# Parametric survival regression (Weibull model)

Parametric survival regression (Weibull model)

### Principle of parametric survival model

The principle of the parametric survival regression is to link the survival time of an individual to covariates using a specified probability distribution (generally the Weibull distribution). For example, in the medical domain, we are seeking to find out which covariate has the most important impact on the survival time of a patient.

### Parametric survival models or Weibull models

A parametric survival model is a well-recognized statistical technique for exploring the relationship between the survival of a patient, a parametric distribution and several explanatory variables. It allows us to estimate the parameters of the distribution.

### Variables selection for the parametric survival regression

It is possible to improve the parametric survival model by selecting the variables being part of the model. XLSTAT offers two options to select the variables:

- Forward selection: The selection process starts by adding the variable with the largest contribution to the model. If a second variable is such that its entry probability is greater than the entry threshold value, then it is added to the model. This process is iterated until no new variable can be entered in the model.
- Backward selection: This method is similar to the previous one but starts from a complete model.

### Results for the parametric survival model in XLSTAT

#### Goodness of fit coefficients for the parametric survival regression

The goodness of fit coefficients table displays a series of statistics for the independent model (corresponding to the case where there is no impact of covariates, beta=0) and for the adjusted model.

- Observations: The total number of observations taken into;
- DF: Degrees of freedom;
- -2 Log(Like.): The logarithm of the likelihood function associated with the model;
- AIC: Akaike’s Information Criterion;
- SBC: Schwarz’s Bayesian Criterion;
- Iterations: Number of iterations until convergence.

#### Model parameters

The parameter estimate, corresponding standard deviation, Wald's Chi², the corresponding p-value and the confidence interval are displayed for each variable of the model.

The predictions and residuals table shows, for each observation, the time variable, the censoring variable, the value of the residuals, the estimated cumulative survival distribution, the empirical cumulative distribution function and the hazard function.

#### Available charts for the parametric survival regression

XLSTAT offers the following charts for the parametric survival regression:

- Cumulative Survival distribution function (SDF),
- -Log(SDF),
- Log(-Log(SDF)),
- hazard function,
- residuals.

On each chart, the empirical and theoretical distribution function is displayed.

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