What Is Survival Analysis?

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Parametric and Semi-Parametric models

Parametric and Semi-Parametric models are the models that can include other dependent variables in the analysis to determine how those dependent variables affect the hazard rate in each point of time.

Semi Parametric model

Baseline hazard is based on observed data without any assumption, parametric assumptions describe the role of covariate

Cox proportional hazards regression model

The cox model hazard function is h(t) = ho(t) exp(β1xi1+ β2xi2 + ……… βkxik) where baseline hazard ho(t) can take any form

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Exp(coef) provides the hazard rate and if it is greater than 1 then it implies and increase in the covariate “age” increase the hazard rate. If the Exp(coef) is negative then it implies that decrease in the covariate “age” decrease the hazard rate.

Here in this data set for an increase in 1 year of age increase the hazard rate by (1.005498-1)*100 i.e. by 0.54%

Syntax to create the plot of survival curve for Cox proportional hazards regression model

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The below figure shows the survival probability for Cox proportional hazards regression model

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Parametric model

These models need to make assumptions about the distributions about the survival data. Distribution observed can be “Weibull”, ”Gaussian”, ”Logistic”, ”Lognormal” and “Loglogistic”. Baseline hazard assumed to vary in a specific manner with time.

Survival distribution functions with two covariates are

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We assume the survival data is distributed in Weibull manner and created the parametric model

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The coefficients are logarithms of ratios of survival times, so a positive coefficient mean longer survival time and negative coefficient means lesser survival time

We calculate the k intercept and α estimates from the summary of Weibull regression model.

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We are predicting the survivability rate of the patient for the time parameter from 0 to 50000 with increment by 1. To create the time vector we are using [pic 30] function

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We are calculating the survival probability for Weibull function by using the k intercept and α estimates and substituted them in the survival probability equation of Weibull distribution

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In the similar way we have calculated the survival probability for our Loglogistic distribution model

Assumed that data is distributed in Loglogistic manner and created the parametric model

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Calculating the survival probability for Weibull function by using the k intercept and α estimates and substituted them in the survival probability equation of Loglogistic distribution

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Plotting the survival probability for both Weibull and Loglogistic distribution to understand how they are predicted for different models

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Selecting between competing parametric models

We have to calculate the AIC value of each model to understand which model is the best one. We use [pic 38] function to get the AIC value from our parametric models

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We conclude that Loglogistic model is the better one than Weibull based on AIC value.