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SHW WS7 - Lineaire Trend

*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 07:41:08 -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/t1258555346gkzdesxfd7ffgnx.htm/, Retrieved Wed, 18 Nov 2009 15:42:38 +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/t1258555346gkzdesxfd7ffgnx.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 «
8.6 1.59 8.5 1.26 8.3 1.13 7.8 1.92 7.8 2.61 8 2.26 8.6 2.41 8.9 2.26 8.9 2.03 8.6 2.86 8.3 2.55 8.3 2.27 8.3 2.26 8.4 2.57 8.5 3.07 8.4 2.76 8.6 2.51 8.5 2.87 8.5 3.14 8.5 3.11 8.5 3.16 8.5 2.47 8.5 2.57 8.5 2.89 8.5 2.63 8.5 2.38 8.5 1.69 8.5 1.96 8.6 2.19 8.4 1.87 8.1 1.6 8 1.63 8 1.22 8 1.21 8 1.49 7.9 1.64 7.8 1.66 7.8 1.77 7.9 1.82 8.1 1.78 8 1.28 7.6 1.29 7.3 1.37 7 1.12 6.8 1.51 7 2.24 7.1 2.94 7.2 3.09 7.1 3.46 6.9 3.64 6.7 4.39 6.7 4.15 6.6 5.21 6.9 5.8 7.3 5.91 7.5 5.39 7.3 5.46 7.1 4.72 6.9 3.14 7.1 2.63
 
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] = + 8.98296836521457 -0.0409587337748366X[t] -0.0776609807441892M1[t] -0.087485820924576M2[t] -0.093542457597681M3[t] -0.139681011738336M4[t] -0.0795938383452158M5[t] -0.0872069069017645M6[t] + 0.0255896118794349M7[t] + 0.0680645297493756M8[t] + 0.0170109275557401M9[t] -0.0119947379491530M10[t] -0.0686187279361657M11[t] -0.0300113248845106t + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)8.982968365214570.22903239.221500
X-0.04095873377483660.051288-0.79860.428620.21431
M1-0.07766098074418920.265648-0.29230.7713370.385668
M2-0.0874858209245760.265209-0.32990.7429920.371496
M3-0.0935424575976810.264958-0.3530.7256670.362833
M4-0.1396810117383360.264781-0.52750.6003580.300179
M5-0.07959383834521580.265289-0.30.7655080.382754
M6-0.08720690690176450.265171-0.32890.7437460.371873
M70.02558961187943490.265150.09650.9235350.461767
M80.06806452974937560.2640490.25780.7977320.398866
M90.01701092755574010.2637440.06450.9488530.474427
M10-0.01199473794915300.263628-0.04550.9639070.481953
M11-0.06861872793616570.263256-0.26070.7955230.397761
t-0.03001132488451060.003565-8.419200


Multiple Linear Regression - Regression Statistics
Multiple R0.828915594493535
R-squared0.68710106279457
Adjusted R-squared0.598673102279992
F-TEST (value)7.77017878503821
F-TEST (DF numerator)13
F-TEST (DF denominator)46
p-value7.69727117599928e-08
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation0.416179568730165
Sum Squared Residuals7.9674499377076


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
18.68.8101716728839-0.210171672883898
28.58.78385188996468-0.283851889964681
38.38.7531085637978-0.453108563797793
47.88.6446012850905-0.844601285090507
57.88.64641560729448-0.84641560729448
688.62312677067461-0.623126770674613
78.68.69976815450508-0.0997681545050762
88.98.718375557556730.181624442443269
98.98.64673113924680.253268860753202
108.68.553718399824280.0462816001757201
118.38.47978029242296-0.179780292422955
128.38.52985614093157-0.229856140931564
138.38.42259342264061-0.122593422640613
148.48.370060050105520.029939949894482
158.58.313512721660480.186487278339517
168.48.250060050105520.149939949894483
178.68.290375582057840.309624417942164
188.58.238006044457840.261993955542164
198.58.309732380235320.190267619764682
208.58.3234247352340.176575264766006
218.58.24031187146710.259688128532894
228.58.209556407382340.290443592617661
238.58.118825219133330.381174780866667
248.58.144325827377040.355674172622960
258.58.04730279252980.452697207470202
268.58.01770631090860.482293689091391
278.58.009899875655630.490100124344369
288.57.922691138511260.577308861488741
298.67.943346478251660.656653521748343
308.47.918828879618550.481171120381455
318.18.012672931634440.08732706836556
3288.02390776260662-0.023907762606625
3387.959635916376160.0403640836238379
3487.90102851332450.0989714866754936
3587.802924752996030.197075247003971
367.97.835388345981460.0646116540185419
377.87.726896865677260.0731031343227378
387.87.682555239897130.117444760102867
397.97.644439341650780.255560658349225
408.17.56992781197660.530072188023397
4187.620483027372630.379516972627369
427.67.582449046593820.0175509534061760
437.37.66195754178853-0.361957541788525
4477.68466081821767-0.684660818217665
456.87.58762198496733-0.787621984967333
4677.4987051189223-0.498705118922298
477.17.38339869040839-0.283398690408389
487.27.41586228339382-0.215862283393819
497.17.29303524626843-0.193035246268429
506.97.24582650912406-0.345826509124061
516.77.17903949723532-0.479039497235318
526.77.11271971431611-0.412719714316113
536.67.0993793050234-0.499379305023397
546.97.03758925865518-0.137589258655183
557.37.115868991836640.18413100816336
567.57.149631126384990.350368873615015
577.37.06569908794260.234300912057399
587.17.036991560546580.0630084394534236
596.97.0150710450393-0.115071045039294
607.17.074567402316120.0254325976838834


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
170.6032969000804960.7934061998390080.396703099919504
180.4559993503989630.9119987007979260.544000649601037
190.4115352819883480.8230705639766970.588464718011652
200.4706285042096530.9412570084193060.529371495790347
210.4355873021237840.8711746042475680.564412697876216
220.4023192778241790.8046385556483590.597680722175821
230.2968643726832740.5937287453665490.703135627316726
240.2223690672460970.4447381344921930.777630932753903
250.1521715130525450.3043430261050910.847828486947455
260.1000261142528690.2000522285057380.899973885747131
270.06582656569035870.1316531313807170.934173434309641
280.0417989213192870.0835978426385740.958201078680713
290.02771569917855880.05543139835711760.972284300821441
300.01756995434283890.03513990868567780.98243004565716
310.0345139119247940.0690278238495880.965486088075206
320.05926241472432540.1185248294486510.940737585275675
330.05909896277823590.1181979255564720.940901037221764
340.04065561340833790.08131122681667570.959344386591662
350.02500511726692810.05001023453385610.974994882733072
360.01501168485596680.03002336971193370.984988315144033
370.01201627611473470.02403255222946940.987983723885265
380.01060577777313870.02121155554627750.989394222226861
390.01334728025255920.02669456050511840.98665271974744
400.03937365448649060.07874730897298110.96062634551351
410.3384708933632760.6769417867265510.661529106636724
420.8639983368368920.2720033263262150.136001663163108
430.9759433029310.04811339413799850.0240566970689992


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level00OK
5% type I error level60.222222222222222NOK
10% type I error level120.444444444444444NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258555346gkzdesxfd7ffgnx/10o9ol1258555264.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258555346gkzdesxfd7ffgnx/10o9ol1258555264.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258555346gkzdesxfd7ffgnx/10q441258555264.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258555346gkzdesxfd7ffgnx/10q441258555264.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258555346gkzdesxfd7ffgnx/2jom91258555264.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258555346gkzdesxfd7ffgnx/2jom91258555264.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258555346gkzdesxfd7ffgnx/3n0k51258555264.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/18/t1258555346gkzdesxfd7ffgnx/48ltf1258555264.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/18/t1258555346gkzdesxfd7ffgnx/5inh21258555264.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/18/t1258555346gkzdesxfd7ffgnx/67yt51258555264.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/18/t1258555346gkzdesxfd7ffgnx/7venm1258555264.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/18/t1258555346gkzdesxfd7ffgnx/89coh1258555264.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258555346gkzdesxfd7ffgnx/89coh1258555264.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258555346gkzdesxfd7ffgnx/9tw3w1258555264.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258555346gkzdesxfd7ffgnx/9tw3w1258555264.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 = 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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