**displaystyle**- In the simplest case of stationary coefficients, for example, a treatment with a drug may, say, halve a subject's hazard at any given time t {\displaystyle t} , while the baseline hazard may vary. (wikipedia.org)
- each unit increase in x {\displaystyle x} results in proportional scaling of the hazard. (wikipedia.org)
- Under these circumstances, for a general linear model y = X β + ϵ {\displaystyle y=X\beta +\epsilon } , the ordinary least-squares estimator β ^ O L S = ( X T X ) − 1 X T y {\displaystyle {\hat {\beta }}_{OLS}=(X^{\mathsf {T}}X)^{-1}X^{\mathsf {T}}y} does not exist. (wikipedia.org)
- In nonparametric models, the set of possible values of the parameter θ {\displaystyle \theta } is a subset of some space, not necessarily finite-dimensional. (wikipedia.org)
- Thus, Θ ⊂ F {\displaystyle \Theta \subset \mathbb {F} } for some possibly infinite-dimensional space F {\displaystyle \mathbb {F} } . In semiparametric models, the parameter has both a finite-dimensional component and an infinite-dimensional component (often a real-valued function defined on the real line). (wikipedia.org)
- Thus the parameter space Θ {\displaystyle \Theta } in a semiparametric model satisfies Θ ⊂ R k × F {\displaystyle \Theta \subset \mathbb {R} ^{k}\times \mathbb {F} } , where F {\displaystyle \mathbb {F} } is an infinite-dimensional space. (wikipedia.org)
- λ 0 ( u ) {\displaystyle \lambda _{0}(u)} is an unknown non-negative function of time (known as the baseline hazard function) and is often a nuisance parameter. (wikipedia.org)
- A hazard is the probability that a patient fails between t {\displaystyle t} and t + Δ t {\displaystyle t+\Delta t} , given that he has survived up to time t {\displaystyle t} , divided by Δ t {\displaystyle \Delta t} , as Δ t {\displaystyle \Delta t} approaches zero. (wikipedia.org)
- For two groups that differ only in treatment condition, the ratio of the hazard functions is given by e β {\displaystyle e^{\beta }} , where β {\displaystyle \beta } is the estimate of treatment effect derived from the regression model. (wikipedia.org)
- For instance, the β {\displaystyle \beta } from the Cox-model or the log-rank test might then be used to assess the significance of any differences observed in these survival curves. (wikipedia.org)
- This reduces the accelerated failure time model to regression analysis (typically a linear model) where − log ( θ ) {\displaystyle -\log(\theta )} represents the fixed effects, and ϵ {\displaystyle \epsilon } represents the noise. (wikipedia.org)
- These right-censored observations can pose technical challenges for estimating the model, if the distribution of T 0 {\displaystyle T_{0}} is unusual. (wikipedia.org)
- The interpretation of θ {\displaystyle \theta } in accelerated failure time models is straightforward: θ = 2 {\displaystyle \theta =2} means that everything in the relevant life history of an individual happens twice as fast. (wikipedia.org)

**Kaplan-Meier**- Groups were compared by Cox proportional hazard models and Kaplan-Meier survival plots. (ahajournals.org)
- Survival analysis includes Cox regression (Proportional hazards model) and Kaplan-Meier survival analysis. (wikipedia.org)
- The hazard ratio is simply the relationship between the instantaneous hazards in the two groups and represents, in a single number, the magnitude of distance between the Kaplan-Meier plots. (wikipedia.org)

**parametric models**- The course focuses on the Cox proportional hazards model, not the parametric models, and is not designed for predictive modelers. (sas.com)
- the best known being the Cox semiparametric proportional hazards model, and the exponential, Gompertz and Weibull parametric models. (wikipedia.org)
- A third class, semi-parametric models, includes features of both. (wikipedia.org)
- Parametric models make "specific assumptions with regard to one or more of the population parameters that characterize the underlying distribution(s)", while non-parametric regressions make fewer assumptions than their parametric counterparts. (wikipedia.org)

**ratios**- The effect of covariates estimated by any proportional hazards model can thus be reported as hazard ratios. (wikipedia.org)
- Hazards ratios (HRs) and 95% confidence intervals (CIs) were calculated among the larger cohort (n=27,035). (pubmedcentralcanada.ca)
- Hazard ratios differ from relative risks and odds ratios in that RRs and ORs are cumulative over an entire study, using a defined endpoint, while HRs represent instantaneous risk over the study time period, or some subset thereof. (wikipedia.org)
- Hazard ratios suffer somewhat less from selection bias with respect to the endpoints chosen and can indicate risks that happen before the endpoint. (wikipedia.org)
- Regression models are used to obtain hazard ratios and their confidence intervals. (wikipedia.org)
- page needed] Hazard ratios do not reflect a time unit of the study. (wikipedia.org)
- Hazard ratios become meaningless when this assumption of proportionality is not met. (wikipedia.org)
- Hazard ratios can prove harder to explain in layman's terms. (wikipedia.org)

**Proportionality**- This hazard ratio, that is, the ratio between the predicted hazard for a member of one group and that for a member of the other group, is given by holding everything else constant, i.e. assuming proportionality of the hazard functions. (wikipedia.org)
- 1991). Compensation law of mortality is a paradoxical empirical observation, and it represents a challenge for methods of survival analysis based on proportionality assumption (proportional hazard models). (wikipedia.org)

**assumption holds**- Sir David Cox observed that if the proportional hazards assumption holds (or, is assumed to hold) then it is possible to estimate the effect parameter(s) without any consideration of the hazard function. (wikipedia.org)
- page needed] If the proportional hazard assumption holds, a hazard ratio of one means equivalence in the hazard rate of the two groups, whereas a hazard ratio other than one indicates difference in hazard rates between groups. (wikipedia.org)

**Cox's**- In the most popular of these models - Cox's proportional hazards design - no hidden circulation of failure times is presumed. (statshelponline.com)

**ratio**- In the majority of scenarios, we have an interest in comparing groups with regard to their hazards, and we utilize a danger ratio, which is comparable to a chances ratio in the setting of several logistic regression analysis . (statshelponline.com)
- Hughes (1993) shows that the regression dilution ratio methods apply approximately in survival models. (wikipedia.org)
- Rosner (1992) shows that the ratio methods apply approximately to logistic regression models. (wikipedia.org)
- 1995) give more detail on regression dilution in nonlinear models, presenting the regression dilution ratio methods as the simplest case of regression calibration methods, in which additional covariates may also be incorporated. (wikipedia.org)
- In survival analysis, the hazard ratio (HR) is the ratio of the hazard rates corresponding to the conditions described by two levels of an explanatory variable. (wikipedia.org)
- The hazard ratio would be 2, indicating higher hazard of death from the treatment. (wikipedia.org)
- Or in another study, men receiving the same treatment may suffer a certain complication ten times more frequently per unit time than women, giving a hazard ratio of 10. (wikipedia.org)
- In its simplest form, the hazard ratio can be interpreted as the chance of an event occurring in the treatment arm divided by the chance of the event occurring in the control arm, or vice versa, of a study. (wikipedia.org)
- When a study reports one hazard ratio per time period, it is assumed that difference between groups was proportional. (wikipedia.org)

**Regression Models**- Multiple linear regression models were used to identify anthropometric, clinical, behavioral, and dietary factors associated with fasting insulin and glucose in a subcohort of non-diabetics in the Alpha-Tocopherol, Beta-Carotene Cancer Prevention Study (n=366). (pubmedcentralcanada.ca)
- 1978). The case of multiple predictor variables subject to variability (possibly correlated) has been well-studied for linear regression, and for some non-linear regression models. (wikipedia.org)
- Poisson regression models are generalized linear models with the logarithm as the (canonical) link function, and the Poisson distribution function as the assumed probability distribution of the response. (wikipedia.org)

**explanatory variables**- We present a bootstrap resampling plan for the Cox partial likelihood estimator for proportional hazards models with nonrandom explanatory variables. (springer.com)
- This expression gives the hazard function at time t for subject i with covariate vector (explanatory variables) Xi. (wikipedia.org)
- In an alternative group of models, the explanatory variables act multiplicatively straight on the failure time. (statshelponline.com)
- Multicollinearity refers to a situation in which two or more explanatory variables in a multiple regression model are highly linearly related. (wikipedia.org)

**baseline hazard function**- The Cox partial likelihood, shown below, is obtained by using Breslow's estimate of the baseline hazard function, plugging it into the full likelihood and then observing that the result is a product of two factors. (wikipedia.org)

**multiplicatively**- The proportional hazards condition states that covariates are multiplicatively related to the hazard. (wikipedia.org)
- The Cox PH model assumes that predictors act multiplicatively on the hazard function but does not assume that the hazard function is constant (i.e., exponential model), Weibull, or any other particular form. (springer.com)

**semiparametric model**- In statistics, a semiparametric model is a statistical model that has parametric and nonparametric components. (wikipedia.org)
- A well-known example of a semiparametric model is the Cox proportional hazards model. (wikipedia.org)

**Weibull**- and 2) to examine the relationship between genetic parameters from a Weibull model, a discrete proportional hazard model, and a binary data analysis using a linear model. (pubfacts.com)
- Data were simulated using the Weibull frailty model with two different shapes of the Weibull distribution. (pubfacts.com)
- Three different statistical models were investigated in this study: a Weibull model, a discrete-time model (a proportional hazard model assuming that the survival data are measured on a discrete scale with few classes), and a linear model based upon binary data. (pubfacts.com)
- An alternative derivation using basic expressions of reliabilities in sire models suggests a simple equation for the heritability on the original scale (effective heritability) that is not dependent on the Weibull parameters. (pubfacts.com)
- In general, the estimates of reliability from either the discrete model or the binary data analysis were close to estimates from the Weibull model for a given number of uncensored records in this simplified case of a balanced design. (pubfacts.com)
- Although selection response from the binary data analysis depends on the end of interval point, there is a relatively good agreement between selection responses in the Weibull model and the binary data analysis. (pubfacts.com)
- Comparison between a Weibull proportional hazards model and a linear model for predicting the genetic merit of US Jersey sires for daughter longevity. (pubfacts.com)
- Predicted transmitting abilities (PTA) of US Jersey sires for daughter longevity were calculated using a Weibull proportional hazards sire model and compared with predictions from a conventional linear animal model. (pubfacts.com)
- The effect of ignoring individual heterogeneity in Weibull log-normal sire frailty models. (pubfacts.com)
- The objective of this study was, by means of simulation, to quantify the effect of ignoring individual heterogeneity in Weibull sire frailty models on parameter estimates and to address the consequences for genetic inferences. (pubfacts.com)
- Data were simulated according to balanced half-sib designs using Weibull log-normal animal frailty models with a normally distributed residual effect on the log-frailty scale. (pubfacts.com)
- In another design - the Weibull proportional hazards design - the failure times are presumed to follow a theoretical circulation understood as the Weibull circulation. (statshelponline.com)

**logistic**- The exact method computes the exact partial likelihood, which is equivalent to a conditional logistic model. (psu.edu)
- While the WC model provided more accurate estimates of the effect of time-dependent covariates than standard logistic regression, the robust sandwich variance estimates were lower than the empirical variance, resulting in a low coverage probability of confidence intervals. (biomedcentral.com)
- The objectives of the present study were to investigate through simulations a new variance estimator and to compare the estimates from the WC model and standard logistic regression for estimating the effects of correlated temporal aspects of exposure with detailed information on exposure history. (biomedcentral.com)
- The performances of the WC model using the two variance estimators were compared to standard logistic regression. (biomedcentral.com)
- The log-logistic distribution provides the most commonly used AFT model. (wikipedia.org)
- Both of these are examples of a simple, binary logistic regression model. (wikipedia.org)

**linearly related**- that is, the regressors are linearly related to log hazard or log cumulative hazard. (springer.com)

**subset**- a missing-data filter function, applied to the model.frame, after any subset argument has been used. (psu.edu)
- Since these models generate a small prioritized list of potential defectors, they are effective at focusing customer retention marketing programs on the subset of the customer base who are most vulnerable to churn. (wikipedia.org)

**estimates**- The superpopulation variance estimator provided better estimates than the robust sandwich variance estimator and the WC model provided accurate estimates of the effects of correlated aspects of temporal patterns of exposure. (biomedcentral.com)
- We use a semiparametric proportional hazard model which is augmented by an ARMA structure very similar to the wide spread ACD model to obtain consistent estimates of the baseline survivor function and to capture well known serial dependencies in the trade intensity process. (uni-konstanz.de)
- In this situation the coefficient estimates of the multiple regression may change erratically in response to small changes in the model or the data. (wikipedia.org)
- Other HR models have different formulations and the interpretation of the parameter estimates differs accordingly. (wikipedia.org)
- Unlike proportional hazards models, the regression parameter estimates from AFT models are robust to omitted covariates. (wikipedia.org)
- The log-rank test statistic compares estimates of the hazard functions of the two groups at each observed event time. (wikipedia.org)

**whereas**- Whereas a proportional hazards model assumes that the effect of a covariate is to multiply the hazard by some constant, an AFT model assumes that the effect of a covariate is to accelerate or decelerate the life course of a disease by some constant. (wikipedia.org)

**predictors**- it makes a parametric assumption concerning the effect of the predictors on the hazard function, but makes no assumption regarding the nature of the hazard function λ( t ) itself. (springer.com)
- That is, a multiple regression model with colinear predictors can indicate how well the entire bundle of predictors predicts the outcome variable, but it may not give valid results about any individual predictor, or about which predictors are redundant with respect to others. (wikipedia.org)
- Generalized additive model is a smoothing method for multiple predictors that allows for non-parametric predictions. (wikipedia.org)

**covariate**- In a proportional hazards model, the unique effect of a unit increase in a covariate is multiplicative with respect to the hazard rate. (wikipedia.org)
- Since the covariate is not measured for all participants, the nested case control model is both less expensive than a full cohort analysis and more efficient than taking a simple random sample from the full cohort. (wikipedia.org)
- parametric survival models and the Cox proportional hazards model may be useful to estimate covariate-adjusted survival. (wikipedia.org)

**covariates**- Survival models relate the time that passes before some event occurs to one or more covariates that may be associated with that quantity of time. (wikipedia.org)
- and the effect parameters, describing how the hazard varies in response to explanatory covariates. (wikipedia.org)
- This is a partial likelihood: the effect of the covariates can be estimated without the need to model the change of the hazard over time. (wikipedia.org)
- The proportional hazards design was presented in 1972 by D. R. Cox in order to approximate the results of various covariates affecting the times to the failures of a system. (statshelponline.com)

**assumes**- Poisson regression assumes the response variable Y has a Poisson distribution, and assumes the logarithm of its expected value can be modeled by a linear combination of unknown parameters. (wikipedia.org)

**probability**- In many cases the model is chosen on the basis of detection theory to try to guess the probability of an outcome given a set amount of input data, for example given an email determining how likely that it is spam. (wikipedia.org)
- In biostatistics, the researcher may be interested in trying to model the probability of a patient being diagnosed with a certain type of cancer based on knowing, say, the incidence of that cancer in his or her family. (wikipedia.org)
- In business, the marketer may be interested in modelling the probability of an individual purchasing a product based on the price of that product. (wikipedia.org)

**mathematical**- Then, a model based on mathematical form of Proportional hazards model is developed where it represents stress condition of the WT. (comillas.edu)

**estimate**- Further, a life table of under-five children under study is constructed using the estimate of survival function obtained from Coxâ s regression model. (thescipub.com)
- A weighted Cox (WC) model has been proposed to estimate the effects of time-dependent exposures. (biomedcentral.com)
- In general, methods for the structural model require some estimate of the variability of the x variable. (wikipedia.org)
- Predictive models can either be used directly to estimate a response (output) given a defined set of characteristics (input), or indirectly to drive the choice of decision rules. (wikipedia.org)

**predictive models**- For example, predictive models are often used to detect crimes and identify suspects, after the crime has taken place. (wikipedia.org)
- Broadly speaking, there are two classes of predictive models: parametric and non-parametric. (wikipedia.org)
- In business, predictive models exploit patterns found in historical and transactional data to identify risks and opportunities. (wikipedia.org)
- Modelling : Predictive modelling provides the ability to automatically create accurate predictive models about future. (wikipedia.org)
- Generally, the term predictive analytics is used to mean predictive modeling, "scoring" data with predictive models, and forecasting. (wikipedia.org)
- Predictive models are models of the relation between the specific performance of a unit in a sample and one or more known attributes or features of the unit. (wikipedia.org)

**frailty models**- Supplied penalty functions include ridge regression ( ridge ), smoothing splines ( pspline ), and frailty models ( frailty ). (psu.edu)

**represents**- The curve represents the odds of an endpoint having occurred at each point in time (the hazard). (wikipedia.org)

**interpretation**- However, Cox also noted that biological interpretation of the proportional hazards assumption can be quite tricky. (wikipedia.org)

**survival time**- The proportional hazards model is usually expressed in terms of a single survival time value for each person, with possible censoring. (psu.edu)
- One of the most popular regression methods for survival analysis is Cox proportional hazards regression, which is utilized to relate a number of threat elements or direct exposures, thought about concurrently, to survival time. (statshelponline.com)

**statistical**- In the statistical area of survival analysis, an accelerated failure time model (AFT model) is a parametric model that provides an alternative to the commonly used proportional hazards models. (wikipedia.org)
- Predictive analytics encompasses a variety of statistical techniques from predictive modelling, machine learning, and data mining that analyze current and historical facts to make predictions about future or otherwise unknown events. (wikipedia.org)
- Data Analysis : Data Analysis is the process of inspecting, cleaning and modelling data with the objective of discovering useful information, arriving at conclusion Statistics : Statistical Analysis enables to validate the assumptions, hypothesis and test them using standard statistical models. (wikipedia.org)

**time**- Other types of survival models such as accelerated failure time models do not exhibit proportional hazards. (wikipedia.org)
- The accelerated failure time model describes a situation where the biological or mechanical life history of an event is accelerated. (wikipedia.org)
- The proportional hazards model included time-dependent effects of herd-year-season contemporary group and parity by stage of lactation interaction, as well as time-independent effects of sire and age at first calving. (pubfacts.com)
- Cox regression (or proportional hazards regression) is approach for examining the result of numerous variables upon the time a defined occasion takes to occur. (statshelponline.com)
- In a Cox proportional hazards regression design, the procedure of impact is the risk rate, which is the threat of failure (i.e., the danger or possibility of suffering the occasion of interest), provided that the individual has actually made it through up to a particular time. (statshelponline.com)
- These are understood as the sped up time failure models, and usually do not presume proportional hazards. (statshelponline.com)
- Different models with selected time-dependent aspects of exposure such as intensity, duration, and time since cessation were considered. (biomedcentral.com)
- The WC model with the superpopulation variance estimator provides an alternative analytical approach for estimating the effects of time-varying exposures with detailed history exposure information in case-control studies, especially if many subjects have time-varying exposure intensity over lifetime, and if only one control is available for each case. (biomedcentral.com)
- The WC model consists in using age as the time axis and weighting cases and controls according to their case-control status and the age conditional probabilities of developing the disease in the source population. (biomedcentral.com)
- The weights proposed in the WC model are therefore time-dependent and estimated from data of the source population. (biomedcentral.com)
- A person's increased risk of contracting disease is directly proportional to the length of time that a person continues to smoke as well as the amount smoked. (wikipedia.org)
- The instantaneous hazard rate is the limit of the number of events per unit time divided by the number at risk, as the time interval approaches 0. (wikipedia.org)
- The difference between hazard-based and time-based measures is akin to the difference between the odds of winning a race and the margin of victory. (wikipedia.org)
- For example, if the model concerns the development of a tumor, it means that all of the pre-stages progress twice as fast as for the unexposed individual, implying that the expected time until a clinical disease is 0.5 of the baseline time. (wikipedia.org)
- Scoring models process a customer's credit history, loan application, customer data, etc., in order to rank-order individuals by their likelihood of making future credit payments on time. (wikipedia.org)

**Fits**- Fits a Cox proportional hazards regression model. (psu.edu)
- Fits multiple variable mixtures of various parametric proportional hazard models using the EM-Algorithm. (rdrr.io)

**function**- In many situations, either the form of the true hazard function is unknown or it is complex, so the Cox model has definite advantages. (springer.com)
- This paper puts a focus on the hazard function of inter-trade durations to characterize the intraday trading process. (uni-konstanz.de)
- We show, based on an exogenous information process, that the way traders aggregate information has implications for the shape of the hazard function. (uni-konstanz.de)

**propensity**- More sophisticated predictive analytics software use churn prediction models that predict customer churn by assessing their propensity of risk to churn. (wikipedia.org)

**approaches**- Robust regression include a number of modelling approaches to handle high leverage observations or violation of assumptions. (wikipedia.org)

**assess**- The objectives of the proposed approach are: to provide a stress condition model for health monitoring, to assess the WT s maintenance strategies, and to provide recommendations on current maintenance schemes for future operations of the wind farm. (comillas.edu)

**analytical**- In most applications, involuntary reasons for churn are excluded from the analytical models. (wikipedia.org)
- Deployment : Predictive model deployment provides the option to deploy the analytical results into everyday decision making process to get results, reports and output by automating the decisions based on the modelling. (wikipedia.org)
- However, people are increasingly using the term to refer to related analytical disciplines, such as descriptive modeling and decision modeling or optimization. (wikipedia.org)

**quantile**- Models can be both parametric (e.g. regression with Huber, White, Sandwich variance estimators) as well as non-parametric(e.g. quantile regression). (wikipedia.org)

**odds**- Semiparametric regression includes the proportional odds model and the Cox proportional hazards model where the response is a rank. (wikipedia.org)

**1992**- in a 1992 paper, Wei pointed out that the Buckley-James model has no theoretical justification and lacks robustness, and reviewed alternatives. (wikipedia.org)

**predictor**- They explain the survival inning accordance with one aspect under examination, however overlook the effect of other.An option approach is the Cox proportional hazards regression analysis , which works for both quantitative predictor variables and for categorical variables. (statshelponline.com)
- Other non-linear models, such as proportional hazards models for survival analysis, have been considered only with a single predictor subject to variability. (wikipedia.org)
- In statistics, multicollinearity (also collinearity) is a phenomenon in which one predictor variable in a multiple regression model can be linearly predicted from the others with a substantial degree of accuracy. (wikipedia.org)
- The model is "simple" in that each has only one independent, or predictor, variable, and it is "binary" in that the dependent variable can take on only one of two values: cancer or no cancer, and purchase or does not purchase. (wikipedia.org)

**variability**- The reply to Frost & Thompson by Longford (2001) refers the reader to other methods, expanding the regression model to acknowledge the variability in the x variable, so that no bias arises. (wikipedia.org)

**observations**- fit Cox proportional hazards models for repeated events and clustered observations. (sas.com)

**prediction**- By deploying new technologies such churn prediction models coupled with effective retention programs, customer attrition could be better managed to stem the significant revenue loss from defecting customers. (wikipedia.org)
- The log-rank test has been shown to be too permissive a test, allowing significant results for survivorship prediction models that have low accuracy. (wikipedia.org)
- Nearly any regression model can be used for prediction purposes. (wikipedia.org)

**form**- In statistics, Poisson regression is a generalized linear model form of regression analysis used to model count data and contingency tables. (wikipedia.org)

**Data**- In: Jewell N.P., Kimber A.C., Lee ML.T., Whitmore G.A. (eds) Lifetime Data: Models in Reliability and Survival Analysis. (springer.com)
- This approach to survival data is called application of the Cox proportional hazards model, sometimes abbreviated to Cox model or to proportional hazards model. (wikipedia.org)
- The Cox proportional hazards model 92 is the most popular model for the analysis of survival data. (springer.com)
- The results of the different models were finally compared for estimating the effects of correlated aspects of occupational exposure to asbestos on the risk of mesothelioma, using population-based case-control data. (biomedcentral.com)
- From an inspection of conditional transaction probabilities based on Bund future transaction data of the DTB we find a decreasing hazard shape providing evidence for the use of non-trading intervals as an indication for the absence of price information among market participants. (uni-konstanz.de)
- Applied longitudinal data analysis: Modeling change and event occurrence New York, USA: Oxford University Press. (pixnet.net)

**Applications**- Bootstrap Applications in Proportional Hazards Models , Ph.D. Dissertation, Iowa State University. (springer.com)
- These models are utilized in environmental applications where the proportional hazards presumption might not be fulfilled. (statshelponline.com)
- in predictive modelling applications, correction is neither necessary nor appropriate. (wikipedia.org)
- Financial services such as banking and insurance use applications of predictive analytics for churn modeling, because customer retention is an essential part of most financial services' business models. (wikipedia.org)

**methods**- However, their usage is limited to those methods that use this type of modelling approach which can have several drawbacks. (wikipedia.org)

**statistics**- Proportional hazards models are a class of survival models in statistics. (wikipedia.org)
- Predictive modeling uses statistics to predict outcomes. (wikipedia.org)

**Theta**- theta )} is always twice as high - that would be the proportional hazards model. (wikipedia.org)

**heterogeneity**- This model is popular because it models the Poisson heterogeneity with a gamma distribution. (wikipedia.org)