How to specify the non-linear interaction of two factor variables in generalised additive models [R]

Question

I have a time-series data set that contains an outcome variable which is continuous and two factor predictors (one with 6 levels and one with 2 levels).

I would like to model the non-linear interaction of the two factor variables on the continuous variable.

This is the model I have so far:

library(mgcv)

model <- bam(
    outcome ~
        factor_1 + factor_2 +
        s(time, k = 9) +
        s(time, by = factor_1, k = 9) +
        s(time, by = factor_2, k = 9),
    data = df
)

summary(model)
Family: gaussian 
Link function: identity 

Formula:
outcome ~ factor_1 + factor_2 + s(time, k = 9) + s(time, by = factor_1, 
    k = 9) + s(time, by = factor_2, k = 9)

Parametric coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  2612.72      23.03 113.465   <2e-16 ***
factor_1b      33.19      27.00   1.229     0.22    
factor_2z    -488.52      27.00 -18.093   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Approximate significance of smooth terms:
                    edf Ref.df      F  p-value    
s(time)           2.564  3.184  6.408 0.000274 ***
s(time):factor_1b 1.000  1.001  0.295 0.587839    
s(time):factor_2z 2.246  2.792 34.281  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

R-sq.(adj) =  0.679   Deviance explained = 69.1%
fREML = 1359.6  Scale est. = 37580     n = 207

Now I would like to add a non-linear interaction of factor_1 and factor_2 with time for the effect on outcome, so that the smoothers in every combination could differ (for example: factor_2 has a stronger non-linear effect for some levels of factor_1). Something like s(time, factor_1, factor_2) or s(time, factor_1, by = factor_2) does not work.

Answer 1

Including an interaction of two factors using interaction() seems to do the job.

library(mgcv)

# The following assumes factors are ordered with treatment contrast.    
model <- bam(
    outcome ~
        interaction(factor_1, factor_2) +
        s(time, k = 9) +
        s(time, by = interaction(factor_1, factor_2), k = 9),
    data = df
)