Implementing Mobility Contrast Model (MCM) in STATA
Luo, Liying. "Heterogeneous effects of intergenerational social mobility: An improved method and new evidence." American Sociological Review 87.1 (2022): 143-173.
MCM model supports generalized linear model. All the models are under sum-to-zero coding, which means that the omitted coefficient for a categorical variable equals to the negative sum of all the other levels. It is possible to use lincom in STATA to recover the omitted coefficients after model estimation (see below for examples using lincom instead of mcm).
When applying generalized linear models, the generated effects are on the transformed scale instead of the natural scale. For example, in logistics regression, the generated effects are on the scale of log odds instead of probability. Please pay attention to the interpretation when reading the results.
The below shows examples for linear models, linear models with weighting, linear models with clustering, and logitsitc regression models with MCM.
xi3: mcm dependentvar independentvars, origin() destination()
Data Source: National Youth Longitudinal Study (1979 cohort)
Note: luck143 is the outcome variable representing the Rotter score collected in 2014. paredef3 is parents' educational level and red14ef3 is respondent'd educational level.
* run linear model under sum-to-zero coding
xi3: reg luck143 e.paredef3*e.red14ef3
* extract origin effect, destination effect, immobility effect, and mobility effect
xi3: mcm luck143 e.paredef3*e.red14ef3, origin(paredef3) destination(red14ef3)
Output
. xi3: reg luck143 e.paredef3*e.red14ef3
. xi3: mcm luck143 e.paredef3*e.red14ef3, origin(paredef3) destination(red14ef3)
e.paredef3 _Iparedef3_1-3 (naturally coded; _Iparedef3_1 omitted)
e.red14ef3 _Ired14ef3_1-3 (naturally coded; _Ired14ef3_1 omitted)
(316 missing values generated)
(316 missing values generated)
(316 missing values generated)
(316 missing values generated)
Source | SS df MS Number of obs = 3,864
-------------+---------------------------------- F(8, 3855) = 10.24
Model | 14.3994348 8 1.79992935 Prob > F = 0.0000
Residual | 677.573391 3,855 .175764823 R-squared = 0.0208
-------------+---------------------------------- Adj R-squared = 0.0188
Total | 691.972826 3,863 .179128353 Root MSE = .41924
------------------------------------------------------------------------------
luck143 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_Iparedef3_2 | .0010803 .0167979 0.06 0.949 -.0318533 .0340139
_Iparedef3_3 | .043264 .0259892 1.66 0.096 -.0076899 .0942178
_Ired14ef3_2 | -.0265313 .0157429 -1.69 0.092 -.0573965 .0043339
_Ired14ef3_3 | .0557952 .0154307 3.62 0.000 .0255422 .0860483
_Ipa2Xre2 | -.0044744 .0188698 -0.24 0.813 -.0414701 .0325212
_Ipa2Xre3 | -.0174969 .0187949 -0.93 0.352 -.0543458 .0193521
_Ipa3Xre2 | -.0134211 .0285653 -0.47 0.638 -.0694258 .0425835
_Ipa3Xre3 | -.0044618 .0272711 -0.16 0.870 -.057929 .0490055
_cons | .7681171 .0141526 54.27 0.000 .7403698 .7958644
------------------------------------------------------------------------------
origin[3,3]
estimates Std Err P>|t|
Orig 1 -.04434426 .01562588 .00456528
Orig 2 .00108031 .01679789 .94872515
Orig 3 .04326395 .02598919 .09605579
destination[3,3]
estimates Std Err P>|t|
Desti 1 -.02926393 .026755 .27412235
Desti 2 -.02653132 .01574291 .09201419
Desti 3 .05579525 .01543068 .00030318
immobility[3,3]
estimates Std Err P>|t|
Orig 1 :Desti 1 -.11346242 .02471486 4.555e-06
Orig 2 :Desti 2 -.02992544 .01961592 .1271993
Orig 3 :Desti 3 .09459744 .02033632 3.403e-06
mobility_effect[3,3]
Desti 1 Desti 2 Desti 3
Orig 1 0 .05774979 .06181288
Orig 2 .02644575 0 -.01302246
Orig 3 .02234467 -.00895937 0
mobility_effect_se[3,3]
Desti 1 Desti 2 Desti 3
Orig 1 0 .04398869 .04455394
Orig 2 .04823156 0 .02085512
Orig 3 .07521516 .02502905 0
mobility_effect_p[3,3]
Desti 1 Desti 2 Desti 3
Orig 1 . .18931743 .16540863
Orig 2 .58351219 . .53238557
Orig 3 .76642391 .7203936 .
Alternatively, use lincom to extract main effects:
* origin
lincom -_Iparedef3_2 -_Iparedef3_3
lincom _Iparedef3_2
lincom _Iparedef3_3
* destinatioin
lincom -_Ired14ef3_2 -_Ired14ef3_3
lincom _Ired14ef3_2
lincom _Ired14ef3_3
Use lincom to extract immobility effects:
* origin == 1 & destination == 1
lincom -_Iparedef3_2-_Iparedef3_3 + _Ipa2Xre2 + _Ipa2Xre3 + _Ipa3Xre2 + _Ipa3Xre3
* origin == 2 & destination == 2
lincom _Iparedef3_2+_Ired14ef3_2 + _Ipa2Xre2
* origin == 3 & destination == 3
lincom _Iparedef3_3+_Ired14ef3_3 + _Ipa3Xre3
Output from lincom
. lincom -_Iparedef3_2-_Iparedef3_3
( 1) - _Iparedef3_2 - _Iparedef3_3 = 0
------------------------------------------------------------------------------
luck143 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) | -.0443443 .0156259 -2.84 0.005 -.07498 -.0137085
------------------------------------------------------------------------------
. lincom _Iparedef3_2
( 1) _Iparedef3_2 = 0
------------------------------------------------------------------------------
luck143 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) | .0010803 .0167979 0.06 0.949 -.0318533 .0340139
------------------------------------------------------------------------------
. lincom _Iparedef3_3
( 1) _Iparedef3_3 = 0
------------------------------------------------------------------------------
luck143 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) | .043264 .0259892 1.66 0.096 -.0076899 .0942178
------------------------------------------------------------------------------
. lincom -_Ired14ef3_2 -_Ired14ef3_3
( 1) - _Ired14ef3_2 - _Ired14ef3_3 = 0
------------------------------------------------------------------------------
luck143 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) | -.0292639 .026755 -1.09 0.274 -.0817192 .0231914
------------------------------------------------------------------------------
. lincom _Ired14ef3_2
( 1) _Ired14ef3_2 = 0
------------------------------------------------------------------------------
luck143 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) | -.0265313 .0157429 -1.69 0.092 -.0573965 .0043339
------------------------------------------------------------------------------
. lincom _Ired14ef3_3
( 1) _Ired14ef3_3 = 0
------------------------------------------------------------------------------
luck143 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) | .0557952 .0154307 3.62 0.000 .0255422 .0860483
------------------------------------------------------------------------------
. * origin == 1 & destination == 1
. lincom -_Iparedef3_2-_Iparedef3_3 - _Ired14ef3_2 - _Ired14ef3_3 + _Ipa2Xre2 + _Ipa2Xre3 + _Ipa3Xre2 + _Ipa3Xre3
( 1) - _Iparedef3_2 - _Iparedef3_3 - _Ired14ef3_2 - _Ired14ef3_3 + _Ipa2Xre2 + _Ipa2Xre3 + _Ipa3Xre2 + _Ipa3Xre3 = 0
------------------------------------------------------------------------------
luck143 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) | -.1134624 .0247149 -4.59 0.000 -.1619179 -.065007
------------------------------------------------------------------------------
. * origin == 2 & destination == 2
. lincom _Iparedef3_2+_Ired14ef3_2 + _Ipa2Xre2
( 1) _Iparedef3_2 + _Ired14ef3_2 + _Ipa2Xre2 = 0
------------------------------------------------------------------------------
luck143 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) | -.0299254 .0196159 -1.53 0.127 -.068384 .0085331
------------------------------------------------------------------------------
. * origin == 3 & destination == 3
. lincom _Iparedef3_3+_Ired14ef3_3 + _Ipa3Xre3
( 1) _Iparedef3_3 + _Ired14ef3_3 + _Ipa3Xre3 = 0
------------------------------------------------------------------------------
luck143 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
(1) | .0945974 .0203363 4.65 0.000 .0547265 .1344684
------------------------------------------------------------------------------
xi3: reg luck143 e.paredef3*e.red14ef3 [aw=weight]
xi3: mcm luck143 e.paredef3*e.red14ef3, origin(paredef3) destination(red14ef3)
Output
. xi3: reg luck143 e.paredef3*e.red14ef3 [aw=weight]
. xi3: mcm luck143 e.paredef3*e.red14ef3, origin(paredef3) destination(red14ef3)
e.paredef3 _Iparedef3_1-3 (naturally coded; _Iparedef3_1 omitted)
e.red14ef3 _Ired14ef3_1-3 (naturally coded; _Ired14ef3_1 omitted)
(316 missing values generated)
(316 missing values generated)
(316 missing values generated)
(316 missing values generated)
(sum of wgt is 1.1480e+09)
Source | SS df MS Number of obs = 3,864
-------------+---------------------------------- F(8, 3855) = 8.48
Model | 11.3259035 8 1.41573793 Prob > F = 0.0000
Residual | 643.263583 3,855 .166864743 R-squared = 0.0173
-------------+---------------------------------- Adj R-squared = 0.0153
Total | 654.589486 3,863 .169451071 Root MSE = .40849
------------------------------------------------------------------------------
luck143 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_Iparedef3_2 | .0031781 .015474 0.21 0.837 -.0271599 .0335161
_Iparedef3_3 | .0386188 .0226057 1.71 0.088 -.0057015 .0829392
_Ired14ef3_2 | -.0274769 .0142736 -1.93 0.054 -.0554615 .0005077
_Ired14ef3_3 | .0445164 .0142643 3.12 0.002 .0165501 .0724828
_Ipa2Xre2 | -.0073068 .0173625 -0.42 0.674 -.0413475 .0267338
_Ipa2Xre3 | -.0201436 .0175538 -1.15 0.251 -.0545593 .014272
_Ipa3Xre2 | -.0203045 .0249549 -0.81 0.416 -.0692305 .0286215
_Ipa3Xre3 | .0055419 .0239311 0.23 0.817 -.0413768 .0524607
_cons | .7771642 .0127686 60.87 0.000 .7521304 .802198
------------------------------------------------------------------------------
origin[3,3]
estimates Std Err P>|t|
Orig 1 -.04179696 .01509156 .00564009
Orig 2 .00317813 .015474 .83728185
Orig 3 .03861883 .02260573 .08764919
destination[3,3]
estimates Std Err P>|t|
Desti 1 -.01703957 .02389582 .47584109
Desti 2 -.02747687 .01427365 .05430194
Desti 3 .04451644 .01426435 .00181683
immobility[3,3]
estimates Std Err P>|t|
Orig 1 :Desti 1 -.10104952 .02848051 .00039276
Orig 2 :Desti 2 -.03160555 .01766992 .07374741
Orig 3 :Desti 3 .08867722 .01765602 5.332e-07
mobility_effect[3,3]
Desti 1 Desti 2 Desti 3
Orig 1 0 .06982428 .05681469
Orig 2 .03475727 0 -.01283684
Orig 3 .00922059 -.02584643 0
mobility_effect_se[3,3]
Desti 1 Desti 2 Desti 3
Orig 1 0 .04138926 .04287663
Orig 2 .04414669 0 .01980183
Orig 3 .06529925 .02280814 0
mobility_effect_p[3,3]
Desti 1 Desti 2 Desti 3
Orig 1 . .09168152 .18522532
Orig 2 .43114699 . .51685188
Orig 3 .88771532 .25719621 .
xi3: reg luck143 e.paredef3*e.red14ef3 [aw=weight], cluster(HHID)
xi3: mcm luck143 e.paredef3*e.red14ef3, origin(paredef3) destination(red14ef3)
xi3: logit luck143 e.paredef3*e.red14ef3
xi3: mcm luck143 e.paredef3*e.red14ef3, origin(paredef3) destination(red14ef3)
Output
. xi3: logit luck143 e.paredef3*e.red14ef3
. xi3: mcm luck143 e.paredef3*e.red14ef3, origin(paredef3) destination(red14ef3)
e.paredef3 _Iparedef3_1-3 (naturally coded; _Iparedef3_1 omitted)
e.red14ef3 _Ired14ef3_1-3 (naturally coded; _Ired14ef3_1 omitted)
(316 missing values generated)
(316 missing values generated)
(316 missing values generated)
(316 missing values generated)
Iteration 0: log likelihood = -2100.87
Iteration 1: log likelihood = -2060.4833
Iteration 2: log likelihood = -2059.7292
Iteration 3: log likelihood = -2059.7284
Iteration 4: log likelihood = -2059.7284
Logistic regression Number of obs = 3864
LR chi2(8) = 82.28
Prob > chi2 = 0.0000
Log likelihood = -2059.7284 Pseudo R2 = 0.0196
------------------------------------------------------------------------------
luck143 | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_Iparedef3_2 | -.0143869 .0977362 -0.15 0.883 -.2059463 .1771726
_Iparedef3_3 | .2546631 .1544784 1.65 0.099 -.048109 .5574352
_Ired14ef3_2 | -.1677565 .0917548 -1.83 0.068 -.3475926 .0120797
_Ired14ef3_3 | .3303276 .0918126 3.60 0.000 .1503782 .5102771
_Ipa2Xre2 | -.0068398 .1086452 -0.06 0.950 -.2197805 .2061008
_Ipa2Xre3 | -.1076951 .1107078 -0.97 0.331 -.3246785 .1092882
_Ipa3Xre2 | -.0960851 .1683443 -0.57 0.568 -.4260339 .2338637
_Ipa3Xre3 | .0274566 .1637476 0.17 0.867 -.2934829 .3483961
_cons | 1.225574 .0833271 14.71 0.000 1.062256 1.388892
------------------------------------------------------------------------------
origin[3,3]
estimates Std Err P>|t|
Orig 1 -.24027624 .09079903 .00817223
Orig 2 -.01438687 .09773621 .88298107
Orig 3 .2546631 .15447841 .09932437
destination[3,3]
estimates Std Err P>|t|
Desti 1 -.16257115 .15751794 .30209927
Desti 2 -.1677565 .09175482 .06757989
Desti 3 .33032765 .09181265 .00032489
immobility[3,3]
estimates Std Err P>|t|
Orig 1 :Desti 1 -.58601092 .1314324 8.483e-06
Orig 2 :Desti 2 -.18898321 .11124056 .08942444
Orig 3 :Desti 3 .61244733 .13110399 3.093e-06
mobility_effect[3,3]
Desti 1 Desti 2 Desti 3
Orig 1 0 .28608851 .26340208
Orig 2 .12137482 0 -.10085528
Orig 3 .04117197 -.12354172 0
mobility_effect_se[3,3]
Desti 1 Desti 2 Desti 3
Orig 1 0 .25379124 .26042904
Orig 2 .27903357 0 .1219254
Orig 3 .44871318 .14925759 0
mobility_effect_p[3,3]
Desti 1 Desti 2 Desti 3
Orig 1 . .25970306 .31188088
Orig 2 .66359921 . .40818145
Orig 3 .92689694 .40788705 .