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WS 7: Multiple Regression 2

*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 03:32:04 -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/t1258713153oaraz214qb980su.htm/, Retrieved Fri, 20 Nov 2009 11:32:45 +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/t1258713153oaraz214qb980su.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 «
79.8 109.87 83.4 95.74 113.6 123.06 112.9 123.39 104 120.28 109.9 115.33 99 110.4 106.3 114.49 128.9 132.03 111.1 123.16 102.9 118.82 130 128.32 87 112.24 87.5 104.53 117.6 132.57 103.4 122.52 110.8 131.8 112.6 124.55 102.5 120.96 112.4 122.6 135.6 145.52 105.1 118.57 127.7 134.25 137 136.7 91 121.37 90.5 111.63 122.4 134.42 123.3 137.65 124.3 137.86 120 119.77 118.1 130.69 119 128.28 142.7 147.45 123.6 128.42 129.6 136.9 151.6 143.95 110.4 135.64 99.2 122.48 130.5 136.83 136.2 153.04 129.7 142.71 128 123.46 121.6 144.37 135.8 146.15 143.8 147.61 147.5 158.51 136.2 147.4 156.6 165.05 123.3 154.64 104.5 126.2 139.8 157.36 136.5 154.15 112.1 123.21 118.5 113.07 94.4 110.45 102.3 113.57 111.4 122.44 99.2 114.93 87.8 111.85 115.8 126.04
 
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
Investgoed[t] = + 2.26371392062023 + 0.970890252831042Uitvoer[t] -27.0259952474602M1[t] -18.0960455070252M2[t] -10.3481032400426M3[t] -13.9322023492286M4[t] -13.4373301649736M5[t] -0.228784107182265M6[t] -14.9263279733971M7[t] -8.48247154905135M8[t] -4.74716796666329M9[t] -9.9347654845262M10[t] -11.4879879092140M11[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)2.263713920620237.5520770.29970.7656920.382846
Uitvoer0.9708902528310420.05159218.818600
M1-27.02599524746023.190163-8.471700
M2-18.09604550702523.432269-5.27233e-062e-06
M3-10.34810324004263.120221-3.31650.0017640.000882
M4-13.93220234922863.117429-4.46914.9e-052.5e-05
M5-13.43733016497363.149149-4.2679.5e-054.8e-05
M6-0.2287841071822653.295157-0.06940.9449420.472471
M7-14.92632797339713.232022-4.61833e-051.5e-05
M8-8.482471549051353.210537-2.64210.0111560.005578
M9-4.747167966663293.116377-1.52330.1343850.067193
M10-9.93476548452623.169961-3.1340.0029680.001484
M11-11.48798790921403.159798-3.63570.0006860.000343


Multiple Linear Regression - Regression Statistics
Multiple R0.969120209849607
R-squared0.939193981138947
Adjusted R-squared0.923669040153146
F-TEST (value)60.4958165057075
F-TEST (DF numerator)12
F-TEST (DF denominator)47
p-value0
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation4.92674642846338
Sum Squared Residuals1140.82302740770


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
179.881.9094307517059-2.10943075170587
283.477.12070121963886.27929878036116
3113.6111.3933651939662.20663480603439
4112.9108.1296598682144.77034013178617
5104105.605063366164-1.60506336616429
6109.9114.007702672442-4.10770267244194
79994.523669859774.47633014022993
8106.3104.9384674181951.36153258180521
9128.9125.7031860352393.19681396476068
10111.1111.903791974765-0.80379197476507
11102.9106.136905852791-3.23690585279057
12130126.8483511638993.15164883610054
138784.2104406509162.78955934908395
1487.585.65482654202371.84517345797628
15117.6120.626531498389-3.0265314983888
16103.4107.284985348251-3.8849853482508
17110.8116.789719078778-5.9897190787779
18112.6122.959310803544-10.3593108035442
19102.5104.776270929666-2.27627092966585
20112.4112.812387368655-0.412387368654536
21135.6138.80049554593-3.20049554593010
22105.1107.447405714271-2.34740571427059
23127.7121.1177424539746.58225754602643
24137134.9844114826242.01558851737642
259193.0746686592635-2.07466865926346
2690.592.5481473371241-2.04814733712411
27122.4122.422678466126-0.0226784661262075
28123.3121.9745548735841.32544512641552
29124.3122.6733140109341.62668598906599
30120118.3184553950121.68154460498822
31118.1114.2230330897123.87696691028810
32119118.3270440047350.672955995265134
33142.7140.6743137338942.02568626610601
34123.6117.0106747046566.58932529534365
35129.6123.6906016239765.90939837602416
36151.6142.0233658156499.57663418435136
37110.4106.9292725671623.47072743283759
3899.2103.082306580341-3.88230658034092
39130.5124.7625239754495.73747602455095
40136.2136.916555864654-0.716555864654212
41129.7127.3821317371652.31786826283543
42128121.9010404279586.09895957204168
43121.6127.504811748441-5.90481174844057
44135.8135.6768528228260.123147177174423
45143.8140.8296561743472.97034382565304
46147.5146.2247624123421.27523758765761
47136.2133.8849492787022.31505072129821
48156.6162.509150150384-5.90915015038364
49123.3125.376187370952-2.07618737095221
50104.5106.694018320872-2.1940183208724
51139.8144.694900866070-4.89490086607033
52136.5137.994244045297-1.49424404529667
53112.1108.4497718069593.65022819304077
54118.5111.8134907010446.68650929895621
5594.494.5722143724116-0.172214372411607
56102.3104.045248385590-1.74524838559023
57111.4116.392348510590-4.99234851058963
5899.2103.913365193966-4.7133651939656
5987.899.3698007905582-11.5698007905582
60115.8124.634721387445-8.83472138744469


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
160.425679275372330.851358550744660.57432072462767
170.2782051418201660.5564102836403330.721794858179833
180.2507011336688460.5014022673376910.749298866331154
190.1543075490998680.3086150981997370.845692450900132
200.09200354258702730.1840070851740550.907996457412973
210.04977211422702820.09954422845405650.950227885772972
220.03388400997234980.06776801994469960.96611599002765
230.2820279845853770.5640559691707550.717972015414623
240.2072924833079930.4145849666159850.792707516692007
250.1393214621063800.2786429242127600.86067853789362
260.1044065984542360.2088131969084720.895593401545764
270.06723098832767810.1344619766553560.932769011672322
280.05050265782884770.1010053156576950.949497342171152
290.05574844223013610.1114968844602720.944251557769864
300.08156843135463970.1631368627092790.91843156864536
310.07232234218642740.1446446843728550.927677657813573
320.04469218709039540.08938437418079080.955307812909605
330.02828042142364560.05656084284729120.971719578576354
340.04895140566106330.09790281132212650.951048594338937
350.06250160579232790.1250032115846560.937498394207672
360.3478753994331270.6957507988662530.652124600566873
370.3405464947805740.6810929895611470.659453505219426
380.2987969018214190.5975938036428390.70120309817858
390.57415960426030.85168079147940.4258403957397
400.4587893121923270.9175786243846550.541210687807673
410.385085294235090.770170588470180.61491470576491
420.3532110782014320.7064221564028650.646788921798568
430.6341708562921490.7316582874157020.365829143707851
440.5400251370826650.919949725834670.459974862917335


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level00OK
5% type I error level00OK
10% type I error level50.172413793103448NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/10rxqy1258713120.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/10rxqy1258713120.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/1b9dm1258713120.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/1b9dm1258713120.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/2z7ao1258713120.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/2z7ao1258713120.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/32vsl1258713120.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/49axs1258713120.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/53vh11258713120.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/6povh1258713120.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/7qqzi1258713120.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/7qqzi1258713120.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/8ccgq1258713120.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/8ccgq1258713120.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/93e1c1258713120.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258713153oaraz214qb980su/93e1c1258713120.ps (open in new window)


 
Parameters (Session):
par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No 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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Software written by Ed van Stee & Patrick Wessa


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