implementation of anova-pca in r factorab: pca score plot...
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Implementation of ANOVA-PCA in R forMultivariate Data Exploration
Matthew J. Keinsley & Bryan A. Hanson
Dept. of Chemistry & BiochemistryDePauw University, Greencastle Indiana USA
Analysis of Variance
Analysis of Variance (ANOVA) is a significance testwhich considers whether or not two samples comefrom the same population. In the diagram below,each curve represents a series of measurements ontwo samples, with each described by a mean andstandard deviation. A null hypothesis is definedwhich typically states that the two samples are fromthe same population. ANOVA can be thought ofas measuring how much the two samples can driftapart and still come from the same population (i.e.the null hypothesis is true). The key values froman ANOVA calculation are the test statistic and thep-value. The p-value represents the probability ofachieving a test statistic more extreme than the oneobserved if the two samples are from the same pop-ulation. The smaller the p-value, the more confidentone is about rejecting the null hypothesis at a se-lected confidence level, usually 95%. Internally, thecalculation compares the between sample variationto the within sample variation to generate the teststatistic. This is referred to as ”partioning the vari-ance.”
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0.0
0.1
0.2
0.3
0.4
sample values
Principal Component Analysis
Principal Component Analysis (PCA) is a commonmultivariate data analysis method which eliminatesuninformative variables (noise) and re-expresses thedata in terms of abstract ”principal components.”Thediagram below illustrates the matrix algebra used inthe computation. The PCA results are scores andloadings. Scores represent the samples in the newdata space, while loadings are the weights which eachvariable should be multiplied to obtain the score.While score plots are usually color-coded by groupmembership, PCA is blind to group membership.
1 ... j ... m
n
.
..
i
...
1
obje
cts
orsp
ecim
ens
measured variables
n
.
..
i
...
1
T
1 ... k ... a
m
...
j
...
1
P
1 ... k ... a
T = X • P or XP
PC1
scor
es fo
r eac
h ob
ject
...
PC2
scor
es fo
r eac
h ob
ject
...
etc
load
ings
for P
C1 ..
.lo
adin
gs fo
r PC2
...
etc
loadings
scoresoriginal data
X
Data Reduction method of PCA
P
T
ANOVA-PCA
ANOVA-PCA is a combination of both methods de-veloped by Harrington. The data is partitionedinto submatrices (shown below) corresponding toeach experimental factor in a manner reminiscent ofANOVA. The submatrices are then separately sub-jected to PCA after adding back the residual error. Ifthe effect of a factor is large compared to the residualerror, separation along the 1st PC in the score plotshould be evident. With this method, the significanceof a factor can be visually determined. ANOVA-PCAis not blind to group membership.
RawData = Grand
MeanMeans foreach levelof factor 1
Means foreach levelof factor 2
Means foreach level
offactor 1 x factor 2
ResidualError+
+
+
+
ANOVA-PCA con’t
The method to create the submatrices is shown be-low. Data for all samples belonging to each level of aparticular factor are replaced by their group averages.
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original data factor matrix
group membership
rows replaced byaverage of allsamples of a givenfactor level
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original data factor matrix
group membership
Computational Strategy
We implemented the ANOVA-PCA concept using aseries of R functions as shown below. The functionswere integrated into the package ChemoSpec, whichuses a Spectra object to store the spectral data andassociated information.
aovPCA(spectra, facs, ...)
avgFacLvls(spectra, facs)aovPCAscores(spectra, LM, plot, type, ...)
Spectra object factors to beanalyzed
aovPCAloadings(spectra, LM, pca, plot, ...)
Simulated Data
For testing purposes, a simulated data set was cre-ated by generating data which simulate UV-Vis spec-tra or chromatograms. Four different sample typeswere created in such a way as to represent groupmembership and the effects of the factors. The lev-els of the first factor are A and B; the second factorhas levels X and Y. The entire data set contains 5 ofeach sample type, and noise was added to the spectrain order to make them more realistic. The prototypesamples are shown below; type BY represents a inter-action factor as a peak is missing entirely comparedto the other sample types.
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0.0
0.2
0.4
Sample Type AX
Wavelength or Time
Pea
k in
tens
ity
0 5 10 15
0.0
1.0
2.0
Sample Type AY
Wavelength or Time
Pea
k in
tens
ity
0 5 10 15
0.0
1.0
2.0
Sample Type BX
Wavelength or Time
Pea
k in
tens
ity
0 5 10 15
0.0
1.0
2.0
Sample Type BY
Wavelength or Time
Pea
k in
tens
ity
Standard PCA
The results of a traditional PCA analysis on thesesimulated data are shown below. As expected,this data set does not present a challenge andeach group is clustered away from the others.
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−10 −5 0 5 10
−5
05
10
Original Data: PCA Score Plot
robust ellipses by group
PC1 score (30%)
PC
2 sc
ore
(8.6
%)
centered/noscale/classical KeyAXAYBXBY
aovPCA Score Plots
The following figures show the results of aovPCA
on the simulated data. Factor 1 has levelsA and B. Because the 1st factor is significant,there is separation between levels along PC1.
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−5 0 5 10
−10
−5
05
1015
FactorAB: PCA Score Plot
robust ellipses by group
PC1 score (32%)
PC
2 sc
ore
(5.5
%)
centered/noscale/classical KeyAXAYBXBY
Factor 2 has levels X and Y. As before, separation isobserved along PC1, but it is not as good as for Fac-tor 1. Note the groups here are reversed comparedto the 1st factor.
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−6 −4 −2 0 2 4 6
−10
010
20
FactorXY: PCA Score Plot
robust ellipses by group
PC1 score (9.1%)
PC
2 sc
ore
(7.4
%)
centered/noscale/classical KeyAXAYBXBY
The following figure shows the interaction betweenFactors 1 and 2. In this case, AX and BY aregrouped separately from AY and BX. This sep-aration therefore shows that the interaction be-tween the two factors is significant as expected.
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−6 −4 −2 0 2 4 6
−10
010
20
FactorAB x FactorXY: PCA Score Plot
robust ellipses by group
PC1 score (15%)
PC
2 sc
ore
(6.9
%)
centered/noscale/classical KeyAXAYBXBY
Finally, aovPCA on the residual error is shown. Theresidual error is the unexplained variance, so thereshould be no separation on PC1 nor any clustering.This is exactly what is observed.
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−20 −10 0 10 20 30
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−20
020
40
Res.Error: PCA Score Plot
robust ellipses by group
PC1 score (8.1%)
PC
2 sc
ore
(7.5
%)
centered/noscale/classical KeyAXAYBXBY
aovPCA Loadings Plots
The loading for Factor AB is shown below. Peaks onthe PC1 loadings show which wavelengths contributethe most to the separation in the data. The referencespectrum clearly shows the noise that was added tothe data.
FactorAB: Loadings Plot
centered/noscale/classical
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50.
00.
51.
0
0 5 10 15
Ref
eren
ce S
pect
rum
−0.
050.
000.
050.
10
PC
1 L
oadi
ngs
aovPCA of Real Data
The data in this example is composed of infrared (IR)spectra of the leaf surface of the common ”weed”purslane (Portulaca oleracea). Varieties collected inSouth Carolina and Wisconsin were grown in low andhigh water conditions as a part of a larger study onclimate change. The aovPCA score plot with Treat-ment as the factor is shown below; aovPCA is notable to separate the groups based upon this factor.
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−0.02 −0.01 0.00 0.01 0.02 0.03
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010
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005
0.00
00.
005
0.01
0
Treatment: PCA Score Plot
robust ellipses by group
PC1 score (77%)
PC
2 sc
ore
(13%
)
centered/noscale/classical KeySC.HSC.LWI.HWI.L
However, if the first 3 PCs from standardPCA are plotted, we can see separation be-tween the treatments. With this difficultdata set standard PCA outperforms aovPCA.
Cuticle IR Spectra: PCA Score Plot
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PC1 (83%)
PC2 (14%)
PC3 (1.1%)
SC.H SC.L WI.H WI.L
EB water trial week 3 IR Spectra Summer 2010
References
Pinto, Bosc, Nocairi, Barros, and Rutledge. ”UsingANOVA-PCA for Discriminant Analysis. . . ” Ana-lytica Chimica Acta 629.1-2 (2008): 47-55.
Harrington, Vieira, Espinoza, Nien, Romero, andYergey. ”Analysis of Variance–Principal Compo-nent Analysis. . . ”Analytica Chimica Acta 544.1-2(2005): 118-27.
Software: R packages ChemoSpec and HandyStuff
github.com/bryanhanson
Acknowledgements
Support was provided by the Science Research Fel-lows program and Chemistry Department researchfunds.
Summer 2011
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