Continuous interaction variables and margins

[Christopher Hartwell]

Hi all,

I've got an analysis set-up that includes several interaction variables, all of which are continuous but some of which are bounded (i.e. one variable is between 0 and 1). I've been searching for a way in stata to garner marginal effects, but the "margins" command has been throwing up categorical/factor blocks left and right. I've searched high and low and cannot find if the margins command can be used to analyze marginal effects of two continuous variables.

To give a sense of what I'm doing, I have institutional variables and production of good X interacted, and I want to show that the marginal effects of my Y variable change in the presence of the institutional variable and the production of good X. That is, does the institutional variable change the effect that good X and my Y variable show and by how much.
My question is, how can one ascertain marginal effects using stata for an interaction variable containing two continuous variables? And does the fact of bounding one of the variables change the interaction?

[Carlo Lazzaro]

Chistopher.
as per FAQ, please post what you typed and what Stata gave you back. Thanks.

[Phil Bromiley]

If you have continuous variables, you need to run the interactions with c. before the variables. Once you do that, it should work just fine. Bounds on a variable don't matter to Stata (but might be important for interpretation). If c. doesn't fix everything, then provide what you typed what Stata gave back.

[Clyde Schechter]

Something like this?

. sysuse auto, clear
(1978 Automobile Data)

. regress price c.mpg##c.headroom

      Source |       SS           df       MS      Number of obs   =        74
-------------+----------------------------------   F(3, 70)        =      7.14
       Model |   148844287         3  49614762.3   Prob > F        =    0.0003
    Residual |   486221109        70  6946015.85   R-squared       =    0.2344
-------------+----------------------------------   Adj R-squared   =    0.2016
       Total |   635065396        73  8699525.97   Root MSE        =    2635.5

----------------------------------------------------------------------------------
           price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-----------------+----------------------------------------------------------------
             mpg |  -42.41371   273.6704    -0.15   0.877     -588.232    503.4046
        headroom |   1224.462   1963.955     0.62   0.535    -2692.523    5141.446
                 |
c.mpg#c.headroom |    -76.377   94.22537    -0.81   0.420    -264.3036    111.5496
                 |
           _cons |   8119.723   6001.821     1.35   0.180    -3850.533    20089.98
----------------------------------------------------------------------------------

. margins, dydx(mpg) at(mpg = (20(10)40) headroom = (2(1)5))

Conditional marginal effects                    Number of obs     =         74
Model VCE    : OLS

Expression   : Linear prediction, predict()
dy/dx w.r.t. : mpg

1._at        : mpg             =          20
               headroom        =           2

2._at        : mpg             =          20
               headroom        =           3

3._at        : mpg             =          20
               headroom        =           4

4._at        : mpg             =          20
               headroom        =           5

5._at        : mpg             =          30
               headroom        =           2

6._at        : mpg             =          30
               headroom        =           3

7._at        : mpg             =          30
               headroom        =           4

8._at        : mpg             =          30
               headroom        =           5

9._at        : mpg             =          40
               headroom        =           2

10._at       : mpg             =          40
               headroom        =           3

11._at       : mpg             =          40
               headroom        =           4

12._at       : mpg             =          40
               headroom        =           5

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
mpg          |
         _at |
          1  |  -195.1677   98.24459    -1.99   0.051    -391.1104    .7749436
          2  |  -271.5447   60.54361    -4.49   0.000    -392.2951   -150.7943
          3  |  -347.9217   124.2413    -2.80   0.007    -595.7132   -100.1303
          4  |  -424.2987   212.0451    -2.00   0.049    -847.2093    -1.38817
          5  |  -195.1677   98.24459    -1.99   0.051    -391.1104    .7749436
          6  |  -271.5447   60.54361    -4.49   0.000    -392.2951   -150.7943
          7  |  -347.9217   124.2413    -2.80   0.007    -595.7132   -100.1303
          8  |  -424.2987   212.0451    -2.00   0.049    -847.2093    -1.38817
          9  |  -195.1677   98.24459    -1.99   0.051    -391.1104    .7749436
         10  |  -271.5447   60.54361    -4.49   0.000    -392.2951   -150.7943
         11  |  -347.9217   124.2413    -2.80   0.007    -595.7132   -100.1303
         12  |  -424.2987   212.0451    -2.00   0.049    -847.2093    -1.38817
------------------------------------------------------------------------------

.

And you could do a similar command to get marginal effects of headroom at specified values of mpg and headroom.

[Erika Gonzalez]

This thread have been quite useful to me. But I have a question, what do the P values in the dy/dx table mean?

[Clyde Schechter]

They test the null hypotheses that the corresponding marginal effects are zero. For example, since 1.at corresponds to mpg = 20 and headroom = 2, the first row of

 ------------------------------------------------------------------------------              |            Delta-method              |      dy/dx   Std. Err.      t    P>|t|     [95% Conf. Interval] -------------+---------------------------------------------------------------- mpg          |          _at |           1  |  -195.1677   98.24459    -1.99   0.051    -391.1104    .7749436           2  |  -271.5447   60.54361    -4.49   0.000    -392.2951   -150.7943           3  |  -347.9217   124.2413    -2.80   0.007    -595.7132   -100.1303           4  |  -424.2987   212.0451    -2.00   0.049    -847.2093    -1.38817           5  |  -195.1677   98.24459    -1.99   0.051    -391.1104    .7749436           6  |  -271.5447   60.54361    -4.49   0.000    -392.2951   -150.7943           7  |  -347.9217   124.2413    -2.80   0.007    -595.7132   -100.1303           8  |  -424.2987   212.0451    -2.00   0.049    -847.2093    -1.38817           9  |  -195.1677   98.24459    -1.99   0.051    -391.1104    .7749436          10  |  -271.5447   60.54361    -4.49   0.000    -392.2951   -150.7943          11  |  -347.9217   124.2413    -2.80   0.007    -595.7132   -100.1303          12  |  -424.2987   212.0451    -2.00   0.049    -847.2093    -1.38817 ------------------------------------------------------------------------------

tells us that the marginal effect of mpg on price, conditional on mpg = 20 and headroom = 2, estimated at -195.1677, is not statistically significant at the 0.05 level (although it is a very near miss).

I hasten to point out that null hypothesis tests about the values of such highly conditional marginal effects are often of no interest, unless they are explicitly part of the research goals. Usually these -margins, dydx()- outputs for continuous by continuous interactions are generated with the intent to graph them so as to get an overall understanding of the model.

Added: Somehow the code box above got mangled. I don't have the time and patience to fix it just now. Suffice it to say that it was just a copy of the -margins, dydx()- output shown in #4.

[Mahnoor Sohail]

schoolenroll=male+heightforagezscore+(heightforage *male)
school enroll is a binary variable. Height for age is a continuous var.

i want the effect of height of girls vs boys on the school enrollment.

prob schoolenroll i.male HeightZscore i.male#c.HeightZscore
margins,dydx ( HeightZscoreP) at (male=(1 0)) atmeans

Clyde Schechter can you pls check is this is correct

[Dart Stater]

Dear Clyde, maybe you find this question interesting and make some advice

We analyse how the radiation dose affects the binary outcome (cancer) using logit with a mixture of predictors interacting each other.
The effect of radiation dose is of primary interest, but we suggest the interaction of the dose with sex and age at exposure. So the Stata command might be written as

logit cr c.dose##sex c.dose#c.age c.age##c.age sex#c.age

which gives us both "pure" dose component, and the dose# interactions with sex and the age along with the "pure" age and sex coeffs, as well.

This should be consistent with basic model (please correct if I'm wrong)

y(_cr=1)=const + b1*sex + b2*dose + b3*(sex*dose) + b4*age + b5*(age*age) + b6*(age*dose) + b7*(sex*age)

Should the mutual interactions of the dose, sex and age exist, can we use the following short equation to describe the modified effect of the dose?

logit cr c.age##sex c.dose#c.age#c.age#sex

What is (in theory) the difference between the dose#age#sex,and the dose#age + dose#sex + sex#age components?

[Clyde Schechter]

It is almost certainly a serious mis-specification of the model to include interaction of dose or sex with the linear age variable but not the quadratic component. In a quadratic model, the linear component by itself is fairly exotic in its interpretation and I have great difficulty imagining any circumstance where interaction with the linear component alone would make any sense. A valid model for your situation would be:

logit cr c.dose##c.age##c.age##i.sex

This will expand to include the interactions of both dose and sex with both linear and quadratic age terms, as well as the dose#sex interaction.

What is (in theory) the difference between the dose#age#sex,and the dose#age + dose#sex + sex#age components?

There is no difference between these. However, as already indicated, because the quadratic component of age is omitted, both versions are equally wrong.

[Dart Stater]

Dear Clyde, thank you for the clarification. Of no doubts, the polynomial age interaction must be included in the model.

Assuming that

There is no difference between these

the question is if the same model you provided

Code:

logit cr c.dose##c.age##c.age##i.sex

can be shown as the following:

 
 logit cr c.dose i.sex c.age c.age#c.age c.dose#i.sex c.dose#c.age c.dose#c.age#c.age i.sex#c.age i.sex#c.age#c.age

Here, we have all possible interactions accounted. However, the output (and the dose coeffs) will be quite different in both models.

I'm trying to interpret the sense of # and ## operators: How does the dose##sex and the dose#sex will be reflected in the general model

 y (cr) = const + b1*sex + b2*age + ... + bi*xk 

In the other words, what do # and ## mathematically do with the variables?

[Clyde Schechter]

logit cr c.dose i.sex c.age c.age#c.age c.dose#i.sex c.dose#c.age c.dose#c.age#c.age i.sex#c.age i.sex#c.age#c.age
// IS, INDEED, EQUIVALENT TO
logit cr c.dose##c.age##c.age##i.sex

However, the output (and the dose coeffs) will be quite different in both models.

No, they should not be different. Please show complete, exact output of both regressions and the exact code that was used to generate them. The order in which the coefficients are presented might be different, but the values of corresponding coefficients should be exactly the same.

How does the dose##sex and the dose#sex will be reflected in the general model

-dose##sex- is equivalent to -dose sex dose#sex-. In terms of the regression equation, dose#sex corresponds to the dose*sex term, whereas dose##sex corresponds to the dose, sex, and dose*sex terms.

[Dart Stater]

Dear Clyde, I acknowledge your help, thank you! The datasample has 10 000 individual records. The results are shown below:

Output of the 1st (short) model @Clyde

. logit cr c.dose##c.age##c.age##i.sex,or

Iteration 0:   log likelihood =  -4562.032  
Iteration 1:   log likelihood = -4216.0365  
Iteration 2:   log likelihood = -4148.5849  
Iteration 3:   log likelihood = -4141.1218  
Iteration 4:   log likelihood = -4140.9693  
Iteration 5:   log likelihood = -4140.9692  

Logistic regression                                     Number of obs = 10,000
                                                        LR chi2(11)   = 842.13
                                                        Prob > chi2   = 0.0000
Log likelihood = -4140.9692                             Pseudo R2     = 0.0923

----------------------------------------------------------------------------------------
                    cr | Odds ratio   Std. err.      z    P>|z|     [95% conf. interval]
-----------------------+----------------------------------------------------------------
                  dose |    1.20182   .0615702     3.59   0.000     1.087005    1.328762
                   age |   1.156728   .0241439     6.98   0.000     1.110362    1.205031
                       |
          c.dose#c.age |    .994604   .0017923    -3.00   0.003     .9910974    .9981231
                       |
           c.age#c.age |   .9991984    .000174    -4.60   0.000     .9988574    .9995395
                       |
    c.dose#c.age#c.age |   1.000043   .0000156     2.73   0.006     1.000012    1.000073
                       |
                 1.sex |   15.30515   11.94568     3.50   0.000     3.314928    70.66446
                       |
            sex#c.dose |
                    1  |   .7285395   .0752805    -3.07   0.002     .5949738    .8920893
                       |
             sex#c.age |
                    1  |    .926632   .0247154    -2.86   0.004     .8794352    .9763619
                       |
      sex#c.dose#c.age |
                    1  |   1.010385   .0034988     2.98   0.003     1.003551    1.017266
                       |
       sex#c.age#c.age |
                    1  |   1.000433   .0002241     1.93   0.053     .9999935    1.000872
                       |
sex#c.dose#c.age#c.age |
                    1  |   .9999203   .0000287    -2.78   0.005     .9998641    .9999765
                       |
                 _cons |   .0007362   .0004531   -11.72   0.000     .0002204    .0024598
----------------------------------------------------------------------------------------
Note: _cons estimates baseline odds.

! please note incredibly high Odds Ratio coeff at sex=1 (female)

and the output of the 2nd (long) model @me

logit cr c.dose i.sex c.age c.age#c.age c.dose#i.sex c.dose#c.age c.dose#c.age#c.age i.sex#c.age i.sex#c.age#c.age,or

Iteration 0:   log likelihood =  -4562.032  
Iteration 1:   log likelihood = -4219.3374  
Iteration 2:   log likelihood = -4154.2363  
Iteration 3:   log likelihood = -4147.4464  
Iteration 4:   log likelihood = -4147.3435  
Iteration 5:   log likelihood = -4147.3435  

Logistic regression                                     Number of obs = 10,000
                                                        LR chi2(9)    = 829.38
                                                        Prob > chi2   = 0.0000
Log likelihood = -4147.3435                             Pseudo R2     = 0.0909

------------------------------------------------------------------------------------
                cr | Odds ratio   Std. err.      z    P>|z|     [95% conf. interval]
-------------------+----------------------------------------------------------------
              dose |   1.090039   .0438321     2.14   0.032     1.007427    1.179424
             1.sex |   3.784193   2.479557     2.03   0.042     1.047687    13.66832
               age |   1.134298   .0211647     6.75   0.000     1.093565    1.176547
                   |
       c.age#c.age |   .9993472   .0001564    -4.17   0.000     .9990407    .9996538
                   |
        sex#c.dose |
                1  |   1.006671    .005437     1.23   0.218     .9960707    1.017384
                   |
      c.dose#c.age |   .9977421   .0013835    -1.63   0.103     .9950342    1.000457
                   |
c.dose#c.age#c.age |   1.000018   .0000118     1.55   0.122     .9999951    1.000041
                   |
         sex#c.age |
                1  |   .9688044   .0217323    -1.41   0.158     .9271327    1.012349
                   |
   sex#c.age#c.age |
                1  |   1.000093   .0001884     0.49   0.623     .9997236    1.000462
                   |
             _cons |   .0013602   .0007444   -12.06   0.000     .0004653     .003976
------------------------------------------------------------------------------------

[Clyde Schechter]

Sorry, my mistake in #11. I didn't read your extended model carefully enough. Compared to c.dose##c.age##c.age##i.sex, it is missing the following terms:

c.dose#c.age#i.sex and c.dose#c.age#c.age#i.sex.

Those omissions in the second model account for the difference. You need to decide which model is the one you want. In the ## model, dose is fully interact with combinations of age and sex, whereas in the extended model that omits the two terms shown above, dose is separately interacted with age and with sex. So, in the extended model, the extent to which that sex modifies dose effect is the same at all ages and the extent to which age modifies dose effect is the same in both sexes. In the ## model, the extent to which age modifies the dose effect does (i.e. can) differ across the sexes, and the extent to which sex modifies dose effect varies by age.

The ease with which it is possible to mistakenly omit terms like this, and the difficulty of fully understanding a long command with the separate terms written out, is one of the strongest arguments for using the ## notation in preference to the # notation in most circumstances.

[Dart Stater]

Well, Clyde, you're welcome to suggest the best ## model in this particular case. I'll test it and (I think it will be useful for readers) show the results.
Is that updated model correct?

logit cr c.dose##c.age##c.age##i.sex c.dose#c.age#i.sex c.dose#c.age#c.age#i.sex

Output:

 logit cr c.dose##c.age##c.age##i.sex c.dose#c.age#i.sex c.dose#c.age#c.age#i.sex

Iteration 0:   log likelihood =  -4562.032  
Iteration 1:   log likelihood = -4216.0365  
Iteration 2:   log likelihood = -4148.5849  
Iteration 3:   log likelihood = -4141.1218  
Iteration 4:   log likelihood = -4140.9693  
Iteration 5:   log likelihood = -4140.9692  

Logistic regression                                     Number of obs = 10,000
                                                        LR chi2(11)   = 842.13
                                                        Prob > chi2   = 0.0000
Log likelihood = -4140.9692                             Pseudo R2     = 0.0923

----------------------------------------------------------------------------------------
                    cr | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-----------------------+----------------------------------------------------------------
                  dose |    .183837   .0512308     3.59   0.000     .0834266    .2842475
                   age |   .1455955   .0208726     6.98   0.000     .1046861     .186505
                       |
          c.dose#c.age |  -.0054106    .001802    -3.00   0.003    -.0089425   -.0018787
                       |
           c.age#c.age |  -.0008019   .0001742    -4.60   0.000    -.0011433   -.0004606
                       |
    c.dose#c.age#c.age |   .0000425   .0000156     2.73   0.006      .000012    .0000731
                       |
                 1.sex |   2.728189   .7805008     3.50   0.000     1.198436    4.257943
                       |
            sex#c.dose |
                    1  |  -.3167135   .1033307    -3.07   0.002    -.5192378   -.1141891
                       |
             sex#c.age |
                    1  |  -.0761987   .0266723    -2.86   0.004    -.1284754    -.023922
                       |
      sex#c.dose#c.age |
                    1  |   .0103314   .0034629     2.98   0.003     .0035443    .0171185
                       |
       sex#c.age#c.age |
                    1  |   .0004327   .0002241     1.93   0.053    -6.46e-06    .0008718
                       |
sex#c.dose#c.age#c.age |
                    1  |  -.0000797   .0000287    -2.78   0.005    -.0001359   -.0000235
                       |
                 _cons |   -7.21395   .6154648   -11.72   0.000    -8.420239   -6.007661
----------------------------------------------------------------------------------------

[Clyde Schechter]

When you have c.dose##c.age##c.age##i.sex, you don't need to also mention c.dose#c.age#i.sex c.dose#c.age#c.age#i.sex: Stata will create those for you "for free." Whenever you use the ## operator, you automatically get the interaction term plus the "main" terms. When it's a 3-or-higher level interaction specified with ##s, you get the highest level interaction, all of the "main" terms, and all of the lower level interactions that are included.

It's not my place to tell you which model to use: you have to pick the model that corresponds to how you think the real data generating process works. That difference is not a technical coding matter, nor a statistical issue: it's a substantive difference in what you think is going on in the real world. I don't even know what your outcome variable cr is, so I couldn't begin to guess how it relates to those things. Even if I did know what cr is, chances are I do not know enough about it and whatever it is you are dosing, to guess how these things all inter-relate. But as the researcher in this area, you must have some understanding in this area that would lean you towards one model or the other. If you believe that the way in which the effect of dose depends on age is also sex-specific, then you want the full-blown -c.dose##c.age##c.age##i.sex- model. If you think that the way in which the of dose depends on age is the same for both sexes, even though sex itself modifies the dose effect, then you want a smaller model, which could be specified compactly with -c.dose##(c.age##c.age i.sex)-.