Given the formula from the paper/vignette for mu, I was expecting mu to be on a log normalized scale:

The mean values μ_ij = s_j * q_ij

where log2(q_ij) = X_j * B_i

head(assays(dds)[["mu"]])

##                treated1     treated2     treated3  untreated1 untreated2
## FBgn0000008  154.396031   71.8609656   78.6055308  107.292909  169.04844
## FBgn0000014    1.501799    0.6989863    0.7645902    1.473255    2.32123

But, when I looked at this on my own data, I noticed that it is definitely not on a log scale, and has the highest correlation with un-normalized counts.

> assays(dds)$mu[1:3,1:3]
                   subject_12207_visit_1 subject_12207_visit_2 subject_12507_visit_1
ENSG00000000419.14             243.76985              218.3846             171.21118
ENSG00000000457.14             257.88875              239.1162             183.68763
ENSG00000000460.17              75.09165              100.5722              70.53956
> assays(normTransform(dds))[[1]][1:3,1:3]
                   subject_12207_visit_1 subject_12207_visit_2 subject_12507_visit_1
ENSG00000000419.14              7.559637              7.277101              7.514149
ENSG00000000457.14              7.594060              7.321629              8.115484
ENSG00000000460.17              6.895476              6.438979              5.774912
> counts(dds, normalized = TRUE)[1:3,1:3]
                   subject_12207_visit_1 subject_12207_visit_2 subject_12507_visit_1
ENSG00000000419.14              187.6589             154.10494             181.80345
ENSG00000000457.14              192.2146             158.96685             276.33463
ENSG00000000460.17              118.0543              85.76122              53.75474
> counts(dds)[1:3,1:3]
                   subject_12207_visit_1 subject_12207_visit_2 subject_12507_visit_1
ENSG00000000419.14                   260                   191                   175
ENSG00000000457.14                   253                   194                   259
ENSG00000000460.17                   112                   109                    48

What is the correct scale for mu?

> sessionInfo()
R version 4.1.2 (2021-11-01)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 20.04.4 LTS

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/atlas/libblas.so.3.10.3
LAPACK: /usr/lib/x86_64-linux-gnu/atlas/liblapack.so.3.10.3

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C               LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8     LC_MONETARY=en_US.UTF-8   
 [6] LC_MESSAGES=en_US.UTF-8    LC_PAPER=en_US.UTF-8       LC_NAME=C                  LC_ADDRESS=C               LC_TELEPHONE=C            
[11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C       

attached base packages:
[1] grid      stats4    stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
 [1] gridExtra_2.3               here_1.0.1                  foreach_1.5.2               forcats_0.5.1               stringr_1.4.0              
 [6] dplyr_1.0.8                 purrr_0.3.4                 readr_2.1.2                 tidyr_1.2.0                 tibble_3.1.6               
[11] ggplot2_3.3.5               tidyverse_1.3.1             DESeq2_1.34.0               SummarizedExperiment_1.24.0 Biobase_2.54.0             
[16] MatrixGenerics_1.6.0        matrixStats_0.61.0          GenomicRanges_1.46.1        GenomeInfoDb_1.30.1         IRanges_2.28.0             
[21] S4Vectors_0.32.3            BiocGenerics_0.40.0        

loaded via a namespace (and not attached):
 [1] bitops_1.0-7           fs_1.5.2               lubridate_1.8.0        bit64_4.0.5            RColorBrewer_1.1-2     httr_1.4.2            
 [7] rprojroot_2.0.2        tools_4.1.2            backports_1.4.1        utf8_1.2.2             R6_2.5.1               DBI_1.1.2             
[13] colorspace_2.0-3       withr_2.4.3            tidyselect_1.1.2       bit_4.0.4              compiler_4.1.2         cli_3.2.0             
[19] rvest_1.0.2            xml2_1.3.3             DelayedArray_0.20.0    scales_1.1.1           genefilter_1.76.0      XVector_0.34.0        
[25] pkgconfig_2.0.3        dbplyr_2.1.1           fastmap_1.1.0          limma_3.50.1           rlang_1.0.1            readxl_1.3.1          
[31] rstudioapi_0.13        RSQLite_2.2.10         generics_0.1.2         jsonlite_1.8.0         BiocParallel_1.28.3    RCurl_1.98-1.6        
[37] magrittr_2.0.2         GenomeInfoDbData_1.2.7 Matrix_1.4-0           Rcpp_1.0.8             munsell_0.5.0          fansi_1.0.2           
[43] lifecycle_1.0.1        stringi_1.7.6          edgeR_3.36.0           zlibbioc_1.40.0        blob_1.2.2             parallel_4.1.2        
[49] crayon_1.5.0           lattice_0.20-45        Biostrings_2.62.0      haven_2.4.3            splines_4.1.2          annotate_1.72.0       
[55] hms_1.1.1              KEGGREST_1.34.0        locfit_1.5-9.4         pillar_1.7.0           codetools_0.2-18       geneplotter_1.72.0    
[61] reprex_2.0.1           XML_3.99-0.9           glue_1.6.2             modelr_0.1.8           png_0.1-7              vctrs_0.3.8           
[67] tzdb_0.2.0             cellranger_1.1.0       gtable_0.3.0           assertthat_0.2.1       cachem_1.0.6           xtable_1.8-4          
[73] broom_0.7.12           survival_3.2-13        iterators_1.0.14       AnnotationDbi_1.56.2   memoise_2.0.1          ellipsis_0.3.2

See the equations above. q is inside the log(). So it is original scale.