Cox regression is a statistical technique that is used to determine the relationship between survival and several independent exploratory variables. Cox regression is useful for modeling the time to a specific event based upon the value of a given covariate. Cox regression is a method of survival analysis. Survival analysis is also known as the time to event study. Survival analysis is a method of analyzing whether or not an event will happen. Cox regression provides an estimate of the treatment on the survival rate, after adjustment of the exploratory variable. In Cox regression, we estimate the coefficient of the exploratory variable. The basic model for Cox regression produces the proportional hazard function, which can be extended through the specifications of a strata variable or time-dependent covariates.
Hazard function: Hazard is the event of interests occurring. For example, in medical research death is a hazard. Another example would be in the industrial field, if an engine breaks down, it is a hazard. In other words, we can say that the probability of the endpoint of an event of interest is called the hazard. Hazard function is also called the Cox proportional hazard function.
Regression: Regression is a statistical technique that is used to describe the relationship between two or more than two variables, or the study of dependent and independent variables. When independent variables are more than two, then it is said to be multiple linear regression. Cox regression is similar to multiple linear regressions. In Cox regression, we can describe the Cox hazard function or risk of t as:
Or,
Cox regression log relative hazard function is defined as:
Where,
hi (t) = is the hazard rate for the i case at t time
h0 (t) = is the baseline hazard at time t
b = Cox regression coefficient
e=natural log
X= exploratory variable
Key term and concepts in Cox Regression:
Status variable: In Cox regression, the status variable is the dependent variable. In Cox regression, the status variable is binary in nature. For example, we will assign code 1 for events that happen, and 0 for events that do not happen.
Time variable: In Cox regression, the time variable measures the duration of the status variable. Time variable is simply the counter unit of time since the series started.
Covariate: In Cox regression, covariates are the independent variables. In Cox regression, covariates can be categorical or a dummy.
Hazard: In Cox regression, hazard is the event of interest occurring.
Hazard rate or hazard ratio: In Cox regression, hazard ratio is also called the odd ratio. Hazard ratio is the probability of events happening in time t+1.
Regression coefficient: Most statistical software displays regression coefficient beta and exponential beta value with standard error and Wald statistics significance. Exponential beta value is used to predicate the odd ratio.
Cox Regression and SPSS:
Most of the statistical software produces the output for Cox regression. To perform Cox regression in SPSS, we have to perform the following steps:
- Click on the “SPSS” icon from the start menu.
- Click on the “open” icon and select the “data.”
- Click on the “analysis” menu and select the “survival analysis option.”
- Select “Cox regression” from the survival analysis option. As we click on the Cox regression option, the following window will appear:
From this window, select the time dependent variables and insert them into the time box. Select the status or dependent variable and insert them into the status variable box. Select the covariates or the independent variables and insert them into the covariate box. Click on the “method box” and select the appropriate method for analysis. Click on the “plot box” and select “survival and hazard” from the plot option. Click on the “ok” button. SPSS will produce the result of Cox regression. Chi-square test will be used to test the goodness of fit model. In Cox Regression, the output exponential beta is used as a regression coefficient to predict the hazard function.
