Inconsistency between odds ratios and marginals in growth curve model

[Nicola Pensiero]

Hi all,
I am attempting to study the gender gap in the life-course evolution of political interest using growth curve modelling (Stata 16.0). The problem is that I cannot reconcile the results of the regression (melogit) and the marginals (margins). In detail, the coefficient of sex##age is an odds ratio > 1, which suggests a steeper growth over time for women, whilst the marginals (margins) show that men have a steeper growth. I suspect this has to do with the fact that sex##age is computed at specific values of other control variables. Any ideas to explain this apparent contradiction? Thank you, Nic

melogit interest i.sex##c.age c.isced_parents##i.sex##c.age c.gender_attitudes##i.sex##c.age ib3.casmin##i.sex##c.age i.year_1  || pidp: age, diff or cov(unstr)

Third row: sex#c.age has a 1.18 odds ratio, implying women having a steeper growth.

melogit results

                     interest | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
--------------------------------------+----------------------------------------------------------------
                                  sex |
                              Female  |   .0797479   .1236744    -1.63   0.103     .0038165    1.666392
                                  age |   .9183841   .1291689    -0.61   0.545     .6971152    1.209885
                                      |
                            sex#c.age |
                              Female  |    1.17762   .2371548     0.81   0.417     .7935718    1.747529
                                      |
                        isced_parents |    1.10222   .0494202     2.17   0.030     1.009493    1.203466
                                      |
                  sex#c.isced_parents |
                              Female  |   1.169129    .074481     2.45   0.014     1.031895    1.324614
                                      |
                c.isced_parents#c.age |   1.006333   .0058582     1.08   0.278      .994916     1.01788
                                      |
            sex#c.isced_parents#c.age |
                              Female  |   .9871253   .0081886    -1.56   0.118     .9712057    1.003306
                                      |
                     gender_attitudes |   .7252891   .1985224    -1.17   0.241      .424155    1.240217
                                      |
               sex#c.gender_attitudes |
                              Female  |   .7648504   .2973847    -0.69   0.491     .3569582    1.638837
                                      |
             c.gender_attitudes#c.age |   1.031944   .0343485     0.94   0.345     .9667716    1.101511
                                      |
         sex#c.gender_attitudes#c.age |
                              Female  |   .9962939   .0485382    -0.08   0.939     .9055616    1.096117
                                      |
                               casmin |
                   Less than level 2  |   .0421242   .0207223    -6.44   0.000     .0160619    .1104753
                             Level 2  |   .1315831    .046803    -5.70   0.000     .0655292    .2642196
                Level 3 (vocational)  |   .4611131   .2711349    -1.32   0.188     .1456465    1.459872
       Higher education (vocational)  |   .2908482    .174043    -2.06   0.039     .0900138    .9397745
          Higher education (general)  |   .7632587   .3877851    -0.53   0.595      .281972    2.066034
                    Studying level 2  |   .7163233   .2019592    -1.18   0.237     .4122138    1.244789
                                      |
                           casmin#sex |
            Less than level 2#Female  |   7.644757   5.476532     2.84   0.005     1.877509    31.12757
                      Level 2#Female  |   3.890349   1.922813     2.75   0.006      1.47666    10.24936
         Level 3 (vocational)#Female  |   5.605182   4.573374     2.11   0.035     1.132605    27.73966
Higher education (vocational)#Female  |   4.526671   3.648537     1.87   0.061     .9326163    21.97126
   Higher education (general)#Female  |   1.093604   .7399108     0.13   0.895     .2903715    4.118757
             Studying level 2#Female  |    .903155   .3519893    -0.26   0.794      .420747    1.938669
                                      |
                         casmin#c.age |
                   Less than level 2  |   1.077859   .0765448     1.06   0.291     .9378067    1.238827
                             Level 2  |   1.133095   .0662414     2.14   0.033     1.010426    1.270656
                Level 3 (vocational)  |   1.041306   .0986981     0.43   0.669     .8647666    1.253885
       Higher education (vocational)  |   1.021982   .0774086     0.29   0.774     .8809886     1.18554
          Higher education (general)  |   1.087357   .0731461     1.24   0.213     .9530424    1.240601
                    Studying level 2  |      1.287    .343928     0.94   0.345     .7622719    2.172937
                                      |
                     casmin#sex#c.age |
            Less than level 2#Female  |   .8207434    .085831    -1.89   0.059     .6686382     1.00745
                      Level 2#Female  |   .7887889   .0659503    -2.84   0.005     .6695641    .9292432
         Level 3 (vocational)#Female  |   .7503382   .1027247    -2.10   0.036     .5737513    .9812743
Higher education (vocational)#Female  |   .8901057   .0908325    -1.14   0.254     .7287509    1.087186
   Higher education (general)#Female  |   .9380943   .0843286    -0.71   0.477     .7865548     1.11883
             Studying level 2#Female  |   1.175099   .4388004     0.43   0.666     .5652268    2.443017
                                      |
                               year_1 |
                                1995  |   .9087041   .2070425    -0.42   0.674     .5814087    1.420245
                                1998  |   1.209628   .2963029     0.78   0.437     .7484225    1.955046
                                2001  |   .9578809    .252838    -0.16   0.870     .5709948    1.606908
                                2004  |   1.029336   .2907509     0.10   0.918     .5917289     1.79057
                                2007  |   .7295872   .2233028    -1.03   0.303     .4004537    1.329236
                                2010  |   1.015031   .3713182     0.04   0.967     .4955568    2.079053
                                2013  |   .8386817   .3510156    -0.42   0.674     .3692679    1.904815
                                2016  |   .9891315   .5252397    -0.02   0.984     .3493467    2.800602
                                2019  |   1.528287   1.167477     0.56   0.579     .3419492    6.830434
                                      |
                                _cons |    .388696    .433558    -0.85   0.397     .0436682    3.459828
--------------------------------------+----------------------------------------------------------------
pidp                                  |
                              var(age)|   .0437968   .0064901                      .0327571    .0585571
                            var(_cons)|   8.008383   .7122384                      6.727316      9.5334
--------------------------------------+----------------------------------------------------------------
pidp                                  |
                        cov(age,_cons)|  -.1669749   .0457278    -3.65   0.000    -.2565997   -.0773501
-------------------------------------------------------------------------------------------------------
Note: Estimates are transformed only in the first equation.
Note: _cons estimates baseline odds (conditional on zero random effects).
LR test vs. logistic model: chi2(3) = 3362.19             Prob > chi2 = 0.0000

Note: LR test is conservative and provided only for reference.

The marginals, instead, show that man have a steeper growth (age is centred (age-16) so that 0=16).

 margins sex, at(age=(0(1)14))

Predictive margins    Number    of    obs     =    12,159
Model VCE    : OIM

Expression   : Marginal predicted mean,    predict()

1._at        : age             =    0

2._at        : age             =    1

3._at        : age             =    2

4._at        : age             =    3

5._at        : age             =    4

6._at        : age             =    5

7._at        : age             =    6

8._at        : age             =    7

9._at        : age             =    8

10._at       : age             =    9

11._at       : age             =    10

12._at       : age             =    11

13._at       : age             =    12

14._at       : age             =    13

15._at       : age             =    14

                
Delta-method
Margin   Std. Err.    z    P>z        [95% Conf.    Interval]
                
_at#sex 
1#Male     .2834477   .0173198    16.37   0.000        .2495015    .3173939
1#Female     .2224946   .0145547    15.29   0.000        .1939679    .2510212
2#Male     .2983585   .0161681    18.45   0.000        .2666696    .3300474
2#Female     .2256212   .0132417    17.04   0.000        .199668    .2515744
3#Male     .3147989   .0157898    19.94   0.000        .2838516    .3457463
3#Female     .2300119   .0126628    18.16   0.000        .2051932    .2548305
4#Male     .3327152   .0163698    20.32   0.000        .300631    .3647994
4#Female     .2357405    .013263    17.77   0.000        .2097455    .2617354
5#Male     .3518121   .0176267    19.96   0.000        .3172644    .3863598
5#Female     .2428226   .0150916    16.09   0.000        .2132437    .2724016
6#Male     .3716529   .0191098    19.45   0.000        .3341983    .4091075
6#Female      .251041   .0174754    14.37   0.000        .2167899    .2852922
7#Male     .3918779   .0206967    18.93   0.000        .3513131    .4324426
7#Female     .2600038   .0196706    13.22   0.000        .2214501    .2985575
8#Male     .4122473    .022445    18.37   0.000        .3682559    .4562387
8#Female     .2693942   .0214995    12.53   0.000        .2272559    .3115325
9#Male     .4325343   .0242777    17.82   0.000        .3849508    .4801178
9#Female     .2790412   .0230641    12.10   0.000        .2338364    .324246
10#Male     .4524836    .026015    17.39   0.000        .4014951    .5034721
10#Female      .288811   .0243535    11.86   0.000        .2410789    .336543
11#Male     .4718567   .0275408    17.13   0.000        .4178777    .5258358
11#Female      .298551   .0252958    11.80   0.000        .2489722    .3481299
12#Male     .4904753   .0288469    17.00   0.000        .4339363    .5470143
12#Female     .3081192   .0259088    11.89   0.000        .2573389    .3588995
13#Male     .5082308   .0299683    16.96   0.000        .449494    .5669676
13#Female     .3174089   .0262743    12.08   0.000        .2659122    .3689056
14#Male     .5250719    .030921    16.98   0.000        .4644678    .585676
14#Female     .3263432   .0264707    12.33   0.000        .2744617    .3782248
15#Male     .5409722   .0316926    17.07   0.000        .4788559    .6030885
15#Female     .3348735   .0265998    12.59   0.000        .2827388    .3870082

Dataset

* Example generated by -dataex-. To install: ssc install dataex
clear
input float interest double sex byte age float(isced_parents gender_attitudes casmin year_1)
. 2  0 15 3.0714285 7 1998
. 2  1 15 3.0714285 7 1998
. 2  2 15 3.0714285 3 1998
. 2  3 15 3.0714285 3 2001
0 2  4 15 3.0714285 3 2001
. 2  5 15 3.0714285 3 2001
0 2  6 15 3.0714285 . 2004
. 2  7 15 3.0714285 . 2004
0 2  8 15 3.0714285 . 2004
. 2  9 15 3.0714285 . 2007
. 2 10 15 3.0714285 . 2007
. 2 11 15 3.0714285 . 2007
. 2 12 15 3.0714285 . 2010
. 2 13 15 3.0714285 . 2010
. 2 14 15 3.0714285 . 2010
. 2  0 15 3.7142856 7 2001
0 2  1 15 3.7142856 7 2001
0 2  2 15 3.7142856 3 2001
0 2  3 15 3.7142856 3 2004
. 2  4 15 3.7142856 3 2004
end
label values sex bm_sex
label def bm_sex 2 "Female", modify
label values isced_parents isced_parents
label values casmin casmin
label def casmin 3 "Level 3 (general)", modify
label def casmin 7 "Studying level 2", modify

[Clyde Schechter]

The -margins- output you show gives you the predicted outcomes at various ages for men and women. You did not calculate the marginal effects. To do that you would run -margins sex, dydx(age) at(age = (0(1)14))-.

In logistic models, marginal effects do not have a simple relationship to the odds ratios because of the non-linearity of the logistic function. When starting from a smaller base, the same odds ratio will have a smaller associated marginal effect than it will if you start from a larger base (until you pass a base rate of 0.50, after which the associated marginal effect decreases again. Think about the logistic curve: it is steep in the model and quickly gets flat as you move to the tails. So you can't just look at odds ratios and say much about what the rates of change in the probability metric will look like.

[Nicola Pensiero]

Thanks Clyde for the explanation - very helpful as always. "So you can't just look at odds ratios and say much about what the rates of change in the probability metric will look like". Therefore - I would add - I need the margins command to understand the rate of change. I do not think there are better ways of doing it? Thanks, Nic

[Clyde Schechter]

I agree that to really understand this kind of model you need the -margins- command to understand the rate of change.

[Richard Williams]

Keep in mind, too, that there are around 10 interactions that involve sex and age, e.g.sex#c.age, sex#c.isced_parents#c.age, casmin#sex#c.age. They are all going to affect what you get with

margins sex, at(age=(0(1)14))

I can't imagine keeping the overall sex/age relationship straight without margins and without a graph.

The model may be correct, but it has so many interactions I would find it incredibly hard to explain. I would think about whether the model really needs to be that complicated. A parsimonious model that tells 99% of the story may be more useful than a far more complicated model that gets things right down to the 12th decimal place.

With large N like yours substantively trivial effects may still be statistically significant. BIC and AIC tests might help you identify models that are more parsimonious but still fine.

[Nicola Pensiero]

These are great points, thank you. The model with all possible interactions is indeed overly complicated (and it could be further complicated by adding age^2......) and I will present it along with more parsimonious ones to address potential criticisms from different perspectives. Thank you Clyde and Richard.