Installation and loading of the limpca package

limpca can be installed from Bioconductor:

if (!requireNamespace("BiocManager", quietly = TRUE))
    install.packages("BiocManager")

BiocManager::install("limpca")

And then loaded into your R session:

library("limpca")

Data and model presentation

This data set comes from the study of the modulation of immunity in rainbow trout (Oncorhynchus mykiss) by exposure to cadmium (Cd) combined with polyunsaturated fatty acids (PUFAs) enriched diets [Cornet et al., 2018].

The responses were quantified by measuring the modification of the expression of 15 immune-related genes (m = 15) by RT-qPCR (reverse transcription quantitative polymerase chain reaction). The experiment was carried out on 72 trouts and 3 factors were considered in the experimental design:

• Day: Measurements on trouts were collected on days 28, 70 and 72
• Treatment : Four polyunsaturated fatty acid diets: alpha-linolenic acid (ALA), linoleic acid (LA), eicosapentaenoic acid (EPA) and docosahexaenoic acid (DHA)
• Exposure: Trouts were exposed (level = 2) or not (level = 0) to high cadmium concentrations.

This gives a 3 × 4 × 2 factorial design. Each of the 24 trials corresponds to a different aquarium. Three fishes were analysed (3 replicates) for each condition, giving a total of 72 observations.

In this limpca vignette, the data are first explored in order to prepare an appropriate data set for ASCA/APCA analysis. Data are first represented by PCA and outliers removed. The remaining observations are then log transformed. Next, the data of each aquarium are mean aggregated in order to avoid the inclusion of an aquarium random factor in the statistical model because limpca is not yet able to handle mixed linear models. The data are finally centered and scaled by column.

The estimated model in then a (general) linear model for fixed factors including main effects and all two way interactions. The three way interaction is not included because the aggregated design has no replicate.

A detailed presentation and analysis of this dataset is also available in [Benaiche, 2022].

Data import and exploration

data("trout")
# print number of and response names
cat("\n Nb of Responses :  ", ncol(trout$outcomes), "\n ")

 Nb of Responses :   15 
 

cat("\nResponses :\n", colnames(trout$outcomes), "\n ")

Responses :
 IL.1b IL6 IL8 Lysozyme IgM MCSFR.a MPO TGF.b TLR3 TLR9 MyD88 SOD Elov5 C3 Cox2 
 

# Order responses by alphabetic order
trout$outcomes <- trout$outcomes[, order(dimnames(trout$outcomes)[[2]])]
cat("\n Ordered responses :\n  ", colnames(trout$outcomes), "\n ")

 Ordered responses :
   C3 Cox2 Elov5 IL.1b IL6 IL8 IgM Lysozyme MCSFR.a MPO MyD88 SOD TGF.b TLR3 TLR9 
 

# print factor names
cat("\nDesign factors :  ", colnames(trout$design), "\n ")

Design factors :   Day Treatment Exposure Aquarium 
 

# plot the design with plotDesign function
limpca::plotDesign(
    design = trout$design, x = "Treatment",
    y = "Day", cols = "Exposure",
    title = "Initial design of the trout dataset"
)

resPCA <- limpca::pcaBySvd(trout$outcomes)
limpca::pcaScreePlot(resPCA, nPC = 8)

limpca::pcaScorePlot(
    resPcaBySvd = resPCA, axes = c(1, 2),
    title = "Score plot of original data ",
    design = trout$design, color = "Aquarium",
    points_labs_rn = TRUE
)

Warning: ggrepel: 64 unlabeled data points (too many overlaps). Consider
increasing max.overlaps

# Log Transformation
trout_log <- trout
trout_log$outcomes <- as.matrix(log10(trout$outcomes))

# new PCA
resPCA1 <- limpca::pcaBySvd(trout_log$outcomes)
limpca::pcaScreePlot(resPCA1, nPC = 8)

limpca::pcaScorePlot(
    resPcaBySvd = resPCA1, axes = c(1, 2),
    title = "Score plot of Log10 data ",
    design = trout_log$design, color = "Aquarium",
    drawShapes = "polygon", points_labs_rn = TRUE
)

Warning: ggrepel: 71 unlabeled data points (too many overlaps). Consider
increasing max.overlaps

# Remove outliers and create new dataset
trout_clean <- trout_log
outliers <- match(
    c("D72EPA2.1", "D28EPA2.2"),
    rownames(trout_log$outcomes)
)
trout_clean$outcomes <- trout_log$outcomes[-outliers, ]
trout_clean$design <- trout_log$design[-outliers, ]

# PCA
resPCA2 <- limpca::pcaBySvd(trout_clean$outcomes)
limpca::pcaScreePlot(resPCA2, nPC = 8)

# Score plot Components 1 and 2
limpca::pcaScorePlot(
    resPcaBySvd = resPCA2, axes = c(1, 2),
    title = "Score plot of Log10 data without outliers (PC 1&2)",
    design = trout_clean$design, color = "Aquarium",
    drawShapes = "polygon",
    points_labs_rn = FALSE
)

# Score plot Components 3 and 4
limpca::pcaScorePlot(
    resPcaBySvd = resPCA2, axes = c(3, 4),
    title = "Score plot of Log10 data without outliers (PC 3&4)",
    design = trout_clean$design, color = "Aquarium",
    drawShapes = "polygon", points_labs_rn = FALSE
)

# Mean aggregation
mean_outcomes <- matrix(0, nrow = 24, ncol = 15)
mean_design <- matrix(0, nrow = 24, ncol = 3)
y <- list(
    trout_clean$design[["Day"]],
    trout_clean$design[["Treatment"]],
    trout_clean$design[["Exposure"]]
)
for (i in 1:15) {
    mean_outcomes[, i] <- aggregate(trout_clean$outcomes[, i], by = y, mean)[, 4]
}
mean_design <- aggregate(trout_clean$outcomes[, 1], by = y, mean)[, c(1:3)]

# Set row and col names
colnames(mean_outcomes) <- colnames(trout_clean$outcomes)
colnames(mean_design) <- colnames(trout_clean$design)[1:3]
trout_mean_names <- apply(mean_design, 1, paste, collapse = "")
rownames(mean_outcomes) <- trout_mean_names
rownames(mean_design) <- trout_mean_names

# Outcomes centering and Scaling
mean_outcomes <- scale(mean_outcomes, center = TRUE, scale = TRUE)
# New data object creation
trout_mean <- list(
    "outcomes" = mean_outcomes,
    "design" = mean_design,
    "formula" = trout$formula
)
# Clean objects
rm(
    resPCA, resPCA1, resPCA2, y, mean_design, mean_outcomes,
    trout_mean_names
)

Exploration of aggregated data

Aggregated data are now explored.

pander(head(trout_mean$design))

limpca::plotDesign(
    design = trout_mean$design,
    title = "Design of mean aggregated trout dataset"
)

limpca::plotLine(trout_mean$outcomes,
    rows = c(1, 24),
    xaxis_type = "character", type = "s"
) +
    ggplot2::theme(axis.text.x = element_text(angle = 45, hjust = 1))

resPCA_mean <- limpca::pcaBySvd(trout_mean$outcomes)
pcaScreePlot(resPCA_mean, nPC = 6)

limpca::pcaScorePlot(
    resPcaBySvd = resPCA_mean, axes = c(1, 2),
    title = "PCA score plot by Exposure and Day",
    design = trout_mean$design,
    shape = "Exposure", color = "Day",
    points_labs_rn = FALSE
)

limpca::pcaScorePlot(
    resPcaBySvd = resPCA_mean, axes = c(1, 2),
    title = "PCA scores plot by Treatment and Day",
    design = trout_mean$design,
    shape = "Day", color = "Treatment",
    points_labs_rn = FALSE
)

limpca::pcaScorePlot(
    resPcaBySvd = resPCA_mean, axes = c(1, 2),
    title = "PCA scores plot by Exposure and Treatment",
    design = trout_mean$design,
    shape = "Treatment", color = "Exposure",
    points_labs_rn = FALSE
)

limpca::pcaLoading1dPlot(
    resPcaBySvd = resPCA_mean, axes = c(1, 2),
    title = "PCA loadings plot trout", xlab = " ",
    ylab = "Expression", xaxis_type = "character", type = "s"
) +
    ggplot2::theme(axis.text.x = element_text(angle = 45, hjust = 1))

limpca::pcaLoading2dPlot(
    resPcaBySvd = resPCA_mean, axes = c(1, 2),
    title = "PCA loadings plot trout", addRownames = TRUE
)

limpca::plotScatterM(
    Y = trout_mean$outcomes, cols = c(1:15),
    design = trout_mean$design,
    varname.colorup = "Day",
    vec.colorup = c("CadetBlue4", "pink", "orange"),
    varname.colordown = "Day",
    vec.colordown = c("CadetBlue4", "pink", "orange"),
    varname.pchup = "Treatment",
    varname.pchdown = "Exposure"
)

GLM decomposition

The estimated model is the following :

outcomes ~ Day + Treatment + Exposure + Day:Treatment + Day:Exposure + Treatment:Exposure

Since the design has only one replicate, the three way interaction has been removed because it is confounded with residuals.

resLmpModelMatrix <- limpca::lmpModelMatrix(trout_mean)
pander::pander(head(resLmpModelMatrix$modelMatrix))

resLmpEffectMatrices <- lmpEffectMatrices(resLmpModelMatrix)
resLmpEffectMatrices$varPercentagesPlot

resLmpBootstrapTests <- lmpBootstrapTests(
    resLmpEffectMatrices = resLmpEffectMatrices,
    nboot = 1000
)

# Print p-values
pander::pander(t(resLmpBootstrapTests$resultsTable))

ASCA and APCA

Visualization of single or combined effect matrices using ASCA and APCA. ASCA-E is also provided in limpca but not shown here.

resASCA <- lmpPcaEffects(
    resLmpEffectMatrices = resLmpEffectMatrices,
    method = "ASCA",
    combineEffects = list(c(
        "Day", "Treatment",
        "Day:Treatment"
    ))
)

resLmpContributions <- lmpContributions(resASCA)

pander::pander(resLmpContributions$totalContribTable)

pander::pander(resLmpContributions$effectTable)

pander::pander(resLmpContributions$contribTable)

pander::pander(resLmpContributions$combinedEffectTable)

## Visualize the more important contributions
resLmpContributions$plotContrib

A <- lmpScorePlot(resASCA,
    effectNames = "Day",
    color = "Day", shape = "Day"
)
B <- lmpLoading2dPlot(resASCA,
    effectNames = "Day",
    points_labs = colnames(trout$outcomes)
)
grid.arrange(A, B, ncol = 2)

A <- lmpScorePlot(resASCA,
    effectNames = "Treatment",
    color = "Treatment", shape = "Treatment"
)
B <- lmpLoading2dPlot(resASCA,
    effectNames = "Treatment",
    points_labs = colnames(trout$outcomes)
)
grid.arrange(A, B, ncol = 2)

A <- lmpScorePlot(resASCA,
    effectNames = "Day:Treatment",
    color = "Treatment", shape = "Day"
)
B <- lmpLoading2dPlot(resASCA,
    effectNames = "Day:Treatment",
    points_labs = colnames(trout$outcomes)
)
grid.arrange(A, B, ncol = 2)

A <- lmpScorePlot(resASCA,
    effectNames = "Day+Treatment+Day:Treatment",
    color = "Treatment", shape = "Day"
)
B <- lmpLoading2dPlot(resASCA,
    effectNames = "Day+Treatment+Day:Treatment",
    points_labs = colnames(trout$outcomes)
)
grid.arrange(A, B, ncol = 2)

A <- lmpScorePlot(resASCA,
    effectNames = "Residuals",
    color = "Treatment", shape = "Day"
)
B <- lmpLoading2dPlot(resASCA,
    effectNames = "Residuals",
    points_labs = colnames(trout$outcomes)
)
grid.arrange(A, B, ncol = 2)

A <- lmpEffectPlot(resASCA,
    effectName = "Day:Treatment",
    x = "Day", z = "Treatment", axes = c(1, 2)
)
A$PC1 <- A$PC1 + ggtitle("PC1: Day:Treatment effect alone")
A$PC2 <- A$PC2 + ggtitle("PC2: Day:Treatment effect alone")
grid.arrange(A$PC1, A$PC2, ncol = 2)

A <- lmpEffectPlot(resASCA,
    effectName = "Day+Treatment+Day:Treatment",
    x = "Day", z = "Treatment", axes = c(1, 2)
)
A$PC1 <- A$PC1 + ggtitle("PC1: Combined D+T+D:T effects")
A$PC2 <- A$PC2 + ggtitle("PC2: Combined D+T+D:T effects")
grid.arrange(A$PC1, A$PC2, ncol = 2)

resAPCA <- lmpPcaEffects(
    resLmpEffectMatrices =
        resLmpEffectMatrices, method = "APCA"
)

# Day Effect
lmpScorePlot(resAPCA,
    effectNames = "Day",
    color = "Day", shape = "Day", drawShapes = "ellipse"
)

# Treatment Effect
lmpScorePlot(resAPCA,
    effectNames = "Treatment",
    color = "Treatment", shape = "Treatment", drawShapes = "ellipse"
)

Warning: Removed 30 rows containing missing values or values outside the scale range
(`geom_path()`).

# Exposure Effect
lmpScorePlot(resAPCA,
    effectNames = "Exposure",
    color = "Exposure", shape = "Exposure", drawShapes = "ellipse"
)

# Day:Treatment Effect
lmpScorePlot(resAPCA,
    effectNames = "Day:Treatment",
    color = "Treatment", shape = "Day", drawShapes = "polygon"
)

Univariate ANOVA

This part of the vignette links the results of the ASCA/APCA analysis to a more classical analysis applied in -omics transcriptomic data analysis (see e.g. limma package).

An ANOVA model is fitted to each response separately and then, for each model effect, p-values of effect significance are corrected by FDR and ordered. These results allow to detect for which responses each effect of interest is significant (e.g. find which gene is a potential biomarker to differentiate patients with or without a given disease).

The code below applies such analysis and compares then responses importance based on FDR p-values to response loadings obtained by ASCA/APCA.

# Creation of a matrix to store the p-values
m <- ncol(trout_mean$outcomes)
mat_pval <- matrix(nrow = m, ncol = 6)
dimnames(mat_pval) <- list(
    dimnames(trout_mean$outcomes)[[2]],
    c(
        "Day", "Treatment", "Exposure", "Day:Treatment",
        "Day:Exposure", "Treatment:Exposure"
    )
)

# Parallel ANOVA modeling
for (i in 1:m) {
    data <- cbind(y = trout_mean$outcomes[, i], trout_mean$design)
    Modl <- lm(y ~ Day + Treatment + Exposure + Day:Treatment + Day:Exposure + Treatment:Exposure,
        contrasts = list(Day = contr.sum, Treatment = contr.sum, Exposure = contr.sum),
        data = data
    )
    tabanova <- Anova(Modl, type = 3)
    mat_pval[i, ] <- tabanova[2:7, 4]
}

# FDR p-values correction
for (i in 1:6) mat_pval[, i] <- p.adjust(mat_pval[, i], method = "BH")

pander(mat_pval)

heatmap(-log10(mat_pval),
    Rowv = NA, Colv = "Rowv",
    cexCol = 0.8, scale = "none", main = "Heatmap of -log10(q-values)"
)

Effects <- c("Day", "Treatment", "Day:Treatment")

for (i in 1:3) {
    Pval_log <- -log10(mat_pval[, Effects[i]])
    resA <- resASCA[[Effects[i]]]
    PC12Load <- as.vector(sqrt(resA$loadings[, 1:2]^2 %*% resA$singvar[1:2]^2))
    matres <- cbind(PC12Load, Pval_log)
    A[[i]] <- plotScatter(
        Y = matres, xy = c(1, 2),
        points_labs = rownames(matres),
        xlab = "PC1&2 ASCA loadings", ylab = "-log10(p-values)",
        title = paste(Effects[i], "effect")
    )
}

A[[1]]

Warning: ggrepel: 1 unlabeled data points (too many overlaps). Consider
increasing max.overlaps

A[[2]]

A[[3]]

Warning: ggrepel: 6 unlabeled data points (too many overlaps). Consider
increasing max.overlaps

sessionInfo

sessionInfo()

R version 4.4.1 (2024-06-14)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 24.04.1 LTS

Matrix products: default
BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3 
LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0

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

time zone: Etc/UTC
tzcode source: system (glibc)

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

other attached packages:
[1] car_3.1-3        carData_3.0-5    pander_0.6.5     gridExtra_2.3   
[5] limpca_1.3.0     ggplot2_3.5.1    BiocStyle_2.35.0

loaded via a namespace (and not attached):
 [1] tidyselect_1.2.1            dplyr_1.1.4                
 [3] farver_2.1.2                fastmap_1.2.0              
 [5] digest_0.6.37               lifecycle_1.0.4            
 [7] magrittr_2.0.3              compiler_4.4.1             
 [9] rlang_1.1.4                 sass_0.4.9                 
[11] tools_4.4.1                 utf8_1.2.4                 
[13] yaml_2.3.10                 knitr_1.48                 
[15] S4Arrays_1.7.1              labeling_0.4.3             
[17] DelayedArray_0.33.1         plyr_1.8.9                 
[19] abind_1.4-8                 withr_3.0.2                
[21] purrr_1.0.2                 BiocGenerics_0.53.1        
[23] sys_3.4.3                   grid_4.4.1                 
[25] stats4_4.4.1                fansi_1.0.6                
[27] colorspace_2.1-1            scales_1.3.0               
[29] iterators_1.0.14            SummarizedExperiment_1.37.0
[31] cli_3.6.3                   rmarkdown_2.28             
[33] crayon_1.5.3                generics_0.1.3             
[35] httr_1.4.7                  reshape2_1.4.4             
[37] cachem_1.1.0                stringr_1.5.1              
[39] zlibbioc_1.52.0             parallel_4.4.1             
[41] BiocManager_1.30.25         XVector_0.47.0             
[43] matrixStats_1.4.1           vctrs_0.6.5                
[45] Matrix_1.7-1                jsonlite_1.8.9             
[47] IRanges_2.41.0              S4Vectors_0.45.0           
[49] ggrepel_0.9.6               Formula_1.2-5              
[51] maketools_1.3.1             foreach_1.5.2              
[53] jquerylib_0.1.4             tidyr_1.3.1                
[55] glue_1.8.0                  codetools_0.2-20           
[57] stringi_1.8.4               gtable_0.3.6               
[59] GenomeInfoDb_1.43.0         GenomicRanges_1.59.0       
[61] UCSC.utils_1.3.0            munsell_0.5.1              
[63] tibble_3.2.1                pillar_1.9.0               
[65] htmltools_0.5.8.1           GenomeInfoDbData_1.2.13    
[67] R6_2.5.1                    doParallel_1.0.17          
[69] evaluate_1.0.1              tidyverse_2.0.0            
[71] lattice_0.22-6              Biobase_2.67.0             
[73] highr_0.11                  ggsci_3.2.0                
[75] bslib_0.8.0                 Rcpp_1.0.13-1              
[77] SparseArray_1.7.0           xfun_0.49                  
[79] MatrixGenerics_1.19.0       buildtools_1.0.0           
[81] pkgconfig_2.0.3            

References

Benaiche, N., (2022), Stabilisation of the R package LMWiRe – Linear Models for Wide Responses. UCLouvain, http://hdl.handle.net/2078.1/thesis:33996

Cornet, V., Ouaach, A., Mandiki, S., Flamion, E., Ferain, A., Van Larebeke, M., Lemaire, B., Reyes Lopez F., Tort, L., Larondelle, Y. and Kestemont, P., (2018), Environmentally-realistic concentration of cadmium combined with polyunsaturated fatty acids enriched diets modulated non-specific immunity in rainbow trout. Aquatic Toxicology, 196, 104–116. https://doi.org/10.1016/j.aquatox.2018.01.012

Thiel, M., Feraud, B. and Govaerts, B. (2017), ASCA+ and APCA+: exten- sions of ASCA and APCA in the analysis of unbalanced multifactorial designs. Journal of Chemometrics. 31 https://doi.org/10.1002/cem.2895