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I am answering a reviewer's comments on a manuscript (medical). In one analysis I run a univariate Cox regression analysis on the variable "has disease vs hasn't disease" as stated by a clinical doctor and then ran the same as stated by a radiologist. The results are:

             Clinician HR 5.30, p<0.001 95% CI 2.26-12.41
             Radiologist HR 3.99, P=0.006 95% CI 1.49-10.66

I state in my manuscript that the prognostic significance of the clinician's diagnosis is greater that that for the radiologist (based on the higher HR). The reviewer says

    What basis did the author use to conclude a greater prognostic
    significance between clinician vs. radiologists’ diagnosis? 
    The p-value of testing HR=0? This needs to be at least clarified 
     if not corrected.

I am not fully sure what that means. My answer would be that the HR is higher (keep in mind I have tested the assumptions of proportionality etc.).

Am I correct in saying this? Is the reviewer driving at something else?

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What is HR? .......... – kjetil b halvorsen Mar 15 at 14:37
    
I think the title might benefit from editing: the key point to this question seems to be about a comparison of the HRs. – Silverfish Mar 15 at 14:50
2  
The upshot here is that while you've shown both HRs are significantly different from 0, you haven't shown they're significantly different from each other. Imagine if you had found the radiologist's HR to be 3.99 and the clinician's to be 4.00 - while the clinician's HR is nominally higher, you most likely can't reject the null hypothesis that the HRs are the same. – Matt Mar 15 at 17:14

I am guessing that the reviewer wanted to see a test of the null-hypothesis that the two hazard ratios are the same. A rejection of that null-hypothesis would support your conclusion. Typically, you do so by estimating one model for both groups, add an interaction term, and look at the test whether that interaction term is 0.

With a Cox model there is a complication that by estimating two separate models you allowed the baseline hazard function to be different for both groups. So combining the two models and adding an interaction term is not exactly equivalent to estimating the two models separately.

You could use stratification for this, this allows separate baseline hazard functions. So a silly example using Stata:

. sysuse cancer, replace
(Patient Survival in Drug Trial)

. stcox i.drug##c.age, strata(drug) nolog

         failure _d:  died
   analysis time _t:  studytime

 Stratified Cox regr. -- Breslow method for ties

 No. of subjects =           48                  Number of obs    =          48
 No. of failures =           31
 Time at risk    =          744
                                                LR chi2(3)       =       10.73
 Log likelihood  =   -59.372185                  Prob > chi2      =      0.0133

------------------------------------------------------------------------------
          _t | Haz. Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
 -------------+----------------------------------------------------------------
        drug |
          2  |          1  (omitted)
          3  |          1  (omitted)
             |
         age |   1.096463   .0533264     1.89   0.058     .9967717    1.206124
             |
  drug#c.age |
          2  |   1.110106   .1132812     1.02   0.306     .9088726    1.355893
          3  |   .9860279   .0992645    -0.14   0.889     .8094646    1.201104
------------------------------------------------------------------------------
                                                            Stratified by drug

The main effect of drug are omitted because they are captured by the stratification. The main effect of age is the effect of age when you get drug 1, and the interaction effects tell us whether the effect of age differs when you get drug 2 or drug 3 respectively. The tests behind those interaction terms are the ones you would be looking for.

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The approach from @MaartenBuis will be very good if there is no overlap in the patients diagnosed by the two types of physicians. If instead the data are paired, one good way to proceed is by comparing, always using the same set of patients in all models, the model with both diagnoses to each of the individual (univariate) models. Then all the models are nested or the usual likelihood ratio $\chi^2$ statistic can be computed. Denoting the clinician and radiologist by C and R, you would be testing whether R adds to the prognostic ability of C and whether C adds to R. If one adds to the other and the reverse is not true, you know that the one is better than the other. If both add to each other you can quantify by how much. This is related to the adequacy index discussed in my book Regression Modeling Strategies and used in a couple of clinical papers.

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Many thanks Prof. The patients are the same set of 70 cases. The C and R independently looked at the data for each case and assigned a diagnosis of "disease present" or "disease not present". Would it be possible to clarify your comment..." Then all the models are nested or the usual likelihood ratio χ2χ2 statistic can be computed." My apologies, I am a doc not a statistician. I use STATA. Many thanks for you input. – GhostRider Mar 15 at 12:24
    
Nested models are required for likelihood ratio tests. This just means that every pair of models used has one of the models being a proper subset of the other. Example of non-nested models: compare a model with just C to a model with just R. – Frank Harrell Mar 15 at 12:54

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