which is the correct way to do three-way interaction

[Alice Yang]

Dear statalist,

I have 3 variables, say x, y, and z. x and y are dummies, z is continuous. I want to do a three-way interaction and I wonder which is the correct way? My variable of interest is x*y*z, and I expect it to be significantly positive. As you can see from the results, one methodology shows it is significantly positively associated with my DV, while the other has insignificant relationship.

reg DV i.x#i.y#c.z control1 control2 control3 i.industry i.year, vce(cluster industry)
-------------------------------------------------------------------------------------------------
                                |               Robust
                             DV |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------------------------+----------------------------------------------------------------
                       x#y#c.z  |
                           0 0  |   .3318569   .0373612     8.88   0.000     .2530317     .410682
                           0 1  |   .1726281   .0372657     4.63   0.000     .0940044    .2512519
                           1 0  |   .1279661   .0340517     3.76   0.002     .0561234    .1998088
                           1 1  |   .0865039   .0337273     2.56   0.020     .0153455    .1576623

reg DV i.x##i.y##c.z control1 control2 control3 i.industry i.year, vce(cluster industry)
-------------------------------------------------------------------------------------------------
                                |               Robust
                             DV |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
--------------------------------+----------------------------------------------------------------
                           1.x  |   .0992311   .0556105     1.78   0.092    -.0180967     .216559
                           1.y  |   .0987155   .0266313     3.71   0.002     .0425284    .1549026
                                |
                           x#y  |
                           1 1  |  -.0726428   .0677343    -1.07   0.298    -.2155498    .0702641
                                |
                             z  |   .1693954   .0403849     4.19   0.001     .0841908       .2546
                                |
                         x#c.z  |
                             1  |    .004427    .097611     0.05   0.964    -.2015142    .2103682
                                |
                         y#c.z  |
                             1  |   .0496641   .0587866     0.84   0.410    -.0743649     .173693
                                |
                       x#y#c.z  |
                           1 1  |  -.0344931     .16652    -0.21   0.838    -.3858197    .3168335

Thanks a lot for any help : )

[Carlo Lazzaro]

Alice:

 
 reg DV i.x##i.y##c.z control1 control2 control3 i.industry i.year, vce(cluster industry)

is the way to go.
It remains to be seen whether or not challenging yourself and your audience with a three-way interaction is a good choice.

[Alice Yang]

Hi Carlo,

Thanks for your reply. I also find 3 way interaction hard to explain, given it is insignificant, I'll probably think of other ways.

[Carlo Lazzaro]

Alice:
usually, I try to stop at two-way interactions (whose results are often difficult to disseminate, though).

[Bruce Weaver]

In some fields (e.g., experimental psychology), three-way interactions are not all that uncommon. And especially when one (or more) of the variables is/are dichotomous, it is not all that difficult to explain the nature of a significant interaction. E.g., if A and B are metric variables and C is dichotomous, and if the primary interest is in the A*B interaction, a significant 3-way interaction means that that the nature of the A*B interaction depends on the level of C.

Several chapters in IVRMUS (2e) by Michael Mitchell have good examples of how to decompose various types of 3-way interactions (see below).

HTH.

6 Continuous by continuous by continuous interactions

6.1 Chapter overview
6.2 Overview
6.3 Examples using the GSS data 6.3.1 A model without a three-way interaction
6.3.2 A three-way interaction model
6.4 Summary


9 Categorical by categorical by categorical interactions

9.1 Chapter overview
9.2 Two by two by two models 9.2.1 Simple interactions by season
9.2.2 Simple interactions by depression status
9.2.3 Simple effects
9.3 Two by two by three models 9.3.1 Simple interactions by depression status
9.3.2 Simple partial interaction by depression status
9.3.3 Simple contrasts
9.3.4 Partial interactions
9.4 Three by three by three models and beyond 9.4.1 Partial interactions and interaction contrasts
9.4.2 Simple interactions
9.4.3 Simple effects and simple comparisons
9.5 Summary


13 Continuous by continuous by categorical interactions

13.1 Chapter overview
13.2 Linear by linear by categorical interactions 13.2.1 Fitting separate models for males and females
13.2.2 Fitting a combined model for males and females
13.2.3 Interpreting the interaction focusing in the age slope
13.2.4 Interpreting the interaction focusing on the educ slope
13.2.5 Estimating and comparing adjusted means by gender
13.3 Linear by quadratic by categorical interactions 13.3.1 Fitting separate models for males and females
13.3.2 Fitting a common model for males and females
13.3.3 Interpreting the interaction
13.3.4 Estimating and comparing adjusted means by gender
13.4 Summary


14 Continuous by categorical by categorical interactions

14.1 Chapter overview
14.2 Simple effects of gender on the age slope
14.3 Simple effects of education on the age slope
14.4 Simple contrasts on education for the age slope
14.5 Partial interaction on education for the age slope
14.6 Summary

[Alice Yang]

Thanks Bruce, for the useful resources!