This question is based on hypothesis testing using fitted lines and Factors that are not of interest as described by Law et al 2021, page 21-29.

The questions are, (1) "What genes have a similar slope across all 3 populations in response to dose and not influenced by lane?" (2) "What genes have different slopes across all 3 populations in response to dose and not influenced by lane?"

> library(limma)
> class (Data$samples$Pop)
[1] "factor"
> class (Data$samples$lane)
[1] "factor"
> class (Data$samples$dose)
[1] "numeric"

> Design <- model.matrix(~ 0 + Pop + Pop:dose + lane , data = Data$samples)
> colnames (MHEE.design) 
[1] "Pop1"  "Pop2" "Pop3" "lane2" "lane3" "lane4" "lane5" "Pop1:dose" "Pop2:dose" "Pop3:dose"

> colnames(Design) <- gsub ("Pop1", "Pop1.Intercept", colnames(Design))
> colnames(Design) <- gsub ("Pop2", "Pop2.Intercept", colnames(Design))
> colnames(Design) <- gsub ("Pop3", "Pop3.Intercept", colnames(Design))
> colnames(Design) <- gsub ("Pop1:dose", "Pop1.Slope", colnames(Design))
> colnames(Design) <- gsub ("Pop2:dose", "Pop2.Slope", colnames(Design))
> colnames(Design) <- gsub ("Pop3:dose", "Pop3.Slope", colnames(Design))
> colnames(Design)

[1] "Pop1.Intercept" "Pop2.Intercept" "Pop3.Intercept" "lane2" "lane3" "lane4" "lane5" "Pop1.Slope" "Pop2.Slope" "Pop3.Slope"

> v <- voom(Data, Design)
> vfit <- lmFit(v, Design)
#"vfit <- contrasts.fit(vfit, contrasts=my.contrasts)"  # if needed
> efit <- eBayes(vfit)

DE genes in columns "Pop1.Slope" "Pop2.Slope" "Pop3.Slope" will be only in reference to "lane 1" so the output will need to be evaluated with topTable() or contrasts before it can be interpreted. However, I haven not been able to find examples where 3 factors have been compared (not contrasted) in one limma voom program, especially in fitted line examples with factors that aren't of interest.

To answer question (1), what genes have the same slope across all 3 populations I have tried topTable(). However, these results clearly show us an average slope between the 3 factors. I do not believe that this answer the question.

topTable(efit, coef = c(8:10), number = Inf, adjust.method = "BH")

Another option that used in the limma handbook would be to look at the overlap in the ven diagram and look at DE genes for all 3 populations, however this approach does not block by all lanes and so I don't it can't be used before the lane effects have been accounted for.

vennDiagram(dt[,8:10], circle.col=c("red", "orange", "turquoise"))
de.common <- which(dt[,8]!=0 & dt[,9]!=0 & dt[,10]!=0)

Another approach that is commonly recommended is making contrasts. However contrasts look for differences between 2 factors. I could see how this could be used help answer question (2), genes where populations respond differently by slope, however the model doesn't answer the primary question (1).

And as a side question, if I wished to look for relationships that were logarithmic would this be the correct code for the model.matrix? It looks correct, but I still feel the need to ask.

> Design <- model.matrix(~ 0 + Pop + Pop:log(dose, 2) + lane , data = Data$samples)

I appreciate any help that can be offered.

> sessionInfo( )
R version 4.1.0 (2021-05-18)
Platform: x86_64-apple-darwin17.0 (64-bit)
Running under: macOS Catalina 10.15.7

Matrix products: default
BLAS:   /System/Library/Frameworks/Accelerate.framework/Versions/A/Frameworks/vecLib.framework/Versions/A/libBLAS.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.1/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

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

other attached packages:
[1] gplots_3.1.1       ggplot2_3.3.5      RColorBrewer_1.1-2 Glimma_2.2.0       pheatmap_1.0.12    edgeR_3.34.0       limma_3.48.3      

loaded via a namespace (and not attached):
 [1] MatrixGenerics_1.4.2        Biobase_2.52.0              httr_1.4.2                  jsonlite_1.7.2              bit64_4.0.5                
 [6] splines_4.1.0               gtools_3.9.2                assertthat_0.2.1            BiocManager_1.30.16         stats4_4.1.0               
[11] blob_1.2.2                  GenomeInfoDbData_1.2.6      pillar_1.6.2                RSQLite_2.2.8               lattice_0.20-44            
[16] glue_1.4.2                  digest_0.6.27               GenomicRanges_1.44.0        XVector_0.32.0              colorspace_2.0-2           
[21] htmltools_0.5.1.1           Matrix_1.3-3                DESeq2_1.32.0               XML_3.99-0.7                pkgconfig_2.0.3            
[26] genefilter_1.74.0           zlibbioc_1.38.0             purrr_0.3.4                 xtable_1.8-4                scales_1.1.1               
[31] BiocParallel_1.26.2         tibble_3.1.3                annotate_1.70.0             KEGGREST_1.32.0             generics_0.1.0             
[36] IRanges_2.26.0              ellipsis_0.3.2              withr_2.4.2                 cachem_1.0.6                SummarizedExperiment_1.22.0
[41] BiocGenerics_0.38.0         survival_3.2-11             magrittr_2.0.1              crayon_1.4.1                memoise_2.0.0              
[46] fansi_0.5.0                 tools_4.1.0                 lifecycle_1.0.0             matrixStats_0.60.1          S4Vectors_0.30.0           
[51] munsell_0.5.0               locfit_1.5-9.4              DelayedArray_0.18.0         AnnotationDbi_1.54.1        Biostrings_2.60.2          
[56] compiler_4.1.0              GenomeInfoDb_1.28.1         caTools_1.18.2              rlang_0.4.11                grid_4.1.0                 
[61] RCurl_1.98-1.4              rstudioapi_0.13             htmlwidgets_1.5.3           bitops_1.0-7                gtable_0.3.0               
[66] DBI_1.1.1                   R6_2.5.1                    dplyr_1.0.7                 fastmap_1.1.0               bit_4.0.4                  
[71] utf8_1.2.2                  KernSmooth_2.23-20          parallel_4.1.0              Rcpp_1.0.7                  vctrs_0.3.8                
[76] geneplotter_1.70.0          png_0.1-7                   tidyselect_1.1.1

The article you are reading is probably this one:

Law CW, Zeglinski K, Dong X, Alhamdoosh M, Smyth GK, Ritchie ME (2020). A guide to creating design matrices for gene expression experiments. F1000Research 9, 1444.

To compare three or more populations, just take contrasts between them -- see for example Section 9.3 of the limma User's Guide. To compare three slopes you need two contrasts. In your case

Design <- model.matrix(~ 0 + Pop + Pop:dose + lane , data = Data$samples)
Pop2SlopevsPop1Slope <- c(0,0,0,0,0,0,0,-1,1,0)
Pop3SlopevsPop1Slope <- c(0,0,0,0,0,0,0,-1,0,1)
ContMatrix <- cbind(Pop2SlopevsPop1Slope, Pop3SlopevsPop1Slope)

When you run topTable, you will get F-tests on 2 df for differences between the slopes. Genes with different slopes will have small p-values. Genes with similar slopes will have large p-values. Any two contrasts between the slopes will give the same F-tests.

Please be careful about renaming the columns of the design matrix. Your gsub code contains errors and will confuse the results.