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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, 29 Dec 2010 14:23:28 +0000
 
Cite this page as follows:
Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr.htm/, Retrieved Wed, 29 Dec 2010 15:21:09 +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/2010/Dec/29/t1293632469qhy3nb7ga9okadr.htm/},
    year = {2010},
}
@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 = {2010},
    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 «
597141 25 593408 24 590072 21 579799 22 574205 20 572775 24 572942 24 619567 24 625809 24 619916 28 587625 27 565742 18 557274 25 560576 27 548854 25 531673 28 525919 28 511038 27 498662 25 555362 24 564591 24 541657 25 527070 18 509846 22 514258 20 516922 23 507561 23 492622 19 490243 17 469357 15 477580 13 528379 15 533590 17 517945 9 506174 4 501866 1 516141 6 528222 2 532638 2 536322 4 536535 7 523597 8 536214 9 586570 15 596594 15 580523 14 564478 16 557560 11 575093 11 580112 11 574761 13 563250 18 551531 13 537034 17 544686 19 600991 22 604378 22 586111 24 563668 26 548604 24
 
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 time5 seconds
R Server'Sir Ronald Aylmer Fisher' @ 193.190.124.24


Multiple Linear Regression - Estimated Regression Equation
Werkloos[t] = + 515846.69583124 + 1373.48053741838cv[t] + 12236.1428176794M1[t] + 16102.7428176796M2[t] + 11856.0311401306M3[t] -110.841612255150M4[t] -3509.26496735309M5[t] -18083.8416122552M6[t] -14552.5455047715M7[t] + 34857.4934203918M8[t] + 41126.7012054244M9[t] + 25914.0934203918M10[t] + 8958.95838774484M11[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)515846.6958312417403.23481329.640900
cv1373.48053741838590.6498442.32540.0244190.01221
M112236.142817679421124.1356670.57920.5651870.282594
M216102.742817679621124.1356670.76230.4496950.224848
M311856.031140130621105.3000540.56180.5769510.288476
M4-110.84161225515021158.459192-0.00520.9958420.497921
M5-3509.2649673530921110.919452-0.16620.8686890.434345
M6-18083.841612255221158.459192-0.85470.3970610.19853
M7-14552.545504771521148.89381-0.68810.4947720.247386
M834857.493420391821273.8922491.63850.1079930.053997
M941126.701205424421306.6646941.93020.0596250.029813
M1025914.093420391821273.8922491.21810.2292590.11463
M118958.9583877448421158.4591920.42340.6739190.336959


Multiple Linear Regression - Regression Statistics
Multiple R0.584188194038514
R-squared0.34127584605398
Adjusted R-squared0.173090955684783
F-TEST (value)2.02917066631144
F-TEST (DF numerator)12
F-TEST (DF denominator)47
p-value0.0424051193047795
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation33336.9386056947
Sum Squared Residuals52233519353.1933


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
1597141562419.8520843834721.1479156195
2593408564912.97154696128495.0284530387
3590072556545.81825715733526.1817428428
4579799545952.4260421933846.5739578102
5574205539807.04161225534397.9583877449
6572775530726.38711702742048.6128829734
7572942534257.6832245138684.3167754897
8619567583667.72214967435899.2778503265
9625809589936.92993470635872.0700652938
10619916580218.24429934739697.755700653
11587625561889.62872928225735.3712707182
12565742540569.34550477125172.6544952285
13557274562419.85208438-5145.85208437949
14560576569033.413159216-8457.41315921646
15548854562039.740406831-13185.7404068307
16531673554193.3092667-22520.3092667001
17525919550794.885911602-24875.8859116022
18511038534846.828729282-23808.8287292817
19498662535631.163761929-36969.1637619287
20555362583667.722149674-28305.7221496735
21564591589936.929934706-25345.9299347062
22541657576097.802687092-34440.8026870919
23527070549528.303892516-22458.3038925163
24509846546063.267654445-36217.267654445
25514258555552.449397288-41294.4493972876
26516922563539.491009543-46617.4910095429
27507561559292.779331994-51731.7793319939
28492622541831.984429935-49209.9844299347
29490243535686.6-45443.6
30469357518365.062280261-49008.0622802612
31477580519149.397312908-41569.3973129081
32528379571306.397312908-42927.3973129081
33533590580322.566172778-46732.5661727775
34517945554122.114088398-36177.1140883978
35506174530299.576368659-24125.576368659
36501866517220.176368659-15354.176368659
37516141536323.72187343-20182.7218734303
38528222534696.399723757-6474.39972375694
39532638530449.6880462082188.31195379204
40536322521229.77636865915092.223631341
41536535521951.79462581614583.2053741838
42523597508750.69851833314846.3014816675
43536214513655.47516323522558.5248367654
44586570571306.39731290815263.6026870919
45596594577575.60509794119018.3949020592
46580523560989.5167754919533.4832245103
47564478546781.3428176817696.6571823204
48557560530954.98174284326605.0182571572
49575093543191.12456052231901.8754394778
50580112547057.72456052233054.2754394776
51574761545557.9739578129203.0260421898
52563250540458.50389251622791.4961074837
53551531530192.67785032721338.3221496735
54537034521112.02335509815921.9766449021
55544686527390.28053741817295.7194625816
56600991580920.76107483720070.2389251633
57604378587189.96885986917188.0311401306
58586111574724.32214967311386.6778503265
59563668560516.1481918633151.85180813665
60548604548810.228729282-206.228729281760


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
160.1594267093083930.3188534186167870.840573290691607
170.08427190829235950.1685438165847190.91572809170764
180.1005425373677690.2010850747355390.89945746263223
190.2599169080355170.5198338160710340.740083091964483
200.3651540336221050.7303080672442110.634845966377894
210.4189682580558170.8379365161116330.581031741944183
220.6636076052446570.6727847895106870.336392394755343
230.7722039245482010.4555921509035980.227796075451799
240.7427241552879960.5145516894240080.257275844712004
250.7888113329264630.4223773341470740.211188667073537
260.8224825350200770.3550349299598460.177517464979923
270.8805721188455380.2388557623089240.119427881154462
280.9297201312105040.1405597375789920.0702798687894958
290.9586840707744120.08263185845117670.0413159292255884
300.9773829092246630.04523418155067420.0226170907753371
310.9856765289086210.02864694218275710.0143234710913786
320.991830508923680.01633898215263930.00816949107631967
330.998341607194970.003316785610060750.00165839280503038
340.998989984467820.002020031064360120.00101001553218006
350.9983063091288520.003387381742295740.00169369087114787
360.9972412496935960.005517500612808460.00275875030640423
370.9993463114937620.001307377012476450.000653688506238226
380.999847702110530.0003045957789412140.000152297889470607
390.9999619185738357.61628523294494e-053.80814261647247e-05
400.9999500064634159.99870731706652e-054.99935365853326e-05
410.9998697855900870.0002604288198259840.000130214409912992
420.9995586312356050.0008827375287898610.000441368764394930
430.9978270168897450.004345966220509180.00217298311025459
440.99579156451580.008416870968398740.00420843548419937


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level120.413793103448276NOK
5% type I error level150.517241379310345NOK
10% type I error level160.551724137931034NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/10z26y1293632600.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/10z26y1293632600.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/1i9p91293632599.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/1i9p91293632599.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/2i9p91293632599.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/2i9p91293632599.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/3b0ou1293632599.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/3b0ou1293632599.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/4b0ou1293632599.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/4b0ou1293632599.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/5b0ou1293632599.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/5b0ou1293632599.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/63r5e1293632599.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/63r5e1293632599.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/7w1501293632599.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/7w1501293632599.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/8w1501293632599.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/8w1501293632599.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/9w1501293632599.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/29/t1293632469qhy3nb7ga9okadr/9w1501293632599.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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