The Cox Proportional Hazard Model

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Cox Proportional Hazard model is a popular model in survival analysis for detecting the effect of some set of variables on the Hazard. This model is popular largely because there is no need to consider specific distribution function to the hazard function. In Cox proportional hazard model Is Unspecified and non-negative function of time that called a baseline hazard function and is a matrices of covariates related to the ith person. One of the important assumptions in Cox model is that the covariate has a linear effect on the log hazard function. However, Continuous variables can be an influence on the risk with non-linear forms and ignoring this can alter the results.
Adding a nonlinear function to a variable in the Cox model needs
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Unlike polynomials, allow for a more local fit to the data and fitting after the knots can be limited to the linear. Generally, there are three methods to estimate splines: smoothing splines, polynomial splines and penalized splines. Better performance of polynomial splines depends on the number and location of knots. To overcome this problem, smoothing splines uses all of points as knots. But when you have a large number of discrete time points, the number of parameters that must be estimated to be high and this is will be complicated calculations. Smoothing splines like polynomial splines use a large number of knots, while reduce the influence of knots with a penalized term. Penalized spline is very similar to smoothing splines, but use a fewer knot…show more content…
For example, restricted cubic spline, which is also known as natural cubic splines, is the limited cubic spline that the tails are limited to linear. In this method, the number of knots previously known and their positions are based on data quantile. In this paper, three smoothing methods that have been used in the last decade in medicine and epidemiology studies have examined: Penalize spline, restricted cubic spline and natural spline. All these methods can easily include to the Cox and linear models.

In this analytical study to determine the nonlinear effects of covariate three non-parametric methods Penalize spline, restricted cubic spline and natural spline in Cox model were used. The ability of nonparametric methods was evaluated to recover the true functional form of linear, quadratic and nonlinear functions, using different simulated sample sizes. Data analysis was carried out using R 2.11.0 software and significant levels were considered 0.05.
Spline:
The most common method of estimating function of f in equation (1) is the use of splines. Spline linear estimator is as

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