Interaction effects
Testing nonlinear interactions with mecompare: the approach of Mize (2019)
Mize (2019) shows that in nonlinear models – and in linear models with nonlinearities – the coefficient on a product term is not a test of interaction on the metric of interest. Interaction should instead be assessed in the natural metric of the outcome by (1) plotting the predictions, (2) computing the marginal effect of one variable at chosen levels of the other, and (3) testing whether those marginal effects differ: a second difference. It is also important to look at both sides of an interaction – the effect of X across levels of Z, and the effect of Z across levels of X.
mecompare automates these steps, and each second difference is one metest call. The pages in this section reproduce the article’s examples:
- Nonlinear effects in a linear model: age, age², and gender (Table 2, Figures 1–2).
- Two nominal variables: gender × parenthood and gender × education (Tables 3–4, Figures 12–13).
- A nominal and a continuous variable: gender × social roles (Figure 14).
- Two continuous variables: political views × survey year (Figures 15–17).
A bonus page, not in the article, shows joint tests of interaction when the moderator has more than two categories.
The same methods, applied to the examples of the marginal effects chapter of Mize and Han (forthcoming) – including a multi-category outcome and a three-way interaction – are on the Tests of interaction page.
The three ingredients
A nominal moderator: by(). by() reports the marginal effect of a focal variable at each level of by(), e.g. by(woman) reports one marginal effect for each level of woman, computed as a counterfactual: the whole sample set to each level in turn, then averaged. (over(woman) instead uses only the observations in each group; see help mecompare for the distinction.)
A continuous moderator: covariates(z=(numlist)). E.g. covariates(age=(25 55)) reports the marginal effect of the focal variable with everyone set to age 25, then to age 55, as numbered rows in one table. Any numlist works, so covariates(age=(20(5)70)) gives the effect across the range of the moderator – the numbers behind the “marginal effect of X across Z” plots.
Where to evaluate a continuous moderator. When there is no substantive reason to pick particular values, a good default is the 10th, 50th, and 90th percentiles of the moderator: a low, a typical, and a high value that are all in the range of the data. summarize z, detail returns them as r(p10), r(p50), and r(p90), so the pattern is mecompare x, covariates(z=(`r(p10)' `r(p50)' `r(p90)')) followed by metest 1 = 2 = 3 for the joint test and metest 3 - 1 for the change from the 10th to the 90th percentile. The pages with a continuous moderator each show it.
Where and how far the focal variable moves: start(), amount(), uncentered. start(age=25) amount(5) is the change from 25 to 30; amount(5) alone is a five-unit change from each observation’s own value; amount(sd) is a standard-deviation change and amount(rate) the instantaneous rate (the dydx() of margins). By default a change is centered on the starting value (half below, half above); uncentered makes it run upward from the start.
Testing the second difference
Every row of the mecompare table is numbered, and metest tests any combination of rows: metest 1 - 2 is the second difference between the marginal effects in rows 1 and 2, metest 1 = 2 = 3 is a joint test that the first three marginal effects in the table are equal, and metest 2 - 1, add rowname("...") accumulates results into one table.
Plotting the marginal effects
store(stub) saves the table as stored estimates so that esttab can be used to make publication-ready tables and coefplot can be used to plot them. With by() or covariates() there is one coefficient per level, so a plot of the effect of gender across the range of age is mecompare i.woman, covariates(age=(20(5)70)) store(gap) followed by coefplot gap_m1, vertical yline(0). Plots of predictions are made with margins and marginsplot.
Translating the 2019 files
| In the replication files | With mecompare |
|---|---|
margins woman, at(age=(25 30)) post then mlincom 3 - 1, mlincom 4 - 2, mlincom (3 - 1) - (4 - 2) |
mecompare age, start(age=25) amount(5) uncentered by(woman) then metest 1 - 2 |
margins woman, at(age=gen(age)) at(age=gen(age+5)) post then the three mlincom lines |
mecompare age, amount(5) uncentered by(woman) then metest 1 - 2 |
margins, dydx(woman) at(age=(25 55)) post then mlincom 2 - 1 |
mecompare i.woman, covariates(age=(25 55)) then metest 2 - 1 |
margins, dydx(woman) at(age=(18(2)72)) then marginsplot |
mecompare i.woman, covariates(age=(18(2)72)) store(gap) then coefplot gap_m1, vertical |
margins parrole#woman, post then mlincom 3 - 1, mlincom 4 - 2, mlincom (1 - 2) - (3 - 4) |
mecompare i.parrole, by(woman) then metest 1 - 2 |
margins educ, dydx(woman) pwcompare(pve) |
mecompare i.woman, by(educ) then metest 1 = 2 = 3 = 4 = 5 and metest 1 - 2, … |
margins woman, at(role=gen(role)) at(role=gen(role + 1)) post then mlincom |
mecompare role, by(woman) uncentered then metest 1 - 2 |
margins woman, dydx(role) post then mlincom 2 - 1 |
mecompare role, by(woman) amount(rate) then metest 1 - 2 |
mchange year1976 conviewSS |
mecompare year1976 conviewSS, amount(sd) uncentered |
margins, dydx(conviewSS) at(year=(0 40)) post then mlincom 2 - 1 |
mecompare conviewSS, covariates(year1976=(0 40)) amount(rate) then metest 2 - 1 |
margins, at(year1976=gen(year1976) conview=(-1 1)) at(year1976=gen(year1976 + 2) conview=(-1 1)) post then mlincom |
mecompare year1976, covariates(conviewSS=(-1 1)) amount(2) uncentered then metest 1 - 2 |
In each case the models() option names the stored model, or is omitted when the model is the one in memory.
Reference
Mize, Trenton D. 2019. “Best Practices for Estimating, Interpreting, and Presenting Nonlinear Interaction Effects.” Sociological Science 6: 81–117. https://doi.org/10.15195/v6.a4
The article’s data and replication files are at trentonmize.com/research.