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*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: Wed, 18 Nov 2009 12:49:17 -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/18/t1258573827ejpjq95eobgh8v6.htm/, Retrieved Wed, 18 Nov 2009 20:50:39 +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/18/t1258573827ejpjq95eobgh8v6.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:
» Textbox « » Textfile « » CSV «
90269 561000 90398 90390 555000 90269 88219 544000 90390 87032 537000 88219 87175 543000 87032 92603 594000 87175 93571 611000 92603 94118 613000 93571 92159 611000 94118 89528 594000 92159 89955 595000 89528 89587 591000 89955 89488 589000 89587 88521 584000 89488 86587 573000 88521 85159 567000 86587 84915 569000 85159 91378 621000 84915 92729 629000 91378 92194 628000 92729 89664 612000 92194 86285 595000 89664 86858 597000 86285 87184 593000 86858 86629 590000 87184 85220 580000 86629 84816 574000 85220 84831 573000 84816 84957 573000 84831 90951 620000 84957 92134 626000 90951 91790 620000 92134 86625 588000 91790 83324 566000 86625 82719 557000 83324 83614 561000 82719 81640 549000 83614 78665 532000 81640 77828 526000 78665 75728 511000 77828 72187 499000 75728 79357 555000 72187 81329 565000 79357 77304 542000 81329 75576 527000 77304 72932 510000 75576 74291 514000 72932 74988 517000 74291 73302 508000 74988 70483 493000 73302 69848 490000 70483 6646 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 Inputview raw input (R code)
Raw Outputview raw output of R engine
Computing time3 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135


Multiple Linear Regression - Estimated Regression Equation
Y[t] = + 20602.0406882828 + 0.105567859493241X[t] + 0.102513766807805Y1[t] -175.182325785885t + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)20602.04068828283776.4301485.45541e-061e-06
X0.1055678594932410.00714214.781500
Y10.1025137668078050.05891.74050.087370.043685
t-175.18232578588518.383412-9.529400


Multiple Linear Regression - Regression Statistics
Multiple R0.989650661324071
R-squared0.979408431459172
Adjusted R-squared0.978285254993309
F-TEST (value)871.998711891046
F-TEST (DF numerator)3
F-TEST (DF denominator)55
p-value0
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation1134.84075998331
Sum Squared Residuals70832495.2785724


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
19026988917.46703009721351.53296990283
29039088095.65327143362294.34672856639
38821986771.62865700581447.37134299418
48703285634.91392702751397.08607297249
58717585971.45491700021203.5450829998
69260391194.89289402311408.10710597687
79357193370.8089058551200.191094144897
89411893505.9956253257612.004374674345
99215993175.7526109972-1016.75261099716
108952891005.0922046497-1477.09220464968
118995590665.7640178857-710.764017885705
128958790112.0836325538-525.083632553786
138948889688.0405215962-200.040521596147
148852188974.87003543-453.870035430083
158658787539.3104427154-952.310442715398
168515986532.4593349638-1373.45933496377
178491586422.0230691628-1507.02306916282
189137891711.3560779244-333.356077924373
199272993043.2631029633-314.263102963257
209219492901.0090166415-707.009016641473
218966490981.8960737216-1317.89607372155
228628588752.7003065268-2467.70030652682
238685888442.2596816838-1584.25968168385
248718487903.5463063059-719.546306305871
258662987445.0798900196-816.079890019607
268522086157.323828723-937.323828722979
278481685204.2924485455-388.29244854545
288483184882.126701476-51.1267014759702
298495784708.4820821922248.517917807798
309095189507.90588720641443.09411279357
319213490580.5982366261553.40176337403
329179089893.28254001431896.71745998573
338662586304.6639746628320.336025337213
348332483277.505134463346.4948655367136
358271981813.8141290057905.185870994331
368361481998.8824122741615.11758772597
378164080648.6355938622991.364406137768
387866578476.4374810126188.562518987359
397782877362.8695420141465.130457985909
407572875518.3653010115209.634698988544
417218773861.0897510103-1674.08975101029
427935779234.7063085795122.293691420530
438132980850.226285738478.773714262047
447730478449.1403397525-1145.14033975251
457557676277.8222101666-701.822210166596
467293274130.8424839517-1198.84248395173
477429174106.885196699184.11480330103
487498874387.7226584846600.277341515387
497330273333.8816927246-31.8816927245972
507048371402.3432637021-919.343263702137
516984870621.4710508053-773.471050805328
526646668164.2674337384-1698.26743373842
536761068592.4942840477-982.494284047712
547509173812.9806821521278.01931784799
557620775038.11100281481168.88899718524
567345473288.2482888945165.751711105476
577200871564.0312491679443.968750832141
587136270818.3425786049543.657421395076
597425072054.88639236662195.11360763343


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
70.2272529883576260.4545059767152520.772747011642374
80.1680107054040710.3360214108081430.831989294595929
90.2182152113148090.4364304226296170.781784788685191
100.2213574399130980.4427148798261950.778642560086902
110.1378878661972770.2757757323945550.862112133802723
120.1191958964304110.2383917928608230.880804103569589
130.1545436419404110.3090872838808220.845456358059589
140.1420019115290210.2840038230580410.85799808847098
150.09963156327424450.1992631265484890.900368436725756
160.07907635977334330.1581527195466870.920923640226657
170.05965024931168030.1193004986233610.94034975068832
180.05137919471172260.1027583894234450.948620805288277
190.2113453138488670.4226906276977340.788654686151133
200.2937889128046920.5875778256093840.706211087195308
210.2909806903514180.5819613807028360.709019309648582
220.4117408656696360.8234817313392720.588259134330364
230.4510655193773760.9021310387547520.548934480622624
240.5450882325475180.9098235349049640.454911767452482
250.6235458831902620.7529082336194760.376454116809738
260.668501881981240.662996236037520.33149811801876
270.7043758691630730.5912482616738530.295624130836927
280.7392024488584290.5215951022831420.260797551141571
290.7749138100839930.4501723798320140.225086189916007
300.916960001516470.1660799969670610.0830399984835305
310.9782196658258250.04356066834835070.0217803341741754
320.9887223982923020.02255520341539670.0112776017076983
330.9853875917792080.02922481644158490.0146124082207925
340.9792790032629740.04144199347405160.0207209967370258
350.9722248195752150.05555036084956910.0277751804247846
360.979460226053030.04107954789394070.0205397739469704
370.98002969979480.03994060041039910.0199703002051996
380.977649215856460.04470156828708130.0223507841435407
390.9876875447279290.02462491054414250.0123124552720713
400.9992817076435540.001436584712891440.00071829235644572
410.9992500367222720.001499926555455390.000749963277727695
420.9987924971361820.002415005727635950.00120750286381797
430.9972766833913340.005446633217331260.00272331660866563
440.9981296531217680.003740693756463990.00187034687823200
450.9971306477386770.005738704522646360.00286935226132318
460.99681909645710.006361807085801290.00318090354290064
470.9923154491108530.01536910177829470.00768455088914737
480.991205500175620.01758899964876020.00879449982438008
490.9922666794722510.01546664105549820.00773332052774908
500.9884882444475970.02302351110480520.0115117555524026
510.9836785700121430.03264285997571450.0163214299878572
520.9727979689912570.05440406201748650.0272020310087432


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level70.152173913043478NOK
5% type I error level200.434782608695652NOK
10% type I error level220.478260869565217NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/1053w71258573753.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/1053w71258573753.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/1xim81258573753.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/1xim81258573753.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/2hoqy1258573753.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/2hoqy1258573753.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/31efw1258573753.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/31efw1258573753.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/48kzt1258573753.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/5iez41258573753.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/68vgs1258573753.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/7m6zh1258573753.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/7m6zh1258573753.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/8w1mm1258573753.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/8w1mm1258573753.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/9wyrq1258573753.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258573827ejpjq95eobgh8v6/9wyrq1258573753.ps (open in new window)


 
Parameters (Session):
par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = Linear Trend ;
 
Parameters (R input):
par1 = 1 ; par2 = Do not include Seasonal 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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