decomposition of lmdme object
Description
This function calculates the decomposition of variance or
covariance structure using Principal Component Analysis
(PCA) or Partial Least Squared Regression (PLSR), on the
ANOVA decomposed lmdme object. In this context, in a two
factor experimental design with interaction, the linear
model of the ith observation (gene) can be written:
X=μ+X_{A}+X_{B}+X_{AB}+ε
where
X stands for the observed value.

the intercept μ.

X_{A}, X_{B} and X_{AB} are the first, second and interaction coefficients respectively.
The error term ε ~ N(0,σ^2).
The model is
iteratively decomposed in a step by step fashion,
decomposing one term each time by calling
lmdme
constructor:
Step 1: X=μ+E_{1}
Step 2: E_{1}=X_{A}+E_{2}
Step 3: E_{2}=X_{B}+E_{3}
Step 4: E_{3}=X_{AB}+E_{4}
Then, if we apply PCA on the ith step using E_{i1} matrix it is known as APCA but if applied on the coefficients X_{i} it is called ASCA. The same decomposition schema can also be used with PLSR.
Arguments
object 
lmdme class object. 
decomposition 
character to indicate the decomposition to be carried out, i.e., "pca" or "plsr". Default value is "pca". 
term 
character specifying the model term to perform the decomposition (e.g. "time" or "time:concentration" for interaction term). If the term is not specified (i.e. missing) it performs the analysis over all the model terms. 
subset 
subset of individuals (rows) to be included in the analysis. By default all the individuals are included. 
type 
character to indicate on which regression matrix ("coefficient" or "residual") the decomposition will be performed. The intercept term is not included in the results, as it can be directly analyzed with the original M data.frame. Default value is "coefficient" a.k.a. ASCA. Note that "residual" performs PCA or PLS on the ith residual E_{i1}=X_{i}+E_{i} and not the residuals of the ith model E_{i}. 
scale 
character "row", "column" or "none" to indicate if the matrix should be scaled by the row, column or not respectively. Default value is "none". 
Omatrix 
the output matrix for PLSR only. If the parameter is missing, the output matrix will be an identity matrix for the ASCA. Otherwise, is the design matrix corresponding to the specified term for APCA. 
... 
additional parameters for 
Value
Internal update of the "components" slot of the lmdme
object, which is a list of prcomp
or a list
of mvr (plsr
) objects using the given term
parameter. If missing(term), the length of the list
equals the number of decomposed models minus the
Intercept term for coefficients or the length of
decomposed models for residual decomposition.
Author(s)
Cristobal Fresno and Elmer A Fernandez
References
Smilde AK, Jansen JJ, Hoefsloot HCJ, Lamer RAN, Van der Greef J, Timmerman ME (2005) ANOVAsimultaneous component analysis (ASCA): a new tool for analyzing designed metabolomics data, Bioinformatics 21,13,3043 DOI:/10.1093/bioinformatics/bti476

Zwanenburg G, Hoefsloot HCJ, Westerhuis JA, Jansen JJ, Smilde AK (2011) ANOVA Principal component analysis and ANOVAsimultaneous component analysis: a comparison J. Chemometrics 25:561567 DOI:10.1002/cem.1400
Cristobal Fresno, Monica G. Balzarini, Elmer A. Fernandez (2014) lmdme: Linear Models on Designed Multivariate Experiments in R, Journal of Statistical Software, 56(7), 116, http://www.jstatsoft.org/v56/i07/.
See Also
prcomp
, plsr
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45  {
data(stemHypoxia)
##Just to make a balanced dataset in the Fisher sense (2 samples per
## time*oxygen levels)
design<design[design$time %in% c(0.5,1,5) & design$oxygen %in% c(1,5,21), ]
design$time<as.factor(design$time)
design$oxygen<as.factor(design$oxygen)
rownames(M)<M[, 1]
#Keeping appropriate samples only
M<M[, colnames(M) %in% design$samplename]
##ANOVA decomposition
fit<lmdme(model=~time+oxygen+time:oxygen, data=M, design=design)
##Just a copy of the same fit object and to perform analysis on those
##subjects/genes where at least one interaction coefficient is statistically
##different from zero (Ftest on the coefficients).
asca<fit
apca<fit
id<F.p.values(fit, term="time:oxygen")<0.001
##ASCA and APCA decomposition for every available term.
decomposition(asca, decomposition="pca", subset=id, scale="row")
decomposition(apca, decomposition="pca", subset=id, scale="row",
type="residual")
##Let's get the components for asca/apca decomposed objects
asca<components(asca)
apca<components(apca)
##Now let's try the PLSR decomposition for residuals and coefficients
plsr.residuals<fit
plsr.coefficients<fit
decomposition(plsr.coefficients, decomposition="plsr", subset=id,
scale="row")
decomposition(plsr.residuals, decomposition="plsr", subset=id, scale="row",
type="residual")
##Obtain the coefficients for decomposed plsr objects
##(coefficients/residuals)
plsr.coefficients<components(plsr.coefficients)
plsr.residuals < components(plsr.residuals)
}
