I am interested in the effect of the expression of certain genes (example: Gene A and Gene B) on the survival of cancer patients. I am working with TCGA data, which contains RNAseq data of 33 different cancer types with a total of about 6500 cancer patients. I performed survival analysis using a univariate model - separately for each gene and separately for each cancer type, obtaining 33 hazard ratios for Gene A and 33 hazard ratios for Gene B.
My question is, whether I can calculate the correlation (Pearson or Spearman) between the obtained hazard ratios of Gene A and hazard rations of Gene B, in order to see whether their effect on survival is similar and compare correlation of hazard ratios of Gene A and Gene B with the correlation of hazard ratios of Gene A and Gene C?
I am worried about whether it makes sense statistically to correlate hazard ratios? In publications from others, I've only seen people plotting hazard ratios in form of a scatter plot and comment on concordance and differences, without any quantitative statistics. As I have to make quite a lot of different comparisons, a score like Pearson or Spearman correlation would be helpful for comparison. Alternatively, I could use the concordance correlation coefficient (ccc)?
Thank you for your help in advance!