# Partial Least Squares Regression for Beta Regression Models

## Frédéric Bertrand and Myriam Maumy-Bertrand

The goal of plsRbeta is to provide Partial least squares Regression for (weighted) beta regression models (Bertrand 2013, http://journal-sfds.fr/article/view/215) and k-fold cross-validation of such models using various criteria. It allows for missing data in the explanatory variables. Bootstrap confidence intervals constructions are also available.

This website and these examples were created by F. Bertrand and M. Maumy-Bertrand.

## Installation

You can install the released version of plsRbeta from CRAN with:

``install.packages("plsRbeta")``

You can install the development version of plsRbeta from github with:

``devtools::install_github("fbertran/plsRbeta")``

## Example

### Using a model matrix

Fit a plsRbeta model using a model matrix.

``````data("GasolineYield",package="betareg")
yGasolineYield <- GasolineYield\$yield
XGasolineYield <- GasolineYield[,2:5]
library(plsRbeta)
modpls <- plsRbeta(yGasolineYield,XGasolineYield,nt=3,modele="pls-beta")
#> ____************************************************____
#>
#> Model: pls-beta
#>
#>
#>
#> Type: ML
#>
#> ____Component____ 1 ____
#> ____Component____ 2 ____
#> ____Component____ 3 ____
#> ____Predicting X without NA neither in X nor in Y____
#> ****________________________________________________****
print(modpls)
#> Number of required components:
#> [1] 3
#> Number of successfully computed components:
#> [1] 3
#> Coefficients:
#>                   [,1]
#> Intercept -3.324462301
#> gravity    0.001577508
#> pressure   0.072027686
#> temp10    -0.008398771
#> temp       0.010365973
#> Information criteria and Fit statistics:
#>                  AIC        BIC Chi2_Pearson_Y      RSS_Y pseudo_R2_Y
#> Nb_Comp_0  -52.77074  -49.83927       30.72004 0.35640772          NA
#> Nb_Comp_1 -112.87383 -108.47662       30.57369 0.05211039   0.8498691
#> Nb_Comp_2 -136.43184 -130.56889       30.97370 0.02290022   0.9256771
#> Nb_Comp_3 -139.08440 -131.75572       31.08224 0.02022386   0.9385887
#>                R2_Y
#> Nb_Comp_0        NA
#> Nb_Comp_1 0.8537900
#> Nb_Comp_2 0.9357471
#> Nb_Comp_3 0.9432564``````

``````modpls\$pp
#>             Comp_ 1    Comp_ 2    Comp_ 3
#> gravity   0.4590380 -0.4538663 -2.5188256
#> pressure  0.6395524 -0.4733525  0.6488823
#> temp10   -0.5435643  0.5292108 -1.3295905
#> temp      0.5682795  0.5473174 -0.2156423
modpls\$Coeffs
#>                   [,1]
#> Intercept -3.324462301
#> gravity    0.001577508
#> pressure   0.072027686
#> temp10    -0.008398771
#> temp       0.010365973
modpls\$Std.Coeffs
#>                   [,1]
#> Intercept -1.547207760
#> gravity    0.008889933
#> pressure   0.188700277
#> temp10    -0.315301400
#> temp       0.723088387
modpls\$InfCrit
#>                  AIC        BIC Chi2_Pearson_Y
#> Nb_Comp_0  -52.77074  -49.83927       30.72004
#> Nb_Comp_1 -112.87383 -108.47662       30.57369
#> Nb_Comp_2 -136.43184 -130.56889       30.97370
#> Nb_Comp_3 -139.08440 -131.75572       31.08224
#> Nb_Comp_0 0.35640772          NA        NA
#> Nb_Comp_1 0.05211039   0.8498691 0.8537900
#> Nb_Comp_2 0.02290022   0.9256771 0.9357471
#> Nb_Comp_3 0.02022386   0.9385887 0.9432564
modpls\$PredictY[1,]
#>   gravity  pressure    temp10      temp
#>  2.049533  1.686655 -1.371820 -1.821977
rm("modpls")``````

###Formula support

Fit a plsRbeta model using formula support.

``````data("GasolineYield",package="betareg")
modpls <- plsRbeta(yield~.,data=GasolineYield,nt=3,modele="pls-beta", verbose=FALSE)
print(modpls)
#> Number of required components:
#> [1] 3
#> Number of successfully computed components:
#> [1] 3
#> Coefficients:
#>                    [,1]
#> Intercept -4.1210566077
#> gravity    0.0157208676
#> pressure   0.0305159627
#> temp10    -0.0074167766
#> temp       0.0108057945
#> batch1     0.0910284843
#> batch2     0.1398537354
#> batch3     0.2287070465
#> batch4    -0.0008124326
#> batch5     0.1018679027
#> batch6     0.1147971957
#> batch7    -0.1005469609
#> batch8    -0.0447907428
#> batch9    -0.0706292318
#> batch10   -0.1984703429
#> Information criteria and Fit statistics:
#>                  AIC        BIC Chi2_Pearson_Y      RSS_Y pseudo_R2_Y
#> Nb_Comp_0  -52.77074  -49.83927       30.72004 0.35640772          NA
#> Nb_Comp_1  -87.96104  -83.56383       31.31448 0.11172576   0.6879757
#> Nb_Comp_2 -114.10269 -108.23975       33.06807 0.04650238   0.8671800
#> Nb_Comp_3 -152.71170 -145.38302       30.69727 0.01138837   0.9526757
#>                R2_Y
#> Nb_Comp_0        NA
#> Nb_Comp_1 0.6865226
#> Nb_Comp_2 0.8695248
#> Nb_Comp_3 0.9680468``````

``````modpls\$pp
#>              Comp_ 1     Comp_ 2     Comp_ 3
#> gravity   0.37895923 -0.42864981  0.50983922
#> pressure  0.61533000 -0.41618828 -0.01737302
#> temp10   -0.50627633  0.47379983 -0.47750566
#> temp      0.30248369  0.60751756  0.28239621
#> batch1    0.50274128 -0.30221156 -0.25801764
#> batch2   -0.14241033 -0.13859422  0.80068659
#> batch3   -0.04388172 -0.17303214  0.48564161
#> batch4    0.11299471 -0.08302689  0.04755182
#> batch5    0.23341035  0.08396326 -0.51238456
#> batch6    0.07974302  0.07209943 -0.30710455
#> batch7   -0.37365392 -0.02133356  0.81852001
#> batch8   -0.12891598  0.16967195 -0.06904725
#> batch9   -0.02230288  0.19425476 -0.57189134
#> batch10  -0.25409429  0.28587553 -0.61277072
modpls\$Coeffs
#>                    [,1]
#> Intercept -4.1210566077
#> gravity    0.0157208676
#> pressure   0.0305159627
#> temp10    -0.0074167766
#> temp       0.0108057945
#> batch1     0.0910284843
#> batch2     0.1398537354
#> batch3     0.2287070465
#> batch4    -0.0008124326
#> batch5     0.1018679027
#> batch6     0.1147971957
#> batch7    -0.1005469609
#> batch8    -0.0447907428
#> batch9    -0.0706292318
#> batch10   -0.1984703429
modpls\$Std.Coeffs
#>                    [,1]
#> Intercept -1.5526788976
#> gravity    0.0885938394
#> pressure   0.0799466278
#> temp10    -0.2784359925
#> temp       0.7537685874
#> batch1     0.0305865495
#> batch2     0.0414169259
#> batch3     0.0677303525
#> batch4    -0.0002729861
#> batch5     0.0301676274
#> batch6     0.0339965674
#> batch7    -0.0337848600
#> batch8    -0.0132645358
#> batch9    -0.0173701781
#> batch10   -0.0587759166
modpls\$InfCrit
#>                  AIC        BIC Chi2_Pearson_Y
#> Nb_Comp_0  -52.77074  -49.83927       30.72004
#> Nb_Comp_1  -87.96104  -83.56383       31.31448
#> Nb_Comp_2 -114.10269 -108.23975       33.06807
#> Nb_Comp_3 -152.71170 -145.38302       30.69727
#> Nb_Comp_0 0.35640772          NA        NA
#> Nb_Comp_1 0.11172576   0.6879757 0.6865226
#> Nb_Comp_2 0.04650238   0.8671800 0.8695248
#> Nb_Comp_3 0.01138837   0.9526757 0.9680468
modpls\$PredictY[1,]
#>    gravity   pressure     temp10       temp
#>  2.0495333  1.6866554 -1.3718198 -1.8219769
#>     batch1     batch2     batch3     batch4
#>  2.6040833 -0.3165683 -0.3165683 -0.3720119
#>     batch5     batch6     batch7     batch8
#> -0.3165683 -0.3165683 -0.3720119 -0.3165683
#>     batch9    batch10
#> -0.2541325 -0.3165683``````

###Information criteria and cross validation

``````data("GasolineYield",package="betareg")
set.seed(1)
bbb <- PLS_beta_kfoldcv_formula(yield~.,data=GasolineYield,nt=3,modele="pls-beta",verbose=FALSE)
kfolds2CVinfos_beta(bbb)
#> ____************************************************____
#>
#> Model: pls-beta
#>
#>
#>
#> Type: ML
#>
#> ____Component____ 1 ____
#> ____Component____ 2 ____
#> ____Component____ 3 ____
#> ____Predicting X without NA neither in X or Y____
#> ****________________________________________________****
#>
#> NK: 1
#> [[1]]
#>                  AIC        BIC Q2Chisqcum_Y  limQ2 Q2Chisq_Y
#> Nb_Comp_0  -52.77074  -49.83927           NA     NA        NA
#> Nb_Comp_1  -87.96104  -83.56383    -1.121431 0.0975 -1.121431
#> Nb_Comp_2 -114.10269 -108.23975    -5.291744 0.0975 -1.965802
#> Nb_Comp_3 -152.71170 -145.38302   -11.583916 0.0975 -1.000068
#> Nb_Comp_0                NA       30.72004 0.35640772          NA
#> Nb_Comp_1          65.17044       31.31448 0.11172576   0.6879757
#> Nb_Comp_2          92.87255       33.06807 0.04650238   0.8671800
#> Nb_Comp_3          66.13838       30.69727 0.01138837   0.9526757
#>                R2_Y
#> Nb_Comp_0        NA
#> Nb_Comp_1 0.6865226
#> Nb_Comp_2 0.8695248
#> Nb_Comp_3 0.9680468``````

###Bootstrap of the coefficients

Computing bootstrap distributions

``````data("GasolineYield",package="betareg")
set.seed(1)
GazYield.boot <- bootplsbeta(modpls, sim="ordinary", stype="i", R=250)``````

Boxplots of the bootstrap distributions

``plsRglm::boxplots.bootpls(GazYield.boot)``

Confidence intervals for the coefficients of the model based on the bootstrap distributions

``````plsRglm::confints.bootpls(GazYield.boot)
#>
#> Intercept -1.79344379 -1.298344159 -1.820910539
#> gravity    0.01927751  0.203207700  0.009687132
#> pressure  -0.10284957  0.158899010 -0.095122794
#> temp10    -0.50396889 -0.177603439 -0.520653497
#> temp       0.63958439  0.963623548  0.658889299
#> batch1    -0.09222398  0.110336215 -0.099261768
#> batch2    -0.04513338  0.126556137 -0.059420084
#> batch3    -0.04255156  0.157619185 -0.038503175
#> batch4    -0.05292024  0.065545216 -0.047023003
#> batch5    -0.04251041  0.078338536 -0.041090268
#> batch6    -0.02959175  0.082854597 -0.031393204
#> batch7    -0.13595057  0.097613550 -0.125280057
#> batch8    -0.11677718  0.084955543 -0.116202080
#> batch9    -0.06966369  0.023673543 -0.070353628
#> batch10   -0.14231213  0.006802744 -0.142929452
#>
#> Intercept -1.298811257 -1.80654654 -1.28444726
#> gravity    0.207137136 -0.02994946  0.16750055
#> pressure   0.205604518 -0.04571126  0.25501605
#> temp10    -0.184002462 -0.37286952 -0.03621849
#> temp       1.002068184  0.50546899  0.84864788
#> batch1     0.106680396 -0.04550730  0.16043487
#> batch2     0.119437955 -0.03660410  0.14225394
#> batch3     0.177505160 -0.04204445  0.17396388
#> batch4     0.081938870 -0.08248484  0.04647703
#> batch5     0.078484689 -0.01814943  0.10142552
#> batch6     0.104052342 -0.03605921  0.09938634
#> batch7     0.127028728 -0.19459845  0.05771034
#> batch8     0.101367639 -0.12789671  0.08967301
#> batch9     0.028700049 -0.06344040  0.03561327
#> batch10    0.007651461 -0.12520329  0.02537762
#>
#> Intercept -1.80046767 -1.282289852
#> gravity    0.02817987  0.189061113
#> pressure  -0.07071644  0.140622124
#> temp10    -0.42413912 -0.175831698
#> temp       0.64173410  0.941303333
#> batch1    -0.13892632  0.100909676
#> batch2    -0.03921924  0.141666087
#> batch3    -0.08110506  0.157875292
#> batch4    -0.06253836  0.077670517
#> batch5    -0.02899872  0.075247803
#> batch6    -0.06114337  0.066367369
#> batch7    -0.15688005  0.078620504
#> batch8    -0.13986427  0.063261668
#> batch9    -0.08371923  0.016030629
#> batch10   -0.14907794 -0.004437664
#> attr(,"typeBCa")
#> [1] TRUE``````

Plot of the confidence intervals for the coefficients of the model based on the bootstrap distributions

``plsRglm::plots.confints.bootpls(plsRglm::confints.bootpls(GazYield.boot))``