Day-03. Shape Statistics I
PCA, PLS, PACA
Content Description
In geomorph, several functions allow analysis of shape
covariation with other matrices, which contain data or parameters. The
table below indicate functions that are frequently used, plus what the
types of analysis they tend to perform.
| Function | Alignment | Notes |
|---|---|---|
gm.prcomp |
Identity matrix or phylogenetic covariance matrix | This function is a wrapper for the RRPP::ordinate
function, which allows alignment to any kind of matrix.
gm.prcomp only aligns data to an identity matrix (PCA) or
phylogenetic covariance matrix (PACA), but has options for transforming
data or choosing between GLS or OLS estimation. |
two.b.pls |
Any alternative matrix | This is a basic function that allows any two data matrices (or vectors) to be input, but a covariance matrix or matrix of model parameters could also be used. |
integration.test |
A portion of shape data | This is basically the same as two.b.pls used for
comparison of two subsets of shape data. It will perform the
partitioning of data. This function is deferred to the module on
integration and modularity. |
phylo.integration |
A portion of shape data | This is basically the same as integration.test but
transforms data based on phylogenetic covariances. This function
is deferred to the module on integration and modularity. |
Set up
library(geomorph)## Loading required package: RRPP## Loading required package: rgl## Loading required package: Matrixdata(plethspecies)
Y.gpa <- gpagen(plethspecies$land, print.progress = FALSE)
plot(Y.gpa)
Principal component analysis (PCA)
PCA is the same as an alignment of data to an identity matrix
(independence). This is an unconstrained alignment. SVD finds principal
components (eigenvectors) and singular values that can be transformed
into eigenvalues. In gm.prcomp, the set-up is pretty
straightforward.
PCA <- gm.prcomp(Y.gpa$coords, phy = plethspecies$phy,
align.to.phy = FALSE,
GLS = FALSE)A summary will provide basic information about the strength of components.
summary(PCA)##
## Ordination type: Principal Component Analysis
## Centering by OLS mean
## Orthogonal projection of OLS residuals
## Number of observations: 9
## Number of vectors 8
##
## Importance of Components:
## Comp1 Comp2 Comp3 Comp4
## Eigenvalues 0.0002720474 0.0001120524 0.0001084758 0.0000568924
## Proportion of Variance 0.4564029477 0.1879858086 0.1819855633 0.0954461044
## Cumulative Proportion 0.4564029477 0.6443887563 0.8263743196 0.9218204240
## Comp5 Comp6 Comp7 Comp8
## Eigenvalues 0.0000264508 1.260516e-05 5.959622e-06 1.584785e-06
## Proportion of Variance 0.0443754550 2.114717e-02 9.998220e-03 2.658730e-03
## Cumulative Proportion 0.9661958790 9.873431e-01 9.973413e-01 1.000000e+00
##
##
## Dispersion (variance) of points, after projection:
## Comp1 Comp2 Comp3
## Tips Dispersion 2.720474e-04 1.120524e-04 1.084758e-04
## Proportion Tips Dispersion 4.564029e-01 1.879858e-01 1.819856e-01
## Cumulative Tips Dispersion 4.564029e-01 6.443888e-01 8.263743e-01
## Ancestors Dispersion 3.073342e-05 2.320038e-05 2.039005e-05
## Proportion Ancestors Dispersion 3.854327e-01 2.909597e-01 2.557149e-01
## Cumulative Ancestors Dispersion 3.854327e-01 6.763923e-01 9.321072e-01
## Comp4 Comp5 Comp6
## Tips Dispersion 5.689240e-05 2.645080e-05 1.260516e-05
## Proportion Tips Dispersion 9.544610e-02 4.437545e-02 2.114717e-02
## Cumulative Tips Dispersion 9.218204e-01 9.661959e-01 9.873431e-01
## Ancestors Dispersion 3.171758e-06 1.748508e-06 3.945777e-07
## Proportion Ancestors Dispersion 3.977752e-02 2.192831e-02 4.948462e-03
## Cumulative Ancestors Dispersion 9.718847e-01 9.938130e-01 9.987615e-01
## Comp7 Comp8
## Tips Dispersion 5.959622e-06 1.584785e-06
## Proportion Tips Dispersion 9.998220e-03 2.658730e-03
## Cumulative Tips Dispersion 9.973413e-01 1.000000e+00
## Ancestors Dispersion 8.354868e-08 1.520561e-08
## Proportion Ancestors Dispersion 1.047797e-03 1.906960e-04
## Cumulative Ancestors Dispersion 9.998093e-01 1.000000e+00The loadings are found in the rotation matrix.
PCA$rot ## Comp1 Comp2 Comp3 Comp4 Comp5
## 1.X 0.0639937781 0.205429539 0.170984900 0.092240499 -0.243267662
## 1.Y -0.2186270977 -0.001301379 0.016212731 0.203187307 0.232599624
## 2.X -0.0878606406 0.319081044 0.248024539 -0.043348223 -0.422003358
## 2.Y -0.4476470091 0.217392614 0.039671153 -0.203712850 0.192957190
## 3.X -0.1019656118 -0.110250492 0.609109959 -0.431680823 0.150168617
## 3.Y -0.0771373917 -0.000805622 -0.029967224 -0.137763592 0.268182949
## 4.X 0.1992343645 0.151262835 -0.039895153 0.185626140 0.194435236
## 4.Y 0.1213302225 0.209895468 0.057399404 0.251871724 0.111182304
## 5.X -0.1842674107 -0.032308565 -0.090353311 -0.072747314 0.034807610
## 5.Y 0.0425135453 0.254311461 -0.063056292 -0.001214202 0.019992286
## 6.X -0.0220413670 -0.295845336 -0.234744294 -0.022580385 -0.150555325
## 6.Y 0.0736696008 0.279725523 -0.052761729 0.001311960 -0.006695822
## 7.X 0.0243393890 -0.141536784 -0.254900654 -0.110273929 -0.221595979
## 7.Y 0.1658520151 -0.303306114 -0.036556119 -0.243396155 -0.146444364
## 8.X -0.0368710126 0.217236787 -0.156732264 -0.107859034 0.040626986
## 8.Y 0.2086654440 -0.239063444 0.137631794 -0.187091618 -0.188967129
## 9.X 0.1213972184 -0.346758562 0.329744837 0.506442373 0.331171499
## 9.Y -0.2372316829 -0.347693604 0.022022217 0.078551200 -0.083188623
## 10.X 0.0250307386 0.120458956 -0.110704841 0.209168061 -0.046515607
## 10.Y -0.2712050629 -0.124605602 -0.088814356 0.346012061 -0.406222454
## 11.X -0.0009894458 -0.086769422 -0.470533717 -0.204987364 0.332727983
## 11.Y 0.6398174166 0.055450698 -0.001781577 -0.107755835 0.006604039
## Comp6 Comp7 Comp8
## 1.X -0.057696809 -0.082018500 -0.303542573
## 1.Y -0.120021863 0.276412995 -0.077892129
## 2.X -0.280884684 0.258233252 -0.072700886
## 2.Y -0.399470856 0.013087244 0.213232472
## 3.X 0.266994201 -0.229702388 0.273994933
## 3.Y 0.173409908 -0.080136106 -0.428624862
## 4.X 0.193960296 0.364404309 0.028518481
## 4.Y -0.060369636 -0.007322196 0.137180098
## 5.X 0.105224695 -0.120514442 -0.610922851
## 5.Y 0.054936387 -0.188788073 0.047714415
## 6.X -0.059997340 0.188172920 0.306580312
## 6.Y 0.046173917 -0.210147221 0.085048928
## 7.X -0.267197382 -0.358079748 -0.000149742
## 7.Y -0.400044714 0.168529291 -0.087173774
## 8.X 0.230143705 0.202353581 0.192887205
## 8.Y 0.263157045 0.234209239 -0.041482088
## 9.X -0.285451487 -0.183496405 0.011214603
## 9.Y 0.193990843 0.278003063 -0.046542405
## 10.X 0.160016458 0.070132402 0.078512706
## 10.Y 0.285586293 -0.366237418 0.176914090
## 11.X -0.005111653 -0.109484980 0.095607812
## 11.Y -0.037347324 -0.117610818 0.021625255Centered data are inherently projected onto PCs, which can be plotted as scores.
PCAplot <- plot(PCA, pch = 16, phylo = TRUE)
This can be done for interactive plotting:
# Explore shape change in the plot
picknplot.shape(PCAplot)
picknplot.shape(PCAplot, mag = 3)In case one wishes to use results for other reasons:
attributes(PCA)## $names
## [1] "d" "sdev" "rotation" "center" "scale" "GLS"
## [7] "transform" "alignment" "RV" "x" "A" "shapes"
## [13] "ancestors" "phy" "anc.x" "anc.var"
##
## $class
## [1] "gm.prcomp" "ordinate"Note that the result provide d, not “eigenvalues”. These
are the singular values, which gm.prcomp transforms to
eigenvalues. This is verified below.
PCA$d## Comp1 Comp2 Comp3 Comp4 Comp5 Comp6
## 2.720474e-04 1.120524e-04 1.084758e-04 5.689240e-05 2.645080e-05 1.260516e-05
## Comp7 Comp8
## 5.959622e-06 1.584785e-06eigen(var(PCA$x), only.values = TRUE)$values## [1] 2.720474e-04 1.120524e-04 1.084758e-04 5.689240e-05 2.645080e-05
## [6] 1.260516e-05 5.959622e-06 1.584785e-06Note also that some results won’t be available, e.g.,
PCA$RV## NULLPhylogenetically aligned component analysis (PACA)
PACA aligned data with a phylogenetic covariance matrix rather than an identity matrix.
PACA <- gm.prcomp(Y.gpa$coords, phy = plethspecies$phy,
align.to.phy = TRUE,
GLS = FALSE)
summary(PACA)##
## Ordination type: Alignment to an alternative matrix
## Alignment matrix: phy
## Centering by OLS mean
## OLS residuals
## Alignment to phy means residual projection is not orthogonal.
## Number of observations: 9
## Number of vectors 8
##
## Importance of Components:
## Comp1 Comp2 Comp3 Comp4
## Singular Value 0.0003931629 0.0001155198 5.091258e-05 2.660786e-05
## Proportion of Covariance 0.6595936040 0.1938028696 8.541399e-02 4.463894e-02
## Cumulative Proportion 0.6595936040 0.8533964737 9.388105e-01 9.834494e-01
## RV by Component 0.0993211764 0.0291827102 1.286158e-02 6.721702e-03
## Cumulative RV 0.0993211764 0.1285038867 1.413655e-01 1.480872e-01
## Comp5 Comp6 Comp7 Comp8
## Singular Value 6.598675e-06 2.351469e-06 7.108871e-07 2.042572e-07
## Proportion of Covariance 1.107033e-02 3.944966e-03 1.192627e-03 3.426742e-04
## Cumulative Proportion 9.945197e-01 9.984647e-01 9.996573e-01 1.000000e+00
## RV by Component 1.666963e-03 5.940304e-04 1.795850e-04 5.159965e-05
## Cumulative RV 1.497541e-01 1.503482e-01 1.505278e-01 1.505794e-01
##
##
## Dispersion (variance) of points, after projection:
## Comp1 Comp2 Comp3
## Tips Dispersion 2.301865e-04 1.124810e-04 1.133102e-04
## Proportion Tips Dispersion 3.861747e-01 1.887049e-01 1.900960e-01
## Cumulative Tips Dispersion 3.861747e-01 5.748795e-01 7.649755e-01
## Ancestors Dispersion 4.069158e-05 2.437196e-05 8.067595e-06
## Proportion Ancestors Dispersion 5.103196e-01 3.056526e-01 1.011770e-01
## Cumulative Ancestors Dispersion 5.103196e-01 8.159722e-01 9.171492e-01
## Comp4 Comp5 Comp6
## Tips Dispersion 7.526878e-05 4.046474e-05 1.571235e-05
## Proportion Tips Dispersion 1.262754e-01 6.788606e-02 2.635999e-02
## Cumulative Tips Dispersion 8.912509e-01 9.591370e-01 9.854970e-01
## Ancestors Dispersion 6.244464e-06 3.539330e-07 3.508799e-09
## Proportion Ancestors Dispersion 7.831282e-02 4.438730e-03 4.400441e-05
## Cumulative Ancestors Dispersion 9.954620e-01 9.999008e-01 9.999448e-01
## Comp7 Comp8
## Tips Dispersion 6.396433e-06 2.248360e-06
## Proportion Tips Dispersion 1.073104e-02 3.771984e-03
## Cumulative Tips Dispersion 9.962280e-01 1.000000e+00
## Ancestors Dispersion 2.594059e-09 1.810230e-09
## Proportion Ancestors Dispersion 3.253251e-05 2.270238e-05
## Cumulative Ancestors Dispersion 9.999773e-01 1.000000e+00In the summary above, the dispersion statistics can be compared to PCA. It should be clear that the tips dispersion (which for PCA is the same as portion of eigenvalue sum) is larger for PCA but the dispersion of estimate ancestral values is large for PACA, in the first component.
Some other typical steps:
PACA$rot # loadings## Comp1 Comp2 Comp3 Comp4 Comp5 Comp6
## 1.X 0.02817716 -0.238265109 0.215048829 -0.02421074 -0.228692597 0.08795239
## 1.Y -0.24529137 0.031301382 -0.162045684 -0.17684791 -0.044540749 -0.18288472
## 2.X -0.07608018 -0.402254103 0.336789063 0.19554593 -0.311415307 -0.08246873
## 2.Y -0.34291440 -0.281460733 -0.146787704 0.21637801 0.084164259 -0.48632474
## 3.X -0.33515549 -0.041319928 0.452877791 0.02352182 0.579512548 0.07341471
## 3.Y -0.06807174 0.005443062 -0.137747673 -0.02648898 0.284838026 0.07828795
## 4.X 0.15491526 -0.050339744 -0.103594409 -0.34842667 0.090954174 0.09545823
## 4.Y 0.06978562 -0.153768894 -0.031041240 -0.30836707 -0.099774563 -0.10096432
## 5.X -0.12222934 0.009775925 -0.132847431 0.12270876 0.042970238 0.15119942
## 5.Y 0.10107867 -0.235599520 -0.098219958 -0.04432624 0.019464133 0.03585376
## 6.X 0.05513603 0.308734201 -0.084911436 0.21393490 -0.107816816 -0.01237432
## 6.Y 0.13049059 -0.261295075 -0.074160763 -0.04790042 0.002664178 0.03473008
## 7.X 0.16449200 0.120453856 -0.054584199 0.34610719 -0.214642033 -0.13112828
## 7.Y 0.18035661 0.274492447 0.235970876 0.32103443 -0.074492678 -0.29994426
## 8.X 0.06637459 -0.184563521 -0.191204043 0.02958695 0.173654346 0.15515196
## 8.Y 0.11342100 0.207464913 0.308225727 0.09008401 0.153482781 0.29553566
## 9.X -0.17773034 0.383705890 0.163897772 -0.55254176 -0.180155187 -0.35938197
## 9.Y -0.28554076 0.317790294 -0.006977614 0.05807331 -0.026819790 0.22237835
## 10.X 0.05435288 -0.069410013 -0.141427053 -0.14695228 -0.114758671 0.17491248
## 10.Y -0.23673443 0.085315124 -0.119121918 0.04578262 -0.397601899 0.45339079
## 11.X 0.18774743 0.163482545 -0.460044884 0.14072590 0.270389306 -0.15273588
## 11.Y 0.58342020 0.010317000 0.231905951 -0.12742175 0.098616303 -0.05005855
## Comp7 Comp8
## 1.X -0.13731582 0.24925071
## 1.Y 0.23959689 0.16972804
## 2.X 0.25723193 0.11256032
## 2.Y 0.06242302 -0.14317967
## 3.X -0.19783219 -0.31865053
## 3.Y -0.19059834 0.41235048
## 4.X 0.35704236 0.06421034
## 4.Y 0.02497692 -0.12266972
## 5.X -0.25589244 0.55727765
## 5.Y -0.15758814 -0.08922005
## 6.X 0.26036117 -0.26151817
## 6.Y -0.16793832 -0.13235333
## 7.X -0.33359784 -0.08712275
## 7.Y 0.12574723 0.14005208
## 8.X 0.26009832 -0.14048572
## 8.Y 0.20718122 0.05767784
## 9.X -0.22801585 -0.02371599
## 9.Y 0.25457671 0.08308760
## 10.X 0.10416734 -0.07753820
## 10.Y -0.27561189 -0.33087067
## 11.X -0.08624698 -0.07426765
## 11.Y -0.12276531 -0.04460261PACAplot <- plot(PACA, pch = 16, phylo = TRUE)
And how to do interactive plotting:
# Explore shape change in the plot
picknplot.shape(PACAplot)
picknplot.shape(PACAplot, mag = 3)phylo-PCA
Phylo-PCA is a little interesting in that it aligns data to an identity matrix that has been transformed. The data are transformed for alignment but not projection. Therefore, the ordination looks like a rotation of the data space, specifically to minimize alignment to phylogeny in the first component.
pPCA <- gm.prcomp(Y.gpa$coords, phy = plethspecies$phy,
align.to.phy = FALSE,
GLS = TRUE)
summary(pPCA)##
## Ordination type: Principal Component Analysis
## Centering by GLS mean
## Oblique projection of GLS-centered residuals
## Number of observations: 9
## Number of vectors 8
##
## Importance of Components:
## Comp1 Comp2 Comp3 Comp4
## Eigenvalues 2.761405e-05 9.322868e-06 7.889847e-06 6.605120e-06
## Proportion of Variance 4.855705e-01 1.639351e-01 1.387365e-01 1.161457e-01
## Cumulative Proportion 4.855705e-01 6.495056e-01 7.882421e-01 9.043878e-01
## Comp5 Comp6 Comp7 Comp8
## Eigenvalues 3.017886e-06 1.203779e-06 9.895110e-07 2.262231e-07
## Proportion of Variance 5.306707e-02 2.116747e-02 1.739975e-02 3.977950e-03
## Cumulative Proportion 9.574548e-01 9.786223e-01 9.960221e-01 1.000000e+00
##
##
## Dispersion (variance) of points, after projection:
## Comp1 Comp2 Comp3
## Tips Dispersion 0.0002498092 8.395185e-05 1.057114e-04
## Proportion Tips Dispersion 0.4190949497 1.408427e-01 1.773479e-01
## Cumulative Tips Dispersion 0.4190949497 5.599376e-01 7.372855e-01
## Ancestors Dispersion 0.0000169982 9.862682e-06 2.194379e-05
## Proportion Ancestors Dispersion 0.2131770701 1.236895e-01 2.752006e-01
## Cumulative Ancestors Dispersion 0.2131770701 3.368665e-01 6.120671e-01
## Comp4 Comp5 Comp6
## Tips Dispersion 0.0001027383 2.765308e-05 1.675548e-05
## Proportion Tips Dispersion 0.1723598707 4.639247e-02 2.811000e-02
## Cumulative Tips Dispersion 0.9096453329 9.560378e-01 9.841478e-01
## Ancestors Dispersion 0.0000272722 1.114653e-06 1.772175e-06
## Proportion Ancestors Dispersion 0.3420249711 1.397905e-02 2.222513e-02
## Cumulative Ancestors Dispersion 0.9540920945 9.680711e-01 9.902963e-01
## Comp7 Comp8
## Tips Dispersion 7.598972e-06 1.850019e-06
## Proportion Tips Dispersion 1.274849e-02 3.103703e-03
## Cumulative Tips Dispersion 9.968963e-01 1.000000e+00
## Ancestors Dispersion 7.068511e-07 6.689930e-08
## Proportion Ancestors Dispersion 8.864732e-03 8.389947e-04
## Cumulative Ancestors Dispersion 9.991610e-01 1.000000e+00pPCA$rot # loadings## Comp1 Comp2 Comp3 Comp4 Comp5 Comp6
## 1.X 0.09744783 -0.10546858 0.15935634 0.20253521 0.16056595 -0.13417564
## 1.Y -0.19346734 -0.04546925 0.04748510 0.13314937 -0.31958879 0.03820919
## 2.X -0.04615034 -0.01456001 0.31788816 0.24860337 0.26929078 -0.43940392
## 2.Y -0.40632562 0.26238469 0.31692917 0.02623064 -0.25583030 -0.28523047
## 3.X 0.14597215 0.71972058 0.06953146 0.24024316 0.17047213 0.24372366
## 3.Y -0.06286199 0.16702865 0.06292780 -0.13230253 -0.13204067 0.28321955
## 4.X 0.16761177 -0.22087601 0.15606275 -0.03986716 -0.10015865 0.26602236
## 4.Y 0.13108628 -0.21833246 0.17681955 0.13826261 -0.15014233 0.01532702
## 5.X -0.19670169 0.07135323 -0.02646040 -0.07427358 -0.01608575 0.14216395
## 5.Y 0.01355821 -0.10497217 0.20863595 -0.05094787 0.04326775 0.05069676
## 6.X -0.10559484 -0.05400958 -0.32560160 -0.16347722 0.07146225 -0.14028781
## 6.Y 0.04410573 -0.12131037 0.22569511 -0.04351735 0.06775454 0.02775797
## 7.X -0.06720180 -0.04258799 -0.21825539 -0.19852916 0.05832446 -0.34749618
## 7.Y 0.13500781 0.18471676 -0.23605774 -0.19167470 -0.03912350 -0.45136637
## 8.X -0.10092950 -0.06141568 0.21025616 -0.18671570 0.13523279 0.19077450
## 8.Y 0.23777203 0.17149286 -0.19692014 -0.05050884 0.31388103 0.11569923
## 9.X 0.27018273 -0.04004122 -0.31088594 0.46053345 -0.56988636 -0.03080956
## 9.Y -0.20846592 0.07202673 -0.32992374 0.11684234 0.08257919 0.15881552
## 10.X -0.02381856 -0.27073567 0.04193546 0.02814845 0.06627038 0.13878284
## 10.Y -0.28627926 -0.29458074 -0.32249372 0.28897650 0.35190153 0.12142179
## 11.X -0.14081777 0.01862092 -0.07382701 -0.51720082 -0.24548798 0.11070581
## 11.Y 0.59587007 -0.07298470 0.04690267 -0.23451016 0.03734153 -0.07455021
## Comp7 Comp8
## 1.X 0.06289659 0.33262394
## 1.Y -0.27563621 0.03568874
## 2.X -0.24701380 0.10036333
## 2.Y 0.04160660 -0.22260700
## 3.X 0.20746529 -0.26120406
## 3.Y 0.05878198 0.40440622
## 4.X -0.34342055 -0.07804298
## 4.Y 0.03274855 -0.14596129
## 5.X 0.08042809 0.60954976
## 5.Y 0.22058268 -0.03698628
## 6.X -0.18360502 -0.30657027
## 6.Y 0.24559786 -0.06969995
## 7.X 0.35579585 0.04852852
## 7.Y -0.19590089 0.09546411
## 8.X -0.15320117 -0.21525919
## 8.Y -0.27444452 0.04088870
## 9.X 0.12476439 -0.02278488
## 9.Y -0.32210796 0.02977825
## 10.X -0.05280724 -0.08448917
## 10.Y 0.34576123 -0.11968177
## 11.X 0.14869758 -0.12271501
## 11.Y 0.12301067 -0.01128974pPCAplot <- plot(pPCA, pch = 16, phylo = TRUE)
Notice the difference in dispersion of ancestral points in the summary above, compared to PACA. Interactive plotting can be done as follows:
# Explore shape change in the plot
picknplot.shape(pPCAplot)
picknplot.shape(pPCAplot, mag = 3)To summarize: PCA, performed with OLS estimation and no adjustment for phylogeny, find the maximum dispersion of tip taxa along the first component. PACA finds the maximum dispersion of ancestor values along the first component. pPCA finds a first component most independent of phylogeny.
Partial least-squares (PLS)
PLS aligns shape data or a portion of it to alternative data sets, or
alternative portions of the data. We will consider three examples, with
the plethShapeFood example data.
data("plethShapeFood")
Y.gpa <- gpagen(plethShapeFood$land,
print.progress = FALSE,)
food <- plethShapeFood$food
rownames(food) <- names(Y.gpa$Csize) # just to assure match
PLSfood <- two.b.pls(food, Y.gpa$coords,
print.progress = FALSE,
iter = 9999)
PLSallometry <- two.b.pls(Y.gpa$Csize, Y.gpa$coords,
print.progress = FALSE,
iter = 9999)
PLSintegration <- two.b.pls(Y.gpa$coords[1:5,, ],
Y.gpa$coords[8:13,,],
print.progress = FALSE,
iter = 9999)It is a good idea to look at the results that are available, at least using one example:
attributes(PLSfood)## $names
## [1] "r.pls" "P.value" "Z"
## [4] "left.pls.vectors" "right.pls.vectors" "random.r"
## [7] "XScores" "YScores" "svd"
## [10] "A1" "A2" "A1.matrix"
## [13] "A2.matrix" "permutations" "call"
## [16] "method"
##
## $class
## [1] "pls"Loadings:
PLSfood$left.pls.vectors## [,1] [,2] [,3] [,4] [,5]
## oligoch 0.09004086 0.003112899 0.10080188 0.409722018 0.0582768813
## gastrop -0.01806988 -0.550910355 -0.34547832 -0.241358177 -0.3564841398
## isopoda 0.04248161 0.341360115 0.01758372 0.293072055 -0.3748506890
## diplop 0.05513422 0.382587566 0.20527963 -0.112872870 -0.4054631580
## chilop 0.09336132 0.037443265 0.04266139 0.316443819 0.0076178754
## acar -0.50752475 0.019441095 0.41786654 0.049677139 0.1092069299
## araneida 0.25374179 -0.046295602 0.40983773 0.009393841 0.2707447245
## chelon -0.11208684 -0.285524589 0.47666133 -0.015241158 -0.5235386218
## coleopt 0.49499065 0.224817353 -0.03268055 -0.077491602 0.0603655414
## collemb -0.39472156 0.072040810 0.09579918 -0.047419077 0.1555297654
## diptera 0.16588149 -0.046529118 0.23630937 0.041576950 0.0622706930
## hymen 0.36492582 -0.498143789 0.27865952 0.175828837 0.0951740242
## isopt -0.11941724 -0.133602846 -0.25262551 0.600488142 0.0586785191
## orthopt 0.12637425 0.040163997 -0.02415219 0.269005133 -0.4004394462
## larvae 0.21581022 0.077047426 0.13298564 -0.312276062 -0.0007495818
## eggs -0.07184867 -0.129214952 0.17493561 0.030878610 0.0257905124
## [,6] [,7] [,8] [,9] [,10]
## oligoch -0.24202483 0.02851481 -0.239517142 -0.232884506 0.418584843
## gastrop -0.15914300 0.20501486 -0.266169944 -0.039474721 -0.015680026
## isopoda 0.02903773 0.14733178 -0.303918325 0.041665298 0.263608405
## diplop -0.17429031 0.06717287 0.164805178 0.324361934 0.117775611
## chilop -0.36809050 0.28627052 0.352598285 -0.581366483 -0.316104964
## acar -0.02943342 0.11660887 0.143237081 0.102910457 -0.048845524
## araneida 0.29655012 0.18577798 -0.415454941 0.043022135 -0.414262051
## chelon 0.18801604 0.12901147 0.121758837 -0.073224050 -0.107619955
## coleopt -0.22367816 0.25101931 -0.006875119 0.310950076 -0.227776911
## collemb -0.22716709 0.29108418 -0.486778217 0.006024732 0.067249645
## diptera 0.11005572 -0.52822073 -0.156177214 -0.221902676 0.180546169
## hymen -0.18761468 0.14787130 0.086530482 0.273626576 0.310052658
## isopt 0.41223476 0.13582060 0.203546686 0.337453961 0.001440993
## orthopt 0.07775462 -0.29550360 -0.232465244 -0.077664805 -0.364075591
## larvae 0.33346151 0.16959963 0.213035154 -0.266521430 0.353689680
## eggs -0.43485635 -0.44841242 0.101257615 0.263989508 -0.113547949
## [,11] [,12] [,13] [,14] [,15]
## oligoch -0.073809090 0.35494842 -0.44935824 0.092971814 -0.3194481770
## gastrop -0.376428351 0.14102266 0.20602910 0.036962767 -0.2073974072
## isopoda 0.238901450 0.15759205 0.51432026 0.041546815 0.1271321816
## diplop -0.438738763 -0.15627424 -0.01338258 0.268470087 0.0008448387
## chilop -0.158215519 -0.14197289 0.19625282 0.102156082 0.1056019970
## acar -0.179371960 0.25917707 0.27780352 -0.470800627 -0.3254354928
## araneida -0.168778772 0.22969794 0.12082794 0.331091683 0.0642920590
## chelon 0.307871176 -0.15007295 -0.32126093 0.160199433 -0.1497107285
## coleopt 0.006256427 -0.09004958 -0.17137230 -0.283281382 -0.4018858894
## collemb -0.151682377 -0.45258176 -0.24787451 -0.085434361 0.3074420104
## diptera -0.284030097 -0.47599160 0.25013914 0.002622961 -0.3269114227
## hymen 0.130797966 -0.17056742 0.17728127 -0.244032347 0.2834018726
## isopt -0.339049946 -0.13044200 -0.09732078 0.117080724 -0.0163624479
## orthopt -0.153489518 0.05327849 -0.19455530 -0.543916824 0.2616735580
## larvae -0.374115621 0.23077222 -0.14648704 -0.209320541 0.3162191967
## eggs -0.153318749 0.31708879 -0.06194181 0.219860136 0.2884668802
## [,16]
## oligoch -0.14211855
## gastrop 0.05531295
## isopoda 0.32324965
## diplop -0.40125483
## chilop 0.05384577
## acar -0.03152220
## araneida -0.11827823
## chelon 0.22809193
## coleopt 0.39087742
## collemb 0.18558969
## diptera 0.18660891
## hymen -0.21909274
## isopt 0.20231360
## orthopt -0.18589916
## larvae 0.30385271
## eggs 0.45107578PLSfood$right.pls.vectors## [,1] [,2] [,3] [,4] [,5]
## 1.X 0.13133953 0.016017800 -0.027231576 -0.112364516 0.129185831
## 1.Y 0.14787439 0.064700347 0.002158334 -0.104844026 -0.139666486
## 2.X 0.18997240 -0.085495097 0.016186956 0.007952418 0.330786702
## 2.Y 0.21071931 -0.041299482 0.160306629 0.033354169 -0.186020125
## 3.X -0.10497337 -0.278032904 0.217025988 -0.149685660 -0.134903347
## 3.Y 0.01823770 -0.011813496 0.023272786 -0.033724800 -0.202462108
## 4.X 0.02175516 0.014456289 -0.236713711 0.314917268 0.219072007
## 4.Y -0.29628475 -0.050082411 -0.049278170 0.102674532 -0.050241924
## 5.X -0.03246685 -0.057861386 -0.169413647 0.151929776 0.089086962
## 5.Y -0.30786568 0.049237233 -0.041987632 0.158915948 0.065421091
## 6.X 0.04097434 -0.307022495 0.528304865 0.101642619 -0.273020730
## 6.Y 0.26835518 0.281134207 0.261101618 0.248854586 0.221016678
## 7.X 0.08484467 -0.072600350 -0.258164600 -0.187020943 -0.238917375
## 7.Y 0.10011120 0.351750512 0.199695532 -0.007810346 0.010582978
## 8.X 0.24257232 0.199231902 -0.319392449 0.324262030 -0.217014150
## 8.Y -0.27613187 0.102491653 -0.029819586 0.177903840 -0.253510340
## 9.X -0.52125417 0.365570595 0.248722170 -0.027340940 0.069370015
## 9.Y -0.29871448 -0.283290297 -0.160260576 -0.233598005 0.286075336
## 10.X -0.01313886 0.341927800 -0.085658889 -0.272896141 -0.005650971
## 10.Y 0.13148260 -0.225262939 0.076937140 -0.213945544 0.411705176
## 11.X -0.02765115 0.184773252 -0.070110492 -0.475293264 -0.108854602
## 11.Y 0.12231665 -0.088280534 0.029395633 0.052501203 -0.016808394
## 12.X -0.07705921 -0.329160469 -0.015287345 0.232986308 -0.154279058
## 12.Y -0.04872696 -0.124990718 -0.363320979 0.092203797 0.016443086
## 13.X 0.06508520 0.008195063 0.171732729 0.090911045 0.295138716
## 13.Y 0.22862671 -0.024294075 -0.108200731 -0.272485353 -0.162534968
## [,6] [,7] [,8] [,9] [,10]
## 1.X 0.1401340912 -0.048586352 0.09725704 -0.041795998 -0.0259859498
## 1.Y -0.0736984759 -0.230544172 -0.35830581 0.241577865 0.2472479926
## 2.X 0.0145429832 -0.016363399 -0.18871483 0.012187120 0.0167051248
## 2.Y -0.2727864253 0.013027011 -0.17922975 -0.077428989 -0.0002602969
## 3.X 0.6600265541 -0.154451292 -0.18365324 -0.061848527 0.0409705660
## 3.Y -0.0971932282 -0.031825072 0.33551500 0.489815553 -0.0807367575
## 4.X 0.0049043804 -0.134865382 0.04031652 0.120265222 -0.0012062396
## 4.Y -0.0005993481 0.111958617 -0.20264383 -0.219817838 -0.0403356434
## 5.X -0.1451065442 -0.292990328 0.38950902 -0.345484339 0.1297816765
## 5.Y -0.0379762975 0.075565005 -0.22815702 -0.093276811 0.0199002696
## 6.X -0.2733362037 -0.034916801 0.07597127 -0.253749318 -0.3482123962
## 6.Y 0.1376658654 -0.006660703 -0.04352786 0.069134947 -0.2438002650
## 7.X -0.2464461566 0.359105671 0.06810656 -0.096426255 0.1935743559
## 7.Y 0.1774596405 0.364309863 0.14288292 -0.396245268 0.4087864102
## 8.X 0.0118153692 0.209171840 -0.17363976 0.006963985 -0.3626273055
## 8.Y 0.3130172825 0.123742170 0.35047836 0.175307289 -0.1581562895
## 9.X -0.2930951586 -0.195561085 -0.01384790 0.072847627 0.0993836985
## 9.Y -0.1546430526 0.202741619 -0.17209468 0.057259227 -0.2379504200
## 10.X 0.0710700689 -0.106154034 -0.24784389 -0.053966004 -0.2164264560
## 10.Y 0.0298885562 0.195674987 0.24280391 -0.016980051 -0.1709875287
## 11.X 0.0611013569 0.015472004 0.19771406 0.100094690 -0.1198548821
## 11.Y -0.0277610804 -0.413931336 0.06971667 0.063579576 0.1523240373
## 12.X 0.0840567961 0.230722027 -0.11124199 0.252853177 0.3356064317
## 12.Y 0.0848484590 -0.275679832 0.07617344 -0.239309345 0.0062884495
## 13.X -0.0896675368 0.169417132 0.05006713 0.288058618 0.2582913758
## 13.Y -0.0782218957 -0.128378157 -0.03361136 -0.053616155 0.0976800419
## [,11] [,12] [,13] [,14] [,15]
## 1.X 0.098001968 -0.132809607 0.0296159761 -0.07018878 0.10614965
## 1.Y -0.063502785 -0.077490815 0.2427606510 0.25134265 -0.04812370
## 2.X -0.122221535 -0.106840735 0.1087801115 -0.35715705 0.05313993
## 2.Y 0.029197131 0.084226807 0.0562822333 0.33005632 0.04377936
## 3.X -0.093207900 0.250112936 -0.1310219050 0.18752347 -0.21881788
## 3.Y -0.037232691 0.002995361 -0.3823916733 -0.05493163 0.07587240
## 4.X 0.244664112 0.195199653 0.1669199042 0.04155005 0.06733808
## 4.Y -0.048856580 -0.009287283 0.3432249625 -0.13867484 0.21846293
## 5.X -0.006791954 -0.167391139 -0.0561485876 0.05261210 -0.37814322
## 5.Y -0.114356368 0.152425046 -0.2214391263 -0.08674412 -0.38964011
## 6.X -0.073394328 -0.040270288 0.0116523433 -0.12609423 0.15396931
## 6.Y 0.093967232 0.017299653 -0.2462110150 -0.14563553 -0.21021674
## 7.X 0.128468171 0.524522781 -0.1688203246 -0.05053098 -0.06555358
## 7.Y 0.057040348 -0.080370549 0.0004142777 -0.06530502 0.13402799
## 8.X -0.133392517 -0.064853981 0.1421295402 0.19387024 -0.16234117
## 8.Y -0.014847222 0.009561439 0.2344571276 -0.08167072 0.03600340
## 9.X 0.093649951 -0.075360447 -0.0092038306 0.23630235 -0.04966482
## 9.Y 0.025084775 -0.047652782 -0.0403954064 -0.18963047 -0.08008613
## 10.X 0.276923577 0.003180371 -0.3210945030 -0.03909153 0.36119797
## 10.Y 0.241302666 -0.012914632 0.1193579980 0.50823832 -0.02557086
## 11.X -0.182969833 -0.086221904 0.3501500603 -0.07712712 -0.09379489
## 11.Y 0.270992905 0.373335101 0.2528410196 -0.31227465 0.06524072
## 12.X 0.349501101 -0.462613893 -0.1063628572 -0.01228502 0.07188926
## 12.Y -0.337681343 -0.069170301 -0.2741261832 0.19510503 0.45397999
## 13.X -0.579230812 0.163346253 -0.0165959275 0.02061649 0.15463136
## 13.Y -0.101108067 -0.342957044 -0.0847748654 -0.20987532 -0.27372926
## [,16]
## 1.X 0.133081728
## 1.Y 0.291607576
## 2.X -0.325402375
## 2.Y -0.504965936
## 3.X 0.028041537
## 3.Y 0.168241560
## 4.X 0.290413009
## 4.Y 0.017694500
## 5.X -0.155221862
## 5.Y 0.011037604
## 6.X 0.255966358
## 6.Y 0.041746925
## 7.X -0.008477203
## 7.Y 0.256753734
## 8.X 0.118447971
## 8.Y -0.376450218
## 9.X 0.003952306
## 9.Y 0.154177445
## 10.X -0.230147859
## 10.Y -0.038252546
## 11.X 0.069258173
## 11.Y -0.034486822
## 12.X -0.069663344
## 12.Y 0.070173486
## 13.X -0.110248440
## 13.Y -0.057277307Now let’s summarize the statistical results.
summary(PLSfood)##
## Call:
## two.b.pls(A1 = food, A2 = Y.gpa$coords, iter = 9999, print.progress = FALSE)
##
##
##
## r-PLS: 0.7591
##
## Effect Size (Z): 4.31522
##
## P-value: 1e-04
##
## Based on 10000 random permutationssummary(PLSallometry)##
## Call:
## two.b.pls(A1 = Y.gpa$Csize, A2 = Y.gpa$coords, iter = 9999, print.progress = FALSE)
##
##
##
##
## r-PLS: 0.8293
##
## Effect Size (Z): 5.56658
##
## P-value: 1e-04
##
## Based on 10000 random permutationssummary(PLSintegration)##
## Call:
## two.b.pls(A1 = Y.gpa$coords[1:5, , ], A2 = Y.gpa$coords[8:13,
## , ], iter = 9999, print.progress = FALSE)
##
##
##
## r-PLS: 0.8979
##
## Effect Size (Z): 5.93647
##
## P-value: 1e-04
##
## Based on 10000 random permutationsAs an example, if we wished to look at the distribution made by RRPP, we could so something like the following.
hist(PLSfood$random.r, breaks = 50,
col = "olivedrab1",
main = "Sampling distribution of r")
abline(v = PLSfood$r.pls, lwd = 2, col = "skyblue4")
Or we might make scatter plots between first PLS scores
foodPlot <- plot(PLSfood)
allomPlot <- plot(PLSallometry)
integPlot <- plot(PLSintegration)
Each plot above can be explored interactively with
picknplot.shape.
Finally, we can compare statistically the three PLS effect sizes.
compare.pls(PLSfood,
PLSallometry,
PLSintegration)##
## Effect sizes
##
## PLSfood PLSallometry PLSintegration
## 4.315216 5.566575 5.936474
##
## Effect sizes for pairwise differences in PLS effect size
##
## PLSfood PLSallometry PLSintegration
## PLSfood 0.000000 2.3948850 3.0941152
## PLSallometry 2.394885 0.0000000 0.7911531
## PLSintegration 3.094115 0.7911531 0.0000000
##
## P-values
##
## PLSfood PLSallometry PLSintegration
## PLSfood 1.000000000 0.01662558 0.001974008
## PLSallometry 0.016625581 1.00000000 0.428854632
## PLSintegration 0.001974008 0.42885463 1.000000000Generalized Linear Model (GLM)
GLM is discussed in more detail in Shape Statistics
II. In this module we become a little acquainted with it. The
procD.lm function wraps the RRPP::lm.rrpp
function, making sure to format 3D arrays of coordinates in the
process.
We will fit two simple models with the same data use in the PLS analyses.
fit.null <- procD.lm(coords ~ 1, data = Y.gpa,
iter = 9999)
fit.alt <- procD.lm(coords ~ log(Csize), data = Y.gpa,
turbo = FALSE, iter = 9999)Note that there are some options for speeding up analysis with large data sets.
system.time(fit <- procD.lm(coords ~ log(Csize), data = Y.gpa,
iter = 9999, turbo = FALSE, Parallel = FALSE))## user system elapsed
## 0.955 0.040 0.998system.time(fit <- procD.lm(coords ~ log(Csize), data = Y.gpa,
iter = 9999, turbo = TRUE, Parallel = FALSE))## user system elapsed
## 0.590 0.031 0.622system.time(fit <- procD.lm(coords ~ log(Csize), data = Y.gpa,
iter = 9999, turbo = FALSE, Parallel = TRUE))## user system elapsed
## 1.499 1.202 0.730system.time(fit <- procD.lm(coords ~ log(Csize), data = Y.gpa,
iter = 9999, turbo = FALSE, Parallel = TRUE))## user system elapsed
## 1.634 1.080 0.681We will need coefficients in every RRPP permutation
(turbo=FALSE), so the previous results will suffice.
Let’s look at some of the linear model parts
model.matrix(fit.null)## (Intercept)
## 1 1
## 2 1
## 3 1
## 4 1
## 5 1
## 6 1
## 7 1
## 8 1
## 9 1
## 10 1
## 11 1
## 12 1
## 13 1
## 14 1
## 15 1
## 16 1
## 17 1
## 18 1
## 19 1
## 20 1
## 21 1
## 22 1
## 23 1
## 24 1
## 25 1
## 26 1
## 27 1
## 28 1
## 29 1
## 30 1
## 31 1
## 32 1
## 33 1
## 34 1
## 35 1
## 36 1
## 37 1
## 38 1
## 39 1
## 40 1
## 41 1
## 42 1
## 43 1
## 44 1
## 45 1
## 46 1
## 47 1
## 48 1
## 49 1
## 50 1
## 51 1
## 52 1
## 53 1
## 54 1
## 55 1
## 56 1
## 57 1
## 58 1
## 59 1
## 60 1
## 61 1
## 62 1
## 63 1
## 64 1
## 65 1
## 66 1
## 67 1
## 68 1
## 69 1
## attr(,"assign")
## [1] 0coef(fit.null)## 1.X 1.Y 2.X 2.Y 3.X 3.Y
## (Intercept) 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139
## 4.X 4.Y 5.X 5.Y 6.X 6.Y
## (Intercept) -0.2806804 -0.210035 -0.2896175 -0.1655067 -0.03925596 -0.008809209
## 7.X 7.Y 8.X 8.Y 9.X 9.Y
## (Intercept) 0.1964961 -0.005244766 0.2489937 0.1166791 0.4406422 0.04415754
## 10.X 10.Y 11.X 11.Y 12.X 12.Y
## (Intercept) -0.3173787 0.05311804 -0.27118 0.1681822 -0.06589549 0.198903
## 13.X 13.Y
## (Intercept) -0.1611084 0.1206271fitted(fit.null)## 1.X 1.Y 2.X 2.Y 3.X 3.Y 4.X
## 1 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 2 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 3 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 4 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 5 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 6 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 7 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 8 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
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## 10 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
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## 14 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
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## 17 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 18 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
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## 21 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 22 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 23 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 24 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 25 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 26 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 27 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 28 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 29 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 30 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 31 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 32 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 33 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 34 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 35 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 36 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 37 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 38 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
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## 41 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
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## 43 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 44 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 45 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 46 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 47 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 48 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 49 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 50 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 51 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 52 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 53 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 54 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 55 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
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## 59 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
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## 65 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 66 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 67 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 68 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 69 0.2415079 -0.07332146 0.2627464 -0.1478185 0.03473031 -0.09093139 -0.2806804
## 4.Y 5.X 5.Y 6.X 6.Y 7.X
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## 65 -0.210035 -0.2896175 -0.1655067 -0.03925596 -0.008809209 0.1964961
## 66 -0.210035 -0.2896175 -0.1655067 -0.03925596 -0.008809209 0.1964961
## 67 -0.210035 -0.2896175 -0.1655067 -0.03925596 -0.008809209 0.1964961
## 68 -0.210035 -0.2896175 -0.1655067 -0.03925596 -0.008809209 0.1964961
## 69 -0.210035 -0.2896175 -0.1655067 -0.03925596 -0.008809209 0.1964961
## 7.Y 8.X 8.Y 9.X 9.Y 10.X 10.Y
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## 11.X 11.Y 12.X 12.Y 13.X 13.Y
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## 69 -0.27118 0.1681822 -0.06589549 0.198903 -0.1611084 0.1206271resid(fit.null)## 1.X 1.Y 2.X 2.Y 3.X
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## 2 -0.0024541083 0.0255973042 -5.583594e-03 -0.0003739754 -0.0088774220
## 3 0.0055248394 0.0193237877 1.247894e-02 0.0302408157 0.0128669464
## 4 0.0133751928 -0.0048101939 2.929055e-02 0.0035852290 0.0188411368
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## 18 0.0014242750 0.0062481457 6.204550e-03 0.0113266945 0.0106293377
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## 31 -0.0163062511 0.0147415161 6.897821e-03 0.0008918456 -0.0098505739
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## 17 -8.172433e-03 -0.0029452600 -0.0076576025 -0.0261077071 0.0071093841
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## 31 -1.101135e-02 0.0148648855 0.0327753900 0.0042896603 0.0170627661
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## 44 1.126552e-02 0.0020957496 0.0043154972 0.0064698957 0.0072907804
## 45 1.030607e-02 -0.0053742816 -0.0049048274 -0.0125792109 -0.0023129920
## 46 7.164159e-03 -0.0209617611 0.0166043294 -0.0034599657 0.0245686734
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## 48 -6.622204e-03 0.0111370668 0.0033455788 0.0059539873 0.0038936115
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## 53 -2.029980e-02 0.0175386615 0.0191117499 0.0119871897 0.0187406354
## 54 -1.952140e-02 0.0159574829 0.0168384007 0.0030213731 0.0195580375
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## 57 -3.006097e-02 0.0100335844 0.0281580199 0.0127455841 0.0338972507
## 58 2.625802e-03 -0.0245868441 0.0149106985 -0.0040910065 0.0216843834
## 59 4.914595e-03 0.0018491785 0.0375025646 0.0143535106 0.0225301114
## 60 2.939076e-03 -0.0063361137 0.0120555891 -0.0062703678 0.0084220799
## 61 1.530836e-02 -0.0121627499 0.0062159885 -0.0125376524 0.0041985071
## 62 1.517057e-02 -0.0003414839 0.0080209056 0.0028507701 -0.0029062323
## 63 -1.881510e-02 0.0227101060 0.0177326838 0.0207240837 0.0142479188
## 64 6.398251e-03 0.0191740326 0.0276648636 0.0162913063 0.0298450421
## 65 1.471846e-02 0.0243434077 0.0169215549 0.0220873421 0.0318050388
## 66 -1.071157e-02 0.0149566972 0.0206064531 0.0143765017 0.0170452934
## 67 8.127026e-03 -0.0003784896 0.0064007523 -0.0020753913 0.0123493277
## 68 1.066665e-02 -0.0076650619 0.0173424229 0.0104262557 0.0111425363
## 69 -2.258842e-02 0.0210608233 0.0202540212 0.0288742954 0.0171695283
## 6.X 6.Y 7.X 7.Y 8.X
## 1 0.0038056662 0.0036112005 1.197083e-02 -0.0055651561 1.076173e-02
## 2 -0.0149171236 0.0171585349 -4.152116e-03 0.0208485346 1.034346e-02
## 3 0.0365725748 0.0043677590 -1.871446e-03 0.0090363171 1.133486e-02
## 4 0.0336829645 0.0235154295 -3.447488e-02 0.0102124491 -1.634645e-02
## 5 0.0130059129 0.0115070753 2.415930e-02 0.0313758803 1.757324e-02
## 6 0.0152625799 0.0200585787 4.292115e-04 0.0215321897 1.044020e-02
## 7 0.0129463283 0.0032590936 -1.059478e-02 0.0245657352 9.136135e-03
## 8 -0.0046901412 0.0128981190 9.022997e-04 0.0253920704 3.403027e-05
## 9 0.0169420576 0.0242943428 -9.475014e-03 0.0146604668 5.339133e-03
## 10 -0.0029922472 0.0078986053 1.779075e-02 0.0212206647 1.547784e-02
## 11 0.0082536242 0.0219507742 7.420256e-03 0.0082756095 -3.252857e-03
## 12 -0.0208026393 0.0256722754 -2.182286e-03 0.0116827402 3.543832e-02
## 13 0.0251058455 0.0176545477 2.192662e-02 0.0080531399 3.288263e-02
## 14 0.0093978747 0.0234502719 -1.872742e-03 0.0047215361 2.574908e-02
## 15 0.0253581438 0.0038880801 -1.757498e-03 0.0094174103 -7.617477e-03
## 16 0.0083849809 -0.0074068925 9.649252e-03 0.0056288523 8.056030e-03
## 17 0.0090976662 0.0149350965 1.935696e-03 0.0036827580 1.029762e-02
## 18 0.0248930852 0.0096176345 1.873849e-03 -0.0048961504 -8.308773e-03
## 19 0.0003863133 -0.0096168415 2.179446e-02 -0.0014427826 -7.215368e-03
## 20 0.0331653792 0.0082130890 2.021699e-02 -0.0025285225 1.852591e-03
## 21 -0.0196330630 0.0062588717 -1.409065e-02 0.0144067973 8.168508e-03
## 22 0.0105565316 0.0115656983 9.448355e-03 -0.0076939318 1.465869e-02
## 23 -0.0027391719 -0.0014326702 -3.157066e-04 -0.0046268505 4.403884e-04
## 24 -0.0547620080 0.0212792248 9.991350e-03 0.0139224751 1.934426e-02
## 25 -0.0373473648 0.0014409584 -4.287800e-04 -0.0339438823 1.733711e-02
## 26 -0.0580566200 0.0020406001 -1.234967e-02 -0.0036220586 -2.056958e-04
## 27 0.0011503086 0.0163190735 1.133971e-02 0.0040166042 -3.300241e-03
## 28 -0.0018552649 0.0158599199 -1.027135e-02 -0.0054956855 2.540826e-02
## 29 -0.0322211739 0.0120222720 -5.530387e-03 0.0082488767 -9.657261e-03
## 30 0.0073734792 0.0073805187 1.455929e-02 -0.0035034069 9.614416e-03
## 31 -0.0359005762 -0.0740866839 5.654548e-02 -0.0439109427 3.594656e-02
## 32 -0.0003725345 0.0228726638 -4.464561e-03 0.0058795844 -6.236538e-03
## 33 -0.0173182666 0.0207181277 -5.080025e-03 -0.0042825878 2.769060e-02
## 34 -0.0261599167 0.0093934945 2.350046e-03 0.0126882144 2.711125e-02
## 35 -0.0104128168 0.0195078307 2.698291e-02 -0.0029648831 2.161262e-02
## 36 -0.0031688772 -0.0055268063 3.687293e-03 -0.0057434960 1.665985e-03
## 37 -0.0130072125 -0.0012525315 3.354683e-03 -0.0182117695 -6.370479e-04
## 38 -0.0169270627 0.0007941987 -6.757666e-03 -0.0180331185 1.191820e-02
## 39 0.0080678915 0.0182793884 -9.267811e-03 -0.0049827470 8.899744e-05
## 40 -0.0037160586 0.0038881410 -7.472602e-03 0.0086924142 -2.917644e-02
## 41 0.0248727818 -0.0215577822 9.029214e-03 -0.0121672353 -1.974171e-02
## 42 0.0324150375 0.0111624720 -1.104125e-02 -0.0061707154 -1.472589e-02
## 43 0.0174146326 -0.0128774380 8.518375e-03 -0.0092542490 -9.087749e-03
## 44 0.0224352383 -0.0114241029 -8.423156e-03 -0.0259465319 -2.260219e-02
## 45 0.0446819703 -0.0070313797 -2.169980e-04 -0.0089545865 -1.449381e-02
## 46 0.0148871838 -0.0082376027 -9.558663e-03 -0.0026589334 -4.772689e-03
## 47 0.0162937898 -0.0206012630 -2.156517e-02 -0.0156735997 -3.045260e-02
## 48 -0.0101542802 -0.0155597538 -2.380539e-02 -0.0161405846 -2.228354e-02
## 49 0.0011850985 -0.0318091435 -7.208049e-03 0.0006870900 -3.725687e-02
## 50 -0.0059419212 -0.0157592712 -8.590521e-03 0.0073375264 -2.105180e-02
## 51 -0.0074613522 0.0005195634 9.611926e-05 -0.0074978587 3.151182e-03
## 52 -0.0086237103 -0.0338294804 -1.015336e-02 -0.0056674981 -2.925629e-02
## 53 -0.0269138138 -0.0460550517 -5.126900e-03 -0.0079117380 -2.024442e-02
## 54 -0.0091540072 -0.0166658846 -1.444115e-03 -0.0059240392 -2.624483e-02
## 55 0.0265604018 0.0195695591 -7.503268e-03 0.0047474932 -6.611938e-03
## 56 -0.0199787386 -0.0168850053 2.564581e-04 0.0100364184 1.428953e-02
## 57 -0.0387452973 -0.0502858556 1.404823e-02 -0.0303973628 7.128994e-03
## 58 -0.0026868921 0.0107063033 -3.782352e-03 0.0058888307 -1.022840e-02
## 59 0.0103959524 -0.0041585469 1.834984e-03 0.0058057213 2.972624e-03
## 60 -0.0084166423 0.0010646468 1.184737e-03 0.0197055044 -4.780408e-03
## 61 0.0243298523 0.0077285765 1.208118e-02 0.0065429690 -2.725987e-04
## 62 0.0175588817 0.0163387270 4.883167e-03 0.0105076861 3.449956e-03
## 63 -0.0170652201 -0.0111369108 -7.525217e-03 -0.0271266651 1.529634e-02
## 64 -0.0079941717 -0.0095256630 1.992146e-02 0.0027340265 1.039039e-02
## 65 -0.0368089571 -0.0203357698 -3.911095e-02 -0.0001145865 -2.426703e-04
## 66 0.0222027151 -0.0262901394 -8.312804e-03 -0.0023656623 -2.284693e-02
## 67 -0.0104887420 -0.0029808134 -7.324334e-03 0.0011987790 -1.479035e-02
## 68 0.0050126327 -0.0070725541 -2.551606e-03 -0.0085997598 -4.027292e-02
## 69 0.0037785091 -0.0212595002 -2.452847e-02 -0.0093337877 -1.818698e-02
## 8.Y 9.X 9.Y 10.X 10.Y
## 1 -0.0001455598 6.344929e-03 0.0133663103 3.552330e-03 -0.0071231040
## 2 -0.0018753036 2.102096e-02 -0.0290001409 1.900475e-04 -0.0175206452
## 3 -0.0231689798 -4.341281e-02 -0.0246479834 -5.578931e-03 0.0141127725
## 4 -0.0180297038 -2.689597e-02 -0.0082862361 -1.851804e-02 0.0186496166
## 5 -0.0069364273 -4.211856e-02 -0.0133354446 -5.206406e-03 -0.0101211561
## 6 -0.0101914164 -1.116592e-02 -0.0222391826 -5.607675e-03 0.0003690447
## 7 0.0092829292 2.780653e-02 -0.0178208385 -1.889536e-03 0.0031097960
## 8 -0.0023889935 -8.714471e-03 -0.0163530744 2.184991e-03 -0.0027418725
## 9 -0.0270890206 -5.767509e-03 -0.0185684346 -2.290473e-03 0.0034288541
## 10 -0.0206536231 8.279872e-03 -0.0134996527 -9.365753e-03 0.0049166992
## 11 -0.0279333197 -1.585434e-02 -0.0132805614 -1.970096e-03 0.0107094728
## 12 -0.0144100607 1.516774e-03 -0.0212909451 4.104129e-03 -0.0030609865
## 13 -0.0171250738 -5.249740e-02 -0.0218214685 -4.784343e-03 -0.0223121710
## 14 -0.0121748880 -6.386857e-03 -0.0101701476 2.550007e-03 0.0094979185
## 15 -0.0186164715 -3.908268e-02 -0.0114689470 2.004519e-04 0.0330464851
## 16 0.0276548007 1.918360e-02 -0.0077125609 1.815657e-02 -0.0184658993
## 17 -0.0034010600 -9.166569e-03 0.0358309639 1.989266e-02 0.0069171153
## 18 -0.0228416324 -1.520236e-02 -0.0005295566 -8.114776e-03 0.0168368078
## 19 -0.0352686390 -2.255516e-02 -0.0111861619 6.927865e-03 0.0150402598
## 20 -0.0208274207 -6.664555e-02 -0.0006341101 -8.857682e-03 0.0476985423
## 21 -0.0079068909 -1.367012e-02 -0.0087769579 -2.340072e-03 -0.0110037341
## 22 -0.0155413676 -4.039472e-02 -0.0140163325 -2.500720e-03 0.0247956675
## 23 -0.0027233889 -2.693969e-03 -0.0099775632 1.027187e-02 -0.0011382558
## 24 -0.0151429962 -2.605276e-02 -0.0171347074 1.862073e-02 -0.0056678679
## 25 -0.0044108513 -1.117428e-02 -0.0263700461 1.768658e-02 -0.0057378144
## 26 0.0021877085 1.562595e-02 0.0105402328 1.345386e-02 -0.0136300439
## 27 -0.0051668782 -2.404091e-02 -0.0215646813 -1.553609e-03 0.0182341992
## 28 -0.0076425111 -6.569477e-03 -0.0091830632 -6.542350e-03 -0.0126135252
## 29 -0.0108100155 -1.035295e-02 -0.0135200818 1.237144e-02 0.0133999988
## 30 -0.0170867714 -4.932479e-02 -0.0170348491 -3.320324e-03 0.0106223244
## 31 -0.0167208890 -4.251982e-02 0.0378267515 -9.695108e-04 -0.0230253437
## 32 -0.0201289002 -3.625877e-02 -0.0162392274 7.944602e-03 0.0169846477
## 33 -0.0174567292 -3.014866e-02 -0.0066932339 3.812703e-03 0.0092854109
## 34 0.0052398153 -1.875009e-02 0.0020546528 6.394144e-03 0.0002292428
## 35 -0.0104924502 -3.510166e-02 -0.0196656033 1.522913e-02 0.0118163584
## 36 0.0183297754 -2.675524e-02 -0.0198096120 -1.431993e-03 0.0098457227
## 37 0.0029697630 -3.777488e-03 0.0053736212 1.017927e-03 0.0026299704
## 38 -0.0039856218 3.550344e-03 0.0074642073 -6.110318e-04 0.0170596829
## 39 -0.0009287908 4.545746e-04 -0.0017765215 -5.518147e-03 -0.0143553817
## 40 -0.0060818474 2.123205e-02 0.0098555771 3.010859e-04 -0.0030365518
## 41 0.0034865804 -4.234539e-05 0.0288053905 -1.069699e-02 0.0153859140
## 42 0.0033103450 -1.902499e-02 -0.0098302541 -2.585533e-02 0.0133269696
## 43 0.0147379394 2.969134e-02 -0.0008071017 -8.052777e-03 -0.0404690870
## 44 -0.0021623099 2.948818e-02 0.0113788365 -8.966045e-03 -0.0027786253
## 45 0.0061699493 -5.254962e-03 -0.0039769952 -7.639559e-03 -0.0097797130
## 46 0.0176918585 4.705775e-02 0.0221577768 2.443666e-03 -0.0249188894
## 47 -0.0034147914 4.981199e-02 0.0474967827 -8.941006e-03 -0.0168879038
## 48 -0.0005078331 2.436815e-02 0.0074081898 9.254526e-03 0.0147720007
## 49 0.0183797408 8.092792e-02 0.0386527209 5.562589e-02 -0.0058274675
## 50 0.0161223828 5.602148e-02 0.0004907896 -1.882398e-03 0.0089872183
## 51 0.0161289563 2.938532e-02 -0.0246650400 -3.744074e-03 -0.0078424220
## 52 0.0005713798 5.973209e-02 0.0226136826 -6.535709e-03 0.0042637708
## 53 0.0058380381 6.188352e-02 0.0226696306 1.531693e-02 0.0129665001
## 54 0.0194569872 2.834058e-02 0.0080875247 4.323020e-03 -0.0057026566
## 55 0.0025628886 -3.932150e-02 -0.0162043184 -2.423201e-02 0.0232013451
## 56 0.0229992215 1.968520e-02 -0.0174881955 2.677877e-03 -0.0090969722
## 57 0.0244935446 -2.922080e-02 0.0517052875 -7.721989e-03 0.0010219043
## 58 0.0175615084 4.139706e-02 0.0166154166 1.352325e-03 -0.0214103726
## 59 0.0031504631 4.792527e-02 0.0072438770 -7.810340e-03 -0.0183996486
## 60 0.0264403123 -1.923371e-02 -0.0031107663 -1.176609e-03 -0.0200624409
## 61 -0.0036682034 -2.860727e-02 0.0142429104 -2.339706e-02 -0.0021623152
## 62 0.0062623273 2.790300e-02 0.0087885926 6.463706e-04 -0.0067086645
## 63 0.0301079604 -2.657743e-02 0.0143709136 -8.793014e-03 0.0055562806
## 64 0.0435939193 8.949765e-03 0.0033344439 -2.591407e-05 -0.0126980870
## 65 0.0325207632 4.683752e-02 0.0244241020 -8.405440e-03 -0.0228179560
## 66 0.0312213061 9.818600e-03 0.0230049855 1.345915e-02 -0.0009691971
## 67 0.0036824816 4.199382e-02 0.0063843375 5.239102e-03 -0.0122798250
## 68 0.0027284438 3.532331e-02 0.0200158055 -5.577952e-03 -0.0057717031
## 69 0.0201735414 8.777406e-03 0.0174862852 -8.776289e-03 -0.0005802141
## 11.X 11.Y 12.X 12.Y 13.X
## 1 -0.0043339320 -0.0111659324 -0.0222950050 -0.0154116977 7.267372e-03
## 2 0.0204226961 -0.0019497355 -0.0169724591 -0.0145563055 8.591421e-03
## 3 -0.0163337558 0.0243520346 -0.0019611092 -0.0037815497 9.832558e-03
## 4 0.0010932452 0.0091344741 -0.0010638191 -0.0059182361 9.192458e-03
## 5 0.0174518606 -0.0144446883 -0.0128773295 0.0049234037 1.796679e-02
## 6 0.0057398544 -0.0055406488 -0.0067956524 -0.0050750895 1.117925e-02
## 7 0.0048237231 -0.0109958388 -0.0130214402 -0.0219273171 1.349773e-02
## 8 -0.0060269517 -0.0007675350 0.0101619605 0.0072914966 -1.784445e-02
## 9 -0.0047946253 -0.0052157594 -0.0184402713 -0.0147917795 1.512938e-02
## 10 0.0069734574 -0.0029692919 -0.0116089718 -0.0144570411 -1.006985e-02
## 11 -0.0205993344 0.0155653270 0.0184195066 -0.0131008385 8.322035e-03
## 12 0.0009993082 -0.0092514364 -0.0221302817 -0.0003172478 -8.270200e-03
## 13 -0.0203193772 0.0085070546 -0.0036268407 0.0139390287 -3.259465e-04
## 14 0.0054992380 -0.0048389141 -0.0027248481 -0.0280992400 8.271358e-03
## 15 -0.0049196561 0.0111142670 0.0074926277 -0.0094229572 1.494767e-02
## 16 0.0098656624 -0.0012989566 0.0075437792 -0.0171204685 -7.749511e-03
## 17 0.0098240177 -0.0024847715 -0.0024444013 -0.0170407707 8.681256e-03
## 18 -0.0136493447 0.0001513612 0.0112587304 -0.0040295770 -4.559362e-03
## 19 -0.0038377922 0.0135221083 -0.0099460723 0.0363713213 -8.904629e-03
## 20 -0.0168020008 0.0179027981 -0.0019150905 -0.0147871313 1.313861e-02
## 21 0.0054962051 0.0055736213 -0.0145149866 -0.0055568682 2.425334e-02
## 22 0.0021465474 0.0074084541 -0.0004113500 -0.0118441882 -1.831073e-03
## 23 0.0036444854 -0.0042213073 -0.0159515202 0.0036073748 6.006975e-03
## 24 0.0138088456 -0.0042107880 -0.0043644613 0.0133805954 6.533434e-05
## 25 -0.0073875956 0.0231338131 -0.0161943071 0.0429965197 -3.975689e-02
## 26 0.0146579362 0.0021683374 0.0073311332 0.0074657425 -7.314142e-03
## 27 -0.0162473015 0.0098057826 -0.0002692961 -0.0047892578 1.083501e-02
## 28 -0.0033225374 -0.0063707455 -0.0004749746 -0.0171215951 1.845067e-02
## 29 0.0131475255 0.0043262263 -0.0303037528 -0.0021349195 6.710923e-03
## 30 -0.0129950899 0.0145581182 0.0018376132 -0.0026886380 3.657315e-03
## 31 -0.0054516533 -0.0034870601 0.0078924558 0.0537149943 -1.543848e-02
## 32 0.0054744723 0.0182475704 0.0067517937 -0.0048559521 6.121692e-03
## 33 0.0001178393 0.0109556923 -0.0003832578 -0.0087699928 2.656625e-03
## 34 0.0035090796 0.0034590412 0.0237252646 -0.0099002682 -8.001125e-03
## 35 0.0080464278 0.0085278524 -0.0355104283 0.0055298723 -9.418727e-03
## 36 -0.0070029228 0.0021166433 0.0154956596 0.0164083875 -6.868826e-03
## 37 0.0158081787 0.0048190869 -0.0141629197 0.0068863317 -1.065748e-02
## 38 0.0097186335 0.0085206799 0.0088590104 -0.0195348706 3.177529e-03
## 39 0.0063246615 -0.0064722691 0.0162638481 -0.0020267640 1.145955e-02
## 40 0.0110137676 0.0063029714 0.0085712330 -0.0012820263 1.493466e-04
## 41 -0.0015699639 -0.0030559503 0.0208082386 -0.0150693267 -5.653438e-03
## 42 0.0111879861 0.0023090908 0.0114719252 -0.0070565918 3.752095e-03
## 43 -0.0066122144 0.0110396275 0.0138048386 -0.0062911534 1.014230e-02
## 44 -0.0056787578 0.0105935243 -0.0041077188 0.0036990902 6.099227e-03
## 45 -0.0177049777 0.0199790396 0.0228000155 -0.0090876550 -3.576074e-04
## 46 0.0109076730 -0.0187108028 -0.0063435878 -0.0076539520 5.673959e-03
## 47 0.0003075837 -0.0126811151 0.0279951992 -0.0134506515 9.976320e-03
## 48 0.0029052509 0.0144653209 -0.0032657689 0.0110021489 -1.595751e-02
## 49 0.0189211988 -0.0189742800 -0.0305222692 -0.0070209519 -1.689561e-02
## 50 0.0274152900 0.0001755518 -0.0412234910 -0.0070122949 -9.954487e-03
## 51 -0.0046522685 -0.0106159736 -0.0041628140 0.0041310956 2.094072e-02
## 52 0.0076363278 -0.0047311434 -0.0100296287 0.0024157558 -4.141806e-03
## 53 0.0064201529 0.0044941006 -0.0012009517 0.0046277803 -1.776465e-02
## 54 -0.0027345103 0.0025867059 -0.0082713960 0.0088876586 4.251091e-03
## 55 -0.0063032750 0.0059280130 0.0298801271 -0.0101029733 1.442902e-02
## 56 -0.0049846970 -0.0086036361 0.0079851057 0.0025357703 -2.030949e-02
## 57 -0.0031035960 0.0024462768 0.0120098470 0.0240281784 -1.077350e-02
## 58 0.0154248465 -0.0323378341 0.0145763831 -0.0168747641 1.342028e-02
## 59 -0.0143486870 -0.0279514109 0.0029941933 -0.0052103346 -1.641906e-02
## 60 -0.0050172874 -0.0089578945 0.0254921520 0.0189823390 -3.600434e-03
## 61 -0.0126771764 -0.0065502790 0.0267770423 -0.0054580011 8.350026e-03
## 62 0.0051450751 -0.0167302312 -0.0088879742 -0.0054185622 5.942734e-03
## 63 -0.0243594204 -0.0072336456 0.0103322535 0.0276335974 -5.929803e-03
## 64 -0.0162262832 -0.0200187854 -0.0164535203 0.0064844417 -7.226573e-03
## 65 0.0053030979 0.0040806612 0.0227893361 0.0195765682 -1.409812e-02
## 66 0.0037422309 -0.0008013319 -0.0225021413 0.0198545612 -2.491802e-02
## 67 -0.0044422278 -0.0143815411 0.0120630488 0.0055411618 5.940273e-03
## 68 0.0002633038 -0.0096536488 0.0306595791 0.0073613915 -2.097809e-02
## 69 -0.0167484723 0.0094039548 -0.0067077925 0.0267731936 3.508652e-03
## 13.Y
## 1 0.010076034
## 2 0.004986731
## 3 0.001205342
## 4 0.009815966
## 5 0.003342032
## 6 0.011140253
## 7 -0.001938680
## 8 0.002197345
## 9 0.010862446
## 10 -0.000369201
## 11 -0.005873004
## 12 -0.006846831
## 13 0.016152178
## 14 0.011230938
## 15 0.004767065
## 16 -0.006518731
## 17 -0.007942723
## 18 0.003837594
## 19 0.011346255
## 20 0.006147441
## 21 0.040288033
## 22 0.007975288
## 23 0.014898548
## 24 0.022041533
## 25 0.008889772
## 26 0.012073103
## 27 -0.005282225
## 28 0.008529933
## 29 0.020175564
## 30 0.026387609
## 31 0.015229002
## 32 0.017048306
## 33 0.014558738
## 34 -0.003256323
## 35 0.025940043
## 36 0.004434085
## 37 0.010478256
## 38 -0.006996265
## 39 0.001749359
## 40 -0.023225237
## 41 -0.002715393
## 42 0.012144311
## 43 -0.017318037
## 44 -0.010136602
## 45 -0.002110642
## 46 -0.007465001
## 47 -0.016123317
## 48 -0.015549540
## 49 -0.031902808
## 50 -0.016688119
## 51 -0.005377747
## 52 -0.009362497
## 53 -0.019940694
## 54 -0.007352564
## 55 0.002398625
## 56 -0.010405337
## 57 -0.006879936
## 58 -0.008109675
## 59 -0.009435693
## 60 -0.015747761
## 61 -0.015372947
## 62 -0.005456385
## 63 -0.002536415
## 64 -0.011899197
## 65 -0.034526793
## 66 0.007654714
## 67 -0.009351756
## 68 -0.015731373
## 69 -0.004256995model.matrix(fit.alt)## (Intercept) log(Csize)
## 1 1 1.501023
## 2 1 1.415598
## 3 1 1.709488
## 4 1 1.641995
## 5 1 1.409325
## 6 1 1.631807
## 7 1 1.643086
## 8 1 1.551710
## 9 1 1.635676
## 10 1 1.586874
## 11 1 1.656084
## 12 1 1.476448
## 13 1 1.651565
## 14 1 1.603674
## 15 1 1.748291
## 16 1 1.387986
## 17 1 1.542401
## 18 1 1.550247
## 19 1 1.489332
## 20 1 1.715451
## 21 1 1.587083
## 22 1 1.679572
## 23 1 1.565459
## 24 1 1.307155
## 25 1 1.311999
## 26 1 1.431253
## 27 1 1.642537
## 28 1 1.651642
## 29 1 1.527439
## 30 1 1.650008
## 31 1 1.373178
## 32 1 1.590954
## 33 1 1.639895
## 34 1 1.425089
## 35 1 1.513723
## 36 1 1.526006
## 37 1 1.511235
## 38 1 1.520858
## 39 1 1.369244
## 40 1 1.261754
## 41 1 1.399274
## 42 1 1.519728
## 43 1 1.477706
## 44 1 1.500453
## 45 1 1.629114
## 46 1 1.369610
## 47 1 1.445095
## 48 1 1.330806
## 49 1 1.142248
## 50 1 1.272799
## 51 1 1.311171
## 52 1 1.374731
## 53 1 1.197519
## 54 1 1.523840
## 55 1 1.612086
## 56 1 1.318992
## 57 1 1.213410
## 58 1 1.463730
## 59 1 1.442140
## 60 1 1.236113
## 61 1 1.482794
## 62 1 1.553500
## 63 1 1.289555
## 64 1 1.338859
## 65 1 1.215119
## 66 1 1.354736
## 67 1 1.345498
## 68 1 1.316122
## 69 1 1.413290coef(fit.alt)## 1.X 1.Y 2.X 2.Y 3.X
## (Intercept) 0.20280224 -0.1113977 0.18840275 -0.26478242 0.07183187
## log(Csize) 0.02625426 0.0258273 0.05042762 0.07933718 -0.02516617
## 3.Y 4.X 4.Y 5.X 5.Y
## (Intercept) -0.10382422 -0.25283896 -0.09617182 -0.24909544 -0.04620803
## log(Csize) 0.00874527 -0.01888499 -0.07723395 -0.02748634 -0.08092083
## 6.X 6.Y 7.X 7.Y 8.X
## (Intercept) -0.15189901 -0.12482067 0.17906920 -0.04202222 0.19738280
## log(Csize) 0.07640631 0.07869113 0.01182072 0.02494632 0.03500789
## 8.Y 9.X 9.Y 10.X 10.Y
## (Intercept) 0.22277002 0.6224375 0.14209923 -0.2785311 -0.01523022
## log(Csize) -0.07196199 -0.1233126 -0.06643432 -0.0263505 0.04636095
## 11.X 11.Y 12.X 12.Y 13.X
## (Intercept) -0.23733900 0.12470788 -0.06325867 0.27081297 -0.22896415
## log(Csize) -0.02295454 0.02948886 -0.00178857 -0.04877688 0.04602688
## 13.Y
## (Intercept) 0.04406722
## log(Csize) 0.05193096fitted(fit.alt)## 1.X 1.Y 2.X 2.Y 3.X 3.Y 4.X
## 1 0.2422105 -0.07263033 0.2640958 -0.1456955 0.03405687 -0.09069737 -0.2811858
## 2 0.2399677 -0.07483663 0.2597880 -0.1524729 0.03620669 -0.09144443 -0.2795725
## 3 0.2476836 -0.06724624 0.2746082 -0.1291564 0.02881060 -0.08887428 -0.2851226
## 4 0.2459116 -0.06898941 0.2712046 -0.1345112 0.03050915 -0.08946453 -0.2838480
## 5 0.2398030 -0.07499866 0.2594716 -0.1529706 0.03636457 -0.09149930 -0.2794540
## 6 0.2456441 -0.06925254 0.2706909 -0.1353195 0.03076555 -0.08955363 -0.2836556
## 7 0.2459402 -0.06896124 0.2712596 -0.1344246 0.03048170 -0.08945499 -0.2838686
## 8 0.2435412 -0.07132121 0.2666518 -0.1416741 0.03278126 -0.09025409 -0.2821430
## 9 0.2457457 -0.06915260 0.2708860 -0.1350125 0.03066817 -0.08951979 -0.2837287
## 10 0.2444645 -0.07041302 0.2684251 -0.1388843 0.03189632 -0.08994657 -0.2828071
## 11 0.2462815 -0.06862552 0.2719151 -0.1333934 0.03015457 -0.08934132 -0.2841141
## 12 0.2415653 -0.07326505 0.2628565 -0.1476452 0.03467534 -0.09091229 -0.2807217
## 13 0.2461628 -0.06874225 0.2716872 -0.1337519 0.03026831 -0.08938084 -0.2840288
## 14 0.2449055 -0.06997913 0.2692722 -0.1375514 0.03147353 -0.08979965 -0.2831243
## 15 0.2487023 -0.06624406 0.2765649 -0.1260779 0.02783407 -0.08853494 -0.2858554
## 16 0.2392428 -0.07554976 0.2583956 -0.1546635 0.03690157 -0.09168590 -0.2790511
## 17 0.2432968 -0.07156165 0.2661824 -0.1424127 0.03301555 -0.09033551 -0.2819672
## 18 0.2435028 -0.07135901 0.2665780 -0.1417902 0.03281809 -0.09026689 -0.2821154
## 19 0.2419036 -0.07293227 0.2635062 -0.1466230 0.03435108 -0.09079961 -0.2809650
## 20 0.2478401 -0.06709224 0.2749089 -0.1286834 0.02866054 -0.08882214 -0.2852352
## 21 0.2444699 -0.07040763 0.2684356 -0.1388677 0.03189107 -0.08994475 -0.2828110
## 22 0.2468982 -0.06801890 0.2730996 -0.1315299 0.02956348 -0.08913591 -0.2845577
## 23 0.2439022 -0.07096614 0.2673451 -0.1405833 0.03243528 -0.09013386 -0.2824026
## 24 0.2371206 -0.07763742 0.2543195 -0.1610764 0.03893579 -0.09239280 -0.2775246
## 25 0.2372478 -0.07751231 0.2545637 -0.1606921 0.03881388 -0.09235043 -0.2776161
## 26 0.2403787 -0.07443231 0.2605774 -0.1512309 0.03581273 -0.09130753 -0.2798682
## 27 0.2459258 -0.06897542 0.2712320 -0.1344682 0.03049551 -0.08945979 -0.2838583
## 28 0.2461649 -0.06874025 0.2716911 -0.1337458 0.03026637 -0.08938016 -0.2840302
## 29 0.2429040 -0.07194809 0.2654278 -0.1435998 0.03339210 -0.09046636 -0.2816846
## 30 0.2461220 -0.06878245 0.2716087 -0.1338754 0.03030749 -0.08939445 -0.2839994
## 31 0.2388540 -0.07593222 0.2576489 -0.1558383 0.03727424 -0.09181541 -0.2787714
## 32 0.2445716 -0.07030766 0.2686308 -0.1385606 0.03179365 -0.08991090 -0.2828841
## 33 0.2458565 -0.06904365 0.2710987 -0.1346778 0.03056201 -0.08948290 -0.2838084
## 34 0.2402169 -0.07459151 0.2602666 -0.1517199 0.03596784 -0.09136143 -0.2797518
## 35 0.2425439 -0.07230233 0.2647362 -0.1446879 0.03373727 -0.09058630 -0.2814256
## 36 0.2428664 -0.07198510 0.2653556 -0.1437134 0.03342816 -0.09047889 -0.2816576
## 37 0.2424786 -0.07236659 0.2646107 -0.1448853 0.03379988 -0.09060806 -0.2813786
## 38 0.2427312 -0.07211805 0.2650960 -0.1441218 0.03355770 -0.09052391 -0.2815604
## 39 0.2387507 -0.07603384 0.2574505 -0.1561505 0.03737325 -0.09184981 -0.2786971
## 40 0.2359287 -0.07881000 0.2520300 -0.1646784 0.04007835 -0.09278984 -0.2766672
## 41 0.2395391 -0.07525824 0.2589648 -0.1537680 0.03661751 -0.09158719 -0.2792642
## 42 0.2427016 -0.07214723 0.2650390 -0.1442115 0.03358614 -0.09053379 -0.2815390
## 43 0.2415983 -0.07323255 0.2629199 -0.1475454 0.03464367 -0.09090128 -0.2807454
## 44 0.2421955 -0.07264505 0.2640670 -0.1457407 0.03407121 -0.09070235 -0.2811750
## 45 0.2455734 -0.06932209 0.2705551 -0.1355331 0.03083332 -0.08957718 -0.2836048
## 46 0.2387603 -0.07602437 0.2574689 -0.1561214 0.03736403 -0.09184661 -0.2787040
## 47 0.2407421 -0.07407479 0.2612755 -0.1501326 0.03546436 -0.09118647 -0.2801296
## 48 0.2377416 -0.07702657 0.2555121 -0.1592000 0.03834058 -0.09218596 -0.2779712
## 49 0.2327911 -0.08189652 0.2460036 -0.1741597 0.04308586 -0.09383495 -0.2744103
## 50 0.2362186 -0.07852476 0.2525870 -0.1638022 0.03980041 -0.09269325 -0.2768758
## 51 0.2372261 -0.07753370 0.2545220 -0.1607578 0.03883472 -0.09235768 -0.2776004
## 52 0.2388948 -0.07589212 0.2577271 -0.1557152 0.03723517 -0.09180183 -0.2788007
## 53 0.2342422 -0.08046901 0.2487908 -0.1697746 0.04169490 -0.09335159 -0.2754541
## 54 0.2428095 -0.07204103 0.2652464 -0.1438852 0.03348266 -0.09049783 -0.2816167
## 55 0.2451264 -0.06976189 0.2696964 -0.1368841 0.03126185 -0.08972610 -0.2832832
## 56 0.2374314 -0.07733171 0.2549164 -0.1601373 0.03863790 -0.09228928 -0.2777481
## 57 0.2346594 -0.08005859 0.2495921 -0.1685139 0.04129498 -0.09321262 -0.2757542
## 58 0.2412314 -0.07359350 0.2622152 -0.1486542 0.03499538 -0.09102350 -0.2804815
## 59 0.2406646 -0.07415111 0.2611265 -0.1503671 0.03553872 -0.09121231 -0.2800738
## 60 0.2352555 -0.07947224 0.2507370 -0.1667127 0.04072364 -0.09301408 -0.2761830
## 61 0.2417319 -0.07310115 0.2631765 -0.1471417 0.03451564 -0.09085679 -0.2808415
## 62 0.2435882 -0.07127499 0.2667421 -0.1415321 0.03273623 -0.09023844 -0.2821768
## 63 0.2366585 -0.07809199 0.2534319 -0.1624728 0.03937872 -0.09254671 -0.2771922
## 64 0.2379530 -0.07681859 0.2559182 -0.1585611 0.03813791 -0.09211553 -0.2781233
## 65 0.2347043 -0.08001445 0.2496783 -0.1683783 0.04125197 -0.09319767 -0.2757865
## 66 0.2383698 -0.07640852 0.2567189 -0.1573015 0.03773835 -0.09197668 -0.2784232
## 67 0.2381273 -0.07664712 0.2562530 -0.1580344 0.03797084 -0.09205747 -0.2782487
## 68 0.2373560 -0.07740583 0.2547716 -0.1603650 0.03871013 -0.09231438 -0.2776939
## 69 0.2399071 -0.07489625 0.2596716 -0.1526560 0.03626478 -0.09146462 -0.2795289
## 4.Y 5.X 5.Y 6.X 6.Y 7.X
## 1 -0.2121018 -0.2903531 -0.1676721 -0.03721136 -6.703465e-03 0.1968124
## 2 -0.2055041 -0.2880050 -0.1607594 -0.04373838 -1.342567e-02 0.1958026
## 3 -0.2282024 -0.2960830 -0.1845412 -0.02128331 9.700891e-03 0.1992766
## 4 -0.2229896 -0.2942279 -0.1790796 -0.02644022 4.389765e-03 0.1984788
## 5 -0.2050195 -0.2878326 -0.1602517 -0.04421770 -1.391933e-02 0.1957284
## 6 -0.2222027 -0.2939478 -0.1782552 -0.02721866 3.588051e-03 0.1983583
## 7 -0.2230738 -0.2942578 -0.1791679 -0.02635689 4.475588e-03 0.1984917
## 8 -0.2160166 -0.2917463 -0.1717737 -0.03333853 -2.714821e-03 0.1974115
## 9 -0.2225016 -0.2940542 -0.1785683 -0.02692301 3.892540e-03 0.1984041
## 10 -0.2187324 -0.2927128 -0.1746192 -0.03065178 5.227178e-05 0.1978272
## 11 -0.2240778 -0.2946151 -0.1802197 -0.02536371 5.498473e-03 0.1986453
## 12 -0.2102037 -0.2896776 -0.1656834 -0.03908908 -8.637337e-03 0.1965219
## 13 -0.2237287 -0.2944909 -0.1798540 -0.02570903 5.142828e-03 0.1985919
## 14 -0.2200299 -0.2931746 -0.1759787 -0.02936815 1.374278e-03 0.1980258
## 15 -0.2311993 -0.2971496 -0.1876812 -0.01831850 1.275436e-02 0.1997353
## 16 -0.2033715 -0.2872461 -0.1585250 -0.04584807 -1.559845e-02 0.1954762
## 17 -0.2152976 -0.2914904 -0.1710204 -0.03404982 -3.447388e-03 0.1973015
## 18 -0.2159035 -0.2917061 -0.1716553 -0.03345033 -2.829971e-03 0.1973942
## 19 -0.2111989 -0.2900317 -0.1667260 -0.03810461 -7.623427e-03 0.1966742
## 20 -0.2286629 -0.2962469 -0.1850238 -0.02082771 1.017011e-02 0.1993471
## 21 -0.2187485 -0.2927185 -0.1746361 -0.03063583 6.869861e-05 0.1978297
## 22 -0.2258918 -0.2952607 -0.1821204 -0.02356912 7.346729e-03 0.1989229
## 23 -0.2170784 -0.2921242 -0.1728862 -0.03228808 -1.632963e-03 0.1975740
## 24 -0.1971286 -0.2850243 -0.1519841 -0.05202411 -2.195916e-02 0.1945207
## 25 -0.1975027 -0.2851575 -0.1523761 -0.05165398 -2.157797e-02 0.1945780
## 26 -0.2067131 -0.2884353 -0.1620262 -0.04254227 -1.219379e-02 0.1959876
## 27 -0.2230314 -0.2942428 -0.1791235 -0.02639883 4.432399e-03 0.1984852
## 28 -0.2237347 -0.2944930 -0.1798603 -0.02570312 5.148909e-03 0.1985928
## 29 -0.2141419 -0.2910791 -0.1698096 -0.03519306 -4.624811e-03 0.1971246
## 30 -0.2236085 -0.2944481 -0.1797281 -0.02582796 5.020339e-03 0.1985735
## 31 -0.2022278 -0.2868391 -0.1573267 -0.04697953 -1.676374e-02 0.1953012
## 32 -0.2190475 -0.2928249 -0.1749494 -0.03034006 3.733071e-04 0.1978754
## 33 -0.2228274 -0.2941701 -0.1789097 -0.02660070 4.224492e-03 0.1984539
## 34 -0.2062371 -0.2882659 -0.1615274 -0.04301322 -1.267882e-02 0.1959148
## 35 -0.2130826 -0.2907021 -0.1686997 -0.03624102 -5.704109e-03 0.1969625
## 36 -0.2140313 -0.2910397 -0.1696937 -0.03530255 -4.737574e-03 0.1971077
## 37 -0.2128905 -0.2906337 -0.1684984 -0.03643111 -5.899889e-03 0.1969331
## 38 -0.2136337 -0.2908983 -0.1692771 -0.03569584 -5.142631e-03 0.1970468
## 39 -0.2019239 -0.2867309 -0.1570084 -0.04728014 -1.707334e-02 0.1952546
## 40 -0.1936221 -0.2837764 -0.1483102 -0.05549301 -2.553180e-02 0.1939840
## 41 -0.2042433 -0.2875564 -0.1594384 -0.04498564 -1.471022e-02 0.1956096
## 42 -0.2135464 -0.2908672 -0.1691857 -0.03578219 -5.231560e-03 0.1970335
## 43 -0.2103009 -0.2897122 -0.1657852 -0.03899294 -8.538322e-03 0.1965367
## 44 -0.2120578 -0.2903374 -0.1676260 -0.03725490 -6.748305e-03 0.1968056
## 45 -0.2219947 -0.2938738 -0.1780373 -0.02742441 3.376144e-03 0.1983265
## 46 -0.2019522 -0.2867410 -0.1570380 -0.04725214 -1.704450e-02 0.1952590
## 47 -0.2077823 -0.2888158 -0.1631464 -0.04148459 -1.110448e-02 0.1961513
## 48 -0.1989552 -0.2856744 -0.1538980 -0.05021701 -2.009803e-02 0.1948003
## 49 -0.1843922 -0.2804917 -0.1386397 -0.06462404 -3.493588e-02 0.1925714
## 50 -0.1944751 -0.2840800 -0.1492039 -0.05464916 -2.466272e-02 0.1941146
## 51 -0.1974387 -0.2851347 -0.1523091 -0.05171726 -2.164315e-02 0.1945682
## 52 -0.2023477 -0.2868817 -0.1574524 -0.04686090 -1.664156e-02 0.1953195
## 53 -0.1886610 -0.2820109 -0.1431123 -0.06040096 -3.058652e-02 0.1932247
## 54 -0.2138640 -0.2909802 -0.1695184 -0.03546800 -4.907978e-03 0.1970821
## 55 -0.2206796 -0.2934058 -0.1766593 -0.02872549 2.036164e-03 0.1981252
## 56 -0.1980428 -0.2853497 -0.1529419 -0.05111971 -2.102773e-02 0.1946606
## 57 -0.1898883 -0.2824476 -0.1443982 -0.05918680 -2.933605e-02 0.1934126
## 58 -0.2092215 -0.2893280 -0.1646543 -0.04006076 -9.638069e-03 0.1963715
## 59 -0.2075540 -0.2887346 -0.1629072 -0.04171038 -1.133702e-02 0.1961163
## 60 -0.1916417 -0.2830717 -0.1462353 -0.05745215 -2.754953e-02 0.1936809
## 61 -0.2106938 -0.2898520 -0.1661969 -0.03860421 -8.137965e-03 0.1965969
## 62 -0.2161548 -0.2917955 -0.1719185 -0.03320180 -2.574003e-03 0.1974327
## 63 -0.1957692 -0.2845406 -0.1505599 -0.05336888 -2.334415e-02 0.1943127
## 64 -0.1995772 -0.2858958 -0.1545496 -0.04960170 -1.946432e-02 0.1948955
## 65 -0.1900203 -0.2824946 -0.1445365 -0.05905620 -2.920155e-02 0.1934328
## 66 -0.2008035 -0.2863322 -0.1558344 -0.04838859 -1.821493e-02 0.1950832
## 67 -0.2000900 -0.2860783 -0.1550869 -0.04909445 -1.894190e-02 0.1949740
## 68 -0.1978211 -0.2852708 -0.1527097 -0.05133900 -2.125357e-02 0.1946267
## 69 -0.2053258 -0.2879416 -0.1605726 -0.04391475 -1.360731e-02 0.1957753
## 7.Y 8.X 8.Y 9.X 9.Y 10.X
## 1 -0.0045772126 0.2499305 0.11475340 0.4373424 0.04237979 -0.3180838
## 2 -0.0067082562 0.2469399 0.12090076 0.4478764 0.04805495 -0.3158328
## 3 0.0006232258 0.2572284 0.09975183 0.4116361 0.02853055 -0.3235770
## 4 -0.0010604846 0.2548656 0.10460879 0.4199588 0.03301442 -0.3217985
## 5 -0.0068647535 0.2467203 0.12135221 0.4486500 0.04847171 -0.3156675
## 6 -0.0013146405 0.2545089 0.10534194 0.4212152 0.03369126 -0.3215300
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## 25 -0.0092926645 0.2433131 0.12835594 0.4606515 0.05493746 -0.3131029
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## 28 -0.0008198239 0.2552033 0.10391456 0.4187692 0.03237352 -0.3220527
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## 30 -0.0008605827 0.2551461 0.10403213 0.4189707 0.03248206 -0.3220097
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## 32 -0.0023337647 0.2530787 0.10828178 0.4262528 0.03640528 -0.3204535
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## 58 -0.0055075279 0.2486249 0.11743705 0.4419411 0.04485730 -0.3171011
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## 69 -0.0067658387 0.2468591 0.12106687 0.4481611 0.04820829 -0.3157720
## 10.Y 11.X 11.Y 12.X 12.Y 13.X 13.Y
## 1 0.05435864 -0.2717943 0.1689713 -0.06594335 0.1975977 -0.1598767 0.1220168
## 2 0.05039825 -0.2698334 0.1664523 -0.06579056 0.2017645 -0.1638086 0.1175806
## 3 0.06402328 -0.2765795 0.1751187 -0.06631621 0.1874295 -0.1502817 0.1328426
## 4 0.06089423 -0.2750302 0.1731284 -0.06619549 0.1907216 -0.1533883 0.1293376
## 5 0.05010741 -0.2696894 0.1662673 -0.06577934 0.2020705 -0.1640973 0.1172548
## 6 0.06042190 -0.2747964 0.1728280 -0.06617727 0.1912185 -0.1538572 0.1288085
## 7 0.06094479 -0.2750553 0.1731606 -0.06619744 0.1906684 -0.1533381 0.1293942
## 8 0.05670856 -0.2729578 0.1704661 -0.06603401 0.1951254 -0.1575438 0.1246490
## 9 0.06060129 -0.2748852 0.1729421 -0.06618419 0.1910298 -0.1536791 0.1290095
## 10 0.05833879 -0.2737650 0.1715030 -0.06609690 0.1934102 -0.1559253 0.1264751
## 11 0.06154743 -0.2753536 0.1735439 -0.06622069 0.1900343 -0.1527398 0.1300693
## 12 0.05321930 -0.2712302 0.1682466 -0.06589940 0.1987965 -0.1610079 0.1207406
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## 14 0.05911765 -0.2741506 0.1719984 -0.06612695 0.1925907 -0.1551520 0.1273476
## 15 0.06582224 -0.2774702 0.1762630 -0.06638561 0.1855368 -0.1484958 0.1348577
## 16 0.04911816 -0.2691996 0.1656380 -0.06574118 0.2031113 -0.1650795 0.1161467
## 17 0.05627696 -0.2727441 0.1701915 -0.06601736 0.1955794 -0.1579722 0.1241656
## 18 0.05664072 -0.2729242 0.1704229 -0.06603139 0.1951967 -0.1576111 0.1245730
## 19 0.05381665 -0.2715259 0.1686266 -0.06592244 0.1981680 -0.1604148 0.1214097
## 20 0.06429972 -0.2767164 0.1752946 -0.06632687 0.1871386 -0.1500073 0.1331522
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## 23 0.05734593 -0.2732734 0.1708715 -0.06605860 0.1944548 -0.1569110 0.1253630
## 24 0.04537073 -0.2673441 0.1632544 -0.06559660 0.2070540 -0.1687999 0.1119490
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## 28 0.06134148 -0.2752517 0.1734129 -0.06621274 0.1902510 -0.1529442 0.1298386
## 29 0.05558328 -0.2724006 0.1697503 -0.06599060 0.1963093 -0.1586609 0.1233886
## 30 0.06126573 -0.2752142 0.1733647 -0.06620982 0.1903307 -0.1530194 0.1297537
## 31 0.04843162 -0.2688597 0.1652013 -0.06571469 0.2038336 -0.1657611 0.1153777
## 32 0.05852793 -0.2738586 0.1716233 -0.06610420 0.1932112 -0.1557375 0.1266870
## 33 0.06079686 -0.2749820 0.1730665 -0.06619173 0.1908240 -0.1534849 0.1292285
## 34 0.05083826 -0.2700513 0.1667321 -0.06580754 0.2013016 -0.1633718 0.1180734
## 35 0.05494741 -0.2720858 0.1693458 -0.06596607 0.1969783 -0.1592922 0.1226763
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## 41 0.04964145 -0.2694587 0.1659709 -0.06576137 0.2025607 -0.1645599 0.1167329
## 42 0.05522582 -0.2722237 0.1695229 -0.06597681 0.1966854 -0.1590158 0.1229881
## 43 0.05327763 -0.2712591 0.1682837 -0.06590165 0.1987351 -0.1609500 0.1208059
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## 45 0.06029705 -0.2747346 0.1727486 -0.06617245 0.1913499 -0.1539811 0.1286687
## 46 0.04826621 -0.2687778 0.1650961 -0.06570831 0.2040077 -0.1659253 0.1151924
## 47 0.05176578 -0.2705105 0.1673221 -0.06584332 0.2003257 -0.1624509 0.1191124
## 48 0.04646722 -0.2678870 0.1639518 -0.06563891 0.2059004 -0.1677113 0.1131773
## 49 0.03772549 -0.2635588 0.1583915 -0.06530166 0.2150977 -0.1763900 0.1033853
## 50 0.04377793 -0.2665555 0.1622413 -0.06553516 0.2087298 -0.1703812 0.1101649
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## 53 0.04028792 -0.2648275 0.1600214 -0.06540051 0.2124017 -0.1738461 0.1062555
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## 57 0.04102464 -0.2651923 0.1604900 -0.06542894 0.2116266 -0.1731147 0.1070808
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## 59 0.05162878 -0.2704427 0.1672350 -0.06583803 0.2004699 -0.1625869 0.1189589
## 60 0.04207717 -0.2657134 0.1611595 -0.06546954 0.2105192 -0.1720697 0.1082598
## 61 0.05351351 -0.2713758 0.1684338 -0.06591075 0.1984869 -0.1607158 0.1210701
## 62 0.05679152 -0.2729989 0.1705188 -0.06603721 0.1950381 -0.1574614 0.1247420
## 63 0.04455477 -0.2669401 0.1627354 -0.06556512 0.2079125 -0.1696100 0.1110350
## 64 0.04684057 -0.2680719 0.1641893 -0.06565331 0.2055076 -0.1673406 0.1135955
## 65 0.04110388 -0.2652315 0.1605404 -0.06543199 0.2115432 -0.1730360 0.1071695
## 66 0.04757665 -0.2684363 0.1646575 -0.06568171 0.2047331 -0.1666099 0.1144200
## 67 0.04714836 -0.2682243 0.1643851 -0.06566518 0.2051838 -0.1670351 0.1139402
## 68 0.04578644 -0.2675500 0.1635188 -0.06561264 0.2066167 -0.1683872 0.1124147
## 69 0.05029124 -0.2697804 0.1663842 -0.06578643 0.2018771 -0.1639148 0.1174607resid(fit.alt)## 1.X 1.Y 2.X 2.Y 3.X
## 1 -0.0083739914 -0.0119598088 3.438126e-03 -1.263857e-02 -0.0152509301
## 2 -0.0009138876 0.0271124769 -2.625230e-03 4.280384e-03 -0.0103538093
## 3 -0.0006508121 0.0132485681 6.171174e-04 1.157875e-02 0.0187866522
## 4 0.0089715288 -0.0091422431 2.083225e-02 -9.722108e-03 0.0230622939
## 5 0.0088859792 0.0209661323 2.136953e-02 -1.014408e-02 -0.0229668670
## 6 0.0004856331 -0.0042569853 1.217184e-03 2.120978e-03 -0.0239075544
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## 30 0.0057031253 0.0063758512 8.590294e-03 -3.645299e-03 -0.0033752428
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## 54 -0.0056375132 -0.0172892848 3.560166e-04 -8.173173e-03 -0.0053171672
## 55 0.0107782268 0.0031242948 5.181791e-03 -1.230854e-03 0.0009032264
## 56 -0.0030067766 0.0137669921 -1.434646e-02 5.240402e-03 0.0108946586
## 57 0.0053298253 -0.0113113420 1.798005e-02 -9.382505e-03 0.0237271076
## 58 -0.0075918256 0.0028262814 4.889220e-03 -1.297887e-02 -0.0375501719
## 59 -0.0197976689 -0.0078917021 -1.007749e-02 -4.721775e-03 -0.0122176136
## 60 0.0119050947 0.0010100831 7.605477e-03 -1.769581e-02 0.0209125918
## 61 -0.0043190593 -0.0090416202 6.907532e-03 -1.288101e-02 0.0150884946
## 62 -0.0059050858 -0.0275622774 -1.946204e-02 -8.639314e-03 -0.0378652876
## 63 0.0068773247 -0.0167051580 1.479350e-02 -6.670658e-03 0.0090319108
## 64 -0.0027798653 -0.0158801562 -7.028148e-03 -3.579338e-02 -0.0100170290
## 65 -0.0016736973 -0.0164482547 1.071692e-03 -2.255104e-02 -0.0087425783
## 66 0.0017356014 -0.0418587386 8.988562e-03 -2.382066e-02 -0.0045426382
## 67 -0.0033607550 0.0057667248 -1.274207e-02 3.084886e-03 -0.0030004377
## 68 -0.0004814076 0.0083333345 -9.338277e-04 -1.413064e-02 0.0049225700
## 69 0.0010666500 -0.0311473546 1.352565e-02 -1.568197e-02 -0.0025028781
## 3.Y 4.X 4.Y 5.X 5.Y
## 1 1.381522e-02 -0.0048877444 0.0148230773 0.0078629620 0.0095019678
## 2 1.009706e-02 -0.0022082079 -0.0112723007 -0.0081239528 -0.0109049501
## 3 -1.422871e-02 -0.0088843746 0.0007149564 0.0003393913 -0.0023817502
## 4 -9.165066e-03 -0.0027703559 -0.0012617782 0.0023718996 -0.0023812828
## 5 1.970250e-04 -0.0286920001 -0.0157432733 -0.0082173793 -0.0044596982
## 6 3.538350e-03 -0.0057610023 -0.0033963793 0.0076734192 -0.0010891849
## 7 -4.466178e-03 0.0053741845 0.0110721710 -0.0044306336 0.0157550720
## 8 -1.565770e-03 0.0144487606 0.0051257129 0.0128966011 0.0011642766
## 9 -4.397977e-03 0.0014402711 0.0094809050 -0.0106590069 0.0029995478
## 10 -7.653582e-04 0.0097176119 0.0047960455 0.0105175359 0.0102807505
## 11 -4.121894e-04 -0.0047359828 -0.0025818458 0.0064066130 -0.0012917360
## 12 5.277247e-03 0.0092799632 -0.0006659602 -0.0007039945 -0.0020202760
## 13 -6.865760e-03 0.0104206668 0.0069222145 0.0045382446 0.0092885060
## 14 1.402983e-02 -0.0009996834 -0.0089983672 -0.0183061018 -0.0091331906
## 15 -1.183243e-02 -0.0035149104 0.0024026187 -0.0118803600 -0.0061475196
## 16 1.338817e-02 -0.0220740344 0.0004192401 -0.0132282233 -0.0116590359
## 17 -8.768315e-03 -0.0016584827 -0.0023950702 -0.0242348555 0.0126231310
## 18 9.927748e-04 -0.0053073581 -0.0060201933 0.0023816121 0.0006587753
## 19 -1.730474e-02 0.0140567169 -0.0154329069 0.0096884349 -0.0146804029
## 20 -1.299902e-02 -0.0081620992 -0.0008504764 0.0001246204 -0.0126303719
## 21 -6.276957e-03 0.0095685078 0.0017424418 0.0089626483 -0.0056573549
## 22 4.264229e-03 -0.0005643653 -0.0023213587 -0.0078741814 -0.0097586517
## 23 5.386184e-03 -0.0034930675 -0.0076128802 -0.0062032373 0.0087508474
## 24 2.899989e-03 0.0061677089 -0.0303404216 -0.0008113249 -0.0303202640
## 25 3.702572e-04 0.0338406818 -0.0191134844 0.0245559103 -0.0254149684
## 26 1.062980e-02 0.0069831829 -0.0012728323 0.0132092297 -0.0093221701
## 27 -9.790013e-03 0.0047031222 -0.0014715941 0.0017222754 0.0030741571
## 28 1.083799e-02 -0.0044999044 0.0022328286 -0.0131196476 0.0190932041
## 29 6.624563e-03 -0.0002967594 -0.0170408143 0.0068454150 -0.0049583771
## 30 -8.563092e-05 0.0084907196 -0.0155814340 0.0082850918 0.0020774373
## 31 -1.012733e-02 0.0129558863 0.0249681566 0.0015111900 0.0088828431
## 32 2.107321e-04 0.0009056105 -0.0195703671 -0.0063610531 -0.0260838155
## 33 1.914884e-03 0.0063961453 -0.0065050971 0.0018898440 -0.0094134650
## 34 -1.049657e-02 0.0009232437 -0.0109965600 -0.0049997541 -0.0091798602
## 35 -7.889277e-03 0.0162033897 -0.0192139650 0.0068617053 -0.0167821501
## 36 -1.089286e-02 0.0057809042 -0.0040062290 0.0093949387 -0.0054259022
## 37 3.726751e-03 0.0124362507 -0.0080077714 0.0160721904 0.0083430207
## 38 1.801342e-02 -0.0113299561 -0.0050951264 -0.0053501511 -0.0175348796
## 39 1.934068e-02 -0.0206790032 -0.0174732725 -0.0157555205 -0.0175307459
## 40 1.949485e-02 -0.0142920356 -0.0079352036 -0.0152059717 -0.0227942579
## 41 5.800096e-04 -0.0197912104 0.0016655709 -0.0065674783 -0.0011307942
## 42 1.253508e-02 -0.0118413616 -0.0135106089 -0.0004925693 -0.0171172928
## 43 1.192918e-02 -0.0132392291 0.0193332341 -0.0113754295 0.0129830849
## 44 1.103648e-02 0.0025903438 0.0063382387 0.0071897572 0.0094100804
## 45 8.951865e-03 -0.0024499327 0.0070548809 -0.0083229408 0.0102176303
## 46 8.079381e-03 -0.0229381396 0.0085215351 -0.0063365038 0.0161000351
## 47 1.358917e-03 -0.0191315136 0.0233340985 -0.0196167080 0.0293749454
## 48 -5.367631e-03 0.0084278756 -0.0077341996 0.0020108703 -0.0077150761
## 49 -1.468626e-02 -0.0444938012 0.0032223899 -0.0323183382 0.0157311007
## 50 -9.978347e-03 0.0174071909 -0.0044966953 -0.0002145228 0.0069784012
## 51 2.424627e-04 0.0059759230 -0.0121138475 -0.0120703168 -0.0120386905
## 52 -1.315993e-02 0.0067534299 0.0166071606 0.0049496091 0.0113499421
## 53 -1.787960e-02 0.0123123494 -0.0022622966 0.0043805063 -0.0036537321
## 54 -1.995496e-02 0.0168937350 0.0206673905 0.0043840497 0.0235698095
## 55 -3.195355e-03 -0.0093590319 -0.0100605108 0.0048892237 -0.0146402677
## 56 9.733405e-04 0.0032821228 0.0030177984 0.0043539899 -0.0029608045
## 57 -2.777973e-02 0.0051073725 0.0080112932 0.0055756843 0.0127887908
## 58 2.717917e-03 -0.0247857611 0.0140971876 -0.0043805221 0.0208320384
## 59 5.195521e-03 0.0012425319 0.0350215620 0.0134705612 0.0199306746
## 60 5.021764e-03 -0.0108335814 -0.0063377029 -0.0128162480 -0.0108492424
## 61 1.523376e-02 -0.0120016588 0.0068748028 -0.0123031909 0.0048887708
## 62 1.447762e-02 0.0011548962 0.0141406505 0.0050286901 0.0035056475
## 63 -1.719977e-02 0.0192218822 0.0034668966 0.0156471168 -0.0006988663
## 64 7.582398e-03 0.0166169238 0.0172070567 0.0125695391 0.0188880167
## 65 1.698474e-02 0.0194494737 -0.0030931648 0.0149644215 0.0108348875
## 66 -9.666274e-03 0.0126994280 0.0113749000 0.0110911387 0.0073730588
## 67 9.253114e-03 -0.0028102225 -0.0035443049 -0.0056146790 0.0019295289
## 68 1.204964e-02 -0.0106515691 0.0051285058 0.0060795167 -0.0016544298
## 69 -2.205519e-02 0.0199093333 0.0155447731 0.0271983489 0.0122354775
## 6.X 6.Y 7.X 7.Y 8.X
## 1 1.761063e-03 1.505457e-03 1.165451e-02 -6.232710e-03 0.0098249338
## 2 -1.043471e-02 2.177499e-02 -3.458647e-03 2.231202e-02 0.0123972157
## 3 1.859992e-02 -1.414234e-02 -4.651971e-03 3.168325e-03 0.0031001380
## 4 2.086723e-02 1.031646e-02 -3.645759e-02 6.028167e-03 -0.0222183715
## 5 1.796766e-02 1.661719e-02 2.492692e-02 3.299587e-02 0.0198466158
## 6 3.225277e-03 7.661319e-03 -1.433063e-03 1.760206e-02 0.0049249469
## 7 4.725856e-05 -1.002570e-02 -1.259038e-02 2.035425e-02 0.0032260319
## 8 -1.060758e-02 6.803731e-03 -1.317891e-05 2.346005e-02 -0.0026772240
## 9 4.609107e-03 1.159259e-02 -1.138303e-02 1.063381e-02 -0.0003115853
## 10 -1.159643e-02 -9.628758e-04 1.645961e-02 1.841144e-02 0.0115355683
## 11 -5.638631e-03 7.643092e-03 5.271003e-03 3.739850e-03 -0.0096180180
## 12 -2.096952e-02 2.550040e-02 -2.208104e-03 1.162825e-02 0.0353618565
## 13 1.155891e-02 3.702511e-03 1.983079e-02 3.630125e-03 0.0266756887
## 14 -4.899325e-04 1.326678e-02 -3.402471e-03 1.493211e-03 0.0212186785
## 15 4.420680e-03 -1.767549e-02 -4.996705e-03 2.581421e-03 -0.0172106161
## 16 1.497709e-02 -6.176536e-04 1.066911e-02 7.781148e-03 0.0110764073
## 17 3.891527e-03 9.573276e-03 1.130261e-03 1.982977e-03 0.0079122661
## 18 1.908746e-02 3.638396e-03 9.756681e-04 -6.791663e-03 -0.0109688000
## 19 -7.650398e-04 -1.080262e-02 2.161634e-02 -1.818694e-03 -0.0077428952
## 20 1.473713e-02 -1.076623e-02 1.736598e-02 -8.545264e-03 -0.0065908749
## 21 -2.825320e-02 -2.619036e-03 -1.542426e-02 1.159236e-02 0.0042189292
## 22 -5.130315e-03 -4.590239e-03 7.021464e-03 -1.281562e-02 0.0074712775
## 23 -9.707054e-03 -8.608916e-03 -1.393698e-03 -6.901833e-03 -0.0027521597
## 24 -4.199386e-02 3.442918e-02 1.196669e-02 1.809122e-02 0.0251943742
## 25 -2.494935e-02 1.420972e-02 1.489301e-03 -2.989598e-02 0.0230176391
## 26 -5.477032e-02 5.425177e-03 -1.184125e-02 -2.549095e-03 0.0013000252
## 27 -1.170683e-02 3.077465e-03 9.350597e-03 -1.811934e-04 -0.0091911303
## 28 -1.540811e-02 1.901802e-03 -1.236809e-02 -9.920628e-03 0.0191986068
## 29 -3.628408e-02 7.837874e-03 -6.158953e-03 6.922357e-03 -0.0115188038
## 30 -6.054525e-03 -6.449029e-03 1.248187e-02 -7.887591e-03 0.0034619653
## 31 -2.817701e-02 -6.613215e-02 5.774039e-02 -4.138923e-02 0.0394853512
## 32 -9.288434e-03 1.369015e-02 -5.843927e-03 2.968583e-03 -0.0103216302
## 33 -2.997353e-02 7.684426e-03 -7.037904e-03 -8.414476e-03 0.0218922058
## 34 -2.240266e-02 1.326310e-02 2.931326e-03 1.391494e-02 0.0288327530
## 35 -1.342776e-02 1.640273e-02 2.651647e-02 -3.949249e-03 0.0202312288
## 36 -7.122291e-03 -9.598441e-03 3.075666e-03 -7.034268e-03 -0.0001453920
## 37 -1.583206e-02 -4.161852e-03 2.917654e-03 -1.913407e-02 -0.0019313382
## 38 -2.048718e-02 -2.872379e-03 -7.308447e-03 -1.919548e-02 0.0102870278
## 39 1.609207e-02 2.654352e-02 -8.026401e-03 -2.362888e-03 0.0037655201
## 40 1.252099e-02 2.061073e-02 -4.960590e-03 1.399374e-02 -0.0217369466
## 41 3.060246e-02 -1.565677e-02 9.915645e-03 -1.029652e-02 -0.0171164804
## 42 2.894127e-02 7.584823e-03 -1.157867e-02 -7.304886e-03 -0.0163175040
## 43 1.715161e-02 -1.314832e-02 8.477683e-03 -9.340124e-03 -0.0092082601
## 44 2.043417e-02 -1.348501e-02 -8.732738e-03 -2.659987e-02 -0.0235190442
## 45 3.285042e-02 -1.921673e-02 -2.047441e-03 -1.281753e-02 -0.0199147944
## 46 2.288336e-02 -2.309810e-06 -8.321585e-03 -4.821635e-05 -0.0011089947
## 47 1.852242e-02 -1.830599e-02 -2.122039e-02 -1.494596e-02 -0.0294314872
## 48 8.067674e-04 -4.270933e-03 -2.210962e-02 -1.256185e-02 -0.0172614051
## 49 2.655317e-02 -5.682474e-03 -3.283388e-03 8.969654e-03 -0.0256337141
## 50 9.451276e-03 9.423635e-05 -6.209060e-03 1.236334e-02 -0.0139989386
## 51 4.999950e-03 1.335350e-02 2.023991e-03 -3.429299e-03 0.0088607085
## 52 -1.018771e-03 -2.599713e-02 -8.976812e-03 -3.184519e-03 -0.0257718510
## 53 -5.768811e-03 -2.427774e-02 -1.855585e-03 -1.007988e-03 -0.0105561950
## 54 -1.294197e-02 -2.056712e-02 -2.030145e-03 -7.160791e-03 -0.0279803949
## 55 1.602993e-02 8.724186e-03 -9.132424e-03 1.309340e-03 -0.0114367967
## 56 -8.114988e-03 -4.666487e-03 2.091883e-03 1.390988e-02 0.0197252662
## 57 -1.881446e-02 -2.975902e-02 1.713170e-02 -2.389003e-02 0.0162609144
## 58 -1.882099e-03 1.153516e-02 -3.657843e-03 6.151592e-03 -0.0098596609
## 59 1.285037e-02 -1.630735e-03 2.214703e-03 6.607077e-03 0.0040971900
## 60 9.779547e-03 1.980497e-02 3.999845e-03 2.564648e-02 0.0035567320
## 61 2.367810e-02 7.057332e-03 1.198035e-02 6.330174e-03 -0.0005712199
## 62 1.150472e-02 1.010352e-02 3.946535e-03 8.531028e-03 0.0006760546
## 63 -2.952305e-03 3.398030e-03 -5.341827e-03 -2.251886e-02 0.0217626018
## 64 2.351569e-03 1.129452e-03 2.152204e-02 6.111865e-03 0.0151306037
## 65 -1.700871e-02 5.656887e-05 -3.604768e-02 6.350105e-03 0.0088294157
## 66 3.133534e-02 -1.688441e-02 -6.899907e-03 6.161002e-04 -0.0186625358
## 67 -6.502560e-04 7.151878e-03 -5.802236e-03 4.411001e-03 -0.0102825498
## 68 1.709567e-02 5.371804e-03 -6.822561e-04 -4.654703e-03 -0.0347367115
## 69 8.437293e-03 -1.646140e-02 -2.380771e-02 -7.812715e-03 -0.0160524146
## 8.Y 9.X 9.Y 10.X 10.Y
## 1 0.0017801150 0.0096447251 0.0151440666 4.257459e-03 -0.0083637049
## 2 -0.0060969937 0.0137867615 -0.0328975466 -1.355819e-03 -0.0148008543
## 3 -0.0062417384 -0.0144066423 -0.0090209890 6.193568e-04 0.0032075295
## 4 -0.0059594173 -0.0062125803 0.0028568851 -1.409823e-02 0.0108734288
## 5 -0.0116095609 -0.0501263445 -0.0176496166 -6.917578e-03 -0.0071105263
## 6 0.0011457134 0.0082611524 -0.0117729016 -1.456332e-03 -0.0069348126
## 7 0.0214316998 0.0486244098 -0.0066052620 2.559008e-03 -0.0047169546
## 8 0.0031842438 0.0008357123 -0.0112079390 4.225757e-03 -0.0063323870
## 9 -0.0154734397 0.0141367077 -0.0078450915 1.962832e-03 -0.0040543932
## 10 -0.0125499163 0.0221662149 -0.0060184229 -6.398400e-03 -0.0003040507
## 11 -0.0148491338 0.0065664443 -0.0012014225 2.820971e-03 0.0022800882
## 12 -0.0142528858 0.0017861059 -0.0211458434 4.161682e-03 -0.0031622452
## 13 -0.0043661209 -0.0306339246 -0.0100425802 -1.123668e-04 -0.0305320267
## 14 -0.0028622240 0.0095711299 -0.0015728255 5.960047e-03 0.0034983075
## 15 0.0011031247 -0.0052915891 0.0067359108 7.421224e-03 0.0203422892
## 16 0.0214461318 0.0085445582 -0.0134443179 1.588312e-02 -0.0144660128
## 17 0.0015022540 -0.0007643525 0.0403576354 2.168812e-02 0.0037581935
## 18 -0.0173736984 -0.0058326245 0.0045183642 -6.112570e-03 0.0133141340
## 19 -0.0341842566 -0.0206969860 -0.0101850751 7.324936e-03 0.0143416549
## 20 -0.0034710858 -0.0369041010 0.0153890176 -2.502272e-03 0.0365168591
## 21 0.0002118381 0.0002419672 -0.0012818600 6.327825e-04 -0.0162341618
## 22 -0.0007669768 -0.0150776293 -0.0003768199 2.909254e-03 0.0152773816
## 23 0.0038391929 0.0085515343 -0.0039190787 1.267491e-02 -0.0053661480
## 24 -0.0271684563 -0.0466593388 -0.0282364453 1.421733e-02 0.0020794408
## 25 -0.0160877152 -0.0311835126 -0.0371499649 1.341084e-02 0.0017849139
## 26 -0.0009074423 0.0103221608 0.0076828326 1.232050e-02 -0.0116360172
## 27 0.0069423969 -0.0032907120 -0.0103855664 2.880473e-03 0.0104328932
## 28 0.0051220029 0.0153035278 0.0026009591 -1.868338e-03 -0.0208369636
## 29 -0.0069834391 -0.0037958064 -0.0099874396 1.377262e-02 0.0109347568
## 30 -0.0044398330 -0.0276532586 -0.0053593711 1.310635e-03 0.0024746331
## 31 -0.0239952038 -0.0549849347 0.0311112049 -3.633164e-03 -0.0183389232
## 32 -0.0117316109 -0.0218693491 -0.0084869663 1.101946e-02 0.0115747596
## 33 -0.0055375823 -0.0097242557 0.0043103572 8.177165e-03 0.0016065936
## 34 0.0017011092 -0.0248139408 -0.0012122316 5.098368e-03 0.0025090266
## 35 -0.0076528770 -0.0302358223 -0.0170441487 1.626891e-02 0.0099869856
## 36 0.0220532314 -0.0203748015 -0.0163721690 -6.856654e-05 0.0074469152
## 37 0.0056302981 0.0007815497 0.0078297903 1.992142e-03 0.0009159415
## 38 -0.0006325849 0.0092960377 0.0105596844 6.167577e-04 0.0148995151
## 39 -0.0084862278 -0.0124956916 -0.0087534426 -8.285472e-03 -0.0094865619
## 40 -0.0213744336 -0.0049730093 -0.0042623264 -5.298637e-03 0.0068155775
## 41 -0.0019098201 -0.0092895046 0.0238235084 -1.267300e-02 0.0188625028
## 42 0.0065820578 -0.0134186560 -0.0068098542 -2.465732e-02 0.0112191942
## 43 0.0149856619 0.0301158346 -0.0005784078 -7.962068e-03 -0.0406286803
## 44 -0.0002776399 0.0327177112 0.0131187377 -8.275931e-03 -0.0039928092
## 45 0.0173132928 0.0138400390 0.0063103851 -3.559174e-03 -0.0169587251
## 46 0.0101607924 0.0341526701 0.0152052010 -3.140027e-04 -0.0200670589
## 47 -0.0055137880 0.0462151944 0.0455590182 -9.709601e-03 -0.0155356417
## 48 -0.0108313103 0.0066780533 -0.0021223012 5.474353e-03 0.0214228203
## 49 -0.0055127523 0.0399862415 0.0165955027 4.687712e-02 0.0095650846
## 50 0.0016245605 0.0311783167 -0.0128933989 -7.191100e-03 0.0183273271
## 51 0.0043924897 0.0092739595 -0.0354999832 -8.041644e-03 -0.0002812950
## 52 -0.0065912032 0.0474584409 0.0160012853 -9.158449e-03 0.0088782090
## 53 -0.0140770247 0.0277574863 0.0042843208 8.024586e-03 0.0257966238
## 54 0.0230246117 0.0344539888 0.0113811061 5.629386e-03 -0.0080010708
## 55 0.0124808387 -0.0223263100 -0.0070482045 -2.060033e-02 0.0168117833
## 56 0.0118255484 0.0005382309 -0.0278035756 -1.413613e-03 -0.0018984205
## 57 0.0057220252 -0.0613872849 0.0343756811 -1.459560e-02 0.0131153091
## 58 0.0168035275 0.0400981977 0.0159156591 1.074773e-03 -0.0209220492
## 59 0.0008388127 0.0439640705 0.0051097934 -8.656803e-03 -0.0169103859
## 60 0.0093025389 -0.0486006373 -0.0189321210 -7.451989e-03 -0.0090215642
## 61 -0.0030543595 -0.0275553980 0.0148096026 -2.317228e-02 -0.0025577793
## 62 0.0119643409 0.0376738472 0.0140526125 2.734291e-03 -0.0103821422
## 63 0.0168159500 -0.0493543461 0.0020999140 -1.366018e-02 0.0141195545
## 64 0.0338499580 -0.0077472837 -0.0056610460 -3.593884e-03 -0.0064206161
## 65 0.0138722400 0.0148818048 0.0072080440 -1.523401e-02 -0.0108037906
## 66 0.0226198951 -0.0049205980 0.0150642821 1.030955e-02 0.0045721948
## 67 -0.0055837300 0.0261154288 -0.0021701005 1.846071e-03 -0.0063101406
## 68 -0.0086517564 0.0158224409 0.0095097626 -9.745067e-03 0.0015599020
## 69 0.0157857447 0.0012585694 0.0134355321 -1.038298e-02 0.0022465897
## 11.X 11.Y 12.X 12.Y 13.X
## 1 -3.719678e-03 -0.0119550427 -2.224714e-02 -0.0141064474 6.035711e-03
## 2 1.907606e-02 -0.0002197552 -1.707739e-02 -0.0174178283 1.129161e-02
## 3 -1.093428e-02 0.0174155250 -1.540393e-03 0.0076919802 -9.941022e-04
## 4 4.943443e-03 0.0041882659 -7.638197e-04 0.0022631794 1.472305e-03
## 5 1.596122e-02 -0.0125297138 -1.299348e-02 0.0017558860 2.095572e-02
## 6 9.356188e-03 -0.0101864211 -6.513875e-03 0.0026093816 3.928025e-03
## 7 8.698955e-03 -0.0159742084 -1.271949e-02 -0.0136927040 5.727376e-03
## 8 -4.249192e-03 -0.0030513575 1.030048e-02 0.0110691177 -2.140910e-02
## 9 -1.089471e-03 -0.0099756365 -1.815157e-02 -0.0069185701 7.700057e-03
## 10 9.558389e-03 -0.0062900599 -1.140756e-02 -0.0089642309 -1.525298e-02
## 11 -1.642572e-02 0.0102036393 1.874471e-02 -0.0042321874 -4.660778e-05
## 12 1.049444e-03 -0.0093158442 -2.212638e-02 -0.0002107124 -8.370729e-03
## 13 -1.624951e-02 0.0032786421 -3.309725e-03 0.0225872322 -8.486570e-03
## 14 8.469805e-03 -0.0086550931 -2.493388e-03 -0.0217869811 2.314981e-03
## 15 1.370529e-03 0.0030334956 7.982746e-03 0.0039432715 2.335018e-03
## 16 7.885212e-03 0.0012452554 7.389467e-03 -0.0213287944 -3.778448e-03
## 17 1.138808e-02 -0.0044940703 -2.322532e-03 -0.0137172333 5.545097e-03
## 18 -1.190518e-02 -0.0020893097 1.139463e-02 -0.0003233319 -8.056652e-03
## 19 -3.491894e-03 0.0130777459 -9.919121e-03 0.0371063314 -9.598200e-03
## 20 -1.126565e-02 0.0107904529 -1.483710e-03 -0.0030227555 2.037506e-03
## 21 8.085929e-03 0.0022466975 -1.431320e-02 -0.0000538758 1.906060e-02
## 22 6.859304e-03 0.0013541478 -4.414172e-05 -0.0018298916 -1.128077e-02
## 23 5.737827e-03 -0.0069105472 -1.578841e-02 0.0080555880 1.809549e-03
## 24 9.972947e-03 0.0007170510 -4.663347e-03 0.0052295640 7.756816e-03
## 25 -1.111230e-02 0.0279188030 -1.648453e-02 0.0350817718 -3.228837e-02
## 26 1.367064e-02 0.0034366801 7.254205e-03 0.0053678044 -5.334484e-03
## 27 -1.238467e-02 0.0048435976 3.167237e-05 0.0034185847 3.089925e-03
## 28 7.491056e-04 -0.0116014368 -1.577206e-04 -0.0084696222 1.028649e-02
## 29 1.436813e-02 0.0027581572 -3.020865e-02 0.0004587896 4.263445e-03
## 30 -8.960951e-03 0.0093756075 2.151945e-03 0.0058836404 -4.431664e-03
## 31 -7.772025e-03 -0.0005061636 7.711657e-03 0.0487843580 -1.078583e-02
## 32 8.153052e-03 0.0148064971 6.960503e-03 0.0008358526 7.507876e-04
## 33 3.919826e-03 0.0060714188 -8.701488e-05 -0.0006910220 -4.966859e-03
## 34 2.380298e-03 0.0049091457 2.363731e-02 -0.0122988546 -5.737770e-03
## 35 8.952199e-03 0.0073642413 -3.543985e-02 0.0074545762 -1.123492e-02
## 36 -5.815209e-03 0.0005908313 1.558820e-02 0.0189322000 -9.250348e-03
## 37 1.665684e-02 0.0037288427 -1.409679e-02 0.0086896810 -1.235916e-02
## 38 1.078819e-02 0.0071466596 8.942348e-03 -0.0172621336 1.032927e-03
## 39 3.913979e-03 -0.0033753537 1.607601e-02 -0.0071493046 1.629328e-02
## 40 6.135717e-03 0.0125696258 8.191145e-03 -0.0116475633 9.930482e-03
## 41 -3.291315e-03 -0.0008445927 2.067411e-02 -0.0187270853 -2.201901e-03
## 42 1.223160e-02 0.0009683958 1.155324e-02 -0.0048389774 1.659509e-03
## 43 -6.533196e-03 0.0109381148 1.381100e-02 -0.0061232435 9.983855e-03
## 44 -5.077583e-03 0.0098212170 -4.060877e-03 0.0049765469 4.893792e-03
## 45 -1.415046e-02 0.0154126777 2.307698e-02 -0.0015345349 -7.484888e-03
## 46 8.505403e-03 -0.0156246938 -6.530768e-03 -0.0127586180 1.049083e-02
## 47 -3.619572e-04 -0.0118209803 2.794303e-02 -0.0148733819 1.131884e-02
## 48 -3.877467e-04 0.0186957148 -3.522352e-03 0.0040047461 -9.354615e-03
## 49 1.129994e-02 -0.0091835230 -3.111610e-02 -0.0232156306 -1.613973e-03
## 50 2.279075e-02 0.0061165248 -4.158383e-02 -0.0168391293 -6.816832e-04
## 51 -8.395984e-03 -0.0058065595 -4.454517e-03 -0.0038240519 2.844736e-02
## 52 5.351597e-03 -0.0017960328 -1.020765e-02 -0.0024391469 4.393810e-04
## 53 6.761753e-05 0.0126549711 -1.695928e-03 -0.0088709384 -5.026978e-03
## 54 -1.596504e-03 0.0011247511 -8.182725e-03 0.0113058462 1.969239e-03
## 55 -3.139633e-03 0.0018637976 3.012663e-02 -0.0033804428 8.085498e-03
## 56 -8.548891e-03 -0.0040248456 7.707391e-03 -0.0050379076 -1.316281e-02
## 57 -9.091362e-03 0.0101385418 1.154329e-02 0.0113045700 1.232763e-03
## 58 1.518306e-02 -0.0320272258 1.455754e-02 -0.0173885346 1.390508e-02
## 59 -1.508606e-02 -0.0270041339 2.936739e-03 -0.0067772047 -1.494053e-02
## 60 -1.048392e-02 -0.0019351123 2.506620e-02 0.0073661074 7.360882e-03
## 61 -1.248137e-02 -0.0068018223 2.679230e-02 -0.0050419288 7.957411e-03
## 62 6.963912e-03 -0.0190668241 -8.746254e-03 -0.0015536546 2.295727e-03
## 63 -2.859932e-02 -0.0017867948 1.000189e-02 0.0186240797 2.571764e-03
## 64 -1.933443e-02 -0.0160258677 -1.669570e-02 -0.0001201567 -9.943373e-04
## 65 -6.454348e-04 0.0117225243 2.232584e-02 0.0069363284 -2.170524e-03
## 66 9.985405e-04 0.0027233872 -2.271592e-02 0.0140243999 -1.941656e-02
## 67 -7.397977e-03 -0.0105843976 1.183274e-02 -0.0007396109 1.186694e-02
## 68 -3.366769e-03 -0.0049902269 3.037673e-02 -0.0003522734 -1.369932e-02
## 69 -1.814810e-02 0.0112020029 -6.816848e-03 0.0237990812 6.315086e-03
## 13.Y
## 1 0.0086863817
## 2 0.0080332890
## 3 -0.0110101031
## 4 0.0011055137
## 5 0.0067143717
## 6 0.0029588794
## 7 -0.0107057695
## 8 -0.0018245490
## 9 0.0024801293
## 10 -0.0062171940
## 11 -0.0153151308
## 12 -0.0069602557
## 13 0.0069447540
## 14 0.0045105079
## 15 -0.0094634668
## 16 -0.0020382809
## 17 -0.0114811716
## 18 -0.0001083091
## 19 0.0105637168
## 20 -0.0063776572
## 21 0.0344291992
## 22 -0.0026865649
## 23 0.0101626985
## 24 0.0307196364
## 25 0.0173163133
## 26 0.0143067009
## 27 -0.0140208136
## 28 -0.0006815047
## 29 0.0174141370
## 30 0.0172610189
## 31 0.0204784690
## 32 0.0109884508
## 33 0.0059573548
## 34 -0.0007026366
## 35 0.0238908812
## 36 0.0017470746
## 37 0.0085582959
## 38 -0.0094159645
## 39 0.0072031395
## 40 -0.0121894306
## 41 0.0011788881
## 42 0.0097832987
## 43 -0.0174968048
## 44 -0.0114966627
## 45 -0.0101521712
## 46 -0.0020302505
## 47 -0.0146085880
## 48 -0.0080996621
## 49 -0.0146609287
## 50 -0.0062258492
## 51 0.0030918065
## 52 -0.0041936608
## 53 -0.0055691044
## 54 -0.0099271196
## 55 -0.0047586056
## 56 -0.0023419209
## 57 0.0066664223
## 58 -0.0075626825
## 59 -0.0077675044
## 60 -0.0033803860
## 61 -0.0158159239
## 62 -0.0095712098
## 63 0.0070556867
## 64 -0.0048675234
## 65 -0.0210691942
## 66 0.0138618723
## 67 -0.0026648480
## 68 -0.0075189175
## 69 -0.0010905671The coefficients in the alternative model could be tested with the following code.
coef(fit.alt, test = TRUE)##
## Linear Model fit with lm.rrpp
##
## Number of observations: 69
## Number of dependent variables: 26
## Data space dimensions: 26
## Sums of Squares and Cross-products: Type I
## Number of permutations: 10000
##
## Statistics (distances) of coefficients with 95 percent confidence intervals,
## effect sizes, and probabilities of exceeding observed values
## based on
## 10000 random permutations using RRPP
##
## d.obs UCL (95%) Zd Pr(>d)
## log(Csize) 0.2734641 0.09937552 5.119583 1e-04We can also perform ANOVA, e.g.,
anova(fit.null, fit.alt, print.progress = FALSE)##
## Analysis of Variance, using Residual Randomization
## Permutation procedure: Randomization of null model residuals
## Number of permutations: 10000
## Estimation method: Ordinary Least Squares
## Effect sizes (Z) based on F distributions
##
## ResDf Df RSS SS MS Rsq F Z
## coords ~ 1 (Null) 68 1 0.45759 0.00000
## coords ~ log(Csize) 67 1 0.35001 0.10758 0.10758 0.23511 20.594 5.3289
## Total 68 0.45759
## P Pr(>F)
## coords ~ 1 (Null)
## coords ~ log(Csize) 1e-04
## Totalanova(fit.alt)##
## Analysis of Variance, using Residual Randomization
## Permutation procedure: Randomization of null model residuals
## Number of permutations: 10000
## Estimation method: Ordinary Least Squares
## Sums of Squares and Cross-products: Type I
## Effect sizes (Z) based on F distributions
##
## Df SS MS Rsq F Z Pr(>F)
## log(Csize) 1 0.10758 0.107583 0.23511 20.594 5.3289 1e-04 ***
## Residuals 67 0.35001 0.005224 0.76489
## Total 68 0.45759
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Call: procD.lm(f1 = coords ~ log(Csize), iter = 9999, turbo = FALSE,
## data = Y.gpa)Looking at the attributes below, what might also be work exploring?
attributes(fit.alt)## $names
## [1] "aov.table" "call" "coefficients" "Y" "X"
## [6] "QR" "fitted" "residuals" "Terms" "term.labels"
## [11] "data" "call" "LM" "ANOVA" "PermInfo"
## [16] "turbo" "Models" "fitted" "residuals" "coefficients"
## [21] "GM"
##
## $class
## [1] "procD.lm" "lm.rrpp"