Generate a Core Collection with Principal Component Scoring Strategy (PCSS) (Hamon and Noirot 1990; Noirot et al. 1996; Noirot et al. 2003) using qualitative and/or quantitative trait data.
Usage
pcss.core(
data,
names,
quantitative,
qualitative,
eigen.threshold = NULL,
size = 0.2,
var.threshold = 0.75
)Arguments
- data
The data as a data frame object. The data frame should possess one row per individual and columns with the individual names and multiple trait/character data.
- names
Name of column with the individual/genotype names as a character string.
- quantitative
Name of columns with the quantitative traits as a character vector.
- qualitative
Name of columns with the qualitative traits as a character vector.
- eigen.threshold
The lower limit of the eigen value of factors to be included in the estimation. The default value is the average of all the eigen values.
- size
The desired core set size proportion.
- var.threshold
The desired proportion of total variability to be
Value
A list of class pcss.core with the following components.
- details
The details of the core set generaton process.
- raw.out
The original output of
PCA,CAandFAMDfunctions ofFactoMineR- eigen
A data frame with eigen values and their partial and cumulative contribution to percentage of variance.
- eigen.threshold
The threshold eigen value used.
- rotation
A matrix of rotation values or loadings.
- scores
A matrix of scores from PCA, CA or FAMD.
- variability.ret
A data frame of individuals/genotypes ordered by variability retained.
- cores.info
A data frame of core set size and percentage variability retained according to the method used.
Details
A core collection is constituted from an entire collection of \(N\) genotypes using quantitative data of \(J\) traits using Principal Component Scoring Strategy (PCSS) (Hamon and Noirot 1990; Noirot et al. 1996; Noirot et al. 2003) as follows:
Principal Component Analysis (PCA) is performed on the standardized genotype \(\times\) trait data. This takes care of multicollinearity between the traits to generate \(J\) standardized and independent variables or factors or principal component.
Considering only a subset of factors \(K\), the Generalized Sum of Squares (GSS) of N individuals in K factorial spaces is computed as \(N \times K\).
\(K\) can be the number of factors for which the eigen value \(\lambda\) is greater than a threshold value such as 1 (Kaiser-Guttman criterion) or the average of all the eigen values.
The contribution of the \(i\)th genotype to GSS (\(P_{i}\)) or total variability is calculated as below.
\[P_{i} = \sum_{j = 1}^{K} x_{ij}^{2}\]
Where \(x_{ij}\) is the component score or coordinate of the \(i\)th genotype on the \(j\)th principal component.
For each genotype, its relative contribution to GSS or total variability is computed as below.
\[CR_{i} = \frac{P_{i}}{N \times K}\]
The genotypes are sorted in descending order of magnitude of their contribution to GSS and then the cumulative contribution of successive genotypes to GSS is computed.
The core collection can then be selected by three different methods.
Selection of fixed proportion or percentage or number of the top accessions.
Selection of the top accessions that contribute up to a fixed percentage of the GSS.
Fitting a logistic regression model of the following form to the cumulative contribution of successive genotypes to GSS (Balakrishnan et al. 2000) .
\[\frac{y}{A-y} = e^{a + bn}\]
The above equation can be reparameterized as below.
\[\log_{e} \left ( {\frac{y}{A-y}} \right ) = a + bn\]
Where, \(a\) and \(b\) are the intercept and regression coefficient, respectively; \(y\) is the cumulative contribution of successive genotypes to GSS; \(n\) is the rank of the genotype when sorted according to the contribution to GSS and \(A\) is the asymptote of the curve (\(A = 100\)).
The rate of increase in the successive contribution of genotypes to GSS can be computed by the following equation to find the point of inflection where the rate of increase starts declining.
\(\frac{\mathrm{d} y}{\mathrm{d} x} = by(A-y)\)
The number of accessions included till the peak or infection point are selected to constitute the core collection.
Similarly for qualitative traits, standardized and independent variables or
factors can be obtained by Correspondence Analysis (CA) on complete
disjunctive table of genotype \(\times\) trait data or to be specific
Multiple Correspondence Analysis (MCA). In rpcss, this has also been
extended for data sets having both quantitative and qualitative traits by
implementing Factor Analysis for Mixed Data (FAMD) for obtaining standardized
and independent variables or factors.
In rpcss, PCA, MCA and FAMD are implemented via the
FactoMineR package.
(Le et al. 2008; Husson et al. 2017)
.
References
Balakrishnan R, Nair NV, Sreenivasan TV (2000).
“A method for establishing a core collection of Saccharum officinarum L. germplasm based on quantitative-morphological data.”
Genetic Resources and Crop Evolution, 47, 1–9.
ISBN: 0925-9864 Publisher: Springer.
Hamon S, Noirot M (1990).
“Some proposed procedures for obtaining a core collection using quantitative plant characterization data.”
In Report of An International Workshop on Okra Genetic Resources held at National Bureau of Plant Genetic Resources (NBPGR), New Delhi, India, 8-12 October,1990, number 5 in International Crop Network Series.
International Board for Plant Genetic Resources, Rome.
Husson F, Le S, Pages J (2017).
Exploratory Multivariate Analysis by Example Using R, Second edition edition.
CRC Press, Boca Raton.
ISBN 978-1-138-19634-6.
Le S, Josse J, Husson F (2008).
“FactoMineR : An R package for multivariate analysis.”
Journal of Statistical Software, 25(1).
Noirot M, Anthony F, Dussert S, Hamon S (2003).
“A method for building core collections.”
In Genetic Diversity of Cultivated Tropical Plants, 81–92.
CRC Press.
Noirot M, Hamon S, Anthony F (1996).
“The principal component scoring: A new method of constituting a core collection using quantitative data.”
Genetic Resources and Crop Evolution, 43(1), 1–6.
Examples
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Prepare example data
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
library(EvaluateCore)
# Get data from EvaluateCore
data("cassava_EC", package = "EvaluateCore")
data = cbind(Genotypes = rownames(cassava_EC), cassava_EC)
quant <- c("NMSR", "TTRN", "TFWSR", "TTRW", "TFWSS", "TTSW", "TTPW", "AVPW",
"ARSR", "SRDM")
qual <- c("CUAL", "LNGS", "PTLC", "DSTA", "LFRT", "LBTEF", "CBTR", "NMLB",
"ANGB", "CUAL9M", "LVC9M", "TNPR9M", "PL9M", "STRP", "STRC",
"PSTR")
rownames(data) <- NULL
# Convert qualitative data columns to factor
data[, qual] <- lapply(data[, qual], as.factor)
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Get core sets with PCSS (quantitative data)
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
out1 <- pcss.core(data = data, names = "Genotypes",
quantitative = quant,
qualitative = NULL, eigen.threshold = NULL, size = 0.2,
var.threshold = 0.75)
out1
#>
#> Method
#> ========================
#> [1] "PCA"
#>
#> Details
#> ========================
#> Detail
#> 1 Total number of individuals/genotypes
#> 2 Quantitative traits
#> 3 Qualitative traits
#> 4 Method
#> 5 Threshold eigen value
#> 6 Number of eigen values selected
#> 7 Threshold size
#> 8 Threshold variance (%)
#> Value
#> 1 1684
#> 2 NMSR, TTRN, TFWSR, TTRW, TFWSS, TTSW, TTPW, AVPW, ARSR, SRDM
#> 3
#> 4 PCA
#> 5 1
#> 6 2
#> 7 0.2
#> 8 75
#>
#> Core sets
#> =========================
#> Method Size VarRet
#> 1 By size specified 337 62.79839
#> 2 By threshold variance 532 75.00000
#> 3 By logistic regression 189 50.03205
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Get core sets with PCSS (qualitative data)
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
out2 <- pcss.core(data = data, names = "Genotypes", quantitative = NULL,
qualitative = qual, eigen.threshold = NULL,
size = 0.2, var.threshold = 0.75)
out2
#>
#> Method
#> ========================
#> [1] "MCA"
#>
#> Details
#> ========================
#> Detail
#> 1 Total number of individuals/genotypes
#> 2 Quantitative traits
#> 3 Qualitative traits
#> 4 Method
#> 5 Threshold eigen value
#> 6 Number of eigen values selected
#> 7 Threshold size
#> 8 Threshold variance (%)
#> Value
#> 1 1684
#> 2
#> 3 CUAL, LNGS, PTLC, DSTA, LFRT, LBTEF, CBTR, NMLB, ANGB, CUAL9M, LVC9M, TNPR9M, PL9M, STRP, STRC, PSTR
#> 4 MCA
#> 5 0.0625
#> 6 24
#> 7 0.2
#> 8 75
#>
#> Core sets
#> =========================
#> Method Size VarRet
#> 1 By size specified 337 51.05444
#> 2 By threshold variance 822 75.00000
#> 3 By logistic regression 322 50.00157
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Get core sets with PCSS (quantitative and qualitative data)
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
out3 <- pcss.core(data = data, names = "Genotypes",
quantitative = quant,
qualitative = qual, eigen.threshold = NULL)
out3
#>
#> Method
#> ========================
#> [1] "FAMD"
#>
#> Details
#> ========================
#> Detail
#> 1 Total number of individuals/genotypes
#> 2 Quantitative traits
#> 3 Qualitative traits
#> 4 Method
#> 5 Threshold eigen value
#> 6 Number of eigen values selected
#> 7 Threshold size
#> 8 Threshold variance (%)
#> Value
#> 1 1684
#> 2 NMSR, TTRN, TFWSR, TTRW, TFWSS, TTSW, TTPW, AVPW, ARSR, SRDM
#> 3 CUAL, LNGS, PTLC, DSTA, LFRT, LBTEF, CBTR, NMLB, ANGB, CUAL9M, LVC9M, TNPR9M, PL9M, STRP, STRC, PSTR
#> 4 FAMD
#> 5 1.63208
#> 6 6
#> 7 0.2
#> 8 75
#>
#> Core sets
#> =========================
#> Method Size VarRet
#> 1 By size specified 337 44.63182
#> 2 By threshold variance 859 75.00000
#> 3 By logistic regression 412 50.02493