Rethinking `degroup()` for cross-classified data

[mattansb]

Follow-up up @jmgirard's #520

I think the current implementation of cross-classified disaggregation is missing a desiderata.

First let's note some desiderata that we do have:

1. The "between" variable are simply the separate group means.

Make some crossed data

mu <- 100
ul <- setNames(c(-1, -3, 0, 4), nm = letters[1:4])
uL <- setNames(c(10, 30, 0, -40), nm = LETTERS[1:4])
um <- setNames(c(100, 150, -250), nm = month.abb[1:3])

dat <- expand.grid(l = letters[1:4], L = LETTERS[1:4], m = month.abb[1:3])

set.seed(111)
e <- rnorm(nrow(dat)-1) |> round(2)
e <- append(e, -sum(e))

dat$y <- mu + ul[dat$l] + uL[dat$L] + um[dat$m] + e
dat$z <- mu + ul[dat$l] + uL[dat$L] + um[dat$m] + 10*e

dat_dem <- datawizard::demean(dat, by = c("l", "L", "m"), select = c("y","z"))

all.equal(c(dat_dem$y_l_between), ave(dat$y, dat$l))
#> TRUE
all.equal(c(dat_dem$y_L_between), ave(dat$y, dat$L))
#> TRUE
all.equal(c(dat_dem$y_m_between), ave(dat$y, dat$m))
#> TRUE

2. The sum of an observation's "between"/"within" variables is equal to the original observation

all.equal(rowSums(dat_dem[grepl("^y_", colnames(dat_dem))]), dat$y)
#> TRUE

What we don't have is that -- unlike with a single grouping variable or with nested designs -- the "within" variable is mean centered:

mean(dat_dem$y_within)
#> -200

This is equal to $(-\bar{Y})\times (\text{number of grouping vars} - 1)$

-mean(dat$y) * (3-1)
#> -200

I think this is something we want, for consistency (typically "within" is considered to be automatically double-centered), however with a crossed design this cannot be achieved without compromising on desiderata 1 or 2.

[jmgirard]

The approach being used leads to correct inference and more interpretable effects in cross-classified data. To me, that justifies its inclusion in the package somewhere. But I could understand if you want it in a separate function.

[mattansb]

Why would grand-centering the "within" component lead to different inference? Grand-centering is a linear transformation, no information loss wrt relationship with any outcome.

mod1 <- lme4::lmer(
  y ~ z_l_between + z_L_between + z_m_between + z_within +
    (z_within | l) + (z_within | L) + (z_within | m), 
  data = dat_dem
)

dat_dem2 <- transform(dat_dem,
                      z_within = datawizard::center(z_within))

mod2 <- update(mod1, data = dat_dem2)

parameters::compare_parameters(mod1, mod2)
#> Parameter    |                       mod1 |                       mod2
#> ----------------------------------------------------------------------
#> (Intercept)  | -134.38 (-145.14, -123.62) | -154.52 (-165.93, -143.10) #< different
#> z l between  |    0.31 (   0.22,    0.39) |    0.31 (   0.21,    0.40) #< same
#> z L between  |    1.22 (   1.16,    1.28) |    1.22 (   1.16,    1.28) #< same
#> z m between  |    1.02 (   1.02,    1.03) |    1.02 (   1.02,    1.03) #< same
#> z within     |    0.10 (   0.10,    0.10) |    0.10 (   0.10,    0.10) #< same
#> ----------------------------------------------------------------------
#> Observations |                         48 |                         48

[jmgirard]

You're right. See the note below Table 1 in Guo, Dhaliwal, & Rights (2024). Grand-mean centering any of the components is fine.

[mattansb]

Thanks for the reference.

How about we add a grand_center= argument that can take the following possible values:

• grand_center = FALSE do nothing
• grand_center = TRUE grand center all components
• grand_center = 1 - only level 1 component (or other numerics for specific levels?)