IntroductionPrincipal Component analysis, ignoring phylogenyUsing PC axes in multiple regressionMultiple Regression, ignoring phylogenyMultiple Regression, using the phylogenyPrincipal Component analysis, using the phylogenyA Comparative Analysis with Oxalis,continuedAbigail Mazie and Cécile AnéApril 26, 2011GoalsComparative analysis of Oxalis data, continued;Using R for comparative analysis;Illustrating multiple regression and principal componentanalysis (PCA).Reading the Data> library(ape)> library(picante)> library(phylobase)> oxalis.dat = read.csv("oxalis-data.csv")> rownames(oxalis.dat) = oxalis.dat$species> oxalis.dat = oxalis.dat[, -1]# add column with (natural) log of # scales/mm> oxalis.dat$log.scales.mm = log(oxalis.dat$scales.mm)> oxalis.datalt precip seasonality scales.mm lat abslat region log.scales.mmOadenophyllaADEPH1 2081 785 72 40 -40.1 40.1 Basal 3.688879ObrasiliensisBRAS2 79 1097 17 7 -30.5 30.5 SESouthAmerican 1.945910OperdicariaMV79 275 1262 63 64 -30.7 30.7 SESouthAmerican 4.158883OdebilisEE171 684 1613 62 10 -20.4 20.4 SESouthAmerican 2.302585OhispidulaMV44MV342 79 1211 31 8 -28.2 28.2 SESouthAmerican 2.079442ObipartitaMV59MV320 803 1510 28 5 -27.0 27.0 SESouthAmerican 1.609438...Visualizing the data7 20 40abslat0 1700 3700altitude17 74 110seasonalityPossible VariablesTemperature (minimum, maximum, mean)PrecipitationLatitudeAltitudeBioclimatic VariablesBioclimatic VariablesBIO1 = Annual Mean TemperatureBIO2 = Mean Diurnal Range (Mean of monthly (max temp - min temp))BIO3 = Isothermality (BIO2/BIO7) (* 100)BIO4 = Temperature Seasonality (standard deviation *100)BIO5 = Max Temperature of Warmest MonthBIO6 = Min Temperature of Coldest MonthBIO7 = Temperature Annual Range (BIO5-BIO6)BIO8 = Mean Temperature of Wettest QuarterBIO9 = Mean Temperature of Driest QuarterBIO10 = Mean Temperature of Warmest QuarterBIO11 = Mean Temperature of Coldest QuarterBIO12 = Annual PrecipitationBIO13 = Precipitation of Wettest MonthBIO14 = Precipitation of Driest MonthBIO15 = Precipitation Seasonality (Coefficient of Variation)BIO16 = Precipitation of Wettest QuarterBIO17 = Precipitation of Driest QuarterBIO18 = Precipitation of Warmest QuarterBIO19 = Precipitation of Coldest QuarterWhy use principal component analysis?Too many variables for direct multiple regression (littlepower)PCA will allow us to reduce number of predictor variablesto a few components, while retaining most information fromall predictor variablesCan use PCA on groups of predictor variables to reduce toone componentBioclimatic Variables: Temperature GroupBIO1 = Annual Mean TemperatureBIO2 = Mean Diurnal Range (Mean of monthly (max temp - min temp))BIO3 = Isothermality (BIO2/BIO7) (* 100)BIO4 = Temperature Seasonality (standard deviation *100)BIO5 = Max Temperature of Warmest MonthBIO6 = Min Temperature of Coldest MonthBIO7 = Temperature Annual Range (BIO5-BIO6)BIO8 = Mean Temperature of Wettest QuarterBIO9 = Mean Temperature of Driest QuarterBIO10 = Mean Temperature of Warmest QuarterBIO11 = Mean Temperature of Coldest QuarterBIO12 = Annual PrecipitationBIO13 = Precipitation of Wettest MonthBIO14 = Precipitation of Driest MonthBIO15 = Precipitation Seasonality (Coefficient of Variation)BIO16 = Precipitation of Wettest QuarterBIO17 = Precipitation of Driest QuarterBIO18 = Precipitation of Warmest QuarterBIO19 = Precipitation of Coldest QuarterBioclimatic Variables: Precipitation GroupBIO1 = Annual Mean TemperatureBIO2 = Mean Diurnal Range (Mean of monthly (max temp - min temp))BIO3 = Isothermality (BIO2/BIO7) (* 100)BIO4 = Temperature Seasonality (standard deviation *100)BIO5 = Max Temperature of Warmest MonthBIO6 = Min Temperature of Coldest MonthBIO7 = Temperature Annual Range (BIO5-BIO6)BIO8 = Mean Temperature of Wettest QuarterBIO9 = Mean Temperature of Driest QuarterBIO10 = Mean Temperature of Warmest QuarterBIO11 = Mean Temperature of Coldest QuarterBIO12 = Annual PrecipitationBIO13 = Precipitation of Wettest MonthBIO14 = Precipitation of Driest MonthBIO15 = Precipitation Seasonality (Coefficient of Variation)BIO16 = Precipitation of Wettest QuarterBIO17 = Precipitation of Driest QuarterBIO18 = Precipitation of Warmest QuarterBIO19 = Precipitation of Coldest QuarterQuestionWith PCA, the author said that sometimes we would like toperform a PCA on the evolutionary correlation matrix,rather than the evolutionary variance-covariance matrix.When should we use one or another?Note: PCA on correlation matrix = PCA on re-scaled variables,so that each one is centered and has variance 1.Principal Component Analysis in R# 1st column = atl, 2nd=precip, 3d=season, 6th=abslat> head(oxalis.dat[,c(1,2,3,6)])alt precip seasonality abslatOadenophyllaADEPH1 2081 785 72 40.1ObrasiliensisBRAS2 79 1097 17 30.5OperdicariaMV79 275 1262 63 30.7OdebilisEE171 684 1613 62 20.4OhispidulaMV44MV342 79 1211 31 28.2ObipartitaMV59MV320 803 1510 28 27.0...> pca.ind = prcomp(oxalis.dat[,c(1,2,3,6)], scale=T)> summary(pca.ind)Importance of components:Comp.1 Comp.2 Comp.3 Comp.4Standard deviation 1.4248 0.9770 0.8254 0.57803Proportion of Variance 0.5075 0.2386 0.1703 0.08353Cumulative Proportion 0.5075 0.7461 0.9165 1.00000# plot variances explained by each axis> plot(pca.ind)How much is explained by the PC axesComp.1 Comp.2 Comp.3 Comp.4pca.indVariances0.0 0.5 1.0 1.5 2.0We could keep 3 components for future multiple regression.Not very useful here, then.4 original variables with clear interpretation versus 3 PCvariables with less clear interpretation.Interpretation of PC axes> pca.ind$roration # loadingsLoadings:Comp.1 Comp.2 Comp.3 Comp.4alt 0.603 0.008 0.374 0.704precip -0.441 -0.716 -0.229 0.491seasonality 0.487 -0.011 -0.872 -0.047abslat -0.452 0.698 -0.217 0.511Interpretation:axis 1 = contrast between average of altitude & seasonalityand average of precip & abslat.axis 2 = contrast between abs.latitude and precipitationaxis 3 = - seasonality mostlyaxis 4 = ∼ average of alt, precip and abslat.Interpretation of PC axes−0.4 0.0 0.2 0.4−0.4 0.0 0.2 0.4Comp.1Comp.2OadenophyllaADEPH1ObrasiliensisBRAS2OperdicariaMV79OdebilisEE171OhispidulaMV44MV342ObipartitaMV59MV320OoreocharisEE583OtriangularisssppapilionaceaREGOtrolliiEE281OmacrocarpaAG49OpinguiculaceaWood22147OalpinaSW976OdiscolorAG35OlatifoliaAG70OlatifoliaEE756COlasiandraAG69OtetraphyllaTETRA2OnelsoniiAG66OcaeruleaAG47OprimaveraAG50OhernandesiiAG56OdivergensAG62OdecaphyllaAR2610AOmorelosiiEPEREZ4856−2 0 2 4−2 0 2 4altprecipseasonalityabslat−0.4 0.0 0.2 0.4−0.4 0.0 0.2
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