multiple regression 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: Wed, 16 Dec 2009 04:03:47 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2009/Dec/16/t1260961661jccysjtvcbf5hua.htm/, Retrieved Wed, 16 Dec 2009 12:07:53 +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/16/t1260961661jccysjtvcbf5hua.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}, }

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Dataseries X:

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101.5 0 99.2 0 107.8 0 92.3 0 99.2 0 101.6 0 87 0 71.4 0 104.7 0 115.1 0 102.5 0 75.3 0 96.7 1 94.6 1 98.6 1 99.5 1 92 1 93.6 1 89.3 1 66.9 1 108.8 1 113.2 1 105.5 1 77.8 1 102.1 1 97 1 95.5 1 99.3 1 86.4 1 92.4 1 85.7 1 61.9 1 104.9 1 107.9 1 95.6 1 79.8 1 94.8 1 93.7 1 108.1 1 96.9 1 88.8 1 106.7 1 86.8 1 69.8 1 110.9 1 105.4 1 99.2 1 84.4 1 87.2 1 91.9 1 97.9 1 94.5 1 85 1 100.3 1 78.7 1 65.8 1 104.8 1 96 1 103.3 1 82.9 1 91.4 1 94.5 1 109.3 1 92.1 1 99.3 1 109.6 1 87.5 1 73.1 1 110.7 1 111.6 1 110.7 1 84 1 101.6 1 102.1 1 113.9 1 99 1 100.4 1 109.5 1 93 1 76.8 1 105.3 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 9 seconds R Server 'Gwilym Jenkins' @ 72.249.127.135 Multiple Linear Regression - Estimated Regression Equation y[t] = + 81.0978124999999 -5.16078124999995x[t] + 16.3589279513888M1[t] + 15.9374317956349M2[t] + 24.1445070684524M3[t] + 15.8372966269841M4[t] + 12.5300861855159M5[t] + 21.3800186011905M6[t] + 6.18709387400795M7[t] -11.3772594246032M8[t] + 26.3012444196429M9[t] + 27.6858494543651M10[t] + 22.1929247271826M11[t] + 0.0929247271825398t + 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) 81.0978124999999 2.446634 33.1467 0 0 x -5.16078124999995 2.000773 -2.5794 0.012098 0.006049 M1 16.3589279513888 2.801308 5.8397 0 0 M2 15.9374317956349 2.799488 5.693 0 0 M3 24.1445070684524 2.798001 8.6292 0 0 M4 15.8372966269841 2.796846 5.6626 0 0 M5 12.5300861855159 2.796024 4.4814 3e-05 1.5e-05 M6 21.3800186011905 2.795535 7.6479 0 0 M7 6.18709387400795 2.79538 2.2133 0.030285 0.015142 M8 -11.3772594246032 2.795558 -4.0698 0.000127 6.3e-05 M9 26.3012444196429 2.79607 9.4065 0 0 M10 27.6858494543651 2.901285 9.5426 0 0 M11 22.1929247271826 2.900803 7.6506 0 0 t 0.0929247271825398 0.030537 3.043 0.003344 0.001672 Multiple Linear Regression - Regression Statistics Multiple R 0.924878815061786 R-squared 0.855400822550093 Adjusted R-squared 0.827344265731454 F-TEST (value) 30.4884461796049 F-TEST (DF numerator) 13 F-TEST (DF denominator) 67 p-value 0 Multiple Linear Regression - Residual Statistics Residual Standard Deviation 5.02406007724201 Sum Squared Residuals 1691.15903720238 Multiple Linear Regression - Actuals, Interpolation, and Residuals Time or Index Actuals Interpolation Forecast Residuals Prediction Error 1 101.5 97.5496651785718 3.95033482142816 2 99.2 97.22109375 1.97890625 3 107.8 105.52109375 2.27890625000002 4 92.3 97.3068080357142 -5.00680803571423 5 99.2 94.0925223214286 5.10747767857144 6 101.6 103.035379464286 -1.43537946428569 7 87 87.9353794642857 -0.935379464285681 8 71.4 70.4639508928571 0.936049107142891 9 104.7 108.235379464286 -3.5353794642857 10 115.1 109.712909226190 5.38709077380959 11 102.5 104.312909226190 -1.81290922619042 12 75.3 82.2129092261904 -6.91290922619044 13 96.7 93.5039806547618 3.19601934523818 14 94.6 93.1754092261905 1.42459077380952 15 98.6 101.475409226190 -2.87540922619048 16 99.5 93.2611235119048 6.23887648809523 17 92 90.046837797619 1.95316220238095 18 93.6 98.9896949404762 -5.38969494047620 19 89.3 83.8896949404762 5.41030505952381 20 66.9 66.4182663690476 0.481733630952386 21 108.8 104.189694940476 4.61030505952381 22 113.2 105.667224702381 7.53277529761904 23 105.5 100.267224702381 5.23277529761904 24 77.8 78.167224702381 -0.367224702380958 25 102.1 94.6190773809523 7.48092261904768 26 97 94.290505952381 2.70949404761905 27 95.5 102.590505952381 -7.09050595238095 28 99.3 94.3762202380952 4.92377976190475 29 86.4 91.1619345238095 -4.76193452380952 30 92.4 100.104791666667 -7.70479166666666 31 85.7 85.0047916666667 0.695208333333334 32 61.9 67.5333630952381 -5.6333630952381 33 104.9 105.304791666667 -0.404791666666663 34 107.9 106.782321428571 1.11767857142857 35 95.6 101.382321428571 -5.78232142857144 36 79.8 79.2823214285714 0.517678571428565 37 94.8 95.7341741071428 -0.934174107142794 38 93.7 95.4056026785714 -1.70560267857143 39 108.1 103.705602678571 4.39439732142857 40 96.9 95.4913169642857 1.40868303571428 41 88.8 92.27703125 -3.47703125000001 42 106.7 101.219888392857 5.48011160714286 43 86.8 86.1198883928571 0.680111607142851 44 69.8 68.6484598214286 1.15154017857142 45 110.9 106.419888392857 4.48011160714286 46 105.4 107.897418154762 -2.49741815476191 47 99.2 102.497418154762 -3.29741815476191 48 84.4 80.3974181547619 4.00258184523809 49 87.2 96.8492708333333 -9.64927083333327 50 91.9 96.520699404762 -4.6206994047619 51 97.9 104.820699404762 -6.9206994047619 52 94.5 96.6064136904762 -2.1064136904762 53 85 93.3921279761905 -8.39212797619048 54 100.3 102.334985119048 -2.03498511904762 55 78.7 87.2349851190476 -8.53498511904762 56 65.8 69.763556547619 -3.96355654761906 57 104.8 107.534985119048 -2.73498511904763 58 96 109.012514880952 -13.0125148809524 59 103.3 103.612514880952 -0.312514880952395 60 82.9 81.5125148809524 1.38748511904762 61 91.4 97.9643675595238 -6.56436755952374 62 94.5 97.6357961309524 -3.13579613095239 63 109.3 105.935796130952 3.36420386904761 64 92.1 97.7215104166667 -5.62151041666668 65 99.3 94.507224702381 4.79277529761904 66 109.6 103.450081845238 6.1499181547619 67 87.5 88.3500818452381 -0.850081845238102 68 73.1 70.8786532738095 2.22134672619046 69 110.7 108.650081845238 2.0499181547619 70 111.6 110.127611607143 1.47238839285712 71 110.7 104.727611607143 5.97238839285714 72 84 82.627611607143 1.37238839285713 73 101.6 99.0794642857142 2.52053571428577 74 102.1 98.7508928571429 3.34910714285713 75 113.9 107.050892857143 6.84910714285714 76 99 98.8366071428571 0.163392857142846 77 100.4 95.6223214285714 4.77767857142857 78 109.5 104.565178571429 4.93482142857142 79 93 89.4651785714286 3.53482142857142 80 76.8 71.99375 4.80624999999999 81 105.3 109.765178571429 -4.46517857142858 Goldfeld-Quandt test for Heteroskedasticity p-values Alternative Hypothesis breakpoint index greater 2-sided less 17 0.584933565008126 0.830132869983748 0.415066434991874 18 0.466688979110695 0.93337795822139 0.533311020889305 19 0.432626591611028 0.865253183222057 0.567373408388972 20 0.304044838479112 0.608089676958223 0.695955161520888 21 0.314714198735596 0.629428397471192 0.685285801264404 22 0.262940899776432 0.525881799552864 0.737059100223568 23 0.247939549239335 0.49587909847867 0.752060450760665 24 0.199337746253199 0.398675492506398 0.800662253746801 25 0.193420874180443 0.386841748360886 0.806579125819557 26 0.155497744350837 0.310995488701674 0.844502255649163 27 0.218522391268997 0.437044782537995 0.781477608731003 28 0.231859487085595 0.46371897417119 0.768140512914405 29 0.284567745427051 0.569135490854102 0.715432254572949 30 0.279313251162901 0.558626502325803 0.720686748837099 31 0.224903134430864 0.449806268861728 0.775096865569136 32 0.205529655740551 0.411059311481103 0.794470344259449 33 0.155483119789080 0.310966239578159 0.84451688021092 34 0.155174949196330 0.310349898392661 0.84482505080367 35 0.146959856118107 0.293919712236214 0.853040143881893 36 0.154300989960944 0.308601979921887 0.845699010039056 37 0.131715025788507 0.263430051577014 0.868284974211493 38 0.0979965278485596 0.195993055697119 0.90200347215144 39 0.217726590875205 0.435453181750410 0.782273409124795 40 0.216576188821209 0.433152377642418 0.783423811178791 41 0.165364338666489 0.330728677332979 0.83463566133351 42 0.342911930323334 0.685823860646667 0.657088069676666 43 0.340815544169974 0.681631088339947 0.659184455830026 44 0.328764348448233 0.657528696896467 0.671235651551767 45 0.552933605313916 0.894132789372168 0.447066394686084 46 0.665676227791506 0.668647544416988 0.334323772208494 47 0.590664463287805 0.818671073424389 0.409335536712195 48 0.725267296223677 0.549465407552647 0.274732703776323 49 0.765725597331916 0.468548805336168 0.234274402668084 50 0.70878796570435 0.582424068591298 0.291212034295649 51 0.703680847613257 0.592638304773486 0.296319152386743 52 0.73731930471502 0.525361390569959 0.262680695284979 53 0.777987635375933 0.444024729248133 0.222012364624067 54 0.719500594005495 0.560998811989009 0.280499405994505 55 0.712323277451198 0.575353445097605 0.287676722548802 56 0.640420041274918 0.719159917450164 0.359579958725082 57 0.582511707241971 0.834976585516057 0.417488292758029 58 0.85175954132453 0.296480917350941 0.148240458675470 59 0.818176148912118 0.363647702175764 0.181823851087882 60 0.759205494604614 0.481589010790771 0.240794505395386 61 0.791866626494876 0.416266747010249 0.208133373505124 62 0.764858828388111 0.470282343223778 0.235141171611889 63 0.68755783097635 0.624884338047301 0.312442169023650 64 0.656569947866097 0.686860104267807 0.343430052133903 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 0 0 OK 10% type I error level 0 0 OK

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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