Paper Multiple regression

*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: Fri, 03 Dec 2010 22:35:40 +0000

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/03/t1291415635d6s1hqi78rtjaor.htm/, Retrieved Fri, 03 Dec 2010 23:34:05 +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/2010/Dec/03/t1291415635d6s1hqi78rtjaor.htm/}, year = {2010}, } @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 = {2010}, 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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1,79 194,9 1,95 195,5 2,26 196 2,04 196,2 2,16 196,2 2,75 196,2 2,79 196,2 2,88 197 3,36 197,7 2,97 198 3,1 198,2 2,49 198,5 2,2 198,6 2,25 199,5 2,09 200 2,79 201,3 3,14 202,2 2,93 202,9 2,65 203,5 2,67 203,5 2,26 204 2,35 204,1 2,13 204,3 2,18 204,5 2,9 204,8 2,63 205,1 2,67 205,7 1,81 206,5 1,33 206,9 0,88 207,1 1,28 207,8 1,26 208 1,26 208,5 1,29 208,6 1,1 209 1,37 209,1 1,21 209,7 1,74 209,8 1,76 209,9 1,48 210 1,04 210,8 1,62 211,4 1,49 211,7 1,79 212 1,8 212,2 1,58 212,4 1,86 212,9 1,74 213,4 1,59 213,7 1,26 214 1,13 214,3 1,92 214,8 2,61 215 2,26 215,9 2,41 216,4 2,26 216,9 2,03 217,2 2,86 217,5 2,55 217,9 2,27 218,1 2,26 218,6 2,57 218,9 3,07 219,3 2,76 220,4 2,51 220,9 2,87 221 3,14 221,8 3,11 222 3,16 222,2 2,47 222,5 2,57 222,9 2,89 223,1

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 7 seconds R Server 'George Udny Yule' @ 72.249.76.132 Multiple Linear Regression - Estimated Regression Equation Xt[t] = + 2.01280333512101 + 0.000841977464621676Yt[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) 2.01280333512101 1.908997 1.0544 0.295336 0.147668 Yt 0.000841977464621676 0.009121 0.0923 0.926714 0.463357 Multiple Linear Regression - Regression Statistics Multiple R 0.0110327518319245 R-squared 0.000121721612984833 Adjusted R-squared -0.0141622537925439 F-TEST (value) 0.00852155016576963 F-TEST (DF numerator) 1 F-TEST (DF denominator) 70 p-value 0.92671364427681 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 0.641553958466168 Sum Squared Residuals 28.8114037136526 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 1.79 2.17690474297577 -0.386904742975770 2 1.95 2.17740992945455 -0.227409929454546 3 2.26 2.17783091818686 0.0821690818131422 4 2.04 2.17799931367978 -0.137999313679782 5 2.16 2.17799931367978 -0.0179993136797817 6 2.75 2.17799931367978 0.572000686320218 7 2.79 2.17799931367978 0.612000686320218 8 2.88 2.17867289565148 0.70132710434852 9 3.36 2.17926227987671 1.18073772012329 10 2.97 2.1795148731161 0.7904851268839 11 3.1 2.17968326860903 0.920316731390975 12 2.49 2.17993586184841 0.310064138151588 13 2.2 2.18002005959487 0.0199799404051263 14 2.25 2.18077783931303 0.0692221606869666 15 2.09 2.18119882804534 -0.0911988280453444 16 2.79 2.18229339874935 0.607706601250648 17 3.14 2.18305117846751 0.956948821532488 18 2.93 2.18364056269275 0.746359437307253 19 2.65 2.18414574917152 0.46585425082848 20 2.67 2.18414574917152 0.48585425082848 21 2.26 2.18456673790383 0.0754332620961688 22 2.35 2.18465093565029 0.165349064349707 23 2.13 2.18481933114322 -0.0548193311432177 24 2.18 2.18498772663614 -0.00498772663614171 25 2.9 2.18524031987553 0.714759680124472 26 2.63 2.18549291311491 0.444507086885085 27 2.67 2.18599809959369 0.484001900406312 28 1.81 2.18667168156539 -0.376671681565385 29 1.33 2.18700847255123 -0.857008472551234 30 0.88 2.18717686804416 -1.30717686804416 31 1.28 2.18776625226939 -0.907766252269393 32 1.26 2.18793464776232 -0.927934647762318 33 1.26 2.18835563649463 -0.928355636494629 34 1.29 2.18843983424109 -0.89843983424109 35 1.1 2.18877662522694 -1.08877662522694 36 1.37 2.1888608229734 -0.818860822973401 37 1.21 2.18936600945217 -0.979366009452175 38 1.74 2.18945020719864 -0.449450207198637 39 1.76 2.1895344049451 -0.429534404945099 40 1.48 2.18961860269156 -0.709618602691561 41 1.04 2.19029218466326 -1.15029218466326 42 1.62 2.19079737114203 -0.570797371142031 43 1.49 2.19104996438142 -0.701049964381418 44 1.79 2.19130255762080 -0.401302557620805 45 1.8 2.19147095311373 -0.391470953113729 46 1.58 2.19163934860665 -0.611639348606653 47 1.86 2.19206033733896 -0.332060337338964 48 1.74 2.19248132607127 -0.452481326071275 49 1.59 2.19273391931066 -0.602733919310661 50 1.26 2.19298651255005 -0.932986512550048 51 1.13 2.19323910578943 -1.06323910578943 52 1.92 2.19366009452175 -0.273660094521745 53 2.61 2.19382849001467 0.41617150998533 54 2.26 2.19458626973283 0.0654137302671707 55 2.41 2.19500725846514 0.214992741534860 56 2.26 2.19542824719745 0.064571752802549 57 2.03 2.19568084043684 -0.165680840436837 58 2.86 2.19593343367622 0.664066566323776 59 2.55 2.19627022466207 0.353729775337927 60 2.27 2.19643862015500 0.0735613798450032 61 2.26 2.19685960888731 0.0631403911126922 62 2.57 2.19711220212669 0.372887797873306 63 3.07 2.19744899311254 0.872551006887457 64 2.76 2.19837516832363 0.561624831676373 65 2.51 2.19879615705594 0.311203842944062 66 2.87 2.1988803548024 0.6711196451976 67 3.14 2.19955393677410 0.940446063225903 68 3.11 2.19972233226702 0.910277667732979 69 3.16 2.19989072775995 0.960109272240055 70 2.47 2.20014332099933 0.269856679000668 71 2.57 2.20048011198518 0.369519888014819 72 2.89 2.20064850747811 0.689351492521895 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 5 0.00689175631862223 0.0137835126372445 0.993108243681378 6 0.0456482487110725 0.091296497422145 0.954351751288927 7 0.0447667407340902 0.0895334814681803 0.95523325926591 8 0.0186259271628986 0.0372518543257973 0.981374072837101 9 0.00871384123546174 0.0174276824709235 0.991286158764538 10 0.00892115701163284 0.0178423140232657 0.991078842988367 11 0.00563078555756071 0.0112615711151214 0.99436921444244 12 0.0271292949758498 0.0542585899516996 0.97287070502415 13 0.0720444654871594 0.144088930974319 0.92795553451284 14 0.103617709303620 0.207235418607241 0.89638229069638 15 0.125407077830078 0.250814155660155 0.874592922169922 16 0.0998979760436006 0.199795952087201 0.9001020239564 17 0.105369263356721 0.210738526713442 0.894630736643279 18 0.102100329719323 0.204200659438646 0.897899670280677 19 0.104889387950633 0.209778775901266 0.895110612049367 20 0.110323017542194 0.220646035084387 0.889676982457806 21 0.138202210640806 0.276404421281612 0.861797789359194 22 0.152354301282079 0.304708602564159 0.84764569871792 23 0.179577875911310 0.359155751822620 0.82042212408869 24 0.200068044799975 0.400136089599951 0.799931955200025 25 0.392990125415328 0.785980250830656 0.607009874584672 26 0.627120286185247 0.745759427629507 0.372879713814753 27 0.933171941277611 0.133656117444777 0.0668280587223886 28 0.984162965898783 0.0316740682024333 0.0158370341012167 29 0.996679818579565 0.00664036284086996 0.00332018142043498 30 0.999612679646247 0.000774640707506062 0.000387320353753031 31 0.99970520597335 0.000589588053298099 0.000294794026649050 32 0.999711830893601 0.0005763382127979 0.00028816910639895 33 0.99964700116242 0.000705997675160694 0.000352998837580347 34 0.999508095392873 0.000983809214253836 0.000491904607126918 35 0.999420859226185 0.00115828154763020 0.000579140773815098 36 0.999037814376303 0.00192437124739406 0.000962185623697031 37 0.998562654383462 0.00287469123307618 0.00143734561653809 38 0.998227192473859 0.00354561505228217 0.00177280752614109 39 0.998060839419226 0.00387832116154843 0.00193916058077421 40 0.996852077354268 0.00629584529146386 0.00314792264573193 41 0.996826462871675 0.00634707425665015 0.00317353712832508 42 0.994776919909271 0.0104461601814576 0.00522308009072879 43 0.99139262913726 0.0172147417254820 0.00860737086274099 44 0.988069432977396 0.023861134045208 0.011930567022604 45 0.983694882398537 0.0326102352029270 0.0163051176014635 46 0.97459715879158 0.0508056824168401 0.0254028412084200 47 0.966730051926095 0.0665398961478105 0.0332699480739053 48 0.952012187057516 0.0959756258849671 0.0479878129424836 49 0.934080932567827 0.131838134864345 0.0659190674321727 50 0.952653454029784 0.0946930919404328 0.0473465459702164 51 0.99352010417963 0.0129597916407385 0.00647989582036924 52 0.993398001944354 0.013203996111292 0.006601998055646 53 0.996277568817302 0.00744486236539627 0.00372243118269813 54 0.994663022795174 0.0106739544096513 0.00533697720482565 55 0.992821083236517 0.0143578335269659 0.00717891676348297 56 0.989394525872445 0.0212109482551095 0.0106054741275547 57 0.991401095362686 0.0171978092746277 0.00859890463731386 58 0.99348476950496 0.0130304609900819 0.00651523049504093 59 0.989084540841044 0.0218309183179121 0.0109154591589560 60 0.984488368648434 0.0310232627031324 0.0155116313515662 61 0.987048360575489 0.0259032788490225 0.0129516394245113 62 0.98353241016013 0.0329351796797411 0.0164675898398705 63 0.9752527482982 0.0494945034035994 0.0247472517017997 64 0.951299915649971 0.0974001687000575 0.0487000843500288 65 0.956903954330832 0.0861920913383357 0.0430960456691678 66 0.947039232478229 0.105921535043543 0.0529607675217715 67 0.87354871882197 0.25290256235606 0.12645128117803 Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity Description # significant tests % significant tests OK/NOK 1% type I error level 14 0.222222222222222 NOK 5% type I error level 36 0.571428571428571 NOK 10% type I error level 45 0.714285714285714 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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