Two-stage predictor substitution estimators

Read in binary outcome data; y outcome, x exposure, w covariate, z* instrumental variables (genotypes).

use ivbinoutdata, clear

Fit the model with a single instrumental variable.

ivtsps y (x = z1)
Final GMM criterion Q(b) =  1.76e-32

note: model is exactly identified.

GMM estimation 

Number of parameters =   4
Number of moments    =   4
Initial weight matrix: Unadjusted                 Number of obs   =      2,500

------------------------------------------------------------------------------
             |               Robust
             | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
    /s1xb_z1 |   .9318626   .0711029    13.11   0.000     .7925034    1.071222
         /a0 |   1.456817   .0466934    31.20   0.000       1.3653    1.548335
         /b1 |   .1433391   .0139152    10.30   0.000     .1160658    .1706123
         /b0 |   .2817003   .0268135    10.51   0.000     .2291468    .3342537
------------------------------------------------------------------------------
Instruments for equation 1: z1 _cons
Instruments for equation 2: predicted _cons

Causal risk difference for: b1

 ( 1)  [b1]_cons = 0

------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
         (1) |   .1433391   .0139152    10.30   0.000     .1160658    .1706123
------------------------------------------------------------------------------

Fit the model with multiple instruments.

ivtsps y (x = z1 z2 z3)
Final GMM criterion Q(b) =  6.65e-33

note: model is exactly identified.

GMM estimation 

Number of parameters =   6
Number of moments    =   6
Initial weight matrix: Unadjusted                 Number of obs   =      2,500

------------------------------------------------------------------------------
             |               Robust
             | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
    /s1xb_z1 |   .9945859   .0614877    16.18   0.000     .8740722      1.1151
    /s1xb_z2 |   .9540576   .0525572    18.15   0.000     .8510474    1.057068
    /s1xb_z3 |   .9554229   .0495052    19.30   0.000     .8583944    1.052451
         /a0 |   .0842244   .0675928     1.25   0.213    -.0482551    .2167039
         /b1 |   .1307856   .0067841    19.28   0.000     .1174891    .1440822
         /b0 |   .3046021   .0148061    20.57   0.000     .2755826    .3336216
------------------------------------------------------------------------------
Instruments for equation 1: z1 z2 z3 _cons
Instruments for equation 2: predicted _cons

Causal risk difference for: b1

 ( 1)  [b1]_cons = 0

------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
         (1) |   .1307856   .0067841    19.28   0.000     .1174891    .1440822
------------------------------------------------------------------------------

Fit the model with multiple instruments, and adjusting for w.

ivtsps y w (x = z1 z2 z3)
Final GMM criterion Q(b) =  1.27e-32

note: model is exactly identified.

GMM estimation 

Number of parameters =   8
Number of moments    =   8
Initial weight matrix: Unadjusted                 Number of obs   =      2,500

------------------------------------------------------------------------------
             |               Robust
             | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
    /s1xb_z1 |   .9882395   .0505573    19.55   0.000     .8891491     1.08733
    /s1xb_z2 |   .9152859   .0431149    21.23   0.000     .8307823    .9997895
    /s1xb_z3 |    .954388   .0404106    23.62   0.000     .8751847    1.033591
     /s1xb_w |   .9801363    .027334    35.86   0.000     .9265627     1.03371
         /a0 |   .0734396   .0541988     1.36   0.175    -.0327882    .1796673
         /b1 |    .129698    .006901    18.79   0.000     .1161722    .1432237
 /s2exogxb_w |   .0638726   .0103505     6.17   0.000     .0435859    .0841593
         /b0 |   .3041477   .0145428    20.91   0.000     .2756443    .3326511
------------------------------------------------------------------------------
Instruments for equation 1: z1 z2 z3 w _cons
Instruments for equation 2: predicted w _cons

Causal risk difference for: b1

 ( 1)  [b1]_cons = 0

------------------------------------------------------------------------------
             | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
         (1) |    .129698    .006901    18.79   0.000     .1161722    .1432237
------------------------------------------------------------------------------

Using the log additive link function.

ivtsps y (x = z1 z2 z3), link(logadd)
Final GMM criterion Q(b) =  8.95e-33

note: model is exactly identified.

GMM estimation 

Number of parameters =   6
Number of moments    =   6
Initial weight matrix: Unadjusted                 Number of obs   =      2,500

------------------------------------------------------------------------------
             |               Robust
             | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
    /s1xb_z1 |   .9945859   .0614877    16.18   0.000     .8740722      1.1151
    /s1xb_z2 |   .9540576   .0525572    18.15   0.000     .8510474    1.057068
    /s1xb_z3 |   .9554229   .0495052    19.30   0.000     .8583944    1.052451
         /a0 |   .0842244   .0675928     1.25   0.213    -.0482551    .2167039
         /b1 |   .2309625   .0125342    18.43   0.000      .206396     .255529
         /b0 |  -1.061422   .0336872   -31.51   0.000    -1.127447    -.995396
------------------------------------------------------------------------------
Instruments for equation 1: z1 z2 z3 _cons
Instruments for equation 2: predicted _cons

Causal risk ratio for: b1

 ( 1)  [b1]_cons = 0

------------------------------------------------------------------------------
             |     exp(b)   Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
         (1) |   1.259812   .0157907    18.43   0.000      1.22924    1.291145
------------------------------------------------------------------------------

Using the log multiplicative link function.

ivtsps y (x = z1 z2 z3), link(logmult)
Final GMM criterion Q(b) =  1.10e-32

note: model is exactly identified.

GMM estimation 

Number of parameters =   6
Number of moments    =   6
Initial weight matrix: Unadjusted                 Number of obs   =      2,500

------------------------------------------------------------------------------
             |               Robust
             | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
    /s1xb_z1 |   .9945859   .0614877    16.18   0.000     .8740722      1.1151
    /s1xb_z2 |   .9540576   .0525572    18.15   0.000     .8510474    1.057068
    /s1xb_z3 |   .9554229   .0495052    19.30   0.000     .8583944    1.052451
         /a0 |   .0842244   .0675928     1.25   0.213    -.0482551    .2167039
         /b1 |   .2410471   .0158072    15.25   0.000     .2100656    .2720286
         /b0 |  -1.081193   .0390212   -27.71   0.000    -1.157673   -1.004713
------------------------------------------------------------------------------
Instruments for equation 1: z1 z2 z3 _cons
Instruments for equation 2: predicted _cons

Causal risk ratio for: b1

 ( 1)  [b1]_cons = 0

------------------------------------------------------------------------------
             |     exp(b)   Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
         (1) |   1.272581   .0201159    15.25   0.000     1.233759    1.312625
------------------------------------------------------------------------------

Using the logit link function.

ivtsps y (x = z1 z2 z3), link(logit)
Final GMM criterion Q(b) =  6.69e-33

note: model is exactly identified.

GMM estimation 

Number of parameters =   6
Number of moments    =   6
Initial weight matrix: Unadjusted                 Number of obs   =      2,500

------------------------------------------------------------------------------
             |               Robust
             | Coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
    /s1xb_z1 |   .9945859   .0614877    16.18   0.000     .8740722      1.1151
    /s1xb_z2 |   .9540576   .0525572    18.15   0.000     .8510474    1.057068
    /s1xb_z3 |   .9554229   .0495052    19.30   0.000     .8583944    1.052451
         /a0 |   .0842244   .0675928     1.25   0.213    -.0482551    .2167039
         /b1 |   .5731411   .0351357    16.31   0.000     .5042763    .6420059
         /b0 |  -.8527558   .0687178   -12.41   0.000    -.9874403   -.7180713
------------------------------------------------------------------------------
Instruments for equation 1: z1 z2 z3 _cons
Instruments for equation 2: predicted _cons

Causal odds ratio for: b1

 ( 1)  [b1]_cons = 0

------------------------------------------------------------------------------
             |     exp(b)   Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
         (1) |    1.77383   .0623248    16.31   0.000     1.655787    1.900289
------------------------------------------------------------------------------

Bootstrap standard errors.

bootstrap, reps(250): ivtsps y (x = z1 z2 z3), estonly
(running ivtsps on estimation sample)

Bootstrap replications (250): .........10.........20.........30.........40.........50.........60.........70.........80.........90.........100.........110..
> .......120.........130.........140.........150.........160.........170.........180.........190.........200.........210.........220.........230.........24
> 0.........250 done

------------------------------------------------------------------------------
             |   Observed   Bootstrap                         Normal-based
           y | coefficient  std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
   predicted |   .1307856   .0064668    20.22   0.000     .1181109    .1434604
       _cons |   .3046021   .0147641    20.63   0.000      .275665    .3335392
------------------------------------------------------------------------------