probit or logit

[saoussen gh]

Dear colleague i need your help , could you please telle me when i should use probit or logit,
doses the use of logit need a specific test
thak you so much

[Richard Williams]

I don't know if it is true, but I heard once that you would need 10 million cases to determine whether logit or probit is better.

Often the choice is just based on what is typically used in your field.

Some advanced techniques may lead you to prefer one over the other. For example, you can estimate a fixed effects logit model but you can't estimate a fixed effects probit model.

[saoussen gh]

tahnk you so much richard, sorry for distrub i have so many question is because i am confused so really why we use fixed effedt logit , i read your article but i have soe difficulties https://www3.nd.edu/~rwilliam/stats3...xedEffects.pdf

[Clyde Schechter]

I don't know about Richard's 10,000,000 cases figure, but the gist of it is correct. The logistic and normal distributions differ only very subtly, far out in the tails. It takes an enormous number of cases to distinguish them. And, frankly, if you had that much data, you would probably find that both models are detectably mis-fit to the data since neither model is likely to be the exact data generating process for anything in the real world and that deviance would overwhelm the minute difference between the logistic and probit models..

That said, you should be aware of the difference in interpreting the coefficients of the two models. The standard deviation of the logistic distribution is pi/sqrt(3), which is approximately 1.8. The standard deviation of the standard normal distribution is 1. Consequently all of the regressions coefficients in a logistic regression will be approximately 1.8 times as large as the coefficients you would see in a probit regression using all of the same variables. The same is also true of the standard errors, so that when you calculate z-statistics and p-values, you get values that are very similar. Similarly, the predicted probabilities from the two models will turn out to be about the same (except perhaps at the far extremes). So any substantive conclusions you draw from the two models are likely to be essentially the same. The choice between them is made either on technical considerations, as Richard suggests, or out of a traditional preference for one or the other in your particular discipline.

. sysuse auto, clear
(1978 Automobile Data)

. 
. logit foreign weight

Iteration 0:   log likelihood =  -45.03321  
Iteration 1:   log likelihood = -30.669507  
Iteration 2:   log likelihood = -29.068209  
Iteration 3:   log likelihood = -29.054005  
Iteration 4:   log likelihood = -29.054002  
Iteration 5:   log likelihood = -29.054002  

Logistic regression                             Number of obs     =         74
                                                LR chi2(1)        =      31.96
                                                Prob > chi2       =     0.0000
Log likelihood = -29.054002                     Pseudo R2         =     0.3548

------------------------------------------------------------------------------
     foreign |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      weight |  -.0025874   .0006094    -4.25   0.000    -.0037817    -.001393
       _cons |   6.282599   1.603967     3.92   0.000     3.138882    9.426316
------------------------------------------------------------------------------

. local logistic_coefficient = _b[weight]

. local logistic_constant = _b[_cons]

. margins, at(weight = (2000 4000))

Adjusted predictions                            Number of obs     =         74
Model VCE    : OIM

Expression   : Pr(foreign), predict()

1._at        : weight          =        2000

2._at        : weight          =        4000

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         _at |
          1  |   .7517233   .0897134     8.38   0.000     .5758882    .9275583
          2  |   .0168411   .0153552     1.10   0.273    -.0132545    .0469367
------------------------------------------------------------------------------

. 
. probit foreign weight

Iteration 0:   log likelihood =  -45.03321  
Iteration 1:   log likelihood = -29.534424  
Iteration 2:   log likelihood = -28.912832  
Iteration 3:   log likelihood = -28.908406  
Iteration 4:   log likelihood = -28.908406  

Probit regression                               Number of obs     =         74
                                                LR chi2(1)        =      32.25
                                                Prob > chi2       =     0.0000
Log likelihood = -28.908406                     Pseudo R2         =     0.3581

------------------------------------------------------------------------------
     foreign |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
      weight |  -.0015049   .0003265    -4.61   0.000    -.0021447   -.0008651
       _cons |   3.655625   .8775791     4.17   0.000     1.935601    5.375648
------------------------------------------------------------------------------

. local probit_coefficient = _b[weight]

. local probit_constant = _b[_cons]

. margins, at(weight = (2000 4000))

Adjusted predictions                            Number of obs     =         74
Model VCE    : OIM

Expression   : Pr(foreign), predict()

1._at        : weight          =        2000

2._at        : weight          =        4000

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         _at |
          1  |   .7408037   .0899293     8.24   0.000     .5645455    .9170619
          2  |     .00904   .0118703     0.76   0.446    -.0142254    .0323055
------------------------------------------------------------------------------

. 
. display `logistic_coefficient'/`probit_coefficient'
1.7193088

. display `logistic_constant'/`probit_constant'
1.7186116

. display c(pi)/sqrt(3)
1.8137994