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First off, let's start with what a significant continuous by continuous interaction means. It means that the slope of one continuous variable on the response variable changes as the values on a second continuous change.
Multiple regression models often contain interaction terms. This FAQ page covers the situation in which there is a moderator variable which influences the regression of the dependent variable on an independent/predictor variable. In other words, a regression model that has a significant two-way interaction of continuous variables.
There are several approaches that one might use to explain an interaction of two continuous variables. The approach that we will demonstrate is to compute simple slopes, i.e., the slopes of the dependent variable on the independent variable when the moderator variable is held constant at different combinations of values from very low to very high.
We will consider a regression model which includes a continuous by continuous interaction of a predictor variable with a moderator variable. In the formula, Y is the response variable, X the predictor (independent) variable with Z being the moderator variable. The term XZ is the interaction of the predictor with the moderator.
Y = b0 + b1X + b2Z + b3XZ
Then, after creating the interaction term, we will run the regression model.
use http://www.ats.ucla.edu/stat/data/hsbdemo, clear
rename read y
rename math x
rename socst z
/* get the high and low values for x */
sum x
global max = r(max)
global min = r(min)
regress y c.x##c.z
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 3, 196) = 78.61
Model | 11424.7622 3 3808.25406 Prob > F = 0.0000
Residual | 9494.65783 196 48.4421318 R-squared = 0.5461
-------------+------------------------------ Adj R-squared = 0.5392
Total | 20919.42 199 105.122714 Root MSE = 6.96
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x | -.1105123 .2916338 -0.38 0.705 -.6856552 .4646307
z | -.2200442 .2717539 -0.81 0.419 -.7559812 .3158928
|
c.x#c.z | .0112807 .0052294 2.16 0.032 .0009677 .0215938
|
_cons | 37.84271 14.54521 2.60 0.010 9.157506 66.52792
------------------------------------------------------------------------------Next, we compute the slope for y on x while holding the value of the moderator variable, z, constant at values running from 30 to 75. To do this we will use the margins command with a range of values for z.
margins, dydx(x) at(z=(30(5)75)) vsquish
Average marginal effects Number of obs = 200
Model VCE : OLS
Expression : Linear prediction, predict()
dy/dx w.r.t. : x
1._at : z = 30
2._at : z = 35
3._at : z = 40
4._at : z = 45
5._at : z = 50
6._at : z = 55
7._at : z = 60
8._at : z = 65
9._at : z = 70
10._at : z = 75
------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x |
_at |
1 | .2279094 .1424924 1.60 0.110 -.0513706 .5071894
2 | .284313 .1195771 2.38 0.017 .0499463 .5186797
3 | .3407166 .0982883 3.47 0.001 .1480752 .533358
4 | .3971202 .0799363 4.97 0.000 .240448 .5537924
5 | .4535238 .0669803 6.77 0.000 .322245 .5848027
6 | .5099274 .0628508 8.11 0.000 .3867422 .6331127
7 | .5663311 .0691477 8.19 0.000 .4308041 .701858
8 | .6227347 .0835458 7.45 0.000 .4589878 .7864815
9 | .6791383 .1026924 6.61 0.000 .4778649 .8804117
10 | .7355419 .1244141 5.91 0.000 .4916947 .9793891
------------------------------------------------------------------------------
mat s=r(b)
mat list s
s[1,10]
x: x: x: x: x: x: x: x: x: x:
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
_at _at _at _at _at _at _at _at _at _at
r1 .22790939 .284313 .34071661 .39712022 .45352383 .50992744 .56633105 .62273466 .67913827 .73554189
margins, at(x=0 z=(30(5)75)) asbalanced noesample noestimcheck force vsquish
Adjusted predictions Number of obs = 200
Model VCE : OLS
Expression : Linear prediction, predict()
1._at : x = 0
z = 30
2._at : x = 0
z = 35
3._at : x = 0
z = 40
4._at : x = 0
z = 45
5._at : x = 0
z = 50
6._at : x = 0
z = 55
7._at : x = 0
z = 60
8._at : x = 0
z = 65
9._at : x = 0
z = 70
10._at : x = 0
z = 75
------------------------------------------------------------------------------
| Delta-method
| Margin Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
_at |
1 | 31.24139 6.873932 4.54 0.000 17.76873 44.71405
2 | 30.14117 5.726198 5.26 0.000 18.91803 41.36431
3 | 29.04095 4.692576 6.19 0.000 19.84367 38.23823
4 | 27.94073 3.865708 7.23 0.000 20.36408 35.51737
5 | 26.84051 3.399947 7.89 0.000 20.17673 33.50428
6 | 25.74028 3.445008 7.47 0.000 18.98819 32.49238
7 | 24.64006 3.983596 6.19 0.000 16.83236 32.44777
8 | 23.53984 4.854122 4.85 0.000 14.02594 33.05375
9 | 22.43962 5.911723 3.80 0.000 10.85286 34.02639
10 | 21.3394 7.072973 3.02 0.003 7.476628 35.20217
------------------------------------------------------------------------------
mat i=r(b)
mat list i
i[1,10]
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
_at _at _at _at _at _at _at _at _at _at
r1 31.241389 30.141168 29.040947 27.940726 26.840505 25.740284 24.640063 23.539842 22.439622 21.339401
twoway (function y = i[1,1] + s[1,1]*x, range($min $max)) ///
(function y = i[1,2] + s[1,2]*x, range($min $max)) ///
(function y = i[1,3] + s[1,3]*x, range($min $max)) ///
(function y = i[1,4] + s[1,4]*x, range($min $max)) ///
(function y = i[1,5] + s[1,5]*x, range($min $max)) ///
(function y = i[1,6] + s[1,6]*x, range($min $max)) ///
(function y = i[1,7] + s[1,7]*x, range($min $max)) ///
(function y = i[1,8] + s[1,8]*x, range($min $max)) ///
(function y = i[1,9] + s[1,9]*x, range($min $max)) ///
(function y = i[1,10] + s[1,10]*x, range($min $max)) ///
(scatter y x, msym(oh) jitter(3)), ///
legend(off) ytitle(Y) xtitle(X) scheme(lean1)
What if you have additional variables in your model? Its not a problem as long as the other variables don't interact with either x or z. The values of the simple slopes will change but because this is a linear model it doesn't matter what values you hold the other variables at. The easiest way is just to hold the other variables at their mean values with the atmeans option in margins. Here is an example adding write to our model.
regress y c.x##c.z write
Source | SS df MS Number of obs = 200
-------------+------------------------------ F( 4, 195) = 62.69
Model | 11768.436 4 2942.109 Prob > F = 0.0000
Residual | 9150.98401 195 46.9281231 R-squared = 0.5626
-------------+------------------------------ Adj R-squared = 0.5536
Total | 20919.42 199 105.122714 Root MSE = 6.8504
------------------------------------------------------------------------------
y | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x | -.2285181 .2903336 -0.79 0.432 -.8011152 .3440789
z | -.3206324 .2700438 -1.19 0.237 -.8532139 .211949
|
c.x#c.z | .0119774 .0051534 2.32 0.021 .0018138 .022141
|
write | .193212 .0713966 2.71 0.007 .0524034 .3340206
_cons | 37.16971 14.31827 2.60 0.010 8.931167 65.40826
------------------------------------------------------------------------------
/* hold write at 20 */
margins , dydx(x) at(z=(30(5)75) write=20) vsquish
Average marginal effects Number of obs = 200
Model VCE : OLS
Expression : Linear prediction, predict()
dy/dx w.r.t. : x
1._at : z = 30
write = 20
2._at : z = 35
write = 20
3._at : z = 40
write = 20
4._at : z = 45
write = 20
5._at : z = 50
write = 20
6._at : z = 55
write = 20
7._at : z = 60
write = 20
8._at : z = 65
write = 20
9._at : z = 70
write = 20
10._at : z = 75
write = 20
------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x |
_at |
1 | .1308037 .1447656 0.90 0.366 -.1529317 .4145391
2 | .1906907 .1226729 1.55 0.120 -.0497438 .4311252
3 | .2505777 .1023138 2.45 0.014 .0500464 .451109
4 | .3104646 .0849439 3.65 0.000 .1439776 .4769517
5 | .3703516 .0727374 5.09 0.000 .227789 .5129143
6 | .4302386 .0685119 6.28 0.000 .2959577 .5645195
7 | .4901256 .0736542 6.65 0.000 .345766 .6344851
8 | .5500125 .0865095 6.36 0.000 .380457 .7195681
9 | .6098995 .1042629 5.85 0.000 .405548 .814251
10 | .6697865 .1248419 5.37 0.000 .4251009 .9144721
------------------------------------------------------------------------------
/* hold write at its mean value */
margins , dydx(x) at(z=(30(5)75)) atmeans vsquish
Conditional marginal effects Number of obs = 200
Model VCE : OLS
Expression : Linear prediction, predict()
dy/dx w.r.t. : x
1._at : x = 52.645 (mean)
z = 30
write = 52.775 (mean)
2._at : x = 52.645 (mean)
z = 35
write = 52.775 (mean)
3._at : x = 52.645 (mean)
z = 40
write = 52.775 (mean)
4._at : x = 52.645 (mean)
z = 45
write = 52.775 (mean)
5._at : x = 52.645 (mean)
z = 50
write = 52.775 (mean)
6._at : x = 52.645 (mean)
z = 55
write = 52.775 (mean)
7._at : x = 52.645 (mean)
z = 60
write = 52.775 (mean)
8._at : x = 52.645 (mean)
z = 65
write = 52.775 (mean)
9._at : x = 52.645 (mean)
z = 70
write = 52.775 (mean)
10._at : x = 52.645 (mean)
z = 75
write = 52.775 (mean)
------------------------------------------------------------------------------
| Delta-method
| dy/dx Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x |
_at |
1 | .1308037 .1447656 0.90 0.366 -.1529317 .4145391
2 | .1906907 .1226729 1.55 0.120 -.0497438 .4311252
3 | .2505777 .1023138 2.45 0.014 .0500464 .451109
4 | .3104646 .0849439 3.65 0.000 .1439776 .4769517
5 | .3703516 .0727374 5.09 0.000 .227789 .5129143
6 | .4302386 .0685119 6.28 0.000 .2959577 .5645195
7 | .4901256 .0736542 6.65 0.000 .345766 .6344851
8 | .5500125 .0865095 6.36 0.000 .380457 .7195681
9 | .6098995 .1042629 5.85 0.000 .405548 .814251
10 | .6697865 .1248419 5.37 0.000 .4251009 .9144721
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