Cox regression, also known as proportional hazards regression, is a type of survival analysis used to model the time-to-event data, where the outcome of interest is the time until an event occurs. This event could be anything, such as death, disease recurrence, or failure of a mechanical component.

Cox regression is named after the statistician Sir David Cox who developed it in 1972. The main assumption of Cox regression is that the hazard function, which describes the probability of an event occurring at a particular time, is proportional over time. This means that the hazard ratio, which measures the effect of a predictor variable on the hazard function, remains constant over time.

Cox regression is a popular method in medical research to analyse survival data, where the outcome of interest is the time until a patient dies or experiences a specific event, such as disease progression or hospitalization. It is also used in other fields, such as engineering, economics, and social sciences, where time-to-event data is common.

Cox regression is useful because it allows researchers to investigate the relationship between predictor variables and the risk of an event occurring over time. By modeling the hazard function, Cox regression can estimate the hazard ratio for each predictor, which represents the change in the hazard function for a one-unit increase in the predictor while holding other predictors constant. The hazard ratio is a useful summary measure for the strength and direction of the association between a predictor and the outcome.

Cox regression works by fitting a Cox proportional hazards model to the data. The model assumes that the hazard function can be written as the product of a baseline hazard function that depends only on time and a function of the predictor variables. The predictor function is assumed to be a linear combination of the predictors, weighted by regression coefficients.

The model is estimated using a method called maximum likelihood estimation, which involves finding the values of the regression coefficients that maximize the likelihood of observing the data given the model. The estimated regression coefficients can be used to calculate the hazard ratios and test the significance of the predictors.

Cox regression has some advantages over other survival analysis methods. It does not assume any particular distribution for the survival times, and it allows for censoring, which occurs when the event of interest has not occurred for some participants by the end of the study. Censoring is common in medical research when some patients are still alive or have not experienced the event of interest at the end of the study period.