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Paper Multiple Regression T2

*Unverified author*
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 06:50:10 -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/t1230126789ezntyi5g7kefpag.htm/, Retrieved Wed, 24 Dec 2008 14:53:19 +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/t1230126789ezntyi5g7kefpag.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:
» Textbox « » Textfile « » CSV «
103,3 107,9 101 94,6 94,2 92,3 107,1 102,6 103,1 104,1 92,7 87 109,3 113,9 103,3 100,8 97,4 98,9 110,8 103,5 99,8 104,9 95,2 85,7 110 113,7 101,1 103,6 96,2 98,3 119,7 109,4 103,5 118,2 98,7 96,8 121,8 124 119,6 122,5 109,7 111,6 131,2 124,4 116,9 131,8 107,4 111 134 126,2 131,2 130,1 123,1 126,3 148,6 130,1 142,3 154,4 121,6 124,8 143,6 146,9 144,6 137,1 134,7 130,8 153,5 137,6 146,5 156,7 137,6 131,4 147,4 158,5 151,5 142,5 131,3 133,4 136,9 143,2 136,4 145,9 138,8 122,9
 
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
Bewerkende_industrie[t] = + 75.479761904762 + 23.2561259920636M1[t] + 25.6679067460318M2[t] + 19.4368303571428M3[t] + 15.7343253968254M4[t] + 8.67467757936508M5[t] + 8.70074404761906M6[t] + 24.6125248015873M7[t] + 15.7814484126984M8[t] + 14.7646577380953M9[t] + 23.7192956349206M10[t] + 5.31679067460318M11[t] + 0.688219246031746t + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)75.4797619047622.81254226.836800
M123.25612599206363.4596896.72200
M225.66790674603183.4570837.424700
M319.43683035714283.4547245.626200
M415.73432539682543.4526114.55722.1e-051.1e-05
M58.674677579365083.4507462.51390.0142110.007106
M68.700744047619063.4491292.52260.0138920.006946
M724.61252480158733.447767.138700
M815.78144841269843.446644.57881.9e-051e-05
M914.76465773809533.4457684.28495.6e-052.8e-05
M1023.71929563492063.4451456.884800
M115.316790674603183.4447711.54340.127170.063585
t0.6882192460317460.02929723.490900


Multiple Linear Regression - Regression Statistics
Multiple R0.949403561963653
R-squared0.901367123469272
Adjusted R-squared0.884696778140135
F-TEST (value)54.070093070827
F-TEST (DF numerator)12
F-TEST (DF denominator)71
p-value0
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation6.44434379787771
Sum Squared Residuals2948.59925595239


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
1103.399.42410714285673.87589285714327
2107.9102.5241071428575.37589285714286
310196.98125000000014.01874999999989
494.693.96696428571430.633035714285726
594.287.59553571428576.6044642857143
692.388.30982142857143.99017857142856
7107.1104.9098214285712.19017857142855
8102.696.76696428571435.8330357142857
9103.196.43839285714286.66160714285715
10104.1106.08125-1.98125000000005
1192.788.36696428571434.33303571428572
128783.73839285714293.26160714285713
13109.3107.6827380952381.61726190476183
14113.9110.7827380952383.11726190476189
15103.3105.239880952381-1.93988095238094
16100.8102.225595238095-1.42559523809525
1797.495.85416666666671.54583333333333
1898.996.56845238095242.33154761904761
19110.8113.168452380952-2.36845238095239
20103.5105.025595238095-1.52559523809525
2199.8104.697023809524-4.89702380952383
22104.9114.339880952381-9.43988095238096
2395.296.6255952380952-1.42559523809524
2485.791.9970238095238-6.29702380952381
25110115.941369047619-5.94136904761912
26113.7119.041369047619-5.34136904761906
27101.1113.498511904762-12.3985119047619
28103.6110.484226190476-6.8842261904762
2996.2104.112797619048-7.91279761904763
3098.3104.827083333333-6.52708333333335
31119.7121.427083333333-1.72708333333333
32109.4113.284226190476-3.88422619047619
33103.5112.955654761905-9.45565476190477
34118.2122.598511904762-4.39851190476191
3598.7104.884226190476-6.18422619047619
3696.8100.255654761905-3.45565476190477
37121.8124.2-2.40000000000007
38124127.3-3.30000000000001
39119.6121.757142857143-2.15714285714285
40122.5118.7428571428573.75714285714286
41109.7112.371428571429-2.67142857142858
42111.6113.085714285714-1.48571428571430
43131.2129.6857142857141.51428571428571
44124.4121.5428571428572.85714285714286
45116.9121.214285714286-4.31428571428571
46131.8130.8571428571430.942857142857157
47107.4113.142857142857-5.74285714285714
48111108.5142857142862.48571428571428
49134132.4586309523811.54136904761898
50126.2135.558630952381-9.35863095238095
51131.2130.0157738095241.1842261904762
52130.1127.0014880952383.09851190476191
53123.1120.6300595238102.46994047619046
54126.3121.3443452380954.95565476190476
55148.6137.94434523809510.6556547619048
56130.1129.8014880952380.298511904761902
57142.3129.47291666666712.8270833333334
58154.4139.11577380952415.2842261904762
59121.6121.4014880952380.198511904761902
60124.8116.7729166666678.02708333333334
61143.6140.7172619047622.88273809523803
62146.9143.8172619047623.0827380952381
63144.6138.2744047619056.32559523809526
64137.1135.2601190476191.83988095238096
65134.7128.8886904761905.81130952380952
66130.8129.6029761904761.19702380952384
67153.5146.2029761904767.29702380952382
68137.6138.060119047619-0.460119047619040
69146.5137.7315476190488.76845238095238
70156.7147.3744047619059.32559523809524
71137.6129.6601190476197.93988095238095
72131.4125.0315476190486.3684523809524
73147.4148.975892857143-1.57589285714291
74158.5152.0758928571436.42410714285715
75151.5146.5330357142864.96696428571432
76142.5143.51875-1.01874999999999
77131.3137.147321428571-5.84732142857141
78133.4137.861607142857-4.46160714285713
79136.9154.461607142857-17.5616071428571
80143.2146.31875-3.11875
81136.4145.990178571429-9.59017857142856
82145.9155.633035714286-9.7330357142857
83138.8137.918750.881250000000013
84122.9133.290178571429-10.3901785714286


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
160.01052477910845320.02104955821690640.989475220891547
170.002760823624833440.005521647249666880.997239176375167
180.0007410448508681270.001482089701736250.999258955149132
190.0001581416197113020.0003162832394226050.999841858380289
200.0001705524003474070.0003411048006948140.999829447599653
210.001269197573543930.002538395147087870.998730802426456
220.0005655277499319050.001131055499863810.999434472250068
230.0001710282286102210.0003420564572204420.99982897177139
240.0001219827944857030.0002439655889714070.999878017205514
253.67468016580961e-057.34936033161921e-050.999963253198342
261.13823226302880e-052.27646452605759e-050.99998861767737
271.82508193296501e-053.65016386593001e-050.99998174918067
289.18659193965268e-061.83731838793054e-050.99999081340806
295.19754627772188e-061.03950925554438e-050.999994802453722
301.71351594730862e-063.42703189461724e-060.999998286484053
318.67128546842385e-061.73425709368477e-050.999991328714532
323.87980190056814e-067.75960380113628e-060.9999961201981
332.32962775626846e-064.65925551253691e-060.999997670372244
343.17709484388431e-056.35418968776862e-050.999968229051561
351.59841952233015e-053.19683904466029e-050.999984015804777
362.01415295323012e-054.02830590646024e-050.999979858470468
374.33214603274093e-058.66429206548186e-050.999956678539673
383.51821014126516e-057.03642028253032e-050.999964817898587
390.0001661394456799480.0003322788913598960.99983386055432
400.001489398286547090.002978796573094170.998510601713453
410.001087892900381080.002175785800762160.998912107099619
420.0008433895953400730.001686779190680150.99915661040466
430.000950722829496950.00190144565899390.999049277170503
440.001003578100471840.002007156200943680.998996421899528
450.001153466482773940.002306932965547880.998846533517226
460.002811444309793090.005622888619586180.997188555690207
470.004664429371235810.009328858742471630.995335570628764
480.006125808336425180.01225161667285040.993874191663575
490.005799158029393220.01159831605878640.994200841970607
500.03324898923667820.06649797847335630.966751010763322
510.07164828127670820.1432965625534160.928351718723292
520.0800164915548230.1600329831096460.919983508445177
530.08540180573668680.1708036114733740.914598194263313
540.08330292627184610.1666058525436920.916697073728154
550.1178179683428070.2356359366856140.882182031657193
560.1166221557706960.2332443115413920.883377844229304
570.1798380647206840.3596761294413680.820161935279316
580.2777186381702930.5554372763405860.722281361829707
590.4729055960732350.945811192146470.527094403926765
600.4255395329488170.8510790658976330.574460467051183
610.3585258024662660.7170516049325330.641474197533734
620.470274594765790.940549189531580.52972540523421
630.5168017527510020.9663964944979960.483198247248998
640.543269184767360.913461630465280.45673081523264
650.4301261269767690.8602522539535390.569873873023231
660.4152809465803520.8305618931607050.584719053419648
670.5097364079457240.9805271841085520.490263592054276
680.7001450762146440.5997098475707120.299854923785356


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level310.584905660377358NOK
5% type I error level340.641509433962264NOK
10% type I error level350.660377358490566NOK
 
Charts produced by software:
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http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Dec/24/t1230126789ezntyi5g7kefpag/11z341230126604.ps (open in new window)


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http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Dec/24/t1230126789ezntyi5g7kefpag/847431230126604.ps (open in new window)


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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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Software written by Ed van Stee & Patrick Wessa


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