Multiple Regression (a)

*The author of this computation has been verified*

R Software Module: /rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Thu, 19 Nov 2009 09:26:46 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Nov/19/t12586497742aih0ljd2ajpv0i.htm/, Retrieved Thu, 19 Nov 2009 17:56:26 +0100

BibTeX entries for LaTeX users:

@Manual{KEY, author = {{YOUR NAME}}, publisher = {Office for Research Development and Education}, title = {Statistical Computations at FreeStatistics.org, URL http://www.freestatistics.org/blog/date/2009/Nov/19/t12586497742aih0ljd2ajpv0i.htm/}, year = {2009}, } @Manual{R, title = {R: A Language and Environment for Statistical Computing}, author = {{R Development Core Team}}, organization = {R Foundation for Statistical Computing}, address = {Vienna, Austria}, year = {2009}, note = {{ISBN} 3-900051-07-0}, url = {http://www.R-project.org}, }

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

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4 7.2 102.9 271244 4.1 7.4 97.4 269907 4 8.8 111.4 271296 3.8 9.3 87.4 270157 4.7 9.3 96.8 271322 4.3 8.7 114.1 267179 3.9 8.2 110.3 264101 4 8.3 103.9 265518 4.3 8.5 101.6 269419 4.8 8.6 94.6 268714 4.4 8.5 95.9 272482 4.3 8.2 104.7 268351 4.7 8.1 102.8 268175 4.7 7.9 98.1 270674 4.9 8.6 113.9 272764 5 8.7 80.9 272599 4.2 8.7 95.7 270333 4.3 8.5 113.2 270846 4.8 8.4 105.9 270491 4.8 8.5 108.8 269160 4.8 8.7 102.3 274027 4.2 8.7 99 273784 4.6 8.6 100.7 276663 4.8 8.5 115.5 274525 4.5 8.3 100.7 271344 4.4 8 109.9 271115 4.3 8.2 114.6 270798 3.9 8.1 85.4 273911 3.7 8.1 100.5 273985 4 8 114.8 271917 4.1 7.9 116.5 273338 3.7 7.9 112.9 270601 3.8 8 102 273547 3.8 8 106 275363 3.8 7.9 105.3 281229 3.3 8 118.8 277793 3.3 7.7 106.1 279913 3.3 7.2 109.3 282500 3.2 7.5 117.2 280041 3.4 7.3 92.5 282166 4.2 7 104.2 290304 4.9 7 112.5 283519 5.1 7 122.4 287816 5.5 7.2 113.3 285226 5.6 7.3 100 287595 6.4 7.1 110.7 289741 6.1 6.8 112.8 289148 7.1 6.4 109.8 288301 7.8 6.1 117.3 etc...

Output produced by software:

Enter (or paste) a matrix (table) containing all data (time) series. Every column represents a different variable and must be delimited by a space or Tab. Every row represents a period in time (or category) and must be delimited by hard returns. The easiest way to enter data is to copy and paste a block of spreadsheet cells. Please, do not use commas or spaces to seperate groups of digits! Summary of computational transaction Raw Input view raw input (R code) Raw Output view raw output of R engine Computing time 4 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation Cons.index[t] = + 25.0581301035609 -1.06842952839769Werkl.graad[t] + 0.0114153233526319Industr.prod.[t] -4.76358601503515e-05BrutoSchuld[t] + e[t] Multiple Linear Regression - Ordinary Least Squares Variable Parameter S.D. T-STAT H0: parameter = 0 2-tail p-value 1-tail p-value (Intercept) 25.0581301035609 5.884439 4.2584 7.2e-05 3.6e-05 Werkl.graad -1.06842952839769 0.278026 -3.8429 0.000292 0.000146 Industr.prod. 0.0114153233526319 0.019279 0.5921 0.555959 0.277979 BrutoSchuld -4.76358601503515e-05 1e-05 -4.584 2.3e-05 1.2e-05 Multiple Linear Regression - Regression Statistics Multiple R 0.623066294095584 R-squared 0.388211606838005 Adjusted R-squared 0.358123653075940 F-TEST (value) 12.9025592736539 F-TEST (DF numerator) 3 F-TEST (DF denominator) 61 p-value 1.24809663959446e-06 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 1.23092396372239 Sum Squared Residuals 92.425602072429 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 4 5.61913302146152 -1.61913302146152 2 4.1 5.40635198236346 -1.30635198236346 3 4 4.0041989597947 -0.00419895979470014 4 3.8 3.25027367984394 0.549726320156060 5 4.7 3.30208194228352 1.39791805771648 6 4.3 4.33798012192557 -0.0379801219255748 7 3.9 4.9754398349272 -1.07543983492720 8 4 4.72803879879754 -0.728038798797539 9 4.3 4.30227015896043 -0.00227015896042693 10 4.8 4.14910322405823 0.650896775941767 11 4.4 4.0912941762099 0.308705823790102 12 4.3 4.70906161851347 -0.409061618513469 13 4.7 4.8025993683697 -0.102599368369699 14 4.7 4.84359123977614 -0.143591239776138 15 4.9 4.1764937311551 0.723506268844896 16 5 3.70080502460329 1.29919497539671 17 4.2 3.97769466932294 0.22230533067706 18 4.3 4.36671153741641 -0.066711537416405 19 4.8 4.40713336013534 0.392866639864664 20 4.8 4.39679817487832 0.403201825121683 21 4.8 3.87706893605491 0.922931063945088 22 4.2 3.85097388300776 0.349026116992238 23 4.6 3.84007924417414 0.759920755825856 24 4.8 4.21771445163431 0.582285548365685 25 4.5 4.41398324283317 0.086016757166831 26 4.4 4.85044168817112 -0.45044168817112 27 4.3 4.70550836991661 -0.405508369916614 28 3.9 4.33073344821149 -0.430733448211489 29 3.7 4.49957977718510 -0.799579777185104 30 4 4.86817281275843 -0.868172812758435 31 4.1 4.92673125802403 -0.826731258024029 32 3.7 5.01601544318607 -1.31601544318607 33 3.8 4.64441022179967 -0.844410221799674 34 3.8 4.60356479317716 -0.803564793177163 35 3.8 4.42298506402813 -0.622985064028128 36 3.3 4.6339257919255 -1.33392579192550 37 3.3 4.70849202034763 -1.40849202034763 38 3.3 5.15600184906594 -1.85600184906594 39 3.2 5.04279062514214 -1.84279062514214 40 3.4 4.87329184119218 -1.47329184119218 41 4.2 4.93971935303372 -0.739719353033716 42 4.9 5.3576758479807 -0.457675847980695 43 5.1 5.26599625810569 -0.165996258105691 44 5.5 5.07180778770661 0.428192212293388 45 5.6 4.70029168158066 0.899708318419343 46 6.4 4.9338949912507 1.46610500874930 47 6.1 5.3066440938797 0.793355906120305 48 7.1 5.74011750872822 1.35988249127178 49 7.8 6.05794440767352 1.74205559232648 50 7.9 5.56111832591909 2.33888167408091 51 7.4 4.42441291963371 2.97558708036629 52 7.5 3.9299241007075 3.57007589929250 53 6.8 4.14420266975708 2.65579733024292 54 5.2 4.47447603791701 0.725523962082986 55 4.7 4.60633826577258 0.0936617342274224 56 4.1 4.0432911423272 0.0567088576727956 57 3.9 2.57203708954851 1.32796291045149 58 2.6 2.03286903125433 0.567130968745668 59 2.7 2.29082682704504 0.40917317295496 60 1.8 2.62066411920809 -0.820664119208088 61 1 3.10106728308513 -2.10106728308513 62 0.3 2.8726973662336 -2.5726973662336 63 1.3 2.29666903760866 -0.996669037608656 64 1 1.67623446106161 -0.676234461061607 65 1.1 1.84956000634913 -0.749560006349129 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 7 0.0382003558547824 0.0764007117095647 0.961799644145218 8 0.0095845116943573 0.0191690233887146 0.990415488305643 9 0.00247411201067749 0.00494822402135498 0.997525887989323 10 0.00313932697883788 0.00627865395767576 0.996860673021162 11 0.000863408896551553 0.00172681779310311 0.999136591103448 12 0.000235297909925155 0.00047059581985031 0.999764702090075 13 0.000162015146896033 0.000324030293792066 0.999837984853104 14 7.55745005460131e-05 0.000151149001092026 0.999924425499454 15 3.8222556100105e-05 7.644511220021e-05 0.9999617774439 16 2.3013328374405e-05 4.602665674881e-05 0.999976986671626 17 7.52489073767691e-06 1.50497814753538e-05 0.999992475109262 18 2.00464060762242e-06 4.00928121524483e-06 0.999997995359392 19 9.11897908905251e-07 1.82379581781050e-06 0.99999908810209 20 4.78122044541163e-07 9.56244089082326e-07 0.999999521877955 21 1.49013558943613e-07 2.98027117887225e-07 0.99999985098644 22 8.39931658719764e-08 1.67986331743953e-07 0.999999916006834 23 2.79434886731486e-08 5.58869773462973e-08 0.999999972056511 24 9.28636646067736e-09 1.85727329213547e-08 0.999999990713634 25 2.38237530749269e-09 4.76475061498538e-09 0.999999997617625 26 5.49099651576913e-10 1.09819930315383e-09 0.9999999994509 27 1.33109219573940e-10 2.66218439147880e-10 0.99999999986689 28 1.18342499234472e-10 2.36684998468943e-10 0.999999999881658 29 2.14242090669190e-10 4.28484181338380e-10 0.999999999785758 30 8.49613240463002e-11 1.69922648092600e-10 0.999999999915039 31 2.53846456762846e-11 5.07692913525693e-11 0.999999999974615 32 1.8894917483247e-11 3.7789834966494e-11 0.999999999981105 33 9.90924436742731e-12 1.98184887348546e-11 0.99999999999009 34 4.8595531073117e-12 9.7191062146234e-12 0.99999999999514 35 1.77723058884518e-12 3.55446117769036e-12 0.999999999998223 36 3.65950236524589e-12 7.31900473049179e-12 0.99999999999634 37 4.49461507795451e-12 8.98923015590901e-12 0.999999999995505 38 7.09215643447724e-12 1.41843128689545e-11 0.999999999992908 39 8.62038375709623e-11 1.72407675141925e-10 0.999999999913796 40 5.14980829705661e-10 1.02996165941132e-09 0.99999999948502 41 9.02213103533153e-09 1.80442620706631e-08 0.99999999097787 42 9.40121662064602e-07 1.88024332412920e-06 0.999999059878338 43 2.14330688363471e-05 4.28661376726942e-05 0.999978566931164 44 0.00057233025933461 0.00114466051866922 0.999427669740665 45 0.0034057364004167 0.0068114728008334 0.996594263599583 46 0.0139572131972380 0.0279144263944759 0.986042786802762 47 0.0408863066433801 0.0817726132867601 0.95911369335662 48 0.100989299600246 0.201978599200491 0.899010700399754 49 0.167613666190583 0.335227332381166 0.832386333809417 50 0.212994049349928 0.425988098699857 0.787005950650072 51 0.234663678068225 0.46932735613645 0.765336321931775 52 0.209570657207026 0.419141314414051 0.790429342792974 53 0.144608949434858 0.289217898869717 0.855391050565142 54 0.122289799653085 0.24457959930617 0.877710200346915 55 0.109741066496070 0.219482132992140 0.89025893350393 56 0.139066131024108 0.278132262048216 0.860933868975892 57 0.49967117542407 0.99934235084814 0.50032882457593 58 0.604665180525155 0.79066963894969 0.395334819474845 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 37 0.711538461538462 NOK 5% type I error level 39 0.75 NOK 10% type I error level 41 0.788461538461538 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;

R code (references can be found in the software module):

library(lattice) library(lmtest) n25 <- 25 #minimum number of obs. for Goldfeld-Quandt test par1 <- as.numeric(par1) x <- t(y) k <- length(x[1,]) n <- length(x[,1]) x1 <- cbind(x[,par1], x[,1:k!=par1]) mycolnames <- c(colnames(x)[par1], colnames(x)[1:k!=par1]) colnames(x1) <- mycolnames #colnames(x)[par1] x <- x1 if (par3 == 'First Differences'){ x2 <- array(0, dim=c(n-1,k), dimnames=list(1:(n-1), paste('(1-B)',colnames(x),sep=''))) for (i in 1:n-1) { for (j in 1:k) { x2[i,j] <- x[i+1,j] - x[i,j] } } x <- x2 } if (par2 == 'Include Monthly Dummies'){ x2 <- array(0, dim=c(n,11), dimnames=list(1:n, paste('M', seq(1:11), sep =''))) for (i in 1:11){ x2[seq(i,n,12),i] <- 1 } x <- cbind(x, x2) } if (par2 == 'Include Quarterly Dummies'){ x2 <- array(0, dim=c(n,3), dimnames=list(1:n, paste('Q', seq(1:3), sep =''))) for (i in 1:3){ x2[seq(i,n,4),i] <- 1 } x <- cbind(x, x2) } k <- length(x[1,]) if (par3 == 'Linear Trend'){ x <- cbind(x, c(1:n)) colnames(x)[k+1] <- 't' } x k <- length(x[1,]) df <- as.data.frame(x) (mylm <- lm(df)) (mysum <- summary(mylm)) if (n > n25) { kp3 <- k + 3 nmkm3 <- n - k - 3 gqarr <- array(NA, dim=c(nmkm3-kp3+1,3)) numgqtests <- 0 numsignificant1 <- 0 numsignificant5 <- 0 numsignificant10 <- 0 for (mypoint in kp3:nmkm3) { j <- 0 numgqtests <- numgqtests + 1 for (myalt in c('greater', 'two.sided', 'less')) { j <- j + 1 gqarr[mypoint-kp3+1,j] <- gqtest(mylm, point=mypoint, alternative=myalt)$p.value } if (gqarr[mypoint-kp3+1,2] < 0.01) numsignificant1 <- numsignificant1 + 1 if (gqarr[mypoint-kp3+1,2] < 0.05) numsignificant5 <- numsignificant5 + 1 if (gqarr[mypoint-kp3+1,2] < 0.10) numsignificant10 <- numsignificant10 + 1 } gqarr } bitmap(file='test0.png') plot(x[,1], type='l', main='Actuals and Interpolation', ylab='value of Actuals and Interpolation (dots)', xlab='time or index') points(x[,1]-mysum$resid) grid() dev.off() bitmap(file='test1.png') plot(mysum$resid, type='b', pch=19, main='Residuals', ylab='value of Residuals', xlab='time or index') grid() dev.off() bitmap(file='test2.png') hist(mysum$resid, main='Residual Histogram', xlab='values of Residuals') grid() dev.off() bitmap(file='test3.png') densityplot(~mysum$resid,col='black',main='Residual Density Plot', xlab='values of Residuals') dev.off() bitmap(file='test4.png') qqnorm(mysum$resid, main='Residual Normal Q-Q Plot') qqline(mysum$resid) grid() dev.off() (myerror <- as.ts(mysum$resid)) bitmap(file='test5.png') dum <- cbind(lag(myerror,k=1),myerror) dum dum1 <- dum[2:length(myerror),] dum1 z <- as.data.frame(dum1) z plot(z,main=paste('Residual Lag plot, lowess, and regression line'), ylab='values of Residuals', xlab='lagged values of Residuals') lines(lowess(z)) abline(lm(z)) grid() dev.off() bitmap(file='test6.png') acf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Autocorrelation Function') grid() dev.off() bitmap(file='test7.png') pacf(mysum$resid, lag.max=length(mysum$resid)/2, main='Residual Partial Autocorrelation Function') grid() dev.off() bitmap(file='test8.png') opar <- par(mfrow = c(2,2), oma = c(0, 0, 1.1, 0)) plot(mylm, las = 1, sub='Residual Diagnostics') par(opar) dev.off() if (n > n25) { bitmap(file='test9.png') plot(kp3:nmkm3,gqarr[,2], main='Goldfeld-Quandt test',ylab='2-sided p-value',xlab='breakpoint') grid() dev.off() } load(file='createtable') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Estimated Regression Equation', 1, TRUE) a<-table.row.end(a) myeq <- colnames(x)[1] myeq <- paste(myeq, '[t] = ', sep='') for (i in 1:k){ if (mysum$coefficients[i,1] > 0) myeq <- paste(myeq, '+', '') myeq <- paste(myeq, mysum$coefficients[i,1], sep=' ') if (rownames(mysum$coefficients)[i] != '(Intercept)') { myeq <- paste(myeq, rownames(mysum$coefficients)[i], sep='') if (rownames(mysum$coefficients)[i] != 't') myeq <- paste(myeq, '[t]', sep='') } } myeq <- paste(myeq, ' + e[t]') a<-table.row.start(a) a<-table.element(a, myeq) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable1.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,hyperlink('http://www.xycoon.com/ols1.htm','Multiple Linear Regression - Ordinary Least Squares',''), 6, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Variable',header=TRUE) a<-table.element(a,'Parameter',header=TRUE) a<-table.element(a,'S.D.',header=TRUE) a<-table.element(a,'T-STAT<br />H0: parameter = 0',header=TRUE) a<-table.element(a,'2-tail p-value',header=TRUE) a<-table.element(a,'1-tail p-value',header=TRUE) a<-table.row.end(a) for (i in 1:k){ a<-table.row.start(a) a<-table.element(a,rownames(mysum$coefficients)[i],header=TRUE) a<-table.element(a,mysum$coefficients[i,1]) a<-table.element(a, round(mysum$coefficients[i,2],6)) a<-table.element(a, round(mysum$coefficients[i,3],4)) a<-table.element(a, round(mysum$coefficients[i,4],6)) a<-table.element(a, round(mysum$coefficients[i,4]/2,6)) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable2.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Regression Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple R',1,TRUE) a<-table.element(a, sqrt(mysum$r.squared)) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'R-squared',1,TRUE) a<-table.element(a, mysum$r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Adjusted R-squared',1,TRUE) a<-table.element(a, mysum$adj.r.squared) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (value)',1,TRUE) a<-table.element(a, mysum$fstatistic[1]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF numerator)',1,TRUE) a<-table.element(a, mysum$fstatistic[2]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'F-TEST (DF denominator)',1,TRUE) a<-table.element(a, mysum$fstatistic[3]) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'p-value',1,TRUE) a<-table.element(a, 1-pf(mysum$fstatistic[1],mysum$fstatistic[2],mysum$fstatistic[3])) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Residual Statistics', 2, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Residual Standard Deviation',1,TRUE) a<-table.element(a, mysum$sigma) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Sum Squared Residuals',1,TRUE) a<-table.element(a, sum(myerror*myerror)) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable3.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a, 'Multiple Linear Regression - Actuals, Interpolation, and Residuals', 4, TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a, 'Time or Index', 1, TRUE) a<-table.element(a, 'Actuals', 1, TRUE) a<-table.element(a, 'Interpolation<br />Forecast', 1, TRUE) a<-table.element(a, 'Residuals<br />Prediction Error', 1, TRUE) a<-table.row.end(a) for (i in 1:n) { a<-table.row.start(a) a<-table.element(a,i, 1, TRUE) a<-table.element(a,x[i]) a<-table.element(a,x[i]-mysum$resid[i]) a<-table.element(a,mysum$resid[i]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable4.tab') if (n > n25) { a<-table.start() a<-table.row.start(a) a<-table.element(a,'Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'p-values',header=TRUE) a<-table.element(a,'Alternative Hypothesis',3,header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'breakpoint index',header=TRUE) a<-table.element(a,'greater',header=TRUE) a<-table.element(a,'2-sided',header=TRUE) a<-table.element(a,'less',header=TRUE) a<-table.row.end(a) for (mypoint in kp3:nmkm3) { a<-table.row.start(a) a<-table.element(a,mypoint,header=TRUE) a<-table.element(a,gqarr[mypoint-kp3+1,1]) a<-table.element(a,gqarr[mypoint-kp3+1,2]) a<-table.element(a,gqarr[mypoint-kp3+1,3]) a<-table.row.end(a) } a<-table.end(a) table.save(a,file='mytable5.tab') a<-table.start() a<-table.row.start(a) a<-table.element(a,'Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity',4,TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'Description',header=TRUE) a<-table.element(a,'# significant tests',header=TRUE) a<-table.element(a,'% significant tests',header=TRUE) a<-table.element(a,'OK/NOK',header=TRUE) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'1% type I error level',header=TRUE) a<-table.element(a,numsignificant1) a<-table.element(a,numsignificant1/numgqtests) if (numsignificant1/numgqtests < 0.01) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'5% type I error level',header=TRUE) a<-table.element(a,numsignificant5) a<-table.element(a,numsignificant5/numgqtests) if (numsignificant5/numgqtests < 0.05) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.row.start(a) a<-table.element(a,'10% type I error level',header=TRUE) a<-table.element(a,numsignificant10) a<-table.element(a,numsignificant10/numgqtests) if (numsignificant10/numgqtests < 0.1) dum <- 'OK' else dum <- 'NOK' a<-table.element(a,dum) a<-table.row.end(a) a<-table.end(a) table.save(a,file='mytable6.tab') }

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