How to Calculate Confidence Intervals in Coxph

What is Cox proportional hazards regression?

Cox proportional hazards regression is a semi-parametric statistical method used to analyze survival data. It models the hazard function, which represents the instantaneous risk of an event occurring at a specific time point, given the values of one or more predictor variables.

The key assumptions of Coxph are:

• The hazard function is proportional over time
• There are no unmeasured confounders
• Censoring is non-informative

The model estimates coefficients for each predictor variable, which represent the log hazard ratio. These coefficients can be used to calculate confidence intervals that quantify the uncertainty around the estimated effects.

Understanding confidence intervals in Coxph

Confidence intervals in Coxph provide a range of plausible values for the true effect of each predictor variable. They are calculated based on the standard error of the coefficient estimate and the desired confidence level (typically 95%).

The formula for the confidence interval is:

These intervals help determine whether the effect is statistically significant (if the interval does not include zero) and provide a measure of the precision of the estimate.

How to calculate confidence intervals in Coxph

To calculate confidence intervals in Coxph, follow these steps:

1. Fit the Cox proportional hazards model to your data using statistical software
2. Extract the coefficient estimates (β) and standard errors (SE) for each predictor
3. Calculate the lower and upper bounds using the formula above
4. Interpret the results in the context of your research question

Most statistical software packages, including R, SAS, and Stata, have built-in functions to calculate these intervals automatically. However, understanding the underlying calculations helps you verify the results and interpret them correctly.

Worked example

Let's consider a simple example where we want to analyze the effect of treatment on survival time in a clinical trial. Suppose we have the following results from a Coxph model:

To calculate a 95% confidence interval for the treatment effect:

1. Calculate the exponent of the coefficient: exp(0.85) = 2.34
2. Calculate the margin of error: 1.96 × 0.20 = 0.392
3. Calculate the lower bound: exp(0.85 - 0.392) = exp(0.458) = 1.58
4. Calculate the upper bound: exp(0.85 + 0.392) = exp(1.242) = 3.46

The 95% confidence interval for the hazard ratio is (1.58, 3.46), indicating that the treatment is associated with a 58% to 246% increase in the hazard of the event, with 95% confidence.

Interpreting the results

When interpreting confidence intervals in Coxph, consider the following:

• If the interval includes 1, the effect is not statistically significant at the chosen confidence level
• If the interval is narrow, the estimate is precise
• If the interval is wide, the estimate is imprecise and may need more data
• Compare intervals across different predictor variables to understand their relative importance

It's important to remember that confidence intervals represent the uncertainty in the estimate, not the probability that the true effect is within the interval.