Editing Survival analysis (section)

====Cox model using a covariate in the melanoma data====

The Cox model extends the log-rank test by allowing the inclusion of additional covariates.<ref>{{Cite journal |last1=Saegusa |first1=Takumi |last2=Di |first2=Chongzhi |last3=Chen |first3=Ying Qing |date=September 2014 |title=Hypothesis testing for an extended cox model with time-varying coefficients |journal=Biometrics |language=en |volume=70 |issue=3 |pages=619–628 |doi=10.1111/biom.12185 |pmid=24888739 |issn=0006-341X|pmc=4247822 }}</ref> This example use the melanoma data set where the predictor variables include a continuous covariate, the thickness of the tumor (variable name = "thick").

[[File:Histograms of melanoma thickness.png|thumb|700px|Histograms of melanoma tumor thickness]]

In the histograms, the thickness values are [[Skewness|positively skewed]] and do not have a [[Normal distribution|Gaussian]]-like, [[Symmetric probability distribution]]. Regression models, including the Cox model, generally give more reliable results with normally-distributed variables.{{Citation needed|date=February 2023}} For this example we may use a [[logarithm]]ic transform. The log of the thickness of the tumor looks to be more normally distributed, so the Cox models will use log thickness. The Cox PH analysis gives the results in the box.

[[File:Cox PH output for melanoma with thickness.png|thumb|500px|Cox PH output for melanoma data set with covariate log tumor thickness]]

The p-value for all three overall tests (likelihood, Wald, and score) are significant, indicating that the model is significant. The p-value for log(thick) is 6.9e-07, with a hazard ratio HR = exp(coef) = 2.18, indicating a strong relationship between the thickness of the tumor and increased risk of death.

By contrast, the p-value for sex is now p=0.088. The hazard ratio HR = exp(coef) = 1.58, with a 95% confidence interval of 0.934 to 2.68. Because the confidence interval for HR includes 1, these results indicate that sex makes a smaller contribution to the difference in the HR after controlling for the thickness of the tumor, and only trend toward significance. Examination of graphs of log(thickness) by sex and a t-test of log(thickness) by sex both indicate that there is a significant difference between men and women in the thickness of the tumor when they first see the clinician.

The Cox model assumes that the hazards are proportional. The proportional hazard assumption may be tested using the R{{nbsp}}function cox.zph(). A p-value which is less than 0.05 indicates that the hazards are not proportional. For the melanoma data we obtain p=0.222. Hence, we cannot reject the null hypothesis of the hazards being proportional. Additional tests and graphs for examining a Cox model are described in the textbooks cited.