Multiple Regression Analysis 2 Paper

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

Title produced by software: Multiple Regression

Date of computation: Sun, 20 Dec 2009 05:20:57 -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/Dec/20/t1261311725mx2kh2cs8od3co1.htm/, Retrieved Sun, 20 Dec 2009 13:22:17 +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/Dec/20/t1261311725mx2kh2cs8od3co1.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:

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Dataseries X:

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-1.2 23.6 -2.4 25.7 0.8 32.5 -0.1 33.5 -1.5 34.5 -4.4 27.9 -4.2 45.3 3.5 40.8 10 58.5 8.6 32.5 9.5 35.5 9.9 46.7 10.4 53.2 16 36.1 12.7 54 10.2 58.1 8.9 41.8 12.6 43.1 13.6 76 14.8 42.8 9.5 41 13.7 61.4 17 34.2 14.7 53.8 17.4 80.7 9 79.5 9.1 96.5 12.2 108.3 15.9 100.1 12.9 108.5 10.9 127.4 10.6 86.5 13.2 71.4 9.6 88.2 6.4 135.6 5.8 70.5 -1 87.5 -0.2 73.3 2.7 92.2 3.6 61.1 -0.9 45.7 0.3 30.5 -1.1 34.8 -2.5 29.2 -3.4 56.7 -3.5 67.1 -3.9 41.8 -4.6 46.8 -0.1 50.1 4.3 81.9 10.2 115.8 8.7 102.5 13.3 106.6 15 101.4 20.7 136.1 20.7 143.4 26.4 127.5 31.2 113.8 31.4 75.3 26.6 98.5 26.6 113.7 19.2 103.7 6.5 73.9 3.1 52.5 -0.2 63.9 -4 44.9 -12.6 31.3 -13 24.9 -17.6 22.8 -21.7 24.8 -23.2 22.8 -16.8 20.9 -19.8 21.5

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 Energiedragers[t] = -7.36746638506505 + 0.236669034135203Invoer[t] -2.56663152756032M1[t] -0.76835819175296M2[t] -3.97043927651090M3[t] -2.75825331497566M4[t] -2.20191074851504M5[t] -1.28672975866373M6[t] -5.86821153019543M7[t] -1.44912310628503M8[t] -1.18873828155045M9[t] -1.61257552120687M10[t] -0.0488920455135976M11[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) -7.36746638506505 4.346891 -1.6949 0.095283 0.047641 Invoer 0.236669034135203 0.034298 6.9003 0 0 M1 -2.56663152756032 5.312628 -0.4831 0.63077 0.315385 M2 -0.76835819175296 5.521738 -0.1392 0.889796 0.444898 M3 -3.97043927651090 5.558126 -0.7143 0.477782 0.238891 M4 -2.75825331497566 5.528363 -0.4989 0.619655 0.309827 M5 -2.20191074851504 5.519074 -0.399 0.691336 0.345668 M6 -1.28672975866373 5.511063 -0.2335 0.816182 0.408091 M7 -5.86821153019543 5.548183 -1.0577 0.29444 0.14722 M8 -1.44912310628503 5.512721 -0.2629 0.793552 0.396776 M9 -1.18873828155045 5.514891 -0.2156 0.830069 0.415035 M10 -1.61257552120687 5.517568 -0.2923 0.771095 0.385547 M11 -0.0488920455135976 5.510171 -0.0089 0.99295 0.496475 Multiple Linear Regression - Regression Statistics Multiple R 0.667430694342178 R-squared 0.445463731750082 Adjusted R-squared 0.334556478100099 F-TEST (value) 4.0165428057207 F-TEST (DF numerator) 12 F-TEST (DF denominator) 60 p-value 0.000146322219272688 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 9.54356708174346 Sum Squared Residuals 5464.78035862425 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 -1.2 -4.34870870703463 3.14870870703463 2 -2.4 -2.05343039954331 -0.34656960045669 3 0.8 -3.64616205218187 4.44616205218187 4 -0.1 -2.19730705651143 2.09730705651143 5 -1.5 -1.40429545591560 -0.0957045440843955 6 -4.4 -2.05113009135663 -2.34886990864337 7 -4.2 -2.51457066893580 -1.68542933106420 8 3.5 0.839507101366184 2.66049289863382 9 10 5.28893383029384 4.71106616970616 10 8.6 -1.28829829687785 9.88829829687785 11 9.5 0.98539228122103 8.51460771877897 12 9.9 3.68497750904891 6.21502249095109 13 10.4 2.65669470336740 7.7433052966326 14 16 0.407927555462797 15.5920724445372 15 12.7 1.44222218172499 11.257777818275 16 10.2 3.62475118321456 6.57524881678544 17 8.9 0.323388493271377 8.57661150672862 18 12.6 1.54623922749845 11.0537607725015 19 13.6 4.75116867901492 8.84883132098508 20 14.8 1.31284516963659 13.4871548303634 21 9.5 1.14722573292780 8.3527742670722 22 13.7 5.55143678962951 8.14856321037048 23 17 0.67772253684527 16.3222774631547 24 14.7 5.36532765140884 9.33467234859116 25 17.4 9.16509314208547 8.23490685791453 26 9 10.6793636369306 -1.67936363693060 27 9.1 11.5006561324711 -2.40065613247111 28 12.2 15.5055366968017 -3.30553669680173 29 15.9 14.1211931833537 1.77880681664631 30 12.9 17.0243940599407 -4.1243940599407 31 10.9 16.9159570335643 -6.01595703356434 32 10.6 11.6552819613449 -1.05528196134495 33 13.2 8.34196437063796 4.85803562936204 34 9.6 11.8941669044529 -2.29416690445295 35 6.4 24.6759625981548 -18.2759625981548 36 5.8 9.31770052146673 -3.51770052146673 37 -1 10.7744425742049 -11.7744425742049 38 -0.2 9.21201562529234 -9.41201562529234 39 2.7 10.4829792856897 -7.78297928568973 40 3.6 4.33475828562016 -0.734758285620164 41 -0.9 1.24639772639867 -2.14639772639867 42 0.3 -1.43579060260511 1.73579060260511 43 -1.1 -4.99959552735543 3.89959552735543 44 -2.5 -1.90585369460217 -0.594146305397834 45 -3.4 4.86292956885048 -8.26292956885048 46 -3.5 6.90045028420017 -10.4004502842002 47 -3.9 2.47640719627281 -6.37640719627281 48 -4.6 3.70864441246243 -8.30864441246243 49 -0.1 1.92302069754827 -2.02302069754827 50 4.3 11.2473693188551 -6.94736931885508 51 10.2 16.0683684912805 -5.86836849128052 52 8.7 14.1328562988176 -5.43285629881756 53 13.3 15.6595419052325 -2.35954190523251 54 15 15.3440439175808 -0.344043917580766 55 20.7 18.9749776305406 1.72502236945940 56 20.7 25.1217500036380 -4.42175000363798 57 26.4 21.6190971856228 4.78090281437717 58 31.2 17.9528941783141 13.2471058216859 59 31.4 10.4048198398021 20.9951801601979 60 26.6 15.9444334772524 10.6555665227476 61 26.6 16.9751712685472 9.62482873145284 62 19.2 16.4067542630025 2.7932457369975 63 6.5 6.15193596101553 0.348064038984474 64 3.1 2.29940459205742 0.80059540794258 65 -0.2 5.55377414765936 -5.75377414765936 66 -4 1.97224348894181 -5.97224348894181 67 -12.6 -5.82793714682864 -6.77206285317136 68 -13 -2.92353054138354 -10.0764694586165 69 -17.6 -3.16015068833289 -14.4398493116671 70 -21.7 -3.11064985971891 -18.5893501402811 71 -23.2 -2.02030445229605 -21.1796955477040 72 -16.8 -2.42108357163932 -14.3789164283607 73 -19.8 -4.84571367871853 -14.9542863212815 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.187105035532919 0.374210071065838 0.812894964467081 17 0.119839419685664 0.239678839371329 0.880160580314336 18 0.097698044371374 0.195396088742748 0.902301955628626 19 0.0481659448653551 0.0963318897307102 0.951834055134645 20 0.057043981730109 0.114087963460218 0.942956018269891 21 0.0557078339966103 0.111415667993221 0.94429216600339 22 0.0554059022404616 0.110811804480923 0.944594097759538 23 0.071323317191875 0.14264663438375 0.928676682808125 24 0.0493197737763554 0.0986395475527108 0.950680226223645 25 0.0417996422549758 0.0835992845099516 0.958200357745024 26 0.0899444623459329 0.179888924691866 0.910055537654067 27 0.102621707115415 0.20524341423083 0.897378292884585 28 0.0790293370181534 0.158058674036307 0.920970662981847 29 0.0502488918488305 0.100497783697661 0.94975110815117 30 0.0365411736722519 0.0730823473445038 0.963458826327748 31 0.0300847184176347 0.0601694368352694 0.969915281582365 32 0.0211289041044241 0.0422578082088482 0.978871095895576 33 0.0146669327254822 0.0293338654509644 0.985333067274518 34 0.0115871407164519 0.0231742814329037 0.988412859283548 35 0.101505739452581 0.203011478905162 0.898494260547419 36 0.0852658395442071 0.170531679088414 0.914734160455793 37 0.127118927752987 0.254237855505974 0.872881072247013 38 0.117988048429578 0.235976096859156 0.882011951570422 39 0.0964068471918447 0.192813694383689 0.903593152808155 40 0.0692757896454382 0.138551579290876 0.930724210354562 41 0.0604457573722032 0.120891514744406 0.939554242627797 42 0.0541466937055623 0.108293387411125 0.945853306294438 43 0.057368652409913 0.114737304819826 0.942631347590087 44 0.088607461544499 0.177214923088998 0.911392538455501 45 0.0933316519516902 0.186663303903380 0.90666834804831 46 0.107228400987146 0.214456801974292 0.892771599012854 47 0.100223808572451 0.200447617144902 0.899776191427549 48 0.0939408002038881 0.187881600407776 0.906059199796112 49 0.066019540052187 0.132039080104374 0.933980459947813 50 0.0445319441868819 0.0890638883737637 0.955468055813118 51 0.0396499925844138 0.0792999851688275 0.960350007415586 52 0.0412707805789318 0.0825415611578637 0.958729219421068 53 0.0247951161140873 0.0495902322281747 0.975204883885913 54 0.014660361043985 0.02932072208797 0.985339638956015 55 0.0158186767841400 0.0316373535682800 0.98418132321586 56 0.0789044824127695 0.157808964825539 0.921095517587231 57 0.11572749629913 0.23145499259826 0.88427250370087 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 0 0 OK 5% type I error level 6 0.142857142857143 NOK 10% type I error level 14 0.333333333333333 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly 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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