Proportional hazard model

Bendell, A. Proportional Hazards Modeling in Reliability Assessment. Reliability Engineering, Vol. 2, 1983, pp. 175-183. 

Wightman, D. W. and A. Bendell. The Practical Application of Proportional Hazards Modeling. Proceedings of the 5th National Reliability Conference. Birmingham, England, 1985, 2B/3,... 

Note that the term censor is introduced in the preceding table. The log-rank test (invoked in SAS with PROC LIFETEST) and the Cox proportional hazards model (invoked in SAS with PROC PHREG) allow for censoring observations in a time-to-event analysis. These tests adjust for the fact that at some point a patient may no longer be able to experience an event. The censor date is the last known time that the patient did not experience a given event and the point at which the patient is no longer considered able to experience the event. Often the censor date is the last known date of patient follow-up, but a patient could be censored for other reasons, such as having taken a protocol-prohibited medication. 

Hazard ratios are created using the Cox proportional hazards model through PROC PHREG. 

Time-to-event analysis in clinical trials is concerned with comparing the distributions of time to some event for various treatment regimens. The two nonparametric tests used to compare distributions are the log-rank test and the Cox proportional hazards model. The Cox proportional hazards model is more useful when you need to adjust your model for covariates. 

The most popular method for analysis of covariance is the proportional hazards model. This model, originally developed by Cox (1972), is now used extensively in the analysis of survival data to incorporate and adjust for both centre and covariate effects. The model assumes that the hazard ratio for the treatment effect is constant. 

In this study reported by Bedikian et al. (2006), several potential baseline prognostic factors were included in a proportional hazards model. These factors were ... 

The proportional hazards model, as the name suggests, assumes that the hazard ratio is a constant. As such it provides a direct extension of the logrank test, which is a simple two treatment group comparison. Indeed if the proportional hazards model is fitted to data without the inclusion of baseline factors then the p-value for the test Hg c = 0 will be essentially the same as the p-value arising out of the logrank test. 

It is interesting to examine how control for various comorbid factors influences the mortality hazard associated with sleep durations. Comorbidities aside, in Cox proportional hazards models for each gender, controlling only for age and hours of sleep, the sample excess fractions (20) of deaths related to sleep durations other than 7 hr were 16.4% for women and 19.4% for men. These fractions are the percentage of observed deaths that would not have occurred in the 6-year fol-... 

Using regression analysis based on Cox s proportional hazards model, Ott and Zober (1996) found evidence of association between 2,3,7,8-TCDD exposure and digestive cancer (conditional risk ratio of 1.46 95% 0=1.13-1.89) the primary tumor sites were the liver, stomach, and pancreas. 

Platt RW, Joseph KS, Ananth CV, Grondines J, Abrahamowicz M, Kramer MS (2004) A proportional hazards model with time-dependent covariates and time-varying effects for analysis of fetal and infant death. Am J Epidemiol, 160(3) 199-206. 

Dellaportas, P. and Smith, A. F. M. (1993). Bayesian inference for generalized linear and proportional hazards models via Gibbs sampling. Applied Statistics, 42, 443 159. 

Data from a 7-year follow-up study in another Cd-polluted area (Nagasaki) showed that, in both men and women, serum p2-microglobulin and creatinine, as well as urinary total protein and p2-microglobulin were significantly related to mortality independent of age as assessed by the Cox s proportional hazards model [111]. In advanced cases, the excess mortality of subjects with Cd-induced renal tubular dysfunction might, to some extent, be ascribed to a reduction in GFR. 

Methods that involve studying the disposition of some exogenously administered agent (e.g. indocyanine green, antipyrine, galactose or dextromethorphan) have now been superceded by functional (often multicomponent) tests. Mono-ethylglycinexylidide formation has not found wide acceptance. More complicated Cox proportional hazards models may exist for other liver diseases, but are only used specifically for them (e.g. the Mayo Clinic Survival Model for primary biliary cirrhosis see the US FDA Guidance). 

Although we do not cover them in detail, there ate parametric methods to analyze time to event data of this type, the most notable of which is Cox s proportional hazards model. 

Adjusted rate ratios (and 95 percent confidence intervals [Cl] were derived from the Cox proportional-hazards model after adjustment for age, race, branch of service, and type of unit. 

Ford I, Norrie J, Ahmadi S (199 5) Model inconsistency, illustrated by the Cox proportional hazards model. Statistics in Medicine 14 735-746. 

Cox regression. Another name for the proportional hazards model originally proposed by Sir David Cox in 1972. 

Proportional hazards model. Generally, a regression model used in survival analysis whereby it is assumed that the hazard functions of individuals under study are proportional to each other over time. (This is equivalent to assuming that the log-hazard functions differ by a constant.) More specifically, and in the form originally proposed by Cox (1972), no further definite assumption is made about the hazard functions themselves. One of the most commonly used statistical techniques in survival analysis. 

The same issue exists for time-to-event endpoints. While the most common metric for trials with these endpoints is the hazard ratio evaluated using survival analysis (usually a Cox proportional hazards model), absolute measures, such as the difference in event rates at a fixed follow-up time, are sometimes used. 

One hundred twenty-five events would provide 90% power, and 95 events would provide 80% power to illustrate the 95% Cl is < 1.8 based on a Cox proportional hazard model. Therefore, in a clinical trial with an expected event rate of 1 event per 100 patient-years and a 5% annual dropout rate, 90% power would be achieved by enrolling 200 patients per month for 24 months and following all enrolled patients for an additional 24 months. The total trial duration would be 48 months. If, however, the true event rate was actually 0.75%, the power would decrease to 80%—a doubling of the type 2 error. Furthermore, if the event rate were actually 0.5%, then the power would drop to 65%, a type 2 error 3.5 times as high. Because it is hard to predict CV event rates in noncardiac populations, such overestimates are not uncommon. Similarly, if the actual event rate... 

The primary analysis will be a Cox proportional hazard model and the corresponding Cls for the treatment group hazard ratio. We use a Bayesian algorithm, based on the predictive probability of success at the final analysis, throughout the accrual stage to determine when to stop accrual. Thus, we use a Bayesian model to predict the likelihood of success for a frequentist test. A fully Bayesian primary analysis could also be performed. 

The predictive probabilities used to stop the trial for futility or for predicted success are calculated using a combination of the current posterior probability, each individual patient s data (follow-up time and whether they ve experienced an event), and simulated data from the current point forward along with the Cox proportional hazard model. 

Samrout, M., Chatelet, E., Kouta, R. Chebbo, N. 2009. Optimization of maintenance policy using the proportional hazard model. Reliability Engineering and System Safety 94, 44-52. 

Finally, let us mention the proportional hazard model for software vulnerability developed similarly as described by Pham (2003) for software reliability. 

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