Calculation of standard errors after predictive margins: how are they computed?

[Mike Smet]

Hello,

I'm struggling how standard errors after -margins- are calculated, e.g. after -logit-. I am able to reproduce the productive margins themselves, but I am not able to derive the standard errors. They don't seem to be in line with the standard deviations of the predicted probabilities ... See example below

sysuse auto
logit foreign price mpg weight length
* Manually predict probabilities at various levels of weight
generate p1500 = invlogit(_b[weight]*1500 + _b[price]*price +_b[mpg]*mpg + _b[length]*length + _b[_cons])
generate p2000 = invlogit(_b[weight]*2000 + _b[price]*price +_b[mpg]*mpg + _b[length]*length + _b[_cons])
generate p2500 = invlogit(_b[weight]*2500 + _b[price]*price +_b[mpg]*mpg + _b[length]*length + _b[_cons])
generate p3000 = invlogit(_b[weight]*3000 + _b[price]*price +_b[mpg]*mpg + _b[length]*length + _b[_cons])
generate p3500 = invlogit(_b[weight]*3500 + _b[price]*price +_b[mpg]*mpg + _b[length]*length + _b[_cons])
generate p4000 = invlogit(_b[weight]*4000 + _b[price]*price +_b[mpg]*mpg + _b[length]*length + _b[_cons])
generate p4500 = invlogit(_b[weight]*4500 + _b[price]*price +_b[mpg]*mpg + _b[length]*length + _b[_cons])
generate p5000 = invlogit(_b[weight]*5000 + _b[price]*price +_b[mpg]*mpg + _b[length]*length + _b[_cons])
* Calculating predictive margins using margins command
margins, at ( weight =(1500(500)5000))

And here is (part of) the output:

Logistic regression                             Number of obs     =         74
                                                LR chi2(4)        =      55.94
                                                Prob > chi2       =     0.0000
Log likelihood = -17.064729                     Pseudo R2         =     0.6211

------------------------------------------------------------------------------
     foreign |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       price |   .0009392   .0003093     3.04   0.002      .000333    .0015454
         mpg |  -.1155925   .0966509    -1.20   0.232    -.3050248    .0738398
      weight |  -.0078002   .0030342    -2.57   0.010    -.0137471   -.0018534
      length |   .0387482   .0875022     0.44   0.658    -.1327529    .2102493
       _cons |   9.883036   11.26217     0.88   0.380    -12.19042    31.95649
------------------------------------------------------------------------------

. * Calculating predictive margins using margins command
. margins, at ( weight =(1500(500)5000))

Predictive margins                              Number of obs     =         74
Model VCE    : OIM

Expression   : Pr(foreign), predict()

1._at        : weight          =        1500
2._at        : weight          =        2000
3._at        : weight          =        2500
4._at        : weight          =        3000
5._at        : weight          =        3500
6._at        : weight          =        4000
7._at        : weight          =        4500
8._at        : weight          =        5000

------------------------------------------------------------------------------
             |            Delta-method
             |     Margin   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         _at |
          1  |   .9978147   .0039638   251.73   0.000     .9900458    1.005584
          2  |   .9230255   .0393331    23.47   0.000     .8459341    1.000117
          3  |   .5344313   .1674495     3.19   0.001     .2062364    .8626262
          4  |   .2003275     .07624     2.63   0.009     .0508998    .3497552
          5  |   .0874817   .0298857     2.93   0.003     .0289067    .1460567
          6  |   .0129882   .0135501     0.96   0.338    -.0135694    .0395459
          7  |   .0003349   .0007068     0.47   0.636    -.0010504    .0017201
          8  |   6.82e-06   .0000234     0.29   0.771    -.0000391    .0000527
------------------------------------------------------------------------------

Below is the summary of manual predictions: the means are identical to what is predicted by -margins-
However I cannot recreate the standard error of -margins-, e.g. Std. Dev. of manual predections at
values of weight of 2500 and 3000 are close to each other (.3565795 and .3427131), while the
Std. Err. produced by -margins- differs a lot for these values (.1674495 and .07624 )

. sum p1500-p5000

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
       p1500 |         74    .9978147    .0039419   .9830204          1
       p2000 |         74    .9230255    .1224789   .5395426   .9999996
       p2500 |         74    .5344313    .3565795   .0231664   .9999796
       p3000 |         74    .2003275    .3427131   .0004798   .9989911
       p3500 |         74    .0874817    .2395264   9.71e-06    .952476
-------------+---------------------------------------------------------
       p4000 |         74    .0129882    .0509261   1.97e-07   .2885812
       p4500 |         74    .0003349     .001379   3.98e-09   .0081432
       p5000 |         74    6.82e-06    .0000281   8.05e-11   .0001661

Is there anyone who can help on this?

Thanks a lot,
MIke

[Rich Goldstein]

in the manual (r.pdf) which you have available to you, there is a section on methods and formulas and a subsection on SE's - see if that has what you want

[Mike Smet]

Thank you for your suggestion: there is indeed a section on methods and formulas dealing with how to estimate the variance of the predicted probabilities. I assume you refer the section in the screenshot below.
However, I'm still unable to figure out how the standard errors at different values of a covariate are calculated ...