages in which pregnancy status was assumed to be known by the farmer and culling could occur. Median productive life time in culled cows was 780d and 14% of the records were censored due to terminated data collection. An individual calving age of 28.2–30.9 months was associated with the highest culling risk, 1.2-fold higher than calving at ≤25.3 months, whereas the risk decreased almost linearly with a higher herd median age at 1st calving. Housing in slatted pens with >7 calves from 3 to 7 months was associated with a 1.7-fold increase in risk, relative to litter pens. If a cow had changed housing system 4 times before 1st calving it increased the risk of culling 1.4 times, relative to two housing changes. These results show that rearing factors affect the productive life time of dairy cows in Swedish family operations.