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# Re: st: Re: Interpretation of the Coefficients obtained via -stcox-

 From Yuval Arbel To statalist@hsphsun2.harvard.edu Subject Re: st: Re: Interpretation of the Coefficients obtained via -stcox- Date Sun, 9 Sep 2012 22:05:04 +0300

```Steve, thank you for the answer. We indeed get this solution of exp(b)
if we run -stcox- without -nohr-

BTW: I believe there is a gap between the mathematical and actual
solution. This emanates from the fact that the mathematical solution
is correct for a very small (infitisimal) change

On Sun, Sep 9, 2012 at 9:17 PM, Steve Samuels <sjsamuels@gmail.com> wrote:
> You are not correct. Although your calculus is right, the approximation
> to the percentage change will be poor in general. Moreover it isn't
> necessary to approximate as the exact result is available.
>
> Suppose the model is:
> (*) log h(t|x) = a(t) + b x
>
> h(t|x) = exp(a(t) +b x)
>
> Increase x by 1 unit: h(t|x+1) = exp(a(t) +b x +b)
>
> hazard ratio ht(t|x+1)/h(t|x) = exp(b)
>
> Percentage change
>
> 100* (h(t|x+1) - h(t|x))/h(t|x) = 100* (exp(b) -1)
>
> Only if b is close to zero is this ~ 100*b
>
> Your results: For b = .0382773,  exp(b)-1 = 0.0390 so the approximation is not
> bad. On the other hand, the coefficient for "appreciation" is about +11, but
> exp(11)-1 is ~60,000! The approximation is bad even for b = 0.51,  the coefficient
> of your fourth predictor, as exp(0.51)-1 = 0.665.
>
>
> Steve
>
>
> On Sep 9, 2012, at 9:18 AM, Yuval Arbel wrote:
>
> Dear statalisters,
>
> I ask this question because I noted that on one hand some scholars,
> who applied the Cox Regression, seems to  avoid a direct
> interpretation of the coefficients obtained via this procedure. It
> occurred to me there might be a resemblance to -probit-, which does
> not yield the coefficient in terms of marginal probabilities (as
> opposed to
> -dprobit-).
>
> On the other hand, if we take a look at the model's specification
> according to stata's manual:
>
> h(t) = h0(t) exp( b1x1 +...  + bkxk)
>
> and derive the term d(h(t))/d(xk), we get:
>
> d(h(t))/d(xk)=h0(t) exp( b1x1 +...  + bkxk)bk=h(t)bk
>
> and then: bk=[dh(t)/h(t)]/d(xk) implying a percent chance on the
> hazard to survive in the numerator.
>
> I wonder am I correct here?
>
> On Sun, Sep 9, 2012 at 9:26 AM, Yuval Arbel <yuval.arbel@gmail.com> wrote:
>> Dear statalisters,
>>
>>
>> I'm attaching below the estimation results of -stcox-.
>>
>> Do they imply that if we increase mean_reduct by 1 unit the hazard to
>> survival increase by 3.83 percent?
>>
>> . stcox mean_reduct reductcurrent_mean_reduct rent_net8
>> appreciation if nachut==0 & nachutspouse==0 & diff_per>=-5 &
>> diff_per<=5,nohr
>>
>>         failure _d:  fail == 1
>>   analysis time _t:  time_index
>>                 id:  appt
>>
>> Iteration 0:   log likelihood = -56991.691
>> Iteration 1:   log likelihood = -54168.973
>> Iteration 2:   log likelihood = -53930.527
>> Iteration 3:   log likelihood = -53916.083
>> Iteration 4:   log likelihood = -53915.926
>> Iteration 5:   log likelihood = -53915.926
>> Refining estimates:
>> Iteration 0:   log likelihood = -53915.926
>>
>> Cox regression -- Breslow method for ties
>>
>> No. of subjects =         7191                     Number of obs   =    324499
>> No. of failures =         7191
>> Time at risk    =       351446
>>                                                   LR chi2(7)      =   6151.53
>> Log likelihood  =   -53915.926                     Prob > chi2     =    0.0000
>>
>> ------------------------------------------------------------------------------
>>          _t |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
>> -------------+----------------------------------------------------------------
>> mean_reduct |   .0382773   .0006943    55.13   0.000     .0369165    .0396382
>> reductcurr~t |   .0282488   .0007893    35.79   0.000     .0267018    .0297958
>>   rent_net8 |   .0018389   .0001947     9.45   0.000     .0014573    .0022204
>> diff_stdma~a |  -.5076186   .0579597    -8.76   0.000    -.6212176   -.3940197
>> permanent~82 |  -.0005113   .0000862    -5.93   0.000    -.0006803   -.0003423
>> diff_mortg~e |  -7.715171    1.23864    -6.23   0.000    -10.14286   -5.287481
>> appreciation |   10.94379   3.632834     3.01   0.003     3.823562    18.06401
>> ------------------------------------------------------------------------------
>>
>>
>> On Tue, Aug 21, 2012 at 1:14 AM, Yuval Arbel <yuval.arbel@gmail.com> wrote:
>>> Dear Statalisters,
>>>
>>> According to stata manual the command -stcox- estimates the following model:
>>>
>>> h(t) = h0(t) exp( b1x1 +...  + bkxk)
>>>
>>> where h(t) is the hazard to survival.
>>>
>>> Can I infer from this specification that bk in its original form
>>> (nohr) measures the percent change of the hazard to survive with
>>> respect to xk?
>>>
>>>
>>> --
>>> Dr. Yuval Arbel
>>> 4 Shaar Palmer Street,
>>> Haifa 33031, Israel
>>> e-mail1: yuval.arbel@carmel.ac.il
>>> e-mail2: yuval.arbel@gmail.com
>>
>>
>>
>> --
>> Dr. Yuval Arbel
>> 4 Shaar Palmer Street,
>> Haifa 33031, Israel
>> e-mail1: yuval.arbel@carmel.ac.il
>> e-mail2: yuval.arbel@gmail.com
>> *
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>
>
>
> --
> Dr. Yuval Arbel
> 4 Shaar Palmer Street,
> Haifa 33031, Israel
> e-mail1: yuval.arbel@carmel.ac.il
> e-mail2: yuval.arbel@gmail.com
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>
>
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--
Dr. Yuval Arbel