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

 From Steve Samuels To statalist@hsphsun2.harvard.edu Subject Re: st: Re: Interpretation of the Coefficients obtained via -stcox- Date Sun, 9 Sep 2012 14:17:56 -0400

```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 appreciate very much your answer to this question.
>
> 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
>> School of Business
>> Carmel Academic Center
>> 4 Shaar Palmer Street,
>> Haifa 33031, Israel
>> e-mail1: yuval.arbel@carmel.ac.il
>> e-mail2: yuval.arbel@gmail.com
>
>
>
> --
> Dr. Yuval Arbel
> School of Business
> Carmel Academic Center
> 4 Shaar Palmer Street,
> Haifa 33031, Israel
> e-mail1: yuval.arbel@carmel.ac.il
> e-mail2: yuval.arbel@gmail.com
> *
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> *   http://www.ats.ucla.edu/stat/stata/

--
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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```