Box Cox Back transformation of coefficients and confidence intervals

Question

I have made a simple gls model with a continuous variable as the response and a categorical as the predictor. I have used box Cox transformation to the response because it was highly skewed. How can I back transform the coefficients and their CI? I used nlme for gls. A sample dataset:

structure(list(station = structure(c(1L, 1L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 
3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 
8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L, 8L
), levels = c("5027", "5047", "5061", "5117", "5155", "5177", 
"5180", "5192", "5211", "5235", "5243", "5251", "5279", "5340", 
"5359", "5370", "5390", "5415", "5422", "5445", "5465", "5479", 
"5490", "5515", "5540", "5590", "5710", "5750", "5845", "5925", 
"5955", "5980", "5990", "6019", "6031", "6041", "6052", "6058", 
"6065", "6072", "6074", "6082", "6102", "6109", "6116", "6119", 
"6126", "6135", "6136", "6141", "6156", "6159", "6174", "6188", 
"6193", "6197"), class = "factor"), temp = c(10.5, 17.6, 11, 
24, 12.8, 19.37, 16.4, 12.13, 15.5, 22.4, 10.3, 10.6, 12.6, 11.6, 
16.2, 11.43, 16.1, 11.35, 11.73, 16.95, 12.4, 11, 16.15, 14.5, 
16.1, 16.15, 11.3, 17.8, 11.9, 12.03, 11.4, 13.12, 17.6, 12.6, 
13.38, 15.5, 10.6, 15.1, 16.9, 12.6, 14, 13.2, 20.35, 24.4, 13.3, 
17.3, 10.65, 12.2, 13.6, 15.9, 14.4, 11.4, 13.3, 12.95, 25.1, 
10.1, 20.2, 11, 10.6, 11.7, 12.57, 11.2, 11, 16.2, 17.5, 12.63, 
10.9, 10.6, 17.95, 18.9), long = c(10.5258406305778, 10.5258406305778, 
10.5258406305778, 10.5258406305778, 10.5258406305778, 10.5258406305778, 
10.5258406305778, 10.5258406305778, 10.5258406305778, 10.5258406305778, 
10.5258406305778, 10.5258406305778, 10.5258406305778, 10.5258406305778, 
10.5258406305778, 10.5258406305778, 10.5258406305778, 10.5258406305778, 
9.86228239692001, 9.86228239692001, 9.86228239692001, 9.86228239692001, 
9.86228239692001, 9.86228239692001, 9.86228239692001, 9.86228239692001, 
9.86228239692001, 9.86228239692001, 9.86228239692001, 9.86228239692001, 
9.86228239692001, 9.86228239692001, 9.86228239692001, 9.86228239692001, 
9.86228239692001, 9.86228239692001, 9.86228239692001, 9.86228239692001, 
9.86228239692001, 9.86228239692001, 9.86228239692001, 9.86228239692001, 
9.86228239692001, 9.86228239692001, 9.86228239692001, 9.86228239692001, 
9.55890424235299, 9.55890424235299, 9.55890424235299, 9.55890424235299, 
9.55890424235299, 9.55890424235299, 9.55890424235299, 9.55890424235299, 
9.55890424235299, 9.55890424235299, 9.55890424235299, 9.55890424235299, 
9.55890424235299, 9.55890424235299, 9.55890424235299, 9.55890424235299, 
9.55890424235299, 9.55890424235299, 9.55890424235299, 9.55890424235299, 
9.55890424235299, 9.55890424235299, 9.55890424235299, 9.55890424235299
), lat = c(57.4260382548796, 57.4260382548796, 57.4260382548796, 
57.4260382548796, 57.4260382548796, 57.4260382548796, 57.4260382548796, 
57.4260382548796, 57.4260382548796, 57.4260382548796, 57.4260382548796, 
57.4260382548796, 57.4260382548796, 57.4260382548796, 57.4260382548796, 
57.4260382548796, 57.4260382548796, 57.4260382548796, 56.9732372789987, 
56.9732372789987, 56.9732372789987, 56.9732372789987, 56.9732372789987, 
56.9732372789987, 56.9732372789987, 56.9732372789987, 56.9732372789987, 
56.9732372789987, 56.9732372789987, 56.9732372789987, 56.9732372789987, 
56.9732372789987, 56.9732372789987, 56.9732372789987, 56.9732372789987, 
56.9732372789987, 56.9732372789987, 56.9732372789987, 56.9732372789987, 
56.9732372789987, 56.9732372789987, 56.9732372789987, 56.9732372789987, 
56.9732372789987, 56.9732372789987, 56.9732372789987, 56.1603237202265, 
56.1603237202265, 56.1603237202265, 56.1603237202265, 56.1603237202265, 
56.1603237202265, 56.1603237202265, 56.1603237202265, 56.1603237202265, 
56.1603237202265, 56.1603237202265, 56.1603237202265, 56.1603237202265, 
56.1603237202265, 56.1603237202265, 56.1603237202265, 56.1603237202265, 
56.1603237202265, 56.1603237202265, 56.1603237202265, 56.1603237202265, 
56.1603237202265, 56.1603237202265, 56.1603237202265), year = c(1990, 
1991, 1992, 1999, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 
2011, 2012, 2014, 2017, 2018, 2019, 1979, 1981, 1983, 1984, 1987, 
1988, 1998, 1999, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 
2009, 2010, 2011, 2012, 2015, 2016, 2017, 2018, 2019, 2020, 2021, 
2022, 1980, 1982, 1983, 1984, 1988, 1997, 1998, 1999, 2000, 2001, 
2002, 2004, 2005, 2009, 2010, 2011, 2012, 2014, 2016, 2017, 2018, 
2020, 2021, 2022), test.bc = c(0.559839770954862, 0.565284201434495, 
0.560556225709956, 0.566831194324542, 0.562522629786345, 0.565854817249371, 
0.56479772703557, 0.561884127815748, 0.564363080689475, 0.566556040366, 
0.559526043258447, 0.559990542945858, 0.562341806600785, 0.561305665047214, 
0.564706869893158, 0.561104304644858, 0.564660270536769, 0.561006658516524, 
0.561454241893781, 0.565032488906873, 0.562152891994338, 0.560556225709956, 
0.564683669673913, 0.563790715686587, 0.564660270536769, 0.564683669673913, 
0.560944659693946, 0.565356590523383, 0.5616418149681, 0.561780343601122, 
0.561067908733378, 0.562796264833387, 0.565284201434495, 0.562341806600785, 
0.563005411163521, 0.564363080689475, 0.559990542945858, 0.564146653514575, 
0.565012014527668, 0.562341806600785, 0.56346150874382, 0.562861828617158, 
0.566113285190366, 0.566892307593349, 0.56294225811913, 0.565171265617297, 
0.560064468460388, 0.561955384464059, 0.563173825474645, 0.564564637659987, 
0.563727389181159, 0.561067908733378, 0.56294225811913, 0.562653218195187, 
0.56699285221528, 0.55919505643473, 0.566075955195481, 0.560556225709956, 
0.559990542945858, 0.561420358676073, 0.562313997064695, 0.560818365873288, 
0.560556225709956, 0.564706869893158, 0.565247148921889, 0.562369434146698, 
0.56042015881917, 0.559990542945858, 0.565409427883572, 0.565717580038285
)), row.names = c(NA, -70L), class = c("grouped_df", "tbl_df", 
"tbl", "data.frame"), groups = structure(list(station = structure(c(1L, 
3L, 8L), levels = c("5027", "5047", "5061", "5117", "5155", "5177", 
"5180", "5192", "5211", "5235", "5243", "5251", "5279", "5340", 
"5359", "5370", "5390", "5415", "5422", "5445", "5465", "5479", 
"5490", "5515", "5540", "5590", "5710", "5750", "5845", "5925", 
"5955", "5980", "5990", "6019", "6031", "6041", "6052", "6058", 
"6065", "6072", "6074", "6082", "6102", "6109", "6116", "6119", 
"6126", "6135", "6136", "6141", "6156", "6159", "6174", "6188", 
"6193", "6197"), class = "factor"), .rows = structure(list(1:18, 
    19:46, 47:70), ptype = integer(0), class = c("vctrs_list_of", 
"vctrs_vctr", "list"))), row.names = c(NA, -3L), class = c("tbl_df", 
"tbl", "data.frame"), .drop = TRUE))

and the code:

#Box-cox tranformation
#Simple linear regression
lo <- lm(temp ~ station, data=test2)

BCTransform <- function(y, lambda=0) {
  if (lambda == 0L) { log(y) }
  else { (y^lambda - 1) / lambda }
}
bc <-boxcox(lo)
bc.power <- bc$x[which.max(bc$y)]
test2$test.bc <- BCTransform(test2$temp, bc.power)
#gls
model4 <- gls(test.bc ~ station,
              correlation= corExp(1, form = ~ long + lat| year), data=test2)

How can I: (A)Back transform the coefficients and their CI (B)Back transform the emmeans

With no data and no code this is more suited for stats.stackexchange.com. On SO it will probably be considered “not a coding problem”. — IRTFM, May 24 '23 at 16:12
`emmeans::make.tran()` already includes "boxcox" as an option. It looks like `emmeans` handles `gls` models. So if you provide a [mcve], someone can show you how to do this (the section on "special transformations" in `vignette("transformations", "emmeans")` does show a very similar example ...) — Ben Bolker, May 24 '23 at 16:21
I will add that it makes no sense to try to back-transform the model coefficients. You can estimate means and back-transform those, and you can estimate contrasts of those results. See the ["transformations" vignette](https://cran.r-project.org/web/packages/emmeans/vignettes/transformations.html) in **emmeans** — Russ Lenth, May 31 '23 at 20:59

Box Cox Back transformation of coefficients and confidence intervals

0 Answers0