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Workshop 4

*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: Thu, 02 Dec 2010 15:34:35 +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/02/t1291304134tt29t6kzpejmgc1.htm/, Retrieved Thu, 02 Dec 2010 16:35:48 +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/02/t1291304134tt29t6kzpejmgc1.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 «
162556 807 213118 6282154 29790 444 81767 4321023 87550 412 153198 4111912 84738 428 -26007 223193 54660 315 126942 1491348 42634 168 157214 1629616 40949 263 129352 1398893 45187 267 234817 1926517 37704 228 60448 983660 16275 129 47818 1443586 25830 104 245546 1073089 12679 122 48020 984885 18014 393 -1710 1405225 43556 190 32648 227132 24811 280 95350 929118 6575 63 151352 1071292 7123 102 288170 638830 21950 265 114337 856956 37597 234 37884 992426 17821 277 122844 444477 12988 73 82340 857217 22330 67 79801 711969 13326 103 165548 702380 16189 290 116384 358589 7146 83 134028 297978 15824 56 63838 585715 27664 236 74996 657954 11920 73 31080 209458 8568 34 32168 786690 14416 139 49857 439798 3369 26 87161 688779 11819 70 106113 574339 6984 40 80570 741409 4519 42 102129 597793 2220 12 301670 644190 18562 211 102313 377934 10327 74 88577 640273 5336 80 112477 697458 2365 83 191778 550608 4069 131 79804 207393 8636 203 128294 301607 13718 56 96448 345783 4525 89 93 etc...
 
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 time6 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135
R Framework
error message
The field 'Names of X columns' contains a hard return which cannot be interpreted.
Please, resubmit your request without hard returns in the 'Names of X columns'.


Multiple Linear Regression - Estimated Regression Equation
wealth [t] = -289294.338884457 + 15.765558710734costs[t] + 3132.12605876057orders[t] + 3.41235292761755dividends[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)-289294.338884457218626.454153-1.32320.1922970.096148
costs15.7655587107346.7467932.33670.0238620.011931
orders3132.126058760571263.9270352.47810.016940.00847
dividends3.412352927617551.4273732.39070.0209680.010484


Multiple Linear Regression - Regression Statistics
Multiple R0.814831228336522
R-squared0.663949930672405
Adjusted R-squared0.642033621803214
F-TEST (value)30.2947879880337
F-TEST (DF numerator)3
F-TEST (DF denominator)46
p-value5.79654102494942e-11
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation665289.84892443
Sum Squared Residuals20360086821767


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
162821545528351.3835454753802.616454605
243210231850043.487030512470979.51296949
341119122904181.906254811207730.09374519
42231932298452.46570670-2075259.46570670
514913481992241.71409147-500893.71409147
616296161445521.32222322184094.677776783
713988931621433.35410860-222540.354108605
819265172060660.09767092-134143.097670922
99836601225524.93791109-241864.937911093
101443586534506.283005668909079.716994332
1110730891281560.76468968-208471.764689679
12984885456577.746761923528307.253238077
1314052251219796.85331738185428.146682616
142271321103900.78586564-876768.785865639
159291181304228.08638886-375110.086388857
161071292528154.591641306543137.408358694
176388301125818.33695723-486988.336957228
188569561276931.27707271-419975.277072713
199924261165634.44802285-173208.448022846
204444771278449.68421646-833972.684216462
21857217425087.080000106432129.919999894
22711969544912.209039999167056.790960001
23702380808314.683008352-105934.683008352
243585891271394.13125202-912805.131252022
25297978540683.6447223-242705.6447223
26585715353416.707638038232298.292361962
276579541141938.64731639-483984.647316388
28209458233332.252227367-23874.2522273671
2978669062045.8231225722724644.176877428
30439798543477.157569432-103679.157569432
31688779142679.199463853546099.800536147
32574339478382.62983922895956.3701607715
33741409221030.640879877520378.359120123
34597793261999.707541946335793.292458054
35644190812695.221832884-168505.221832884
363779341013352.62538600-635418.625386002
37640273407549.899539154232723.100460846
38697458429211.987336504268246.012663496
39550608662371.890096194-111763.890096194
40207393457483.646242745-250090.646242745
41301607920463.022565603-618856.022565603
42345783431491.269962841-85708.2699628407
43501749380920.274004035120828.725995965
44379983495646.093124119-115663.093124119
45387475141816.499241584245658.500758416
46377305194518.1878765182786.8121235
47370837659821.516911582-288984.516911582
48430866762939.414727448-332073.414727448
49469107385670.04470009483436.9552999055
50194493135845.36634880658647.633651194


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
70.9998925488169680.0002149023660639940.000107451183031997
80.9999992623384621.47532307560704e-067.37661537803519e-07
90.9999979192015774.16159684558364e-062.08079842279182e-06
100.9999999816519373.66961251508855e-081.83480625754428e-08
110.9999999806531493.86937028312172e-081.93468514156086e-08
120.999999982322643.53547217667679e-081.76773608833840e-08
130.9999999998595752.80849614782837e-101.40424807391418e-10
140.9999999999900381.99230623435424e-119.9615311717712e-12
150.9999999999957768.44840946101723e-124.22420473050861e-12
160.999999999999558.97956655539194e-134.48978327769597e-13
170.99999999999976.01054469167839e-133.00527234583919e-13
180.9999999999998393.22642173661611e-131.61321086830805e-13
190.999999999999529.6226720985326e-134.8113360492663e-13
200.9999999999994411.11732441216481e-125.58662206082403e-13
210.9999999999995329.35175383564155e-134.67587691782077e-13
220.9999999999978074.38618377581493e-122.19309188790746e-12
230.9999999999909641.80711660581779e-119.03558302908897e-12
240.9999999999816533.66935159608415e-111.83467579804207e-11
250.999999999942191.15619721469750e-105.78098607348752e-11
260.9999999997324045.35191559120733e-102.67595779560366e-10
270.9999999993097471.38050620334204e-096.90253101671021e-10
280.999999999227981.54404084132423e-097.72020420662115e-10
290.9999999990966541.80669234488757e-099.03346172443785e-10
300.9999999948992491.02015026411145e-085.10075132055723e-09
310.9999999887353572.25292864254602e-081.12646432127301e-08
320.9999999426819871.14636026414715e-075.73180132073575e-08
330.9999999412927631.17414473943641e-075.87072369718203e-08
340.9999998229648083.54070384046758e-071.77035192023379e-07
350.9999990422615781.91547684332016e-069.5773842166008e-07
360.9999956183274328.76334513620044e-064.38167256810022e-06
370.9999924333485781.51333028432779e-057.56665142163897e-06
380.9999973911082125.21778357694502e-062.60889178847251e-06
390.9999889310475032.21379049934618e-051.10689524967309e-05
400.9999594452066588.1109586682929e-054.05547933414645e-05
410.9998613557116290.0002772885767423450.000138644288371172
420.9991857392925640.001628521414871810.000814260707435903
430.9982529782499150.003494043500170690.00174702175008535


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level371NOK
5% type I error level371NOK
10% type I error level371NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/106ir41291304068.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/106ir41291304068.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/1zhdb1291304068.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/1zhdb1291304068.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/2s9uw1291304068.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/2s9uw1291304068.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/3s9uw1291304068.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/3s9uw1291304068.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/4liby1291304068.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/4liby1291304068.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/5liby1291304068.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/5liby1291304068.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/6liby1291304068.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/6liby1291304068.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/7wabk1291304068.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/7wabk1291304068.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/86ir41291304068.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/86ir41291304068.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/96ir41291304068.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Dec/02/t1291304134tt29t6kzpejmgc1/96ir41291304068.ps (open in new window)


 
Parameters (Session):
par1 = 4 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;
 
Parameters (R input):
par1 = 4 ; par2 = Do not include Seasonal 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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