Paper Multiple Regression

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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: Wed, 24 Dec 2008 06:43:47 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Dec/24/t12301264480k9vvoxhz7ds465.htm/, Retrieved Wed, 24 Dec 2008 14:47:38 +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/2008/Dec/24/t12301264480k9vvoxhz7ds465.htm/}, year = {2008}, } @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 = {2008}, 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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105.7 109.5 105.3 102.8 100.6 97.6 110.3 107.2 107.2 108.1 97.1 92.2 112.2 111.6 115.7 111.3 104.2 103.2 112.7 106.4 102.6 110.6 95.2 89 112.5 116.8 107.2 113.6 101.8 102.6 122.7 110.3 110.5 121.6 100.3 100.7 123.4 127.1 124.1 131.2 111.6 114.2 130.1 125.9 119 133.8 107.5 113.5 134.4 126.8 135.6 139.9 129.8 131 153.1 134.1 144.1 155.9 123.3 128.1 144.3 153 149.9 150.9 141 138.9 157.4 142.9 151.7 161 138.5 135.9 151.5 164 159.1 157 142.1 144.8 152.1 154.6 148.7 157.7 146.4 136.5

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 Totale_industrie[t] = + 77.8714285714286 + 20.7964285714286M1[t] + 23.5928571428572M2[t] + 21.1464285714286M3[t] + 21.8M4[t] + 10.2535714285714M5[t] + 9.67857142857144M6[t] + 24.0892857142857M7[t] + 15.2M8[t] + 14.7964285714286M9[t] + 23.3214285714286M10[t] + 2.51785714285714M11[t] + 0.746428571428571t + 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) 77.8714285714286 2.556647 30.4584 0 0 M1 20.7964285714286 3.144915 6.6127 0 0 M2 23.5928571428572 3.142546 7.5076 0 0 M3 21.1464285714286 3.140401 6.7337 0 0 M4 21.8 3.138481 6.946 0 0 M5 10.2535714285714 3.136785 3.2688 0.001667 0.000833 M6 9.67857142857144 3.135315 3.087 0.002884 0.001442 M7 24.0892857142857 3.134071 7.6863 0 0 M8 15.2 3.133052 4.8515 7e-06 4e-06 M9 14.7964285714286 3.13226 4.7239 1.1e-05 6e-06 M10 23.3214285714286 3.131694 7.4469 0 0 M11 2.51785714285714 3.131354 0.8041 0.424035 0.212018 t 0.746428571428571 0.026632 28.0278 0 0 Multiple Linear Regression - Regression Statistics Multiple R 0.962618494753033 R-squared 0.926634366440596 Adjusted R-squared 0.914234541050274 F-TEST (value) 74.7296302384906 F-TEST (DF numerator) 12 F-TEST (DF denominator) 71 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5.85801485387064 Sum Squared Residuals 2436.46000000000 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 105.7 99.4142857142856 6.28571428571442 2 109.5 102.957142857143 6.54285714285714 3 105.3 101.257142857143 4.04285714285703 4 102.8 102.657142857143 0.142857142857155 5 100.6 91.8571428571428 8.74285714285715 6 97.6 92.0285714285714 5.57142857142857 7 110.3 107.185714285714 3.11428571428571 8 107.2 99.0428571428571 8.15714285714286 9 107.2 99.3857142857142 7.81428571428575 10 108.1 108.657142857143 -0.557142857142879 11 97.1 88.6 8.5 12 92.2 86.8285714285714 5.37142857142857 13 112.2 108.371428571429 3.82857142857139 14 111.6 111.914285714286 -0.314285714285722 15 115.7 110.214285714286 5.4857142857143 16 111.3 111.614285714286 -0.314285714285721 17 104.2 100.814285714286 3.38571428571429 18 103.2 100.985714285714 2.21428571428571 19 112.7 116.142857142857 -3.44285714285713 20 106.4 108 -1.6 21 102.6 108.342857142857 -5.74285714285715 22 110.6 117.614285714286 -7.01428571428572 23 95.2 97.5571428571428 -2.35714285714285 24 89 95.7857142857143 -6.78571428571429 25 112.5 117.328571428571 -4.82857142857145 26 116.8 120.871428571429 -4.07142857142858 27 107.2 119.171428571429 -11.9714285714286 28 113.6 120.571428571429 -6.97142857142858 29 101.8 109.771428571429 -7.97142857142858 30 102.6 109.942857142857 -7.34285714285716 31 122.7 125.1 -2.39999999999999 32 110.3 116.957142857143 -6.65714285714286 33 110.5 117.3 -6.8 34 121.6 126.571428571429 -4.97142857142857 35 100.3 106.514285714286 -6.21428571428572 36 100.7 104.742857142857 -4.04285714285714 37 123.4 126.285714285714 -2.88571428571431 38 127.1 129.828571428571 -2.72857142857143 39 124.1 128.128571428571 -4.02857142857142 40 131.2 129.528571428571 1.67142857142856 41 111.6 118.728571428571 -7.12857142857144 42 114.2 118.9 -4.7 43 130.1 134.057142857143 -3.95714285714286 44 125.9 125.914285714286 -0.0142857142857129 45 119 126.257142857143 -7.25714285714286 46 133.8 135.528571428571 -1.72857142857141 47 107.5 115.471428571429 -7.97142857142857 48 113.5 113.7 -0.2 49 134.4 135.242857142857 -0.842857142857158 50 126.8 138.785714285714 -11.9857142857143 51 135.6 137.085714285714 -1.48571428571428 52 139.9 138.485714285714 1.41428571428572 53 129.8 127.685714285714 2.11428571428572 54 131 127.857142857143 3.14285714285714 55 153.1 143.014285714286 10.0857142857143 56 134.1 134.871428571429 -0.771428571428574 57 144.1 135.214285714286 8.8857142857143 58 155.9 144.485714285714 11.4142857142857 59 123.3 124.428571428571 -1.12857142857143 60 128.1 122.657142857143 5.44285714285714 61 144.3 144.2 0.099999999999993 62 153 147.742857142857 5.25714285714287 63 149.9 146.042857142857 3.85714285714289 64 150.9 147.442857142857 3.45714285714286 65 141 136.642857142857 4.35714285714286 66 138.9 136.814285714286 2.08571428571429 67 157.4 151.971428571429 5.42857142857144 68 142.9 143.828571428571 -0.928571428571424 69 151.7 144.171428571429 7.5285714285714 70 161 153.442857142857 7.55714285714286 71 138.5 133.385714285714 5.11428571428572 72 135.9 131.614285714286 4.2857142857143 73 151.5 153.157142857143 -1.65714285714287 74 164 156.7 7.3 75 159.1 155 4.10000000000002 76 157 156.4 0.600000000000001 77 142.1 145.6 -3.50000000000001 78 144.8 145.771428571429 -0.971428571428567 79 152.1 160.928571428571 -8.82857142857143 80 154.6 152.785714285714 1.81428571428570 81 148.7 153.128571428571 -4.42857142857145 82 157.7 162.4 -4.7 83 146.4 142.342857142857 4.05714285714286 84 136.5 140.571428571429 -4.07142857142857 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 16 0.129381924582035 0.25876384916407 0.870618075417965 17 0.0779082911911938 0.155816582382388 0.922091708808806 18 0.0341236049135963 0.0682472098271925 0.965876395086404 19 0.0225297148656784 0.0450594297313568 0.977470285134322 20 0.0352918747582251 0.0705837495164501 0.964708125241775 21 0.0938517367181271 0.187703473436254 0.906148263281873 22 0.0548681437996192 0.109736287599238 0.94513185620038 23 0.050564479884561 0.101128959769122 0.949435520115439 24 0.0509743186068195 0.101948637213639 0.94902568139318 25 0.0288395956030650 0.0576791912061299 0.971160404396935 26 0.0175661408399206 0.0351322816798412 0.98243385916008 27 0.0325097551635528 0.0650195103271056 0.967490244836447 28 0.0241862721576914 0.0483725443153827 0.975813727842309 29 0.0191073564727371 0.0382147129454742 0.980892643527263 30 0.0113512889492651 0.0227025778985302 0.988648711050735 31 0.019624121351755 0.03924824270351 0.980375878648245 32 0.0117650998208259 0.0235301996416517 0.988234900179174 33 0.00763399410023541 0.0152679882004708 0.992366005899765 34 0.0143138178613327 0.0286276357226655 0.985686182138667 35 0.00878178029375542 0.0175635605875108 0.991218219706245 36 0.00834169986322896 0.0166833997264579 0.991658300136771 37 0.00954273376684374 0.0190854675336875 0.990457266233156 38 0.0113937261716173 0.0227874523432345 0.988606273828383 39 0.0129941191417824 0.0259882382835648 0.987005880858218 40 0.0446174379516201 0.0892348759032401 0.95538256204838 41 0.0349963290089924 0.0699926580179847 0.965003670991008 42 0.0278169132772006 0.0556338265544011 0.9721830867228 43 0.0250086364547280 0.0500172729094559 0.974991363545272 44 0.0256664894174395 0.051332978834879 0.97433351058256 45 0.0302749348298417 0.0605498696596833 0.969725065170158 46 0.0480517654725506 0.0961035309451013 0.95194823452745 47 0.0698218163576569 0.139643632715314 0.930178183642343 48 0.079442179770766 0.158884359541532 0.920557820229234 49 0.0705524561035782 0.141104912207156 0.929447543896422 50 0.402815068605833 0.805630137211666 0.597184931394167 51 0.53843235154522 0.923135296909561 0.461567648454780 52 0.600349234612882 0.799301530774235 0.399650765387118 53 0.613952779815127 0.772094440369746 0.386047220184873 54 0.624306836799287 0.751386326401425 0.375693163200713 55 0.776391904442298 0.447216191115404 0.223608095557702 56 0.77798692132923 0.444026157341539 0.222013078670770 57 0.809421603030137 0.381156793939725 0.190578396969863 58 0.863169250658183 0.273661498683635 0.136830749341817 59 0.942969097338568 0.114061805322863 0.0570309026614317 60 0.920540781133704 0.158918437732592 0.079459218866296 61 0.883601794372048 0.232796411255904 0.116398205627952 62 0.89715417800322 0.205691643993561 0.102845821996781 63 0.88957673241378 0.220846535172441 0.110423267586221 64 0.844762196612402 0.310475606775195 0.155237803387598 65 0.75900699032729 0.481986019345419 0.240993009672710 66 0.672189582299946 0.655620835400108 0.327810417700054 67 0.652776973911379 0.694446052177243 0.347223026088621 68 0.770066963887418 0.459866072225164 0.229933036112582 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 14 0.264150943396226 NOK 10% type I error level 25 0.471698113207547 NOK

Charts produced by software:

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Parameters (Session):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;

Parameters (R input):

par1 = 1 ; par2 = Include Monthly Dummies ; par3 = 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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