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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: Fri, 20 Nov 2009 01:50:21 -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/20/t1258707209wb0fv06zi550fe5.htm/, Retrieved Fri, 20 Nov 2009 09:53:41 +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/20/t1258707209wb0fv06zi550fe5.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 «
8.2 267722 8 266003 7.9 262971 7.6 265521 7.6 264676 8.3 270223 8.4 269508 8.4 268457 8.4 265814 8.4 266680 8.6 263018 8.9 269285 8.8 269829 8.3 270911 7.5 266844 7.2 271244 7.4 269907 8.8 271296 9.3 270157 9.3 271322 8.7 267179 8.2 264101 8.3 265518 8.5 269419 8.6 268714 8.5 272482 8.2 268351 8.1 268175 7.9 270674 8.6 272764 8.7 272599 8.7 270333 8.5 270846 8.4 270491 8.5 269160 8.7 274027 8.7 273784 8.6 276663 8.5 274525 8.3 271344 8 271115 8.2 270798 8.1 273911 8.1 273985 8 271917 7.9 273338 7.9 270601 8 273547 8 275363 7.9 281229 8 277793 7.7 279913 7.2 282500 7.5 280041 7.3 282166 7 290304 7 283519 7 287816 7.2 285226 7.3 287595
 
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 time4 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135


Multiple Linear Regression - Estimated Regression Equation
wkh[t] = + 23.7242314611506 -5.62069109049765e-05los[t] -0.0275271564433608M1[t] -0.0940245016618537M2[t] -0.522924687831299M3[t] -0.698702671431273M4[t] -0.82863197409711M5[t] -0.0983733354658894M6[t] + 0.0178126737747341M7[t] + 0.0259354497915659M8[t] -0.324101697078169M9[t] -0.428680101825853M10[t] -0.408762127383254M11[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)23.72423146115062.4506199.680900
los-5.62069109049765e-059e-06-6.320100
M1-0.02752715644336080.263103-0.10460.9171190.458559
M2-0.09402450166185370.261309-0.35980.7205920.360296
M3-0.5229246878312990.26434-1.97820.0537810.026891
M4-0.6987026714312730.262932-2.65730.0107270.005363
M5-0.828631974097110.262406-3.15780.0027760.001388
M6-0.09837333546588940.26151-0.37620.708480.35424
M70.01781267377473410.2612310.06820.9459260.472963
M80.02593544979156590.2610480.09940.9212820.460641
M9-0.3241016970781690.262334-1.23550.2228010.111401
M10-0.4286801018258530.261839-1.63720.108270.054135
M11-0.4087621273832540.263543-1.5510.1276040.063802


Multiple Linear Regression - Regression Statistics
Multiple R0.747816602639393
R-squared0.559229671183124
Adjusted R-squared0.446692565953283
F-TEST (value)4.96929141762602
F-TEST (DF numerator)12
F-TEST (DF denominator)47
p-value3.04602975735868e-05
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation0.412749828750751
Sum Squared Residuals8.00703379328739


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
18.28.6488777034051-0.448877703405109
288.67900003803224-0.679000038032239
37.98.42051920572668-0.520519205726682
47.68.10141359931902-0.501413599319018
57.68.01897913636789-0.418979136367885
68.38.4374580402092-0.137458040209200
78.48.59383199074688-0.193831990746883
88.48.66102823012485-0.261028230124845
98.48.45954594877696-0.0595459487769628
108.48.306292359185570.0937076408144305
118.68.53204004136220.0679599586378067
128.98.588553458103960.311446541896041
138.88.530449742128290.269550257871709
148.38.40313651931061-0.103136519310613
157.58.2028298397917-0.702829839791707
167.27.77974144820984-0.579741448209837
177.47.72496078542395-0.324960785423953
188.88.377148024808160.422851975191839
199.38.557353705569550.742646294430447
209.38.499995430382090.800004569617913
218.78.382823515391670.317176484608329
228.28.4512499824095-0.251249982409505
238.38.39152276409975-0.0915227640997506
248.58.5810217320427-0.0810217320426921
258.68.593120447787340.00687955221266002
268.58.31483546227890.185164537721105
278.28.118126025057910.0818739749420914
288.17.952240457777210.147759542222789
297.97.681850084759840.218149915240164
308.68.294636279599660.305363720400343
318.78.42009642913960.279903570860399
328.78.555584065267110.14441593473289
338.58.176712773103120.323287226896879
348.48.09208782172670.307912178273296
358.58.186817194583830.313182805416173
368.78.322020286592560.377979713407439
378.78.308151409499110.391848590500891
388.68.079834367785190.520165632214811
398.57.771104557130580.728895442869417
408.37.774120757119340.525879242880661
4187.657062837050740.342937162949258
428.28.40513906643884-0.205139066438841
438.18.34635296203227-0.246352962032272
448.18.35031642664213-0.250316426642135
4588.1165151715239-0.116515171523891
467.97.93206674638024-0.0320667463802358
477.98.10582303596976-0.205823035969755
4888.34899960382695-0.348999603826949
4988.21940069718015-0.219400697180151
507.97.823193612593070.0768063874069349
5187.587420372293120.412579627706881
527.77.29248373757460.407516262425405
537.27.017147156397580.182852843602416
547.57.88561858894414-0.385618588944142
557.37.88236491251169-0.58236491251169
5677.43307584758382-0.433075847583823
5777.46440259120435-0.464402591204354
5877.11830309029799-0.118303090297986
597.27.28379696398447-0.0837969639844735
607.37.55940491943384-0.259404919433839


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
160.6105700810410990.7788598379178030.389429918958901
170.4888527096920110.9777054193840210.511147290307989
180.5175609056910020.9648781886179960.482439094308998
190.8067555604828810.3864888790342370.193244439517119
200.9468830332666640.1062339334666730.0531169667333364
210.925320111374930.1493597772501410.0746798886250707
220.9122384519726450.1755230960547110.0877615480273554
230.888695721419760.2226085571604820.111304278580241
240.8615639641461740.2768720717076530.138436035853826
250.8062245758036170.3875508483927660.193775424196383
260.7633680542583090.4732638914833820.236631945741691
270.828601572266160.342796855467680.17139842773384
280.9035529277132680.1928941445734650.0964470722867324
290.873694143070540.2526117138589190.126305856929459
300.8850981135752070.2298037728495870.114901886424793
310.9176394703156070.1647210593687860.082360529684393
320.8884935378291440.2230129243417120.111506462170856
330.9076088008693990.1847823982612030.0923911991306014
340.8707869440780940.2584261118438130.129213055921907
350.8508906887118850.298218622576230.149109311288115
360.9376450180719440.1247099638561130.0623549819280564
370.9813143828150440.03737123436991290.0186856171849565
380.9895506031408140.02089879371837190.0104493968591860
390.9912148852570980.01757022948580410.00878511474290203
400.983539431708690.032921136582620.01646056829131
410.963329906982070.07334018603586020.0366700930179301
420.9292974721761280.1414050556477440.070702527823872
430.9426649843134620.1146700313730750.0573350156865377
440.8930105947717310.2139788104565390.106989405228269


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level00OK
5% type I error level40.137931034482759NOK
10% type I error level50.172413793103448NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/10a4ba1258707016.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/10a4ba1258707016.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/143521258707016.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/143521258707016.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/2xh171258707016.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/2xh171258707016.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/3g49y1258707016.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/3g49y1258707016.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/4wr9n1258707016.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/4wr9n1258707016.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/50gje1258707016.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/50gje1258707016.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/6z3wq1258707016.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/6z3wq1258707016.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/7yvvo1258707016.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/8ryla1258707016.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/8ryla1258707016.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/9kt3p1258707016.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258707209wb0fv06zi550fe5/9kt3p1258707016.ps (open in new window)


 
Parameters (Session):
par1 = 1 ; par2 = Include Monthly Dummies ; par3 = 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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