Cox regression, also known as the proportional hazards regression model, is a statistical technique used to analyze the relationship between the survival time of an individual and one or more predictor variables. It is commonly used in medical research to analyze the effect of different factors on the survival time of patients.
The basic idea behind Cox regression is to model the hazard function, which represents the instantaneous probability of an event (such as death) occurring at a particular time, given that the individual has survived up to that time. The hazard function is assumed to be proportional across different groups or levels of the predictor variables, which means that the effect of each predictor on the hazard is constant over time.
The Cox regression model is a semi-parametric model, which means that it makes some assumptions about the form of the hazard function (such as proportionality) but does not specify a particular parametric form for it. This makes it more flexible than fully parametric models, which assume a specific distribution for the survival time.
To estimate the parameters of the Cox regression model, maximum likelihood estimation is commonly used. The likelihood function is a product of the probabilities of observing the event or censoring times for each individual, given the predictor variables and the model parameters.
The output of a Cox regression analysis includes the estimated coefficients for each predictor variable, along with their standard errors and confidence intervals. These coefficients can be used to calculate hazard ratios, which represent the ratio of the hazard for one group or level of a predictor variable to the hazard for another group or level, holding all other predictors constant.
Cox regression can be used to model the effect of both categorical and continuous predictor variables on survival time. In the case of continuous predictors, it is often necessary to transform the variable (such as taking the logarithm) to ensure that the proportional hazards assumption is met.
In addition to estimating the effect of predictor variables on survival time, Cox regression can also be used to predict the survival probability for new individuals, given their predictor values. This is commonly done by using the estimated coefficients to calculate a predicted hazard function, which can then be integrated over time to obtain the survival probability.
Cox regression is a powerful and widely used technique for analyzing survival data, but it does make some assumptions about the form of the hazard function and the proportionality of hazards. It is important to check these assumptions carefully before interpreting the results of a Cox regression analysis.