Package loading

library(pander)
library(gridExtra)
library(ggplot2)
library(SummarizedExperiment)

1 Introduction

The model used in this example is a three-way ANOVA with fixed effects. This document presents all the usual steps of the analysis, from importing the data to visualising the results. Details on the methods used and the package implementation can be found in the articles of Thiel, Féraud, and Govaerts (2017), Guisset, Martin, and Govaerts (2019) and Thiel et al. (2023).

2 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")

3 Data import

In order to use the limpca core functions, the data need to be formatted as a list (informally called an lmpDataList) with the following elements: outcomes (multivariate matrix), design (data.frame) and formula (character string). The UCH data set is already formatted appropriately and can be loaded from limpca with the data function.

data("UCH")

str(UCH)
List of 3
 $ design  :'data.frame':   34 obs. of  5 variables:
  ..$ Hippurate: Factor w/ 3 levels "0","1","2": 1 1 1 1 1 1 2 2 2 2 ...
  ..$ Citrate  : Factor w/ 3 levels "0","2","4": 1 1 2 2 3 3 1 1 2 2 ...
  ..$ Dilution : Factor w/ 1 level "diluted": 1 1 1 1 1 1 1 1 1 1 ...
  ..$ Day      : Factor w/ 2 levels "2","3": 1 1 1 1 1 1 1 1 1 1 ...
  ..$ Time     : Factor w/ 2 levels "1","2": 1 2 1 2 1 2 1 2 1 2 ...
 $ outcomes: num [1:34, 1:600] 0.0312 0.0581 0.027 0.0341 0.0406 ...
  ..- attr(*, "dimnames")=List of 2
  .. ..$   : chr [1:34] "M2C00D2R1" "M2C00D2R2" "M2C02D2R1" "M2C02D2R2" ...
  .. ..$ X1: chr [1:600] "9.9917004" "9.9753204" "9.9590624" "9.9427436" ...
 $ formula : chr "outcomes ~ Hippurate + Citrate + Time + Hippurate:Citrate + Time:Hippurate + Time:Citrate + Hippurate:Citrate:Time"

Alternatively, the lmpDataList can be created with the function data2LmpDataList :

  • from scratch:
UCH2 <- data2LmpDataList(outcomes = UCH$outcomes, design = UCH$design,
    formula = UCH$formula)
| dim outcomes: 34x600
| formula: ~ Hippurate + Citrate + Time + Hippurate:Citrate + Time:Hippurate + Time:Citrate + Hippurate:Citrate:Time
| design variables (5): 
* Hippurate (factor)
* Citrate (factor)
* Dilution (factor)
* Day (factor)
* Time (factor)
  • or from a SummarizedExperiment:
se <- SummarizedExperiment(assays = list(counts = t(UCH$outcomes)),
    colData = UCH$design, metadata = list(formula = UCH$formula))

UCH3 <- data2LmpDataList(se, assay_name = "counts")
| dim outcomes: 34x600
| formula: ~ Hippurate + Citrate + Time + Hippurate:Citrate + Time:Hippurate + Time:Citrate + Hippurate:Citrate:Time
| design variables (5): 
* Hippurate (factor)
* Citrate (factor)
* Dilution (factor)
* Day (factor)
* Time (factor)

SummarizedExperiment is a generic data container that stores rectangular matrices of experimental results. See Morgan et al. (2023) for more information.

4 Data exploration

The UCH (Urine-Citrate-Hippurate) data set is described in Thiel, Féraud, and Govaerts (2017) and Guisset, Martin, and Govaerts (2019) and is issued form a metabolomics experiment. In this experiment, 36 samples of a pool of rat urine samples were spiked with two molecules Citrate and Hippurate according to a \(3^2\) full factorial design in the quantities of these two molecules. The spiked samples were analyzed by 1H NMR at two different time after defrozing and over two different days. Two of the spectra where finally missing at the end of the experiment.

The UCH data set is a list containing 2 elements:

  • an outcomes matrix with 34 observations of 600 response variables representing the spectra from the 1H NMR spectroscopy,
  • a design matrix with 34 observations and 4 explanatory variables

A formula with the general linear model to be estimated will be added to this list below.

For the purpose of this example, only 3 factors of interest will be studied : concentrations of Hippurate and Citrate and Time after defrozing.

Here are the limpca functions that are available for data exploration.

limpca data exploration functions
limpca data exploration functions

4.1 Design

The design matrix contains the information about each observation for the four variables: Hippurate, Citrate, Day and Time. Only 3 of these variables are used in the model. The function plotDesign is useful to visualise the design.

pander(head(UCH$design))
  Hippurate Citrate Dilution Day Time
M2C00D2R1 0 0 diluted 2 1
M2C00D2R2 0 0 diluted 2 2
M2C02D2R1 0 2 diluted 2 1
M2C02D2R2 0 2 diluted 2 2
M2C04D2R1 0 4 diluted 2 1
M2C04D2R2 0 4 diluted 2 2
plotDesign(design = UCH$design, x = "Hippurate", y = "Citrate",
    rows = "Time", title = "Design of the UCH dataset")

This plot confirms that the design is a full 3x3x2 factorial design replicated twice with 2 missing values. Hence, the design is not balanced.

4.2 Outcomes visualization

The 600 response (outcomes) variables represent, for each observation, the intensities of the 1H NMR spectra. These spectra can be visualized by the plotLine function.

4.2.1 plotLine function

Here, annotations (cit_peaks and hip_peaks) are added to the ggplot objects in order to highlight the Hippurate (green) and Citrate (red) peaks.

p1 <- plotLine(Y = UCH$outcomes, title = "H-NMR spectrum", rows = c(3),
    xlab = "ppm", ylab = "Intensity")

cit_peaks <- annotate("rect", xmin = c(2.509), xmax = c(2.709),
    ymin = -Inf, ymax = Inf, alpha = 0.2, fill = c("tomato"))

hip_peaks <- annotate("rect", xmin = c(7.458, 3.881), xmax = c(7.935,
    4.041), ymin = -Inf, ymax = Inf, alpha = 0.2, fill = c("yellowgreen"))

p1 <- plotLine(Y = UCH$outcomes, title = "H-NMR spectrum", rows = c(3),
    xlab = "ppm", ylab = "Intensity")

p1 + cit_peaks + hip_peaks

4.2.2 plotScatter function

The plotScatter function enables to visualize the values of two outcomes variables with different colors or markers for the values of the design factors. Here, it is used to show that the \(3^2\) factorial design can be recovered from the intensities of the Hippurate and Citrate peaks in the spectra.

# xy corresponds to citrate (453) and hippurate peaks (369)
plotScatter(Y = UCH$outcomes, xy = c(453, 369), design = UCH$design,
    color = "Hippurate", shape = "Citrate")

# Or refering to the variables names (ppm values here)
plotScatter(Y = UCH$outcomes, xy = c("2.6092056", "3.9811536"),
    design = UCH$design, color = "Hippurate", shape = "Citrate")

4.2.3 plotScatterM function

The plotScatter function allows to visualize the values of a series of outcomes variables with different colors or markers for the values of the design factors. Here, it is done for the 4 peaks of Hippurate and single peak of Citrate.

plotScatterM(Y = UCH$outcomes, cols = c(133, 145, 150, 369, 453),
    design = UCH$design, varname.colorup = "Hippurate", varname.colordown = "Citrate")

4.2.4 plotMeans function

The plotMeans represents the mean values of a response variable for different levels of the design factors. Here we show the evolution of the Citrate peak height with respect to the three design factors of interest. Note that the results of this function must be interpreted with caution for unbalanced designs because simple means are biased estimators of expected means.

plotMeans(Y = UCH$outcomes, design = UCH$design, cols = c(453),
    x = c("Citrate"), w = c("Hippurate"), z = c("Time"), ylab = "Intensity",
    title = c("Mean reponse for main Citrate peak"))
$`2.6092056`

4.3 PCA

The function pcaBySvd is useful to compute a Principal Component Analysis (PCA) decomposition of the outcomes matrix. Several functions can be applied to the output value of pcaBySvd:

  • pcaScreePlot to obtaine a scree plot
  • pcaLoading1dPlot for the loading plots
  • pcaScorePlot for the score plots
ResPCA <- pcaBySvd(UCH$outcomes)
pcaScreePlot(ResPCA, nPC = 6)

The score plots below indicate that all tree factors from the design affect the spectral profiles, which will be more clearly highlighted by ASCA and APCA.

pcaScorePlot(resPcaBySvd = ResPCA, axes = c(1, 2), title = "PCA scores plot: PC1 and PC2",
    design = UCH$design, color = "Hippurate", shape = "Citrate",
    points_labs_rn = FALSE)

pcaScorePlot(resPcaBySvd = ResPCA, axes = c(1, 2), title = "PCA scores plot: PC1 and PC2",
    design = UCH$design, color = "Time", shape = "Hippurate",
    points_labs_rn = FALSE)

pcaScorePlot(resPcaBySvd = ResPCA, axes = c(3, 4), title = "PCA scores plot: PC3 and PC4",
    design = UCH$design, color = "Time", shape = "Citrate", points_labs_rn = FALSE)

In the first two loading plots, a mixture of Citrate and Hippurate peaks already appears but they are not separated.

p2 <- pcaLoading1dPlot(resPcaBySvd = ResPCA, axes = c(1, 2),
    title = "PCA loadings plot UCH", xlab = "ppm", ylab = "Intensity")

p2 + hip_peaks + cit_peaks

5 Application of ASCA+ and APCA+

Here below, ASCA+ and APCA+ steps are illustrated on the UCH data set. The following graph represents the steps of the methodology.
ASCA+/APCA+ methodology The next graph presents the functions available in limpca for this purpose. They are all illustrated in the next sections.

limpca ASCA/APCA functions
limpca ASCA/APCA functions

6 Model estimation and effect matrix decomposition

6.1 Model formula

The formula of the ANOVA-GLM model used in this analysis is the 3 ways crossed ANOVA model:

UCH$formula <- "outcomes ~ Hippurate + Citrate + Time + Hippurate:Citrate +
                  Time:Hippurate + Time:Citrate + Hippurate:Citrate:Time"

6.2 Model matrix generation

The first step of ASCA+ is to build the (GLM) model matrix from the experimental design matrix and the model. Each factor is reencoded with multiple binary variables using contr.sum coding. The size of the model matrix is 34xp with p being the total number of parameters in the ANOVA model for one response.

The function lmpModelMatrix encodes the design matrix as a model matrix.

resLmpModelMatrix <- lmpModelMatrix(UCH)
pander::pander(head(resLmpModelMatrix$modelMatrix))
Table continues below
  (Intercept) Hippurate1 Hippurate2 Citrate1 Citrate2
M2C00D2R1 1 1 0 1 0
M2C00D2R2 1 1 0 1 0
M2C02D2R1 1 1 0 0 1
M2C02D2R2 1 1 0 0 1
M2C04D2R1 1 1 0 -1 -1
M2C04D2R2 1 1 0 -1 -1
Table continues below
  Time1 Hippurate1:Citrate1 Hippurate2:Citrate1
M2C00D2R1 1 1 0
M2C00D2R2 -1 1 0
M2C02D2R1 1 0 0
M2C02D2R2 -1 0 0
M2C04D2R1 1 -1 0
M2C04D2R2 -1 -1 0
Table continues below
  Hippurate1:Citrate2 Hippurate2:Citrate2 Hippurate1:Time1
M2C00D2R1 0 0 1
M2C00D2R2 0 0 -1
M2C02D2R1 1 0 1
M2C02D2R2 1 0 -1
M2C04D2R1 -1 0 1
M2C04D2R2 -1 0 -1
Table continues below
  Hippurate2:Time1 Citrate1:Time1 Citrate2:Time1
M2C00D2R1 0 1 0
M2C00D2R2 0 -1 0
M2C02D2R1 0 0 1
M2C02D2R2 0 0 -1
M2C04D2R1 0 -1 -1
M2C04D2R2 0 1 1
Table continues below
  Hippurate1:Citrate1:Time1 Hippurate2:Citrate1:Time1
M2C00D2R1 1 0
M2C00D2R2 -1 0
M2C02D2R1 0 0
M2C02D2R2 0 0
M2C04D2R1 -1 0
M2C04D2R2 1 0
  Hippurate1:Citrate2:Time1 Hippurate2:Citrate2:Time1
M2C00D2R1 0 0
M2C00D2R2 0 0
M2C02D2R1 1 0
M2C02D2R2 -1 0
M2C04D2R1 -1 0
M2C04D2R2 1 0

6.3 Model estimation and effect matrices decomposition

lmpEffectMatrices is the used to estimate the linear model and decomposes the multivariate outcomes into effect matrices for every model term. This function calculates also type III effect contributions (in %) and generates a barpot to visualise these contributions.

resLmpEffectMatrices <- lmpEffectMatrices(resLmpModelMatrix)

6.4 Effects importance

The contributions from each effect is outputted from lmpEffectMatrices.

pander(resLmpEffectMatrices$variationPercentages)
Table continues below
Hippurate Citrate Time Hippurate:Citrate Hippurate:Time
39.31 29.91 16.24 1.543 6.229
Citrate:Time Hippurate:Citrate:Time Residuals
0.5387 1.684 4.298
resLmpEffectMatrices$varPercentagesPlot

6.5 Bootstrap tests and quantification of effects importance

lmpBootstrapTests applies a parametric bootstrap test to determine whether an effect is significant or not. We recommend the user to choose first a small value of nboot (e.g. nboot=100) to develop its code and increase it later on (e.g. nboot=1000) in order to get an accurate value for the p-values.

resLmpBootstrapTests <- lmpBootstrapTests(resLmpEffectMatrices = resLmpEffectMatrices,
    nboot = 100)

# Print P-values
pander::pander(t(resLmpBootstrapTests$resultsTable))
Table continues below
  Hippurate Citrate Time Hippurate:Citrate
% of variance (T III) 39.31 29.91 16.24 1.54
Bootstrap p-values < 0.01 < 0.01 < 0.01 0.19
Table continues below
  Hippurate:Time Citrate:Time
% of variance (T III) 6.23 0.54
Bootstrap p-values < 0.01 0.44
  Hippurate:Citrate:Time Residuals
% of variance (T III) 1.68 4.30
Bootstrap p-values 0.14 -

7 ASCA/APCA/ASCA-E decomposition

The ASCA/APCA/ASCA-E decomposition enables to represent the information from the effect matrices in a space of reduced dimension through PCA. The function lmpPcaEffects has a method argument to define which method to use, namely ASCA, APCA or ASCA-E. Combined effects matrices can also be built and visualized by PCA.

7.1 ASCA

The ASCA method performs PCA on the pure effect matrices. Here a combined effect matrix Hippurate+Time+Hippurate:Time is also built.

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

7.1.1 Contributions

The contribution of each principal component of the effects is calculated and reported in different tables and plots with the function lmpContributions.

resLmpContributions <- lmpContributions(resASCA)

The tables are:

  • totalContribTable: Table of the contribution of each effect to the total variance in percentage as outputted from lmpEffectMatrices.
pander::pander(resLmpContributions$totalContribTable)
  Percentage of Variance
Hippurate 39.31
Citrate 29.91
Time 16.24
Hippurate:Citrate 1.54
Hippurate:Time 6.23
Citrate:Time 0.54
Hippurate:Citrate:Time 1.68
Residuals 4.3
  • effectTable: Table of the percentage of variance explained by each Principal Component in each model effect decomposition.
pander::pander(resLmpContributions$effectTable)
  PC1 PC2 PC3 PC4 PC5 Sum
Hippurate 97.71 2.29 0 0 0 100
Citrate 98.22 1.78 0 0 0 100
Time 100 0 0 0 0 100
Hippurate:Citrate 44.01 38.51 15.13 2.34 0 99.99
Hippurate:Time 93.92 6.08 0 0 0 100
Citrate:Time 90.76 9.24 0 0 0 100
Hippurate:Citrate:Time 47.23 27.49 22.6 2.68 0 100
Residuals 48.54 16.9 10.28 5.93 4.32 85.97
  • combinedEffectTable: Equivalent of the previous effectTable but for the combination of effects mentioned in lmpPcaEffects, here for Hippurate+Time+Hippurate:Time.
pander::pander(resLmpContributions$combinedEffectTable)
  PC1 PC2 PC3 PC4 PC5 Sum
Hippurate+Time+Hippurate:Time 62.95 26.32 10.09 0.48 0.17 100
Residuals 48.54 16.9 10.28 5.93 4.32 85.97
  • contribTable: Table of the percentage of variance explained by each Principal Component of each effect reported to the percentage contribution of the given effect to the total variance. This table gives an overview of the importance of each PC regardless of the effects.
pander::pander(resLmpContributions$contribTable)
  PC1 PC2 PC3 PC4 PC5 Contrib
Hippurate 38.41 0.9 0 0 0 39.31
Citrate 29.37 0.53 0 0 0 29.91
Time 16.24 0 0 0 0 16.24
Hippurate:Citrate 0.68 0.59 0.23 0.04 0 1.54
Hippurate:Time 5.85 0.38 0 0 0 6.23
Citrate:Time 0.49 0.05 0 0 0 0.54
Hippurate:Citrate:Time 0.8 0.46 0.38 0.05 0 1.68
Residuals 2.09 0.73 0.44 0.25 0.19 4.3
  • Moreover lmpContributions also produces a barplot either with the ordered contributions per effect (plotTotal) or across all the PCs of the different effects (plotContrib).
pander("Ordered contributions per effect:")

Ordered contributions per effect:

resLmpContributions$plotTotal

pander("For each PC of the different effects:")

For each PC of the different effects:

resLmpContributions$plotContrib

In the following analysis, we will focus only on the first PC of the three main effects, the interaction Hippurate:Time and the residuals since the other PCs and effects account for less than 1% of the total variance.

7.1.2 Scores and loadings Plots

The loadings can be represented either on a line plot with the function lmpLoading1dPlot to conveniently compare them with the original spectral profiles or on a scatterplot with the function lmpLoading2dPlot.

Here we create an object including the loading plots (as line plots) for all the effects included in the model as well as the combined effect and the residuals.

all_loadings_pl <- lmpLoading1dPlot(resASCA, effectNames = c("Hippurate",
    "Citrate", "Time", "Hippurate:Time", "Hippurate+Time+Hippurate:Time",
    "Residuals"), axes = 1, xlab = "ppm")

The score matrices are represented two components at a time on a scatterplot with the function lmpScorePlot.

7.1.2.1 Main effects

The scores and loadings of the main effects included in the model are represented below.

They show that, thank to the ASCA analysis, Hippurate and Citrate peaks are clearly and separately retrieved in the respective loading plots compared to the classical PCA (see Section 4.3) where peaks of these two chemicals are both present in the two first PCs. The loading profile of the Time effect shows a high peak on the left side of the removed water area.

# Hippurate
hip_scores_pl <- lmpScorePlot(resASCA, effectNames = "Hippurate",
    color = "Hippurate", shape = "Hippurate")

hip_loadings_pl <- all_loadings_pl$Hippurate + hip_peaks

grid.arrange(hip_scores_pl, hip_loadings_pl, ncol = 2)

# Citrate
cit_scores_pl <- lmpScorePlot(resASCA, effectNames = "Citrate",
    color = "Citrate", shape = "Citrate")
cit_loadings_pl <- all_loadings_pl$Citrate + cit_peaks

grid.arrange(cit_scores_pl, cit_loadings_pl, ncol = 2)

# Time
tim_scores_pl <- lmpScorePlot(resASCA, effectNames = "Time",
    color = "Time", shape = "Time")
Warning in FUN(X[[i]], ...): The variance of PC2 is inferior to 1%. Graph
scaled
time_peaks <- annotate("rect", xmin = c(5.955364), xmax = c(6.155364),
    ymin = -Inf, ymax = Inf, alpha = 0.2, fill = c("royalblue"))

tim_loadings_pl <- all_loadings_pl$Time + time_peaks

grid.arrange(tim_scores_pl, tim_loadings_pl, ncol = 2)

7.1.2.2 Interaction Hippurate:Time

The scores and loadings fot the interaction term is represented here. It is not straighforward to interpret the scores plot of such an interaction term but the loadings of PC1 indicate a shift in the spectrum, along the whole spectral profile (but mostly around 3 ppm).

# Hippurate:Time
hiptim_scores_pl <- lmpScorePlot(resASCA, effectNames = "Hippurate:Time",
    color = "Hippurate", shape = "Time")
hiptim_loadings_pl <- all_loadings_pl$`Hippurate:Time` + time_peaks +
    hip_peaks

grid.arrange(hiptim_scores_pl, hiptim_loadings_pl, ncol = 2)

7.1.2.3 Combination of effects Hippurate+Time+Hippurate:Time

The scores and the loadings of a combination of effects, in this case "Hippurate+Time+Hippurate:Time" can also be visualised.

# Hippurate+Time+Hippurate:Time
hiptimInter_scores_pl <- lmpScorePlot(resASCA, effectNames = "Hippurate+Time+Hippurate:Time",
    color = "Hippurate", shape = "Time")

hiptimInter_loadings_pl <- all_loadings_pl$`Hippurate:Time` +
    time_peaks + hip_peaks

grid.arrange(hiptimInter_scores_pl, hiptimInter_loadings_pl,
    ncol = 2)

However, note that a better graphical representation is possible with the function lmpEffectPlot (see Section 7.1.4) to depict interaction terms or combinations of effects.

7.1.2.4 Model residuals

PCA on the model residuals can also be applied to spot outliers or an unexpected underlying structure of the data. The scores plot shows that the effect of Day, which was excluded in our modeling step, could have been included as well as it spans the two first PCs of the residuals.

resid_scores_pl <- lmpScorePlot(resASCA, effectNames = "Residuals",
    color = "Day", shape = "Day", drawShapes = "segment")


resid_loadings_pl <- all_loadings_pl$Residuals

grid.arrange(resid_scores_pl, resid_loadings_pl, ncol = 2)

7.1.3 Other graphs

7.1.3.1 Scores scatter plot

We can also represent the scores as a matrix of plots with lmpScoreScatterPlotM. This graph has the advantage to compare more than 2 variables simultaneously. For example, the PC1 scores of Hippurate and Citrate clearly represent the orthogonal design of this experiment. The interaction Hippurate:Time can also be clearly represented when comparing les PC1s of Hippurate and the interaction term.

lmpScoreScatterPlotM(resASCA, PCdim = c(1, 1, 1, 1, 1, 1, 1,
    2), modelAbbrev = TRUE, varname.colorup = "Citrate", varname.colordown = "Time",
    varname.pchup = "Hippurate", varname.pchdown = "Time", title = "ASCA scores scatterplot matrix")

7.1.3.2 2D Loadings

Finally the loadings can also be represented 2-by-2 as a scatterplot with lmpLoading2dPlot. Such graphic is of course of limited interest for spectral data.

lmpLoading2dPlot(resLmpPcaEffects = resASCA, effectNames = c("Hippurate"),
    axes = c(1, 2), addRownames = TRUE, pl_n = 10)
Warning: ggrepel: 1 unlabeled data points (too many overlaps). Consider
increasing max.overlaps

# adding manually labels to points for the Hippurate peaks
labels <- substr(colnames(UCH$outcomes), 1, 4)
labels[-c(369, 132, 150, 133, 149, 144, 145, 368, 151)] <- ""

lmpLoading2dPlot(resLmpPcaEffects = resASCA, effectNames = c("Hippurate"),
    axes = c(1, 2), points_labs = labels)
Warning: ggrepel: 1 unlabeled data points (too many overlaps). Consider
increasing max.overlaps

7.1.4 Effects plot

The lmpEffectPlot function is particularly interesting to visualise an interaction term or a combination of effects. Note that this function can only be applied with the ASCA method.

7.1.4.1 Main effects for Hippurate

Here the first PC of the Hippurate is represented along the levels of this effect showing an expected linear effect of Hippurate dose on the PC value.

lmpEffectPlot(resASCA, effectName = "Hippurate", x = "Hippurate")

7.1.4.2 Interaction Hippurate:Time

A more interesting application is the visualization of interaction terms as line plots, here Hippurate:Time along the Hippurate or the Time effect. This representation gives the impression of a quite important interaction effect.

lmpEffectPlot(resASCA, effectName = "Hippurate:Time", x = "Hippurate",
    z = "Time")

lmpEffectPlot(resASCA, effectName = "Hippurate:Time", x = "Time",
    z = "Hippurate")

7.1.4.3 Combination of effects Hippurate+Time+Hippurate:Time

An alternative visualisation of this interaction combines the main effects of Hippurate, Time and the interaction Hippurate:Time. Even if significant, the interaction effect is actually quite small compared to the main effects (Hippurate for PC1 and Time for PC2).

lmpEffectPlot(resASCA, effectName = "Hippurate+Time+Hippurate:Time",
    x = "Hippurate", z = "Time")

lmpEffectPlot(resASCA, effectName = "Hippurate+Time+Hippurate:Time",
    axes = c(2), x = "Time", z = "Hippurate")

7.2 APCA

The APCA method performs PCA on the effect matrices augmented by the residuals. The same functions already used for ASCA can be applied.

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

7.2.1 Scores Plot

Different shapes with the drawShapes argument highlight the data structure recovery.

# Hippurate main effect
lmpScorePlot(resAPCA, effectNames = "Hippurate", color = "Hippurate",
    shape = "Hippurate", drawShapes = "ellipse")

# Citrate main effect
lmpScorePlot(resAPCA, effectNames = "Citrate", color = "Citrate",
    shape = "Citrate", drawShapes = "ellipse")

# Time main effect
lmpScorePlot(resAPCA, effectNames = "Time", color = "Time", shape = "Time",
    drawShapes = "ellipse")

lmpScorePlot(resAPCA, effectNames = "Time", color = "Time", shape = "Time",
    drawShapes = "polygon")

lmpScorePlot(resAPCA, effectNames = "Time", color = "Time", shape = "Time",
    drawShapes = "segment")

# Interaction term
lmpScorePlot(resAPCA, effectNames = "Hippurate:Time", color = "Hippurate",
    shape = "Time", drawShapes = "segment")

lmpScorePlot(resAPCA, effectNames = "Hippurate:Time", color = "Hippurate",
    shape = "Time", drawShapes = "polygon")

A scatterplot matrix can also be applied to visualise the relationship between the scores profiles of interest.

lmpScoreScatterPlotM(resAPCA, effectNames = c("Hippurate", "Citrate",
    "Time", "Hippurate:Time"), modelAbbrev = TRUE, varname.colorup = "Citrate",
    varname.colordown = "Time", varname.pchup = "Hippurate",
    varname.pchdown = "Time", title = "APCA scores scatterplot matrix")

7.2.2 Loadings plot

Note that the APCA loadings contain the residuals of the model and differ from the ASCA loadings in that respect.

lmpLoading1dPlot(resAPCA, effectNames = c("Hippurate", "Citrate",
    "Time", "Hippurate:Time"), axes = 1)
$Hippurate


$Citrate


$Time


$`Hippurate:Time`

7.3 ASCA-E

The ASCA-E method performs PCA on the pure effect matrices then projects the effect matrices augmented with residuals on the PC obtained.

resASCAE <- lmpPcaEffects(resLmpEffectMatrices = resLmpEffectMatrices,
    method = "ASCA-E")

The contributions and loadings are identical to the ASCA results.

7.3.1 Scores Plot

For the main effects, all score plots show a clear separation between the different levels of the considered effects on the first PC. This separation of the groups suggests a strong effect of those factors, confirmed by their significance.

lmpScorePlot(resASCAE, effectNames = "Hippurate", color = "Hippurate",
    shape = "Hippurate")

lmpScorePlot(resASCAE, effectNames = "Citrate", color = "Citrate",
    shape = "Citrate")

lmpScorePlot(resASCAE, effectNames = "Time", color = "Time",
    shape = "Time")
Warning in FUN(X[[i]], ...): The variance of PC2 is inferior to 1%. Graph
scaled

lmpScorePlot(resASCAE, effectNames = "Hippurate:Time", color = "Hippurate",
    shape = "Time")

Again, the scores profiles can be compared 2 by 2 with ASCA-E.

lmpScoreScatterPlotM(resASCAE, effectNames = c("Hippurate", "Citrate",
    "Time", "Hippurate:Time"), modelAbbrev = TRUE, varname.colorup = "Citrate",
    varname.colordown = "Time", varname.pchup = "Hippurate",
    varname.pchdown = "Time", title = "ASCA-E scores scatterplot matrix")

8 sessionInfo

sessionInfo()
R Under development (unstable) (2024-10-21 r87258)
Platform: x86_64-pc-linux-gnu
Running under: Ubuntu 24.04.1 LTS

Matrix products: default
BLAS:   /home/biocbuild/bbs-3.21-bioc/R/lib/libRblas.so 
LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.12.0

locale:
 [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C              
 [3] LC_TIME=en_GB              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: America/New_York
tzcode source: system (glibc)

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

other attached packages:
 [1] SummarizedExperiment_1.37.0 Biobase_2.67.0             
 [3] GenomicRanges_1.59.0        GenomeInfoDb_1.43.0        
 [5] IRanges_2.41.0              S4Vectors_0.45.0           
 [7] BiocGenerics_0.53.0         MatrixGenerics_1.19.0      
 [9] matrixStats_1.4.1           car_3.1-3                  
[11] carData_3.0-5               pander_0.6.5               
[13] gridExtra_2.3               limpca_1.3.0               
[15] ggplot2_3.5.1               BiocStyle_2.35.0           

loaded via a namespace (and not attached):
 [1] tidyselect_1.2.1        dplyr_1.1.4             farver_2.1.2           
 [4] fastmap_1.2.0           digest_0.6.37           lifecycle_1.0.4        
 [7] magrittr_2.0.3          compiler_4.5.0          rlang_1.1.4            
[10] sass_0.4.9              tools_4.5.0             utf8_1.2.4             
[13] yaml_2.3.10             knitr_1.48              S4Arrays_1.7.0         
[16] labeling_0.4.3          DelayedArray_0.33.0     plyr_1.8.9             
[19] abind_1.4-8             withr_3.0.2             purrr_1.0.2            
[22] grid_4.5.0              fansi_1.0.6             colorspace_2.1-1       
[25] scales_1.3.0            iterators_1.0.14        cli_3.6.3              
[28] rmarkdown_2.28          crayon_1.5.3            generics_0.1.3         
[31] httr_1.4.7              reshape2_1.4.4          cachem_1.1.0           
[34] stringr_1.5.1           zlibbioc_1.53.0         parallel_4.5.0         
[37] BiocManager_1.30.25     formatR_1.14            XVector_0.47.0         
[40] vctrs_0.6.5             Matrix_1.7-1            jsonlite_1.8.9         
[43] bookdown_0.41           ggrepel_0.9.6           Formula_1.2-5          
[46] foreach_1.5.2           jquerylib_0.1.4         tidyr_1.3.1            
[49] glue_1.8.0              codetools_0.2-20        stringi_1.8.4          
[52] gtable_0.3.6            UCSC.utils_1.3.0        munsell_0.5.1          
[55] tibble_3.2.1            pillar_1.9.0            htmltools_0.5.8.1      
[58] GenomeInfoDbData_1.2.13 R6_2.5.1                doParallel_1.0.17      
[61] evaluate_1.0.1          tidyverse_2.0.0         lattice_0.22-6         
[64] highr_0.11              ggsci_3.2.0             bslib_0.8.0            
[67] Rcpp_1.0.13             SparseArray_1.7.0       xfun_0.48              
[70] pkgconfig_2.0.3        

References

Guisset, Séverine, Manon Martin, and Bernadette Govaerts. 2019. “Comparison of PARAFASCA, AComDim, and AMOPLS Approaches in the Multivariate GLM Modelling of Multi-Factorial Designs.” Chemometrics and Intelligent Laboratory Systems 184: 44–63. https://doi.org/https://doi.org/10.1016/j.chemolab.2018.11.006.
Morgan, Martin, Valerie Obenchain, Jim Hester, and Hervé Pagès. 2023. SummarizedExperiment: SummarizedExperiment Container. https://doi.org/10.18129/B9.bioc.SummarizedExperiment.
Thiel, Michel, Nadia Benaiche, Manon Martin, Sébastien Franceschini, Robin Van Oirbeek, and Bernadette Govaerts. 2023. “Limpca: An r Package for the Linear Modeling of High-Dimensional Designed Data Based on ASCA/APCA Family of Methods.” Journal of Chemometrics 37 (7): e3482. https://doi.org/https://doi.org/10.1002/cem.3482.
Thiel, Michel, Baptiste Féraud, and Bernadette Govaerts. 2017. “ASCA+ and APCA+: Extensions of ASCA and APCA in the Analysis of Unbalanced Multifactorial Designs.” Journal of Chemometrics 31 (6): e2895. https://doi.org/https://doi.org/10.1002/cem.2895.