Interaction with two binary variables
In a regression model with interaction term, people tend to pay attention to only the coefficient of the interaction term.
Let’s start with the simpliest situation: \(x_1\) and \(x_2\) are binary and coded 0/1.
\[ E(y) = \beta_1 x_1 + \beta_2 x_2 + \beta_{12} x_1x_2 \]
In this case, we have a saturated model; that is, we have three coefficients representing additive effects from the baseline situation (both \(x_1\) and \(x_2\) being 0). There are four different situations, with four combinations of \(x_1\) and \(x_2\).
A lot of people just pay attention to the interaction term. In the case of studying treatment effects between two groups, say female and male, that makes sense, the interaction term representing the difference between male and female in terms of treatment effect.
In this model:
\[ E(y) = \beta_1 female + \beta_2 treatment + \beta_{12} female*treatment \]
The two dummy-coded binary variables, female and treatment, form four combinations. The following 2x2 table represents the expected means of the four cells(combinations).
| male | female | |
|---|---|---|
| control | \[ \beta_0 \] | \[ \beta_0 + \beta_1 \] |
| treatment | \[ \beta_0 + \beta_2 \] | \[ \beta_0 + \beta_1 + \beta_2 + \beta_{12}\] |
We can see from this table that, for example,
\[\beta_0=E(Y|(0,0))\];
that is, \(\beta_0\) is the expected mean of the cell (0,0) (male and control).
\[\beta_0 + \beta_1 =E(Y|(1,0))\];
that is ,\(\beta_0 + \beta_1\) is the expected mean of the cell (1,0) (female and control). And so on.
Now,
\[ \beta_{12} = (E(Y|(1,1))-E(Y|(0,1)))-(E(Y|(1,0))-E(Y|(0,0))) \]
that is, the coefficient on the interaction term is actually the difference in difference. That’s why in many situations, people are only interested in the interaction coefficient, since they are only interested in the diff-in-diff estimates. The usually diff-in-diff estimator in causal inference literature refer to something similar, instead of female vs. male, people are interested in the treatment effect difference in before and after treatment. If we simply replace female/male dummy with before/after dummy, we can use the same logic. In those situations, it’s fine to mainly focus on the interaction term coefficient.
In some other situations, the three coefficients are equally important. It depends on your interest. For example, if we are interested in studying differences between union member and non-union member and black vs. non-black, we may not be only interested in the interaction effect. Instead, we might be interested in all four cells, maybe all possible pairwise comparisons. In that case, we should pay attention to all three coefficients. Stata’s “margins” command is of great help if we’d like to compare the cell means.
Let’s take a look from a sample example in Stata:
webuse union3
reg ln_wage i.union##i.black, r
margins union#black
margins union#black, pwcompare
##
## . webuse union3
## (National Longitudinal Survey. Young Women 14-26 years of age in 1968)
##
## . reg ln_wage i.union##i.black, r
##
## Linear regression Number of obs = 1,244
## F(3, 1240) = 34.76
## Prob > F = 0.0000
## R-squared = 0.0762
## Root MSE = .37699
##
## ------------------------------------------------------------------------------
## | Robust
## ln_wage | Coef. Std. Err. t P>|t| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## 1.union | .2045053 .0291682 7.01 0.000 .1472808 .2617298
## 1.black | -.1709034 .0308067 -5.55 0.000 -.2313425 -.1104644
## |
## union#black |
## 1 1 | .0386275 .0516609 0.75 0.455 -.062725 .13998
## |
## _cons | 1.657525 .0138278 119.87 0.000 1.630396 1.684653
## ------------------------------------------------------------------------------
##
## . margins union#black
##
## Adjusted predictions Number of obs = 1,244
## Model VCE : Robust
##
## Expression : Linear prediction, predict()
##
## ------------------------------------------------------------------------------
## | Delta-method
## | Margin Std. Err. t P>|t| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## union#black |
## 0 0 | 1.657525 .0138278 119.87 0.000 1.630396 1.684653
## 0 1 | 1.486621 .027529 54.00 0.000 1.432613 1.54063
## 1 0 | 1.86203 .0256822 72.50 0.000 1.811644 1.912415
## 1 1 | 1.729754 .0325611 53.12 0.000 1.665873 1.793635
## ------------------------------------------------------------------------------
##
## . margins union#black, pwcompare
##
## Pairwise comparisons of adjusted predictions
## Model VCE : Robust
##
## Expression : Linear prediction, predict()
##
## -----------------------------------------------------------------
## | Delta-method Unadjusted
## | Contrast Std. Err. [95% Conf. Interval]
## ----------------+------------------------------------------------
## union#black |
## (0 1) vs (0 0) | -.1709034 .0308067 -.2313425 -.1104644
## (1 0) vs (0 0) | .2045053 .0291682 .1472808 .2617298
## (1 1) vs (0 0) | .0722294 .0353756 .0028268 .141632
## (1 0) vs (0 1) | .3754087 .0376487 .3015466 .4492709
## (1 1) vs (0 1) | .2431328 .0426388 .1594807 .326785
## (1 1) vs (1 0) | -.1322759 .0414705 -.2136359 -.0509159
## -----------------------------------------------------------------
##
## .What we get by using “margins union#black” is the four cell means of \(E(Y)\), in this case, log of wage. Then “margins union#black, pwcompare” tells us all pairwise comparison of these four cell means. Instead of only paying attention to the interaction coefficient, in this case we might be interested in some comparisons of the four different situations of union and black. In fact, in this example, despite the interaction term being insignificant, all six comparisons of the cell means turn out to have 95% confidence intervals that do not include zero.
Interaction with continuous variables
Let’s start with the simpliest situation: \(x_1\) and \(x_2\) are continuous.
\[ E(y) = \beta_1 x_1 + \beta_2 x_2 + \beta_{12} x_1*x_2 \]
In this case, we recommend “centering” \(x_1\) and \(x_2\) if they are continuous; that is, subtracting the mean value from each continuous independent variable when they are involved in the interaction term. There are two reason for it:
- To reduce multi-collinearity. If the range of \(x_1\) and \(x_2\) include only positive numbers, then \(x_1*x_2\) can be highly correlated with both or one of \(x_1\) and \(x_2\). This can lead to numerical problems and unstable coefficient estimates (multi-collinearity problem).
“Centering” can reduce the correlation between the interaction term and the independent variables. If the original variables are normally distributed, interaction term after centering is actually uncorrelated with the original variables. When they are not normally distributed, centering will still reduce the correlation to a large degree.
- To help with interpretation. In a model with interaction, \(\beta_1\) represents the effect of \(x_1\) when \(x_2\) is zero. However, in many situations, zero is not within the range of \(x_2\). After centering, centered \(x_2\) at zero simply means original \(x_2\) at its mean value.
When we have dummy variable interacting with continuous variable, only continuous variable should be centered.
Again, Stata’s margins command is helpful.
sysuse auto
sum mpg
gen mpg_centered=mpg-r(mean)
sum mpg_centered
reg price i.foreign##c.mpg_centered
margins foreign, at(mpg_centered=(-3 (1) 3))
marginsplot
##
## . sysuse auto
## (1978 Automobile Data)
##
## . sum mpg
##
## Variable | Obs Mean Std. Dev. Min Max
## -------------+---------------------------------------------------------
## mpg | 74 21.2973 5.785503 12 41
##
## . gen mpg_centered=mpg-r(mean)
##
## . sum mpg_centered
##
## Variable | Obs Mean Std. Dev. Min Max
## -------------+---------------------------------------------------------
## mpg_centered | 74 -4.03e-08 5.785503 -9.297297 19.7027
##
## . reg price i.foreign##c.mpg_centered
##
## Source | SS df MS Number of obs = 74
## -------------+---------------------------------- F(3, 70) = 9.48
## Model | 183435285 3 61145094.9 Prob > F = 0.0000
## Residual | 451630112 70 6451858.74 R-squared = 0.2888
## -------------+---------------------------------- Adj R-squared = 0.2584
## Total | 635065396 73 8699525.97 Root MSE = 2540.1
##
## ------------------------------------------------------------------------------
## price | Coef. Std. Err. t P>|t| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## foreign |
## Foreign | 1666.519 717.217 2.32 0.023 236.0751 3096.963
## mpg_centered | -329.2551 74.98545 -4.39 0.000 -478.8088 -179.7013
## |
## foreign#|
## c. |
## mpg_centered |
## Foreign | 78.88826 112.4812 0.70 0.485 -145.4485 303.225
## |
## _cons | 5588.295 369.0945 15.14 0.000 4852.159 6324.431
## ------------------------------------------------------------------------------
##
## . margins foreign, at(mpg_centered=(-3 (1) 3))
##
## Adjusted predictions Number of obs = 74
## Model VCE : OLS
##
## Expression : Linear prediction, predict()
##
## 1._at : mpg_centered = -3
##
## 2._at : mpg_centered = -2
##
## 3._at : mpg_centered = -1
##
## 4._at : mpg_centered = 0
##
## 5._at : mpg_centered = 1
##
## 6._at : mpg_centered = 2
##
## 7._at : mpg_centered = 3
##
## ------------------------------------------------------------------------------
## | Delta-method
## | Margin Std. Err. t P>|t| [95% Conf. Interval]
## -------------+----------------------------------------------------------------
## _at#foreign |
## 1#Domestic | 6576.06 370.446 17.75 0.000 5837.229 7314.891
## 1#Foreign | 8005.915 766.8178 10.44 0.000 6476.545 9535.284
## 2#Domestic | 6246.805 354.4734 17.62 0.000 5539.83 6953.78
## 2#Foreign | 7755.548 709.9327 10.92 0.000 6339.632 9171.464
## 3#Domestic | 5917.55 354.0032 16.72 0.000 5211.513 6623.587
## 3#Foreign | 7505.181 658.8306 11.39 0.000 6191.185 8819.177
## 4#Domestic | 5588.295 369.0945 15.14 0.000 4852.159 6324.431
## 4#Foreign | 7254.814 614.9548 11.80 0.000 6028.325 8481.303
## 5#Domestic | 5259.04 397.981 13.21 0.000 4465.292 6052.788
## 5#Foreign | 7004.447 579.9479 12.08 0.000 5847.778 8161.117
## 6#Domestic | 4929.785 437.9413 11.26 0.000 4056.338 5803.231
## 6#Foreign | 6754.081 555.4891 12.16 0.000 5646.192 7861.969
## 7#Domestic | 4600.53 486.253 9.46 0.000 3630.729 5570.331
## 7#Foreign | 6503.714 543.0057 11.98 0.000 5420.723 7586.704
## ------------------------------------------------------------------------------
##
## . marginsplot
##
## Variables that uniquely identify margins: mpg_centered foreign
##
## .In this example, the graph shows the predicted price for foreign and domestic cars at different level of mpg.