In HerveAbdi/R4SPISE2018: A collection of R utilities for the SPISE2018 Advanced Sensory Evaluation meeting
knitr::opts_knit$get()
**NOTE:**
This `pdf` was generated from the vignette
`R4SPISE2018::cheeseMCA` from the `R` Package
`R4SPISE2018`. Check the help for the
very last version of this document.
Prelude
If you want to make sure that you have a clean start,
you can execute the following commands:
rm(list = ls())
graphics.off()
Or, better (see below, preamble), you can use an Rproject for this project.
# Important: Remember
# build the vignettes with devtools::build_vignettes()
knitr::opts_chunk$set(
collapse = TRUE,
fig.width = 9,
comment = "#>"
)
knitr::opts_knit$get()
Preamble
Make sure that you start this analysis as a new Rproject so that
the default directory will be correctly set, and the memory cleared.
Before we start the analysis,
we need to have our standard packages installed (including some from Github)
and the corresponding libraries loaded:
DistatisRPTCA4CTAR4SPISE2018prettyGraphsExPosition
load
# Decomment all/some 0f these lines if the packages are not installed
# devtools::install_github('HerveAbdi/PTCA4CATA')
# devtools::install_github('HerveAbdi/DistatisR')
# install.packages(prettyGraphs)
# install.packages('Matrix')
# install.packages('factoextra')
#
# load the libraries that we will need
suppressMessages(library(Matrix))
suppressMessages(library(DistatisR))
suppressMessages(library(PTCA4CATA))
suppressMessages(library(prettyGraphs))
suppressMessages(library(ExPosition))
suppressMessages(library(factoextra))
Introduction
The data set can be found from the package R4SPISE2018.
The data are stored in an excel file called
beersNovicesExpertsCATA.xlsx
whose location
can be found and stored in the
variable path2file using the R function system.functionfile()
as shown in the following command:
```{R findDataPath}
path2file <- system.file("extdata",
"beersNovicesExpertsCATA.xlsx", package = "R4SPISE2018")
```r", fig.height=3, fig.width=4, include=TRUE, out.width='70%'}
# label.spicesxl = capFigNo
# 
knitr::include_graphics('../man/figures/beerXLS-screen.png')
if you open this excel file, it will look like the
Figure above.
The sorting data are stared in the sheet DataSort.
The first columns gives the names of the products
(here spices) and each column
that were sorted
When you record you own data,
make sure that you follow the same format, this way the script
described in this vignette will apply
to your own analysis with minimal change.
We will first compute the results of the analysis, then create
the graphics, and finally save everything into a powerpoint.
Run the statistical analysis
Read the data
The excel file name and location (i.e., path) are
stored in the variable path2file.
To read the data we will use the function
PTCA4CATA::read.xls.CATAl()
(based upon
the function readxl::read_excel()).
How to save the data file
To save the data file
(under the original name of myDataFile
or maybe a more informative name)
in a directory
(say Downloads). use the following command
saveFile <- file.copy(from = path2file, to = '~/Downloads/myDataFile.xlsx')
Back to the data and norming
In addition we are norming the responses of the Judges
so that each row for each Judges sums to 1.
dataCATA.list <- read.xls.CATA(path2file = path2file,
sheet2read = 'DataCATA',
threshold4cleaning = 5)
# norm
dataCube <- normBrick4PTCA(dataCATA.list$CATA.Brick ,
normingConstant = 100,
normalization = "byRow",
code4Groups = NULL)$normedArray
# get the contingency table from the cube
contingencyTable <- apply(dataCube,c(1,2),sum)
nI <- dim(dataCube)[[1]]
nJ <- dim(dataCube)[[2]]
nK <- dim(dataCube)[[3]]
namesR <- dimnames(dataCube)[[1]]
namesC <- dimnames(dataCube)[[2]]
namesA <- dimnames(dataCube)[[3]]
The variables are organized in block.
First we get the blocks, then we keep only
the variables that passed the threshold for cleaning
(see parameter threshold4cleaning).
The categories of the CATA descriptors are stored in the file
beersNovicesExpertsCATA.xlsx (whose path is stored in the
variable path2file) in the sheet Categories4Descriptors.
We read them with the function readxl::read_excel
catDescriptors.tmp <- as.data.frame(readxl::read_excel(path2file,
'Categories4Descriptors'))
catDescriptors <- catDescriptors.tmp[catDescriptors.tmp[,1] %in% namesC,]
The assessors are also described by several variables.
These data are stored in the file
beersNovicesExpertsCATA.xlsx (whose path is stored in the
variable path2file) in the sheet DescriptionJuges.
We read them with the function readxl::read_excel:
descJudges <- as.data.frame(readxl::read_excel(path2file,
'DescriptionJuges'))
# Color for the descriptors
col4Emotion <- "red4"
col4Context <- "darkmagenta"
col4Taste <- "darkolivegreen"
color4Attributes <- rep("", nJ)
color4Attributes[catDescriptors$Category == 'C'] <- col4Context
color4Attributes[catDescriptors$Category == 'E'] <- col4Emotion
color4Attributes[catDescriptors$Category == 'G'] <- col4Taste
# Color for the products from prettyGraphs
color4Products <- prettyGraphs::prettyGraphsColorSelection(nI)
# Color for the Judges Novices vs Experts
col4Experts <- 'mediumpurple4'
col4Novices <- 'olivedrab3'
color4Participants <- rep(col4Experts, nK)
color4Participants[descJudges$Expertise == 'N'] <- col4Novices
# color4Participants[descJudges$Expertise == 'E'] <- col4Experts
Descriptive approach
Get Cochran Q
# Compute Cochran's Q from the cube of data
Q4CATA <- Q4CATA.Cube(dataCube)
# Create the vector of probability stars
zeStars <- p2Star( Q4CATA['Sidak p(FW)',])
Create a Heat Map (with significant levels)
a.000.zeMapCATA <- makeggHeatMap4CT(
CTstared(contingencyTable,zeStars,'after'),
colorAttributes = color4Attributes,
colorProducts = color4Products)
Print the heat map
print(a.000.zeMapCATA)
CA: descriptive and inferences
Correspondence analysis on the contingency table
# The CA
ResCATACA <- epCA(contingencyTable,
masses = NULL, weights = NULL,
hellinger = FALSE,
symmetric = TRUE, graphs = FALSE)
# Renormalize the results of CA
RenomrFi <- CARenormalization(G = ResCATACA$ExPosition.Data$fi ,
delta = ResCATACA$ExPosition.Data$pdq$Dv,
singularValues = TRUE,
masses = NULL)
Bootstrap Inferences
Bootstrap and bootstrap ratios:
#
bootCA <- Boot4PTCA(ZeDataCube = dataCube,
fi = ResCATACA$ExPosition.Data$fi,
fj = ResCATACA$ExPosition.Data$fj,
eigs = ResCATACA$ExPosition.Data$eigs,
nf2keep = 3,
nBootIter = 500)
# Compute Bootstraped ratios
bootRatios.I <- PTCA4CATA::boot.ratio.test(bootCA$RowsBoot,
critical.value = 2)
bootRatios.J <- PTCA4CATA::boot.ratio.test(bootCA$ColumnsBoot,
critical.value = 2)
# Probabilities
probBR.I <- bootRatios.I$prob.boot.ratios
probBR.J <- bootRatios.J$prob.boot.ratios
Permutation tests
# from PTCA4CATA
#
ResCATACA.inference <- perm4PTCA(aCube = dataCube,
nIter = 1000,
permType = 'byRows' ,
Malinvaud = TRUE)
Ind.Permu <- ResCATACA.inference$permInertia
InertiaFixed <- ResCATACA.inference$fixedInertia
prob.cpt.lst <- ResCATACA.inference$MalinvaudQ['p-perm',]
# Get the p values for the components
prob.cpt <- (unlist(prob.cpt.lst[2:length(prob.cpt.lst)]))
prob.cpt[is.na(prob.cpt)] <- 1
Participants analysis
Cmat <- createCmat4PTCA(dataCube)
# To find the observations Kept:
# dimnames((ZeRawDataCube))[[3]] %in% rownames(Cmat)
eigenCmat <- eigen(Cmat, symmetric = TRUE)
#
eig4Cmat <- eigenCmat$values
tau4Cmat <- round( (100*eig4Cmat) / sum(eig4Cmat))
nk <- 3
F4Cmat <- eigenCmat$vectors[,1:nk] %*% diag(eig4Cmat[1:nk]^(1/2))
rownames(F4Cmat) <- rownames(Cmat)
Graphics
RV analysis on the participants
Scree
# Scree plot with significance
ScreeInf <- PlotScree(ev = eig4Cmat,
max.ev = NULL, alpha = 0.05,
col.ns = "#006D2C", col.sig = "#54278F",
title = "RV Analysis: Explained Variance per Dimension")
a000.00.screeRV <- recordPlot()
Participant maps
Shortnames4Participants <- substr(rownames(F4Cmat),1,1)
F4Plot <- F4Cmat
rownames(F4Plot) <- Shortnames4Participants
# Make labels
labels4RV <- createxyLabels.gen(1,2,lambda = eig4Cmat, tau = tau4Cmat )
#
BaseMap.Participants <- createFactorMap(X = F4Plot ,
axis1 = 1, axis2 = 2,
constraints = NULL,
title = NULL,
col.points = color4Participants,
display.points = TRUE,
pch = 19, cex = 2.5,
display.labels = TRUE,
col.labels = color4Participants,
text.cex = 4, font.face = "bold",
font.family = "sans",
col.axes = "darkorchid",
alpha.axes = 0.2,
width.axes = 1.1,
col.background = adjustcolor("lavender",
alpha.f = 0.2),
force = 1, segment.size = 3)
a.01.Map4Partipants <- BaseMap.Participants$zeMap_background +
BaseMap.Participants$zeMap_dots + labels4RV
a.02.Map4Partipants.labels <- BaseMap.Participants$zeMap_background +
BaseMap.Participants$zeMap_text + labels4RV
```r", fig.height=3, fig.width= 8}
print(a.01.Map4Partipants + ggtitle('The Rv Map')) # How to add a title when printing
```r", fig.height=3, fig.width= 8}
print(a.02.Map4Partipants.labels)
Plot mean and confidence intervals
getMeans <- function(G, factor, FUN = mean){
# A function to get the mean by groups
groupMean.tmp <- aggregate(G, list(factor), FUN)
groupMean <- groupMean.tmp[,2:ncol(groupMean.tmp )]
rownames(groupMean) <- groupMean.tmp[,1]
return(groupMean)
}
# Get the factors from the Cmat analysis
ExpertiseMeans <- getMeans(F4Cmat, descJudges$Expertise)
# Bootstrap for CI
BootCube <- PTCA4CATA::Boot4Mean(F4Cmat, design = descJudges$Expertise,
niter = 100,
suppressProgressBar = TRUE)
# head(BootCube)
# First the means
# A tweak for colors
col4Group <- c(col4Experts,col4Novices)
#
gg.rv.means <- PTCA4CATA::createFactorMap(ExpertiseMeans,
axis1 = 1, axis2 = 2,
constraints = BaseMap.Participants$constraints, title = NULL,
col.points = col4Group ,
alpha.points = 1, # no transparency
display.points = TRUE,
pch = 19, cex = 5,
display.labels = TRUE,
col.labels = col4Group ,
text.cex = 4,
font.face = "bold",
font.family = "sans", col.axes = "darkorchid",
alpha.axes = 0.2, width.axes = 1.1,
col.background = adjustcolor("lavender", alpha.f = 0.2),
force = 1, segment.size = 0)
#
dimnames(BootCube$BootCube)[[2]] <-
paste0('Dimension ',1 : dim(BootCube$BootCube)[[2]])
#c('Dim1','Dim2')
GraphElli.rv <- MakeCIEllipses(BootCube$BootCube[,1:2,],
names.of.factors = c("Dimension 1","Dimension 2"),
col = col4Group,
p.level = .95)
a.03.gg.RVMap.CI <- a.01.Map4Partipants + gg.rv.means$zeMap_dots + GraphElli.rv
As can be seen in the map, Experts and Novices
are significantly different (but not strongly because
the two samples clearly overlap).
```r", fig.height=3, fig.width= 8}
print(a.03.gg.RVMap.CI)
## More description of the Judges
Here we use as additional information the other variables that
we have collected on the participants:
`Age`, `Sex` and, `Consumption.`
We will *bin* age in three levels (with the function
`ggplot2::cut_number()`)
```r
ageBin <- ggplot2::cut_number(descJudges$Age, 4)
levelsAge <- levels(ageBin)
# rename the levels for clarity
levels(ageBin) <- list(A1 = "[25,35]", A2 = "(35,45]",
A3 = "(45,54.5]", A4 = "(54.5,67]")
The variable Consumption is quite unbalanced
as can be seen from its summary
summary(as.factor(descJudges$Consumption))
so we decide to combine the values 4 and 3 (to do so
we recode the value 4 as 3):
descJudges$Consumption[descJudges$Consumption == 4] <- 3
Project on the RV Map
First we compute the means per group:
Age <- getMeans(F4Cmat, ageBin)
Sex <- getMeans(F4Cmat, descJudges$Sex)
Con <- getMeans(F4Cmat, descJudges$Consumption)
rownames(Con) <- c('C1', 'C2', 'C3')
Concatenate:
GroupMeans4RV = rbind(Age,Sex,Con,ExpertiseMeans)
Get colors
col4Means.gr <- prettyGraphs::prettyGraphsColorSelection(n.colors = 4)
col4Means <- c(rep(col4Means.gr[1],4), rep(col4Means.gr[2],2),
rep(col4Means.gr[3],3), rep(col4Means.gr[4],2) )
Project the mean onto the RV map.
colnames(GroupMeans4RV) <- paste0('Dimension ',1:ncol(GroupMeans4RV))
gg.rv.groups <- PTCA4CATA::createFactorMap(GroupMeans4RV,
constraints = BaseMap.Participants$constraints,
title = 'Judges Variables',
col.points = col4Means,
alpha.points = 1, # no transparency
cex = 1,
col.labels = col4Means)
```r", fig.height=3, fig.width= 8}
a.04a.gg.RVMap.Gr <- gg.rv.groups$zeMap + labels4RV
print(a.04a.gg.RVMap.Gr)
The map has the same scale as the participant map,
and it shows that none of the variables under consideration
explain much of the variance of the participants.
## Post-Hoc Analysis: Exploring the individual participants between participants
Here we conduct a hierarchical cluster analysis on the factor score
```r
D <- dist(F4Cmat, method = "euclidean")
fit <- hclust(D, method = "ward.D2")
a05.tree4participants <- fviz_dend(fit, k = 1,
k_colors = 'burlywood4',
label_cols = color4Participants[fit$order],
cex = .4, xlab = 'Participants',
main = 'Cluster Analysis: Participants')
print(a05.tree4participants)
The tree confirms that Novices and Experts do not differ
in an important way because the analysis does not separate one group
from the other.
The tree, however, suggests that there are two groups of participants,
and further analysis should be performed to identify the rouses of
these differences. One way of doing so could be to plot
the tree categories on the map.
Cut the tree in two
grpParticipants <- cutree(fit, k = 2)
Color the RV Plot by participant
Get the colors
col4Gr <- prettyGraphs::createColorVectorsByDesign(
ExPosition::makeNominalData(as.matrix(grpParticipants))
)
Make the plot
BaseMap.Participants.tree <- createFactorMap(X = F4Plot ,
title = 'The RV map colored from the tree',
col.points = col4Gr$oc,
display.labels = TRUE,
col.labels = col4Gr$oc)
a.05.Map4Partipants.tree <- BaseMap.Participants.tree$zeMap_background +
BaseMap.Participants.tree$zeMap_dots + labels4RV
print(a.05.Map4Partipants.tree)
As expected form the very large proportion of explained variance
by the first dimension, the tree splits the participants
according to
their position of Dimension.
A possible interpretation for Dimension could be that
it reflects the number of descriptors chosen,
but this interpretation needs to be confirmed.
A Question: How can we confirm this conjecture?
Plot for Cochran's Q
# Temporary version before a ggplot2 version
UnC.criticalC = .05
CritChi <- qchisq(p = 1 - UnC.criticalC ,NROW(contingencyTable) -1)
Plot4Q <- PrettyBarPlotColor4Q((Q4CATA[1,]),
color4bar = rep('darkorchid4',
ncol(Q4CATA)),
threshold = CritChi , # Critical value,
main = 'Chi2 for Q (uncorrected) ',
ylab = 'Chi2 for Q')
b00.Q.Uncorrected <- recordPlot()
C.criticalC = 1 - (1 - UnC.criticalC)^(1/ncol(Q4CATA))
CritChi.C <- qchisq(p = 1 - C.criticalC, NROW(contingencyTable) - 1)
Plot4Q <- PrettyBarPlotColor4Q((Q4CATA[1,]),
color4bar = rep('darkorchid4',
ncol(Q4CATA)),
threshold = CritChi.C , # Critical value,
main = 'Chi2 for Q (Sidak corrected) ',
ylab ='Chi2 for Q')
b01.Q.Corrected <- recordPlot()
Scree Plot for CA (with significance)
# Scree plot with significance
ScreeInf <- PlotScree(ev = ResCATACA$ExPosition.Data$eigs,
p.ev = prob.cpt,
#ResCATANovicesCA.inference$Inference.Data$components$p.vals,
max.ev = NULL, alpha = 0.05,
col.ns = "#006D2C", col.sig = "#54278F",
title = "Explained Variance per Dimension")
c00.screeCA.inf <- recordPlot()
Factorial Plots
I and J sets together
2. Descriptive Fi and Fj
x_axis = 1
y_axis = 2
xyLabels <- createxyLabels(ResCATACA,x_axis = x_axis, y_axis = y_axis)
BaseMap.Fi <- createFactorMap(X = RenomrFi$G_A,
axis1 = x_axis,
axis2 = y_axis,
constraints = NULL,
title = NULL, col.points = color4Products,
display.points = TRUE,
pch = 19, cex = 2.5,
display.labels = TRUE,
col.labels = color4Products,
text.cex = 4, font.face = "bold",
font.family = "sans",
col.axes = "darkorchid", alpha.axes = 0.2,
width.axes = 1.1,
col.background = adjustcolor("lavender",
alpha.f = 0.2),
force = 1, segment.size = 0)
BaseMap.Fj <- createFactorMap(X = ResCATACA$ExPosition.Data$fj,
axis1 = x_axis,
axis2 = y_axis,
constraints = BaseMap.Fi$constraints,
title = NULL, col.points = color4Attributes,
display.points = TRUE,
pch = 19, cex = 1,
display.labels = TRUE,
col.labels = color4Attributes,
text.cex = 2.5, font.face = "bold",
font.family = "sans",
col.axes = "darkorchid", alpha.axes = 0.2,
width.axes = 1.1,
col.background = adjustcolor("lavender",
alpha.f = 0.2),
force = 2, segment.size = 0)
d01.BaseMap.Fij <- BaseMap.Fi$zeMap + BaseMap.Fj$zeMap_dots + xyLabels
d02.BaseMapNoDoti.Fij <- BaseMap.Fi$zeMap_background +
BaseMap.Fi$zeMap_text +
BaseMap.Fj$zeMap_dots + xyLabels
d03.BaseMapNoLabeli.Fij <- BaseMap.Fi$zeMap_background +
BaseMap.Fi$zeMap_dots +
BaseMap.Fj$zeMap_text + xyLabels
An asymmetric map
```r", fig.height = 9}
print(d01.BaseMap.Fij )
```r", fig.height = 9}
print(d02.BaseMapNoDoti.Fij)
```r", fig.height = 9}
print(d03.BaseMapNoLabeli.Fij, )
### Plot (Q-) Significant Descriptors
To make the graph more readable, we can
highlight the significant descriptors (as identified from Cochran's Q), plot in grey
the non-significant descriptors.
```r
color4AttributesCorrected <- color4Attributes
col4ns <- 'gray80' # Color for NS Attribute
color4AttributesCorrected[Q4CATA[3,] >= UnC.criticalC ] <- col4ns
BaseMap.Fj.cor <- createFactorMap(X = ResCATACA$ExPosition.Data$fj,
axis1 = 1, axis2 = 2,
constraints = BaseMap.Fi$constraints,
title = NULL,
col.points = color4AttributesCorrected,
display.points = TRUE,
pch = 19, cex = 1,
display.labels = TRUE,
col.labels = color4AttributesCorrected,
text.cex = 2.5, font.face = "bold",
font.family = "sans",
col.axes = "darkorchid", alpha.axes = 0.2,
width.axes = 1.1,
col.background = adjustcolor("lavender",
alpha.f = 0.2),
force = 2, segment.size = 0)
e01.BaseMap.Fij.cor <- BaseMap.Fi$zeMap + BaseMap.Fj.cor$zeMap_dots
e02.BaseMapNoDoti.Fij.cor <- BaseMap.Fi$zeMap_background +
BaseMap.Fi$zeMap_text +
BaseMap.Fj.cor$zeMap_dots + xyLabels
e03.BaseMapNoLabeli.Fij.cor <- BaseMap.Fi$zeMap_background +
BaseMap.Fi$zeMap_dots +
BaseMap.Fj.cor$zeMap_text + xyLabels
# Same maps but with names for the products instead of dots
e04.BaseMapLabeli.Fij.cor <- BaseMap.Fi$zeMap_background +
BaseMap.Fi$zeMap_text +
BaseMap.Fj.cor$zeMap_text + xyLabels
```r", fig.height = 9}
print(e04.BaseMapLabeli.Fij.cor)
## Confidence Ellipsoids
Here we add bootstrap derived confidence ellipsoids.
```r
# Former problems here that needed to be fixed
# the vector of names of factors and
# the number of factors need to be of length 2
# this will be fixed in the the next release of PTCA4CATA
graphElli4I <- MakeCIEllipses(data = bootCA$RowsBoot.asym,
axis1 = 1, axis2 = 2,
names.of.factors = # c("Factor 1","Factor 2"),
c('V1','V2'), # Only two here
col = color4Products, # Color for the Products
# centers = Center_Fi,
centers = RenomrFi$G_A[,1:2], # only two here
# Give the coordinates of the centers
# centers = RenomrFi$G_S,
line.size = 1,
line.type = 1,
alpha.ellipse = 0.3,
alpha.line = 0.5, p.level = 0.95)
#
# 2. Descriptive Fi and Fj
BaseMap.Fi.elli <- createFactorMap(X = #RenomrFi$G_S,
RenomrFi$G_A ,
axis1 = 1, axis2 = 2,
constraints =
lapply(BaseMap.Fi$constraints,'*',1.8),
title = NULL, col.points = color4Products,
display.points = TRUE,
pch = 19, cex = 2.5,
display.labels = TRUE,
col.labels = color4Products,
text.cex = 4, font.face = "bold",
font.family = "sans",
col.axes = "darkorchid", alpha.axes = 0.2,
width.axes = 1.1,
col.background = adjustcolor("lavender",
alpha.f = 0.2),
force = 1, segment.size = 0)
f.00.MapElli4I <- BaseMap.Fi.elli$zeMap_background +
BaseMap.Fi.elli$zeMap_dots + graphElli4I +
BaseMap.Fi.elli$zeMap_text + xyLabels
The map can be printed with the following command
```r", fig.height = 9}
print(f.00.MapElli4I)
## Partial Projections
The descriptors comprises three main groups.
In this section we explore the similarities and differences between
these three groups.
### Compute partial projections
```r
partialFi <- partialProj4CA(resCA = ResCATACA,
code4Blocks = catDescriptors$Category)
color4Blocks <- c('chartreuse4','red2','purple4')
Create the map
First we need to create a new background map:
because the partial factor scores have a larger variance than the
the global factor scores.
First create two data.frames (this will make typing easier)
Fi <- ResCATACA$ExPosition.Data$fi
dimnames(Fi)[[2]] <- paste0('Dimension ',1:dim(Fi)[[2]])
pFi <- partialFi$Fk
dimnames(pFi)[[2]] <- paste0('Dimension ',1:dim(pFi)[[2]])
We need to compute explicitly the size (called constraints
in graphics packages)
constraints4PFM <- minmaxHelper4Partial(Fi,pFi)
Create the "background map" with the new constraints
gg.Fi.map <- createFactorMap(Fi,
constraints = constraints4PFM,
title = "Product by Blocks",
col.points = color4Products ,
col.labels = color4Products
)
Create the partial factor scores maps
map4PFS <- createPartialFactorScoresMap(
factorScores = Fi,
partialFactorScores = pFi,
colors4Items = color4Products,
colors4Blocks = color4Blocks)
f01.d1.partialFS.map.byProducts <- gg.Fi.map$zeMap + map4PFS$mapColByItems
f01.d2.partialFS.map.byCategories <- gg.Fi.map$zeMap + map4PFS$mapColByBlocks
Print the partial factor maps
print(f01.d1.partialFS.map.byProducts)
print(f01.d2.partialFS.map.byCategories)
Save the graphics as a powerpoint
The graphics are saved as a powerpoint with the following command
r name4Graphs = 'beersCATA.pptx'
pptx4MCA <- PTCA4CATA::saveGraph2pptx(file2Save.pptx = name4Graphs,
title = 'Novices and Experts Evaluate beers ',
addGraphNames = TRUE)
Note that we could also have created a powerpoint with
Rmarkdown by using the following options in the
preamble:
output:
powerpoint_presentation:
slide_level: 4
instead of (for example):
output:
rmarkdown::html_vignette:
toc: true
number_sections: true
HerveAbdi/R4SPISE2018 documentation built on Sept. 3, 2020, 6:40 a.m.
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knitr::opts_knit$get()
**NOTE:** This `pdf` was generated from the vignette `R4SPISE2018::cheeseMCA` from the `R` Package `R4SPISE2018`. Check the help for the very last version of this document.
Prelude
If you want to make sure that you have a clean start, you can execute the following commands:
rm(list = ls()) graphics.off()
Or, better (see below, preamble), you can use an Rproject for this project.
# Important: Remember # build the vignettes with devtools::build_vignettes() knitr::opts_chunk$set( collapse = TRUE, fig.width = 9, comment = "#>" )
knitr::opts_knit$get()
Preamble
Make sure that you start this analysis as a new Rproject so that
the default directory will be correctly set, and the memory cleared.
Before we start the analysis,
we need to have our standard packages installed (including some from Github)
and the corresponding libraries loaded:
DistatisRPTCA4CTAR4SPISE2018prettyGraphsExPosition
load
# Decomment all/some 0f these lines if the packages are not installed # devtools::install_github('HerveAbdi/PTCA4CATA') # devtools::install_github('HerveAbdi/DistatisR') # install.packages(prettyGraphs) # install.packages('Matrix') # install.packages('factoextra') # # load the libraries that we will need suppressMessages(library(Matrix)) suppressMessages(library(DistatisR)) suppressMessages(library(PTCA4CATA)) suppressMessages(library(prettyGraphs)) suppressMessages(library(ExPosition)) suppressMessages(library(factoextra))
Introduction
The data set can be found from the package R4SPISE2018.
The data are stored in an excel file called
beersNovicesExpertsCATA.xlsx
whose location
can be found and stored in the
variable path2file using the R function system.functionfile()
as shown in the following command:
```{R findDataPath}
path2file <- system.file("extdata",
"beersNovicesExpertsCATA.xlsx", package = "R4SPISE2018")
```r", fig.height=3, fig.width=4, include=TRUE, out.width='70%'}
# label.spicesxl = capFigNo
# 
knitr::include_graphics('../man/figures/beerXLS-screen.png')
if you open this excel file, it will look like the
Figure above.
The sorting data are stared in the sheet DataSort.
The first columns gives the names of the products
(here spices) and each column
that were sorted
When you record you own data, make sure that you follow the same format, this way the script described in this vignette will apply to your own analysis with minimal change.
We will first compute the results of the analysis, then create the graphics, and finally save everything into a powerpoint.
Run the statistical analysis
Read the data
The excel file name and location (i.e., path) are
stored in the variable path2file.
To read the data we will use the function
PTCA4CATA::read.xls.CATAl()
(based upon
the function readxl::read_excel()).
How to save the data file
To save the data file
(under the original name of myDataFile
or maybe a more informative name)
in a directory
(say Downloads). use the following command
saveFile <- file.copy(from = path2file, to = '~/Downloads/myDataFile.xlsx')
Back to the data and norming
In addition we are norming the responses of the Judges so that each row for each Judges sums to 1.
dataCATA.list <- read.xls.CATA(path2file = path2file, sheet2read = 'DataCATA', threshold4cleaning = 5) # norm dataCube <- normBrick4PTCA(dataCATA.list$CATA.Brick , normingConstant = 100, normalization = "byRow", code4Groups = NULL)$normedArray # get the contingency table from the cube contingencyTable <- apply(dataCube,c(1,2),sum)
nI <- dim(dataCube)[[1]] nJ <- dim(dataCube)[[2]] nK <- dim(dataCube)[[3]] namesR <- dimnames(dataCube)[[1]] namesC <- dimnames(dataCube)[[2]] namesA <- dimnames(dataCube)[[3]]
The variables are organized in block.
First we get the blocks, then we keep only
the variables that passed the threshold for cleaning
(see parameter threshold4cleaning).
The categories of the CATA descriptors are stored in the file
beersNovicesExpertsCATA.xlsx (whose path is stored in the
variable path2file) in the sheet Categories4Descriptors.
We read them with the function readxl::read_excel
catDescriptors.tmp <- as.data.frame(readxl::read_excel(path2file, 'Categories4Descriptors')) catDescriptors <- catDescriptors.tmp[catDescriptors.tmp[,1] %in% namesC,]
The assessors are also described by several variables.
These data are stored in the file
beersNovicesExpertsCATA.xlsx (whose path is stored in the
variable path2file) in the sheet DescriptionJuges.
We read them with the function readxl::read_excel:
descJudges <- as.data.frame(readxl::read_excel(path2file, 'DescriptionJuges'))
# Color for the descriptors col4Emotion <- "red4" col4Context <- "darkmagenta" col4Taste <- "darkolivegreen" color4Attributes <- rep("", nJ) color4Attributes[catDescriptors$Category == 'C'] <- col4Context color4Attributes[catDescriptors$Category == 'E'] <- col4Emotion color4Attributes[catDescriptors$Category == 'G'] <- col4Taste # Color for the products from prettyGraphs color4Products <- prettyGraphs::prettyGraphsColorSelection(nI) # Color for the Judges Novices vs Experts col4Experts <- 'mediumpurple4' col4Novices <- 'olivedrab3' color4Participants <- rep(col4Experts, nK) color4Participants[descJudges$Expertise == 'N'] <- col4Novices # color4Participants[descJudges$Expertise == 'E'] <- col4Experts
Descriptive approach
Get Cochran Q
# Compute Cochran's Q from the cube of data Q4CATA <- Q4CATA.Cube(dataCube) # Create the vector of probability stars zeStars <- p2Star( Q4CATA['Sidak p(FW)',])
Create a Heat Map (with significant levels)
a.000.zeMapCATA <- makeggHeatMap4CT( CTstared(contingencyTable,zeStars,'after'), colorAttributes = color4Attributes, colorProducts = color4Products)
Print the heat map
print(a.000.zeMapCATA)
CA: descriptive and inferences
Correspondence analysis on the contingency table
# The CA ResCATACA <- epCA(contingencyTable, masses = NULL, weights = NULL, hellinger = FALSE, symmetric = TRUE, graphs = FALSE) # Renormalize the results of CA RenomrFi <- CARenormalization(G = ResCATACA$ExPosition.Data$fi , delta = ResCATACA$ExPosition.Data$pdq$Dv, singularValues = TRUE, masses = NULL)
Bootstrap Inferences
Bootstrap and bootstrap ratios:
# bootCA <- Boot4PTCA(ZeDataCube = dataCube, fi = ResCATACA$ExPosition.Data$fi, fj = ResCATACA$ExPosition.Data$fj, eigs = ResCATACA$ExPosition.Data$eigs, nf2keep = 3, nBootIter = 500) # Compute Bootstraped ratios bootRatios.I <- PTCA4CATA::boot.ratio.test(bootCA$RowsBoot, critical.value = 2) bootRatios.J <- PTCA4CATA::boot.ratio.test(bootCA$ColumnsBoot, critical.value = 2) # Probabilities probBR.I <- bootRatios.I$prob.boot.ratios probBR.J <- bootRatios.J$prob.boot.ratios
Permutation tests
# from PTCA4CATA # ResCATACA.inference <- perm4PTCA(aCube = dataCube, nIter = 1000, permType = 'byRows' , Malinvaud = TRUE) Ind.Permu <- ResCATACA.inference$permInertia InertiaFixed <- ResCATACA.inference$fixedInertia prob.cpt.lst <- ResCATACA.inference$MalinvaudQ['p-perm',] # Get the p values for the components prob.cpt <- (unlist(prob.cpt.lst[2:length(prob.cpt.lst)])) prob.cpt[is.na(prob.cpt)] <- 1
Participants analysis
Cmat <- createCmat4PTCA(dataCube) # To find the observations Kept: # dimnames((ZeRawDataCube))[[3]] %in% rownames(Cmat) eigenCmat <- eigen(Cmat, symmetric = TRUE) # eig4Cmat <- eigenCmat$values tau4Cmat <- round( (100*eig4Cmat) / sum(eig4Cmat)) nk <- 3 F4Cmat <- eigenCmat$vectors[,1:nk] %*% diag(eig4Cmat[1:nk]^(1/2)) rownames(F4Cmat) <- rownames(Cmat)
Graphics
RV analysis on the participants
Scree
# Scree plot with significance ScreeInf <- PlotScree(ev = eig4Cmat, max.ev = NULL, alpha = 0.05, col.ns = "#006D2C", col.sig = "#54278F", title = "RV Analysis: Explained Variance per Dimension") a000.00.screeRV <- recordPlot()
Participant maps
Shortnames4Participants <- substr(rownames(F4Cmat),1,1) F4Plot <- F4Cmat rownames(F4Plot) <- Shortnames4Participants # Make labels labels4RV <- createxyLabels.gen(1,2,lambda = eig4Cmat, tau = tau4Cmat )
# BaseMap.Participants <- createFactorMap(X = F4Plot , axis1 = 1, axis2 = 2, constraints = NULL, title = NULL, col.points = color4Participants, display.points = TRUE, pch = 19, cex = 2.5, display.labels = TRUE, col.labels = color4Participants, text.cex = 4, font.face = "bold", font.family = "sans", col.axes = "darkorchid", alpha.axes = 0.2, width.axes = 1.1, col.background = adjustcolor("lavender", alpha.f = 0.2), force = 1, segment.size = 3) a.01.Map4Partipants <- BaseMap.Participants$zeMap_background + BaseMap.Participants$zeMap_dots + labels4RV a.02.Map4Partipants.labels <- BaseMap.Participants$zeMap_background + BaseMap.Participants$zeMap_text + labels4RV
```r", fig.height=3, fig.width= 8} print(a.01.Map4Partipants + ggtitle('The Rv Map')) # How to add a title when printing
```r", fig.height=3, fig.width= 8} print(a.02.Map4Partipants.labels)
Plot mean and confidence intervals
getMeans <- function(G, factor, FUN = mean){ # A function to get the mean by groups groupMean.tmp <- aggregate(G, list(factor), FUN) groupMean <- groupMean.tmp[,2:ncol(groupMean.tmp )] rownames(groupMean) <- groupMean.tmp[,1] return(groupMean) }
# Get the factors from the Cmat analysis ExpertiseMeans <- getMeans(F4Cmat, descJudges$Expertise) # Bootstrap for CI BootCube <- PTCA4CATA::Boot4Mean(F4Cmat, design = descJudges$Expertise, niter = 100, suppressProgressBar = TRUE) # head(BootCube)
# First the means # A tweak for colors col4Group <- c(col4Experts,col4Novices) # gg.rv.means <- PTCA4CATA::createFactorMap(ExpertiseMeans, axis1 = 1, axis2 = 2, constraints = BaseMap.Participants$constraints, title = NULL, col.points = col4Group , alpha.points = 1, # no transparency display.points = TRUE, pch = 19, cex = 5, display.labels = TRUE, col.labels = col4Group , text.cex = 4, font.face = "bold", font.family = "sans", col.axes = "darkorchid", alpha.axes = 0.2, width.axes = 1.1, col.background = adjustcolor("lavender", alpha.f = 0.2), force = 1, segment.size = 0) # dimnames(BootCube$BootCube)[[2]] <- paste0('Dimension ',1 : dim(BootCube$BootCube)[[2]]) #c('Dim1','Dim2') GraphElli.rv <- MakeCIEllipses(BootCube$BootCube[,1:2,], names.of.factors = c("Dimension 1","Dimension 2"), col = col4Group, p.level = .95) a.03.gg.RVMap.CI <- a.01.Map4Partipants + gg.rv.means$zeMap_dots + GraphElli.rv
As can be seen in the map, Experts and Novices are significantly different (but not strongly because the two samples clearly overlap). ```r", fig.height=3, fig.width= 8} print(a.03.gg.RVMap.CI)
## More description of the Judges Here we use as additional information the other variables that we have collected on the participants: `Age`, `Sex` and, `Consumption.` We will *bin* age in three levels (with the function `ggplot2::cut_number()`) ```r ageBin <- ggplot2::cut_number(descJudges$Age, 4) levelsAge <- levels(ageBin) # rename the levels for clarity levels(ageBin) <- list(A1 = "[25,35]", A2 = "(35,45]", A3 = "(45,54.5]", A4 = "(54.5,67]")
The variable Consumption is quite unbalanced
as can be seen from its summary
summary(as.factor(descJudges$Consumption))
so we decide to combine the values 4 and 3 (to do so we recode the value 4 as 3):
descJudges$Consumption[descJudges$Consumption == 4] <- 3
Project on the RV Map
First we compute the means per group:
Age <- getMeans(F4Cmat, ageBin) Sex <- getMeans(F4Cmat, descJudges$Sex) Con <- getMeans(F4Cmat, descJudges$Consumption) rownames(Con) <- c('C1', 'C2', 'C3')
Concatenate:
GroupMeans4RV = rbind(Age,Sex,Con,ExpertiseMeans)
Get colors
col4Means.gr <- prettyGraphs::prettyGraphsColorSelection(n.colors = 4) col4Means <- c(rep(col4Means.gr[1],4), rep(col4Means.gr[2],2), rep(col4Means.gr[3],3), rep(col4Means.gr[4],2) )
Project the mean onto the RV map.
colnames(GroupMeans4RV) <- paste0('Dimension ',1:ncol(GroupMeans4RV)) gg.rv.groups <- PTCA4CATA::createFactorMap(GroupMeans4RV, constraints = BaseMap.Participants$constraints, title = 'Judges Variables', col.points = col4Means, alpha.points = 1, # no transparency cex = 1, col.labels = col4Means)
```r", fig.height=3, fig.width= 8} a.04a.gg.RVMap.Gr <- gg.rv.groups$zeMap + labels4RV print(a.04a.gg.RVMap.Gr)
The map has the same scale as the participant map, and it shows that none of the variables under consideration explain much of the variance of the participants. ## Post-Hoc Analysis: Exploring the individual participants between participants Here we conduct a hierarchical cluster analysis on the factor score ```r D <- dist(F4Cmat, method = "euclidean") fit <- hclust(D, method = "ward.D2") a05.tree4participants <- fviz_dend(fit, k = 1, k_colors = 'burlywood4', label_cols = color4Participants[fit$order], cex = .4, xlab = 'Participants', main = 'Cluster Analysis: Participants')
print(a05.tree4participants)
The tree confirms that Novices and Experts do not differ in an important way because the analysis does not separate one group from the other. The tree, however, suggests that there are two groups of participants, and further analysis should be performed to identify the rouses of these differences. One way of doing so could be to plot the tree categories on the map.
Cut the tree in two
grpParticipants <- cutree(fit, k = 2)
Color the RV Plot by participant
Get the colors
col4Gr <- prettyGraphs::createColorVectorsByDesign( ExPosition::makeNominalData(as.matrix(grpParticipants)) )
Make the plot
BaseMap.Participants.tree <- createFactorMap(X = F4Plot , title = 'The RV map colored from the tree', col.points = col4Gr$oc, display.labels = TRUE, col.labels = col4Gr$oc) a.05.Map4Partipants.tree <- BaseMap.Participants.tree$zeMap_background + BaseMap.Participants.tree$zeMap_dots + labels4RV
print(a.05.Map4Partipants.tree)
As expected form the very large proportion of explained variance by the first dimension, the tree splits the participants according to their position of Dimension. A possible interpretation for Dimension could be that it reflects the number of descriptors chosen, but this interpretation needs to be confirmed.
A Question: How can we confirm this conjecture?
Plot for Cochran's Q
# Temporary version before a ggplot2 version UnC.criticalC = .05 CritChi <- qchisq(p = 1 - UnC.criticalC ,NROW(contingencyTable) -1) Plot4Q <- PrettyBarPlotColor4Q((Q4CATA[1,]), color4bar = rep('darkorchid4', ncol(Q4CATA)), threshold = CritChi , # Critical value, main = 'Chi2 for Q (uncorrected) ', ylab = 'Chi2 for Q') b00.Q.Uncorrected <- recordPlot()
C.criticalC = 1 - (1 - UnC.criticalC)^(1/ncol(Q4CATA)) CritChi.C <- qchisq(p = 1 - C.criticalC, NROW(contingencyTable) - 1) Plot4Q <- PrettyBarPlotColor4Q((Q4CATA[1,]), color4bar = rep('darkorchid4', ncol(Q4CATA)), threshold = CritChi.C , # Critical value, main = 'Chi2 for Q (Sidak corrected) ', ylab ='Chi2 for Q') b01.Q.Corrected <- recordPlot()
Scree Plot for CA (with significance)
# Scree plot with significance ScreeInf <- PlotScree(ev = ResCATACA$ExPosition.Data$eigs, p.ev = prob.cpt, #ResCATANovicesCA.inference$Inference.Data$components$p.vals, max.ev = NULL, alpha = 0.05, col.ns = "#006D2C", col.sig = "#54278F", title = "Explained Variance per Dimension") c00.screeCA.inf <- recordPlot()
Factorial Plots
I and J sets together
2. Descriptive Fi and Fj
x_axis = 1 y_axis = 2 xyLabels <- createxyLabels(ResCATACA,x_axis = x_axis, y_axis = y_axis) BaseMap.Fi <- createFactorMap(X = RenomrFi$G_A, axis1 = x_axis, axis2 = y_axis, constraints = NULL, title = NULL, col.points = color4Products, display.points = TRUE, pch = 19, cex = 2.5, display.labels = TRUE, col.labels = color4Products, text.cex = 4, font.face = "bold", font.family = "sans", col.axes = "darkorchid", alpha.axes = 0.2, width.axes = 1.1, col.background = adjustcolor("lavender", alpha.f = 0.2), force = 1, segment.size = 0) BaseMap.Fj <- createFactorMap(X = ResCATACA$ExPosition.Data$fj, axis1 = x_axis, axis2 = y_axis, constraints = BaseMap.Fi$constraints, title = NULL, col.points = color4Attributes, display.points = TRUE, pch = 19, cex = 1, display.labels = TRUE, col.labels = color4Attributes, text.cex = 2.5, font.face = "bold", font.family = "sans", col.axes = "darkorchid", alpha.axes = 0.2, width.axes = 1.1, col.background = adjustcolor("lavender", alpha.f = 0.2), force = 2, segment.size = 0) d01.BaseMap.Fij <- BaseMap.Fi$zeMap + BaseMap.Fj$zeMap_dots + xyLabels d02.BaseMapNoDoti.Fij <- BaseMap.Fi$zeMap_background + BaseMap.Fi$zeMap_text + BaseMap.Fj$zeMap_dots + xyLabels d03.BaseMapNoLabeli.Fij <- BaseMap.Fi$zeMap_background + BaseMap.Fi$zeMap_dots + BaseMap.Fj$zeMap_text + xyLabels
An asymmetric map
```r", fig.height = 9} print(d01.BaseMap.Fij )
```r", fig.height = 9} print(d02.BaseMapNoDoti.Fij)
```r", fig.height = 9} print(d03.BaseMapNoLabeli.Fij, )
### Plot (Q-) Significant Descriptors To make the graph more readable, we can highlight the significant descriptors (as identified from Cochran's Q), plot in grey the non-significant descriptors. ```r color4AttributesCorrected <- color4Attributes col4ns <- 'gray80' # Color for NS Attribute color4AttributesCorrected[Q4CATA[3,] >= UnC.criticalC ] <- col4ns BaseMap.Fj.cor <- createFactorMap(X = ResCATACA$ExPosition.Data$fj, axis1 = 1, axis2 = 2, constraints = BaseMap.Fi$constraints, title = NULL, col.points = color4AttributesCorrected, display.points = TRUE, pch = 19, cex = 1, display.labels = TRUE, col.labels = color4AttributesCorrected, text.cex = 2.5, font.face = "bold", font.family = "sans", col.axes = "darkorchid", alpha.axes = 0.2, width.axes = 1.1, col.background = adjustcolor("lavender", alpha.f = 0.2), force = 2, segment.size = 0) e01.BaseMap.Fij.cor <- BaseMap.Fi$zeMap + BaseMap.Fj.cor$zeMap_dots e02.BaseMapNoDoti.Fij.cor <- BaseMap.Fi$zeMap_background + BaseMap.Fi$zeMap_text + BaseMap.Fj.cor$zeMap_dots + xyLabels e03.BaseMapNoLabeli.Fij.cor <- BaseMap.Fi$zeMap_background + BaseMap.Fi$zeMap_dots + BaseMap.Fj.cor$zeMap_text + xyLabels # Same maps but with names for the products instead of dots e04.BaseMapLabeli.Fij.cor <- BaseMap.Fi$zeMap_background + BaseMap.Fi$zeMap_text + BaseMap.Fj.cor$zeMap_text + xyLabels
```r", fig.height = 9} print(e04.BaseMapLabeli.Fij.cor)
## Confidence Ellipsoids Here we add bootstrap derived confidence ellipsoids. ```r # Former problems here that needed to be fixed # the vector of names of factors and # the number of factors need to be of length 2 # this will be fixed in the the next release of PTCA4CATA graphElli4I <- MakeCIEllipses(data = bootCA$RowsBoot.asym, axis1 = 1, axis2 = 2, names.of.factors = # c("Factor 1","Factor 2"), c('V1','V2'), # Only two here col = color4Products, # Color for the Products # centers = Center_Fi, centers = RenomrFi$G_A[,1:2], # only two here # Give the coordinates of the centers # centers = RenomrFi$G_S, line.size = 1, line.type = 1, alpha.ellipse = 0.3, alpha.line = 0.5, p.level = 0.95) # # 2. Descriptive Fi and Fj BaseMap.Fi.elli <- createFactorMap(X = #RenomrFi$G_S, RenomrFi$G_A , axis1 = 1, axis2 = 2, constraints = lapply(BaseMap.Fi$constraints,'*',1.8), title = NULL, col.points = color4Products, display.points = TRUE, pch = 19, cex = 2.5, display.labels = TRUE, col.labels = color4Products, text.cex = 4, font.face = "bold", font.family = "sans", col.axes = "darkorchid", alpha.axes = 0.2, width.axes = 1.1, col.background = adjustcolor("lavender", alpha.f = 0.2), force = 1, segment.size = 0) f.00.MapElli4I <- BaseMap.Fi.elli$zeMap_background + BaseMap.Fi.elli$zeMap_dots + graphElli4I + BaseMap.Fi.elli$zeMap_text + xyLabels
The map can be printed with the following command
```r", fig.height = 9} print(f.00.MapElli4I)
## Partial Projections The descriptors comprises three main groups. In this section we explore the similarities and differences between these three groups. ### Compute partial projections ```r partialFi <- partialProj4CA(resCA = ResCATACA, code4Blocks = catDescriptors$Category) color4Blocks <- c('chartreuse4','red2','purple4')
Create the map
First we need to create a new background map: because the partial factor scores have a larger variance than the the global factor scores.
First create two data.frames (this will make typing easier)
Fi <- ResCATACA$ExPosition.Data$fi dimnames(Fi)[[2]] <- paste0('Dimension ',1:dim(Fi)[[2]]) pFi <- partialFi$Fk dimnames(pFi)[[2]] <- paste0('Dimension ',1:dim(pFi)[[2]])
We need to compute explicitly the size (called constraints
in graphics packages)
constraints4PFM <- minmaxHelper4Partial(Fi,pFi)
Create the "background map" with the new constraints
gg.Fi.map <- createFactorMap(Fi, constraints = constraints4PFM, title = "Product by Blocks", col.points = color4Products , col.labels = color4Products )
Create the partial factor scores maps
map4PFS <- createPartialFactorScoresMap( factorScores = Fi, partialFactorScores = pFi, colors4Items = color4Products, colors4Blocks = color4Blocks) f01.d1.partialFS.map.byProducts <- gg.Fi.map$zeMap + map4PFS$mapColByItems f01.d2.partialFS.map.byCategories <- gg.Fi.map$zeMap + map4PFS$mapColByBlocks
Print the partial factor maps
print(f01.d1.partialFS.map.byProducts)
print(f01.d2.partialFS.map.byCategories)
Save the graphics as a powerpoint
The graphics are saved as a powerpoint with the following command
r name4Graphs = 'beersCATA.pptx'
pptx4MCA <- PTCA4CATA::saveGraph2pptx(file2Save.pptx = name4Graphs, title = 'Novices and Experts Evaluate beers ', addGraphNames = TRUE)
Note that we could also have created a powerpoint with
Rmarkdown by using the following options in the
preamble:
output: powerpoint_presentation: slide_level: 4
instead of (for example):
output: rmarkdown::html_vignette: toc: true number_sections: true
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