Determining the Fit of a Cox Regression Model

Basic Concepts

To compare two hierarchical models, we use the usual chi-square test on 2(LL₁−LL₀) where LL₁ is the full (i.e. larger) model and LL₀ is the reduced model (i.e. the model with k variables omitted).

2(LL₁ – LL₀) ~ χ²(k)

We can use this as a measure of overall model fit where LL₁ is for the Cox regression model under consideration with k covariates and LL₀ is the reduced model where all the coefficients are 0.

Example

For Example 1 of Cox Regression using Solver, we see from Figure 1 of Cox Regression using Solver that LL0 = -15.273 and from Figure 3 of Cox Regression using Solver we see that LL1 = -11.045. Based on the chi-square test shown in Figure 1 below, we see that there is a significant difference between the complete Cox regression model (i.e. the model with Age and Size covariates) and the baseline model (i.e. the model without any covariates).

Figure 1 – Chi-square Goodness of Fit

Age-only and Size-only

If we rerun the analysis we have done using only the Age covariate we find that the chi-square test shows that the model with only the Age covariate is statistically equivalent to the baseline model (p-value = .08) based on an alpha of .05. See Figure 2.

The model with only the Size covariate does a little better (p-value = .06), but it too is statistically equivalent to the baseline model based on an alpha of .05. Both covariates together give a better fit.

Figure 2 – Reduced models

Also, you should be aware that the chi-square test using the LL values is a better indicator than the p-values of the coefficients.

R-square

There is no commonly accepted value for R² for Cox regression. The best available such value is

Examples Workbook

Click here to download the Excel workbook with the examples described on this webpage.

References

NCSS (2015) Cox regression
https://www.ncss.com/wp-content/themes/ncss/pdf/Procedures/NCSS/Cox_Regression.pdf

Collett, D. (2003) Modelling survival data in medical research. 2nd ed. Chapman & Hall/CRC

Zhang, D. (2011) Modeling survival data with Cox regression models
https://www.scribd.com/document/396594178

Grund, B., Yang, L. (2015) Hazard regression
https://edoc.hu-berlin.de/server/api/core/bitstreams/595cb9b6-0c1a-4121-8a83-8a194f19b3ae/content