How to find effect size from emmeans in GAM?

Question

I would like to get to know effect sizes of my data, but when I use eff_size() it requires sigma and edf as default. I assigned to the latter df.residual(gam1), but I do not know what should be assigned as sigma?

My GAM looks like this:

library(mgcv)
library(emmeans)
set.seed(123)
n = 500
dat <- data.frame(
"y" = rnbinom( n, size = 2.594, mu = 144.629),
"fac" = as.factor(rep(c("x","y"), n)[1:n]),
"fac2" = as.factor(rep(c("a","b","c"), n)[1:n]),
"fac3" = as.factor(rep(c("p","q","r"), n)[1:n]),
"x1" = rnorm(n, mean = 0, sd=1.02),
"x2" = as.factor(rep(1:5, each = n/5)))


gam1 <- mgcv::gam(y ~ fac*fac2*fac3 + 
              s(x2, bs = "re") +
              s(x1, bs = "tp"), data= dat, family = nb, method = "REML")

To analyse specific interactions I used package emmeans:

emm2 <- contrast(pairs(emmeans(gam1, ~ fac|fac2|fac3, nesting = NULL)), by = NULL,
                  method = list( c(1,0,0,0,0,0,0,0,0) - c(0,0,0,0,1,0,0,0,0),
                                 c(1,0,0,0,0,0,0,0,0) - c(0,0,0,0,0,0,0,0,1)))

The code in question is eff_size(emm2, edf = df.residual(gam1), sigma = ???, method = "identity")