Fitting a frailty Cox model: what does the error "flat region resulting in a missing likelihood" mean?

[Simon Walsh]

I am a Stata novice (Stata -V14).

I have a data set which consists of 404 doctors each independently evaluating 60 patients and deciding if they have malignant disease or benign disease. I am then performing Cox regression on the diagnosis (i.e. evaluating to see if the diagnosis is prognostically significant - in my speciality, this is considered a surrogate of diagnostic accuracy). I want to account for the fact that although there are 24240 doctor-patient interactions (404 doctors * 60 patients), there are only 60 patients.

My first approach was to use the "cluster" option but a reviewer has suggested I should try to fit a "frailty model" along the lines of : stcox diagnosis, shared(patient_id)

I have tried this and get the following error:

"flat region resulting in a missing likelihood"

Having looked this error up, I cannot work out what is wrong - is it a problem with my data (i.e. there is an error in the data) or is it telling me that this operation cannot be performed (i.e. the data is fine but the analysis cannot be done). Without this information, I cannot start to rectify the problem. I would appreciate any help that can be given.

[Carlo Lazzaro]

Simon:
the error message that Stata returns is pretty general: it simply means that the MLE stops along the way (and hence cannot find a maximum).
That said, it does not necessarily mean that -shared()-option is unsuitable for your data.
However, I'm not clear about your dependent variable: if it is -diagnosis- it should not be reported in -stcox- code, as you should have already -stset- your data with -diagnosis- as -failure-.
Eventually, I'm also wondering whether you do not have other predictors to plug in the right-hand side of your regression equation.

[Simon Walsh]

Hi Carlo,
Thanks. No - the failure variable is dead. In my speciality, we sometimes use outcome as a surrogate of diagnostic accuracy. For the disease I am analysing, the more accurate a doctors diagnosis of the disease is, the greater the prognostic separation between patients diagnosed with the condition or not. We've done this before (ref below). But this isn't the issue. The issue is the error reported. As mentioned (death is the failure variable). The diagnosis variable is a doctors diagnosis of dead or not dead - the KM curve for all doctors diagnoses on the 60 patients(so therefore, 24240 doctor-patient evaluations) is attached as a simple graphical example . The issue is accounting for the fact that the 404 doctors in the study evaluated the same 60 patients - a reviewer of our study manuscript has suggested using a frailty model to account for this - but so far I can't get it to work. Any help would be useful... thanks


Walsh SLF, Wells AU, Desai SR, Poletti V, Piciucchi S, Dubini A, et al.Multicentre evaluation of multidisciplinary team meeting agreement on diagnosis in diffuse parenchymal lung disease: a case-cohort study. Lancet Respir Med. 2016;4(7):557-65.

[Clyde Schechter]

Wait, what? 60 patients were examined by 404 doctors (avg. nearly 7 doctors per patient) to determine whether or not they were dead? Really? In my experience it is extremely rare for doctors to ever disagree about whether a patient is dead. The situations where that happens usually arise in unusual situations such as a patient who still has a heartbeat while on a respirator and there is some dispute as to whether brain death has occurred. Is that your context here?

[Simon Walsh]

Many thanks for the interest but this is stated in the problem from the beginning and again, the actual study set up isn't in question. I can see where the confusion has been created. Let me clarify more.

This is a trial looking at doctor performance and each doctor was tested by evaluating their diagnostic accuracy on a retrospective set of data coming from 60 individual patients. Again, can I stress, the study design is not in question - I simply wish to understand better what might be the reasons for this specific analysis failing. Perhaps it is not possible to say from the information I have given? If so, let me know what information I should provide in order to help....

FYI - there are many diseases for which there is no reference standard diagnostic test and no "absolute truth". But when those diseases invariably cause death, while their logical differentials do not, a method for testing diagnostic accuracy is to look at outcome distinctions between a doctors diagnostic of "yes you have the disease" versus "no you do not". I even supplied a reference above for the interested reader which utilises this approach. Again - for your information, the disease is idiopathic pulmonary fibrosis, a malignant fibrotic lung disease which kills as many people as breast cancer per year in the US. If you get it, you die quickly. Doctors are notoriously bad at agreeing on who has it and who does not, when a patient first presents to the clinic - misdiagnosis is a massive problem. Often it is only after a year or so of watching the behaviour of the disease that doctors realise - yes, this is IPF because the patient is spirally downward quickly. At this point, the patient has missed a year of treatment which may have slowed the conditions progression. My study is evaluating this and is the largest observational study of its kind, in this particular disease Hope this clarifies....now back to the problem at hand if thats ok.


Also - based on your comment 7 doctors per patient- There were 24240 doctor-patient interactions; each doctor evaluated all 60 patients. So each patient was evaluated by 404 doctors. Doctors had no discussion or interaction which each other...


Many thanks for your time.

[Clyde Schechter]

In #3, you said "The diagnosis variable is a doctors diagnosis of dead or not dead -[emphasis added]." That is what I found not credible and I raised the issue because it suggested that there was something wrong with your data. In general, when these kinds of models have convergence difficulties, the problem is often that the data are invalid or are unsuitable for the model that is being attempted. In fact, in my experience, data problems are far and away the most frequent cause of the kind of situation you find yourself in.

Indeed, you do not really provide enough information to diagnose the problem with your analysis. Here's what I suggest you do. Re-run the model, this time adding the -iterate(#)- option. Set # to a number of iterations that gets you to just before the point where you encounter the ""flat region resulting in a missing likelihood"" error. Stata will halt there and give you interim results. Those results are not usable as answers to your research questions, but they may will indicate the source of the problem. You may find that some of your variables have outlandishly large (in magnitude) coefficients or standard errors. It then becomes a question of exploring those variables, and how they relate to your survival outcome, and deciding whether the data can be repaired in some way, or whether those variables simply have to be omitted from the model.

If you want more concrete guidance, post both the exact command and the exact output you get from the above. It would probably also help to show a brief excerpt of your data (please use the -dataex- command to do that), and briefly explain what each variable represents.

[Simon Walsh]

Thanks Clyde - this is very helpful. I have pasted a the output of dataex (for 180 observations - this is 3 doctors scores for 60 patients). Sorry for the long file -
futime = follow up time
dead:dead = 1, alive: dead=0
age = age
dlco = a lung function variable that reflects severity of disease.
IPF_considered = the doctors diagnosis of IPF (has IPF = 1, doesn't have IPF=0)

The command I want to run is:

stcox ipf_considered, shared(patient_id)

Basically looking at survival based on the doctors diagnoses, but taking into account that its based on the evaluation of the same 60 patients.

The output of the above is:

. stcox ipf_considered, shared(patient_id)

         failure _d:  dead == 1
   analysis time _t:  futime

Fitting comparison Cox model:

Estimating frailty variance:

flat region resulting in a missing likelihood
r(430);

* Example generated by -dataex-. To install: ssc install dataex
clear
input int doctor_id byte(patient_id age) double dlco byte(ipf_considered dead) double futime byte(_st _d) double _t byte _t0
2  1 65 44.4 0 0 4.692 1 0 4.692 0
2  2 49 30.6 0 0 4.583 1 0 4.583 0
2  3 78   40 1 0 4.831 1 0 4.831 0
2  4 78 38.4 0 1 3.233 1 1 3.233 0
2  5 68 47.3 1 1 4.256 1 1 4.256 0
2  6 58 52.1 0 0 4.772 1 0 4.772 0
2  7 70 40.1 1 1 2.244 1 1 2.244 0
2  8 63   32 0 1 4.114 1 1 4.114 0
2  9 54 33.6 0 1 4.058 1 1 4.058 0
2 10 83 46.5 1 1 4.975 1 1 4.975 0
2 11 77 46.7 1 0 4.756 1 0 4.756 0
2 12 44   60 0 0 4.811 1 0 4.811 0
2 13 28 79.5 0 0 4.353 1 0 4.353 0
2 14 60 50.1 0 1  .561 1 1  .561 0
2 15 55 36.6 0 0 4.736 1 0 4.736 0
2 16 73 65.4 0 0 4.653 1 0 4.653 0
2 17 55 32.3 1 1   3.3 1 1   3.3 0
2 18 70 41.4 0 0 4.336 1 0 4.336 0
2 19 42 32.9 0 0 4.856 1 0 4.856 0
2 20 73 38.1 0 1 2.594 1 1 2.594 0
2 21 51    0 1 1 2.117 1 1 2.117 0
2 22 77    0 1 1  2.05 1 1  2.05 0
2 23 32 63.1 0 0 4.692 1 0 4.692 0
2 24 48 42.5 0 0 4.989 1 0 4.989 0
2 25 84 57.8 1 1 1.103 1 1 1.103 0
2 26 38 78.1 0 0 4.989 1 0 4.989 0
2 27 83 38.2 1 1 4.031 1 1 4.031 0
2 28 77 32.2 0 1 5.042 1 1 5.042 0
2 29 79 49.3 1 0 4.392 1 0 4.392 0
2 30 36 76.2 0 0 4.792 1 0 4.792 0
2 31 56 31.4 1 1 3.339 1 1 3.339 0
2 32 67 35.7 0 0 4.467 1 0 4.467 0
2 33 78 65.3 0 0 4.894 1 0 4.894 0
2 34 69 36.5 1 1 4.244 1 1 4.244 0
2 35 52 61.5 0 0 4.856 1 0 4.856 0
2 36 73 47.6 1 1 4.419 1 1 4.419 0
2 37 55 65.2 0 0 4.908 1 0 4.908 0
2 38 71 37.9 1 1 2.633 1 1 2.633 0
2 39 73 47.7 1 1 4.481 1 1 4.481 0
2 40 29 43.8 0 0 4.856 1 0 4.856 0
2 41 54   50 0 0 4.875 1 0 4.875 0
2 42 54 37.1 1 0 4.908 1 0 4.908 0
2 43 60 33.7 0 1 3.872 1 1 3.872 0
2 44 33 39.1 0 0 4.653 1 0 4.653 0
2 45 68 35.2 1 1  .314 1 1  .314 0
2 46 36 55.5 0 0   4.6 1 0   4.6 0
2 47 86 35.5 0 0  4.75 1 0  4.75 0
2 48 84   40 0 0 4.772 1 0 4.772 0
2 49 68 49.8 0 1 4.581 1 1 4.581 0
2 50 38 51.5 0 0 4.681 1 0 4.681 0
2 51 81 39.1 0 1 1.047 1 1 1.047 0
2 52 46 91.6 0 0 4.772 1 0 4.772 0
2 53 82 36.4 1 1 3.803 1 1 3.803 0
2 54 55 43.4 1 0 4.672 1 0 4.672 0
2 55 59 36.3 0 0 4.875 1 0 4.875 0
2 56 55   75 1 0 4.989 1 0 4.989 0
2 57 46 43.7 0 1 5.408 1 1 5.408 0
2 58 73 54.8 0 1 3.494 1 1 3.494 0
2 59 47 54.6 0 0 4.811 1 0 4.811 0
2 60 67 49.4 0 0 4.581 1 0 4.581 0
5  1 65 44.4 0 0 4.692 1 0 4.692 0
5  2 49 30.6 0 0 4.583 1 0 4.583 0
5  3 78   40 1 0 4.831 1 0 4.831 0
5  4 78 38.4 0 1 3.233 1 1 3.233 0
5  5 68 47.3 1 1 4.256 1 1 4.256 0
5  6 58 52.1 1 0 4.772 1 0 4.772 0
5  7 70 40.1 1 1 2.244 1 1 2.244 0
5  8 63   32 0 1 4.114 1 1 4.114 0
5  9 54 33.6 1 1 4.058 1 1 4.058 0
5 10 83 46.5 1 1 4.975 1 1 4.975 0
5 11 77 46.7 0 0 4.756 1 0 4.756 0
5 12 44   60 1 0 4.811 1 0 4.811 0
5 13 28 79.5 0 0 4.353 1 0 4.353 0
5 14 60 50.1 0 1  .561 1 1  .561 0
5 15 55 36.6 0 0 4.736 1 0 4.736 0
5 16 73 65.4 1 0 4.653 1 0 4.653 0
5 17 55 32.3 1 1   3.3 1 1   3.3 0
5 18 70 41.4 1 0 4.336 1 0 4.336 0
5 19 42 32.9 0 0 4.856 1 0 4.856 0
5 20 73 38.1 0 1 2.594 1 1 2.594 0
5 21 51    0 1 1 2.117 1 1 2.117 0
5 22 77    0 1 1  2.05 1 1  2.05 0
5 23 32 63.1 0 0 4.692 1 0 4.692 0
5 24 48 42.5 1 0 4.989 1 0 4.989 0
5 25 84 57.8 1 1 1.103 1 1 1.103 0
5 26 38 78.1 0 0 4.989 1 0 4.989 0
5 27 83 38.2 1 1 4.031 1 1 4.031 0
5 28 77 32.2 0 1 5.042 1 1 5.042 0
5 29 79 49.3 1 0 4.392 1 0 4.392 0
5 30 36 76.2 0 0 4.792 1 0 4.792 0
5 31 56 31.4 1 1 3.339 1 1 3.339 0
5 32 67 35.7 1 0 4.467 1 0 4.467 0
5 33 78 65.3 1 0 4.894 1 0 4.894 0
5 34 69 36.5 1 1 4.244 1 1 4.244 0
5 35 52 61.5 0 0 4.856 1 0 4.856 0
5 36 73 47.6 1 1 4.419 1 1 4.419 0
5 37 55 65.2 0 0 4.908 1 0 4.908 0
5 38 71 37.9 1 1 2.633 1 1 2.633 0
5 39 73 47.7 1 1 4.481 1 1 4.481 0
5 40 29 43.8 0 0 4.856 1 0 4.856 0
5 41 54   50 0 0 4.875 1 0 4.875 0
5 42 54 37.1 1 0 4.908 1 0 4.908 0
5 43 60 33.7 0 1 3.872 1 1 3.872 0
5 44 33 39.1 1 0 4.653 1 0 4.653 0
5 45 68 35.2 1 1  .314 1 1  .314 0
5 46 36 55.5 0 0   4.6 1 0   4.6 0
5 47 86 35.5 1 0  4.75 1 0  4.75 0
5 48 84   40 0 0 4.772 1 0 4.772 0
5 49 68 49.8 1 1 4.581 1 1 4.581 0
5 50 38 51.5 0 0 4.681 1 0 4.681 0
5 51 81 39.1 1 1 1.047 1 1 1.047 0
5 52 46 91.6 0 0 4.772 1 0 4.772 0
5 53 82 36.4 1 1 3.803 1 1 3.803 0
5 54 55 43.4 0 0 4.672 1 0 4.672 0
5 55 59 36.3 1 0 4.875 1 0 4.875 0
5 56 55   75 1 0 4.989 1 0 4.989 0
5 57 46 43.7 0 1 5.408 1 1 5.408 0
5 58 73 54.8 1 1 3.494 1 1 3.494 0
5 59 47 54.6 0 0 4.811 1 0 4.811 0
5 60 67 49.4 0 0 4.581 1 0 4.581 0
6  1 65 44.4 0 0 4.692 1 0 4.692 0
6  2 49 30.6 0 0 4.583 1 0 4.583 0
6  3 78   40 1 0 4.831 1 0 4.831 0
6  4 78 38.4 0 1 3.233 1 1 3.233 0
6  5 68 47.3 1 1 4.256 1 1 4.256 0
6  6 58 52.1 0 0 4.772 1 0 4.772 0
6  7 70 40.1 1 1 2.244 1 1 2.244 0
6  8 63   32 0 1 4.114 1 1 4.114 0
6  9 54 33.6 0 1 4.058 1 1 4.058 0
6 10 83 46.5 1 1 4.975 1 1 4.975 0
6 11 77 46.7 0 0 4.756 1 0 4.756 0
6 12 44   60 1 0 4.811 1 0 4.811 0
6 13 28 79.5 0 0 4.353 1 0 4.353 0
6 14 60 50.1 0 1  .561 1 1  .561 0
6 15 55 36.6 0 0 4.736 1 0 4.736 0
6 16 73 65.4 0 0 4.653 1 0 4.653 0
6 17 55 32.3 1 1   3.3 1 1   3.3 0
6 18 70 41.4 0 0 4.336 1 0 4.336 0
6 19 42 32.9 0 0 4.856 1 0 4.856 0
6 20 73 38.1 0 1 2.594 1 1 2.594 0
6 21 51    0 0 1 2.117 1 1 2.117 0
6 22 77    0 1 1  2.05 1 1  2.05 0
6 23 32 63.1 0 0 4.692 1 0 4.692 0
6 24 48 42.5 0 0 4.989 1 0 4.989 0
6 25 84 57.8 1 1 1.103 1 1 1.103 0
6 26 38 78.1 0 0 4.989 1 0 4.989 0
6 27 83 38.2 1 1 4.031 1 1 4.031 0
6 28 77 32.2 0 1 5.042 1 1 5.042 0
6 29 79 49.3 1 0 4.392 1 0 4.392 0
6 30 36 76.2 0 0 4.792 1 0 4.792 0
6 31 56 31.4 1 1 3.339 1 1 3.339 0
6 32 67 35.7 1 0 4.467 1 0 4.467 0
6 33 78 65.3 1 0 4.894 1 0 4.894 0
6 34 69 36.5 1 1 4.244 1 1 4.244 0
6 35 52 61.5 0 0 4.856 1 0 4.856 0
6 36 73 47.6 1 1 4.419 1 1 4.419 0
6 37 55 65.2 0 0 4.908 1 0 4.908 0
6 38 71 37.9 1 1 2.633 1 1 2.633 0
6 39 73 47.7 1 1 4.481 1 1 4.481 0
6 40 29 43.8 0 0 4.856 1 0 4.856 0
6 41 54   50 0 0 4.875 1 0 4.875 0
6 42 54 37.1 0 0 4.908 1 0 4.908 0
6 43 60 33.7 0 1 3.872 1 1 3.872 0
6 44 33 39.1 0 0 4.653 1 0 4.653 0
6 45 68 35.2 1 1  .314 1 1  .314 0
6 46 36 55.5 0 0   4.6 1 0   4.6 0
6 47 86 35.5 1 0  4.75 1 0  4.75 0
6 48 84   40 0 0 4.772 1 0 4.772 0
6 49 68 49.8 1 1 4.581 1 1 4.581 0
6 50 38 51.5 0 0 4.681 1 0 4.681 0
6 51 81 39.1 1 1 1.047 1 1 1.047 0
6 52 46 91.6 0 0 4.772 1 0 4.772 0
6 53 82 36.4 1 1 3.803 1 1 3.803 0
6 54 55 43.4 0 0 4.672 1 0 4.672 0
6 55 59 36.3 0 0 4.875 1 0 4.875 0
6 56 55   75 1 0 4.989 1 0 4.989 0
6 57 46 43.7 0 1 5.408 1 1 5.408 0
6 58 73 54.8 1 1 3.494 1 1 3.494 0
6 59 47 54.6 0 0 4.811 1 0 4.811 0
6 60 67 49.4 0 0 4.581 1 0 4.581 0
end

[Carlo Lazzaro]

Simon:
thanks for providing an excerpt of your data, that allows me to replicate your problem.
What strikes me is, again, the lack of other predictors in the righ-hand side of your -stcox- equation: this might be (one of) the cuuse(s) of the problem you complain about.

[Simon Walsh]

Hi Carlo,
cheers.
It might be logical to include age and dlco in the equation for clinical reasons - but the same problem occurs when I do this. Because I’m not a statistician I am unsure if there is a “problem” with my data but I don’t think so. Originally I had simply run a cox regression analysis

stcox ipf_considered

but a reviewer has highlighted that since the patient set is the same for each Doctor observations are not independent and suggested a “frailty” model

[Carlo Lazzaro]

Simon:
in the meantime I have run the -stcox with all the predictors (including a squared term for age, just in case...) but the error message still creeps up, no matter the number of predictors.
I have also applied Clyde's suggestion to set the number of iterations explicitly, just to see when Stata starts gasping: immediately.
At this point, my concern switch from the predictors to the regressand: as patients are dead once and for all, it may well be that, due to the absence of variation in the regressand, -shared()- option (that makes theoretically sense in your research) is not suitable for your data (conversely, please see Example n 10, -stcox- entry, Stata .pdf manual).
Interstingly, when we consider a clustered standard errors (ie, we take into account the correlation between patients belonging to the same panel), things are easier:

. stcox ipf_considered dlco i.ipf_considered c.age##c.age i.doctor_id, cluster(patient_id)

         failure _d:  dead
   analysis time _t:  futime

note: 1.ipf_considered omitted because of collinearity
Iteration 0:   log pseudolikelihood = -360.14696
Iteration 1:   log pseudolikelihood = -319.73895
Iteration 2:   log pseudolikelihood = -310.53538
Iteration 3:   log pseudolikelihood = -307.53952
Iteration 4:   log pseudolikelihood = -307.17467
Iteration 5:   log pseudolikelihood = -307.17025
Iteration 6:   log pseudolikelihood = -307.17025
Refining estimates:
Iteration 0:   log pseudolikelihood = -307.17025

Cox regression -- Breslow method for ties

No. of subjects      =          180             Number of obs    =         180
No. of failures      =           78
Time at risk         =      739.398
                                                Wald chi2(6)     =       30.07
Log pseudolikelihood =   -307.17025             Prob > chi2      =      0.0000

                                (Std. Err. adjusted for 60 clusters in patient_id)
----------------------------------------------------------------------------------
                 |               Robust
              _t | Haz. Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-----------------+----------------------------------------------------------------
  ipf_considered |   2.647626   1.118966     2.30   0.021     1.156414    6.061777
            dlco |   .9403424   .0173415    -3.34   0.001     .9069606    .9749529
1.ipf_considered |          1  (omitted)
             age |   1.841074   .4079811     2.75   0.006     1.192465    2.842478
                 |
     c.age#c.age |   .9956304   .0016773    -2.60   0.009     .9923485    .9989232
                 |
       doctor_id |
              5  |   .8457473   .0764816    -1.85   0.064     .7083792    1.009754
              6  |   .9650093   .0684103    -0.50   0.615     .8398256    1.108853
----------------------------------------------------------------------------------

.

[Simon Walsh]

Carlo,
this is excellent assistance. May I ask, is this cluster approach as “legitimate” as a frailty model?

[Carlo Lazzaro]

Simon:
just a second thought on my previous reply.
Assuming that we want to test whether there's an unobservable doctor (not patient)-level random effect (that is probably the meaning of reviewer's suggestion), my previous code should be changed a bit:

. stcox ipf_considered dlco i.ipf_considered c.age##c.age i.doctor_id, shared(doctor_id)
note: 1.ipf_considered omitted because of collinearity

         failure _d:  dead
   analysis time _t:  futime

Fitting comparison Cox model:

Estimating frailty variance:

Iteration 0:   log profile likelihood = -307.17025
numerical derivatives are approximate
flat or discontinuous region encountered
Iteration 1:   log profile likelihood = -307.17025
Iteration 2:   log profile likelihood = -307.17025

Fitting final Cox model:

Iteration 0:   log likelihood = -360.14696
Iteration 1:   log likelihood = -319.73895
Iteration 2:   log likelihood = -310.53538
Iteration 3:   log likelihood = -307.53952
Iteration 4:   log likelihood = -307.17467
Iteration 5:   log likelihood = -307.17025
Iteration 6:   log likelihood = -307.17025
Refining estimates:
Iteration 0:   log likelihood = -307.17025

Cox regression -- Breslow method for ties

Gamma shared frailty                            Number of obs     =        180
Group variable: doctor_id                       Number of groups  =          3
                                                Obs per group:
No. of subjects =          180                                min =         60
No. of failures =           78                                avg =         60
Time at risk    =      739.398                                max =         60

                                                Wald chi2(6)      =      64.96
Log likelihood  =   -307.17025                  Prob > chi2       =     0.0000

----------------------------------------------------------------------------------
              _t | Haz. Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
-----------------+----------------------------------------------------------------
  ipf_considered |   2.647626   .7303335     3.53   0.000     1.541904    4.546279
            dlco |   .9403424    .009442    -6.13   0.000     .9220173    .9590318
1.ipf_considered |          1  (omitted)
             age |   1.841074   .2894438     3.88   0.000     1.352851     2.50549
                 |
     c.age#c.age |   .9956304   .0011615    -3.75   0.000     .9933566    .9979095
                 |
       doctor_id |
              5  |   .8457473    .237502    -0.60   0.551     .4877609    1.466474
              6  |   .9650093   .2680693    -0.13   0.898     .5598565     1.66336
-----------------+----------------------------------------------------------------
           theta |   2.08e-18   6.58e-15
----------------------------------------------------------------------------------
LR test of theta=0: chibar2(01) = 3.0e-12              Prob >= chibar2 = 0.500

Note: Standard errors of hazard ratios are conditional on theta.

.

As far as the excerpt of your data is concerned, the likelihhod ratio appearing as a footnote of the -stcox- outcome table tell us that there's no evidence of a doctor-level unobservable random effect.

[Simon Walsh]

Super. I will look at this closely and evaluate my results accordingly. Many thanks for your rapid responses Carlo.

[Carlo Lazzaro]

Simon:
as per your question on my reply #11.
Not quite: -cluster- option assumes that the observation belonging to the same panel are more similar (in an unspecified sense) that the ones belonging to a different panel, whereas -shared()- option actually models the correlation assuming that is caused by an unobservable effect that follows a given statistical distribution (gamma).