# Multiple regression: Interpreting three-way interaction with two categorical variables and one numeric (in R)

I have two quick questions. I’d like to confirm that I’m interpreting correctly one of my coefficients from a 3-way interaction in a multiple linear regression model. The summary of my model is below. I’m using the lm function in R to fit my model.

The three-way interaction includes the following explanatory variables: a categorical variable with two levels: Language “lan" (English “Eng", Spanish “Span”); a categorical variable with three levels: Consonant “cons" (/p/, /t/, /k/); and a numeric variable Group (“groupCont”). The response variable is called “cog”.

I’ve set the reference level for the factor Consonant to /t/. The reference level for the factor Language is English by default.

Question #1:

I’m particularly interested in knowing whether the slope for /t/ is different between levels of Language (English and Spanish) across changes in Group.

I’ve identified the two-way interaction coefficient “lanSpan:groupCont” (or LanguageSpanish * Group) as the relevant coefficient for this, and I’d like to confirm that this is accurate. I’ve interpreted this coefficient to mean the following:

Compared to cog for the baseline (when Language = English and Consonant = /t/), cog decreases by a factor of ~292 for a one unit increase in Group for Language = Spanish and Consonant = /t/.

Is that accurate?

Question #2:

The three-way interaction coefficient "lanSpan:cons/k/:groupCont" (LanguageSpanish * Consonant /k/ * Group)--I’ve interpreted this to mean the following:

Compared to cog for the baseline (when Language = English and Consonant = /t/), cog increases by a factor of ~341 for a one unit increase in Group for Language = Spanish and Consonant = /k/.

Is that accurate?

lm(formula = cog ~ lan + cons + groupCont + lan * cons * groupCont)

Residuals:
Min      1Q  Median      3Q     Max
-2355.0  -307.0   -48.2   161.0  5080.1

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)                3424.876     94.454  36.260  < 2e-16 ***
lanSpan                   -1874.785    133.578 -14.035  < 2e-16 ***
cons/p/                   -3204.580    133.578 -23.990  < 2e-16 ***
cons/k/                   -1521.898    133.578 -11.393  < 2e-16 ***
groupCont                   -14.546     85.318  -0.170   0.8647
lanSpan:cons/p/            1858.394    188.907   9.838  < 2e-16 ***
lanSpan:cons/k/            1182.706    188.907   6.261 8.51e-10 ***
lanSpan:groupCont          -291.917    120.658  -2.419   0.0159 *
cons/p/:groupCont             7.982    120.658   0.066   0.9473
cons/k/:groupCont           -97.438    120.658  -0.808   0.4197
lanSpan:cons/p/:groupCont   291.181    170.637   1.706   0.0886 .
lanSpan:cons/k/:groupCont   340.980    170.637   1.998   0.0463 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 816.6 on 480 degrees of freedom
Multiple R-squared:  0.6516,    Adjusted R-squared:  0.6436
F-statistic: 81.61 on 11 and 480 DF,  p-value: < 2.2e-16