# Best predictive Cox regression model using c-index and cross-validation

I have a set of 40 genes expression data as binary variables (overexpressed yes or not) for predict recurrence in a kind of cancer in a sample of 77 patients, where 22 has recurred. This set is called discovery set. The objective is to find a subset of these genes to predict the recurrence in cancer. Times to recurrence and follow-up times for censored observations were saved.

In a first step, I have fitted univariate Cox regression model, and I have discarded 13 genes of the original set of 40 genes, removing those that have P-values > 0.25 in the LRT. Then, I work only with 27 genes.

In the second step, I am analyzing all the possible multivariate models of 3 variables, 4 variables, 5 variables, etc. I decide what is the more predictive model of $X$ variables fitting multivariate Cox regression and calculating the average c-index of 100 times 5-folder CV. For calculate the c-index I use the “index.corrected” reported by the validate function of the Harrell’s rms package, and I calculate the average of the 100 times I run the CV.

I will use the AIC for compare the best model with $X+1$ variables and the best model with $X$ variable, to decide what it is the best predictive model.

Finally, I have an independent validation set of patients ($n=105$) where I will check the model. I will use the linear predictor of the best model in the discovery set as prognostic index, dividing in tertiles of risk, and calculating the values of the prognostic index of these new patients. I will analyze statistical differences of the 3 groups with the log-rank test, HR with 95CI% and KM Plots.

• I was reading your question and I thought about why both c-index and AIC. I didn't understand the difference between these two steps. You fitted all the possibles models and selected those with better c-index values obtained by a cross-validation approach, so we can say that you search the betters models in the complete set of solutions. So try to find the best model through the X + 1 models approach using AIC would not be repeat the same search? The difference is: your search will look for the better models in a subset from the same complete set, but this was looked before ... (ctd) – Felipe Prata Lima Mar 28 '14 at 11:53
• (ctd)... in your first search, use of AIC values instead c-index values. So if your final selection criterion is AIC why not apply directly in the first search? Isn't the result be same? I am trying to understand that! – Felipe Prata Lima Mar 28 '14 at 11:53

2. You took the variables that passed the screen and fitted a model, then used cross-validation (CV) to validate the model without informing CV of the univariable screening step. This creates a major bias. CV needs to repeat all modeling steps that made use of $Y$.
3. You are using a non-optimum optimality criterion (the $c$-index; generalized ROC area) whereas the most sensitive criterion will be likelihood-based
7. Validating a model by binning predictions (in your case tertiles of predicted risk) is arbitrary and has very low resolution. The R rms package has a few methods for estimating continuous calibration curves. See for example the function calibrate.