Editing Survival analysis (section)

====Example: Cox proportional hazards regression analysis for melanoma====

This example uses the melanoma data set from Dalgaard Chapter 14.
<ref name="Dalgaard2008">{{Citation
|last1= Dalgaard
|first1= Peter
|title= Introductory Statistics with R
|edition=Second
|year=2008
|publisher= Springer
|isbn= 978-0387790534
}}
</ref>

Data are in the R package ISwR. The Cox proportional hazards regression using{{nbsp}}R gives the results shown in the box.
	
 [[File:Cox proportional hazards regression output for melanoma data set.png|thumb|400px|right|Cox proportional hazards regression output for melanoma data. Predictor variable is sex 1: female, 2: male.]]

The Cox regression results are interpreted as follows.

*Sex is encoded as a numeric vector (1: female, 2: male). The R{{nbsp}}summary for the Cox model gives the hazard ratio (HR) for the second group relative to the first group, that is, male versus female.
*coef = 0.662 is the estimated logarithm of the hazard ratio for males versus females.
*exp(coef) = 1.94 = exp(0.662) - The log of the hazard ratio (coef= 0.662) is transformed to the hazard ratio using exp(coef). The summary for the Cox model gives the hazard ratio for the second group relative to the first group, that is, male versus female. The estimated hazard ratio of 1.94 indicates that males have higher risk of death (lower survival rates) than females, in these data.
*se(coef) = 0.265 is the standard error of the log hazard ratio.
*z = 2.5 = coef/se(coef) = 0.662/0.265. Dividing the coef by its standard error gives the z score.
*p=0.013. The p-value corresponding to z=2.5 for sex is p=0.013, indicating that there is a significant difference in survival as a function of sex.

The summary output also gives upper and lower 95% confidence intervals for the hazard ratio: lower 95% bound = 1.15; upper 95% bound = 3.26.

Finally, the output gives p-values for three alternative tests for overall significance of the model:

*Likelihood ratio test = 6.15 on 1 df, p=0.0131
*Wald test = 6.24 on 1 df, p=0.0125
*Score (log-rank) test = 6.47 on 1 df, p=0.0110

These three tests are asymptotically equivalent. For large enough N, they will give similar results. For small N, they may differ somewhat. The last row, "Score (logrank) test" is the result for the log-rank test, with p=0.011, the same result as the log-rank test, because the log-rank test is a special case of a Cox PH regression. The Likelihood ratio test has better behavior for small sample sizes, so it is generally preferred.