Cox proportional hazards regression assumes the following relationship of the baseline hazard rate (h0(t)), the values of the predictor variables (xi) and the parameter coefficients (βi):
When performing Cox proportional hazards regression, the first step in the analysis process is to estimate the best-fit values of the beta (β) coefficients in the model. This is done by using a process known as maximum likelihood estimation (MLE). The mathematics involved in this technique are well beyond the scope of this guide, but it is important to note that there are three separate techniques that are generally used for MLE depending on the data being analyzed, and if there are ties in the data. For information on interpretation of these beta coefficients, read more on this page.
When collecting survival data, the variable representing elapsed time is treated as continuous, but is typically measured in a discrete manner. The status of each individual in the study is checked hourly, daily, weekly, monthly, etc., and the elapsed time recorded for the analysis is generally truncated at this level. For example, we may record that an individual experienced the event of interest after 6 weeks of a 4 month study instead of knowing the exact number of days (or hours, minutes, seconds, etc.) that elapsed prior to the event of interest for this individual. However, it should be noted that elapsed time is still considered to be a continuous variable, even with this method of data recording. Because of this, it often occurs that multiple individuals in a study will have the same recorded survival time. This is known as “ties” in the elapsed time.
When there are no ties (or a small number of ties) in the data, the MLE using the “exact” method to determine the beta coefficients in the model can be used (and is the default in Prism). However, when there are a large number of ties in event times, this method becomes extremely computationally intense, and an approximation is used to calculate the desired beta coefficient values. There are two common approximations that can be used:
1.Breslow's approximation
2.Efron's approximation
Breslow’s approximation is the older of the two approaches, and is used as the default in a number of statistical packages when there are ties in the data. However, Efron’s approximation is often considered to provide more accurate results. When there are a small number of ties in the data, all three methods should give very similar results. When the number of ties is large, Prism will automatically use Efron’s approximation to determine the values of the beta coefficients. However, controls on the Model tab of the analysis dialog will allow you to specify your preferred method.
Subsequent sections of this guide will provide information on how to perform Cox proportional hazards regression within Prism, and guidance on how to interpret the results that Prism generates from this analysis.
