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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: Sun, 22 Nov 2009 12:23:46 -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/22/t1258917879pt25dq99subbrnd.htm/, Retrieved Sun, 22 Nov 2009 20:24:51 +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/22/t1258917879pt25dq99subbrnd.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 «
17823.2 1.2218 17872 1.249 17420.4 1.2991 16704.4 1.3408 15991.2 1.3119 16583.6 1.3014 19123.5 1.3201 17838.7 1.2938 17209.4 1.2694 18586.5 1.2165 16258.1 1.2037 15141.6 1.2292 19202.1 1.2256 17746.5 1.2015 19090.1 1.1786 18040.3 1.1856 17515.5 1.2103 17751.8 1.1938 21072.4 1.202 17170 1.2271 19439.5 1.277 19795.4 1.265 17574.9 1.2684 16165.4 1.2811 19464.6 1.2727 19932.1 1.2611 19961.2 1.2881 17343.4 1.3213 18924.2 1.2999 18574.1 1.3074 21350.6 1.3242 18594.6 1.3516 19823.1 1.3511 20844.4 1.3419 19640.2 1.3716 17735.4 1.3622 19813.6 1.3896 22160 1.4227 20664.3 1.4684 17877.4 1.457 20906.5 1.4718 21164.1 1.4748 22786.7 1.437 22321.5 1.3322 17842.2 1.2732 16373.5 1.3449 15993.8 1.3239 16446.1 1.2785 17729 1.305 16643 1.319 16196.7 1.365 18252.1 1.4016 17570.4 1.4088 15836.8 1.4268
 
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
EUDO[t] = + 0.962415386308849 + 1.98712515138476e-05UITV[t] -0.0531840779040217M1[t] -0.0467402096762419M2[t] -0.0135028975421446M3[t] + 0.0282457901815517M4[t] + 0.0168342620170412M5[t] + 0.0210981792690236M6[t] -0.0605419433507508M7[t] -0.0384205855434918M8[t] -0.0389194261214411M9[t] -0.0459060463579918M10[t] -0.0156144435369607M11[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)0.9624153863088490.1252517.683900
UITV1.98712515138476e-057e-062.73760.0091110.004556
M1-0.05318407790402170.055948-0.95060.3473820.173691
M2-0.04674020967624190.056097-0.83320.4095610.20478
M3-0.01350289754214460.055636-0.24270.8094470.404724
M40.02824579018155170.0538810.52420.6029440.301472
M50.01683426201704120.0546850.30780.7597640.379882
M60.02109817926902360.0543560.38810.6999140.349957
M7-0.06054194335075080.065578-0.92320.3613030.180651
M8-0.03842058554349180.059074-0.65040.5190760.259538
M9-0.03891942612144110.058203-0.66870.5074450.253723
M10-0.04590604635799180.058888-0.77960.4401320.220066
M11-0.01561444353696070.05642-0.27680.783360.39168


Multiple Linear Regression - Regression Statistics
Multiple R0.472525101596154
R-squared0.223279971638456
Adjusted R-squared-0.00405271958931319
F-TEST (value)0.98217273737699
F-TEST (DF numerator)12
F-TEST (DF denominator)41
p-value0.48127171157826
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation0.0791341574222953
Sum Squared Residuals0.256750809708401


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
11.22181.26340059838643-0.0416005983864301
21.2491.27081418368809-0.0218141836880902
31.29911.295077638638530.00402236136146588
41.34081.322598510278320.0182014897216845
51.31191.297014805534130.0148851944658712
61.30141.31305045218291-0.0116504521829146
71.32011.281881321283160.0382186787168384
81.29381.278472095145430.0153279048545709
91.26941.265468275989820.00393172401018444
101.21651.28584635621298-0.0693463562129846
111.20371.26986973700917-0.0661697370091729
121.22921.26329792823092-0.0340979282309227
131.22561.29080106709888-0.065201067098879
141.20151.26832034162310-0.0668203416231023
151.17861.32825666729121-0.149656667291205
161.18561.34914451517566-0.163544515175664
171.21031.32730455421669-0.117004554216687
181.19381.33626404820139-0.142464048201391
191.2021.3206084033585-0.118608403358499
201.22711.26518418925812-0.0380841892581193
211.2771.30978315399085-0.0327831539908471
221.2651.30986871216807-0.0448687121680749
231.26841.29603620100261-0.0276362010026074
241.28111.2836421155308-0.00254211553079997
251.27271.29601727062126-0.023317270621264
261.26111.31175094893177-0.0506509489317674
271.28811.34556651448492-0.0574665144849179
281.32131.33529623999566-0.0139962399956642
291.29991.35529718622424-0.0553971862242437
301.30741.35260417832123-0.0452041783212281
311.32421.32613658552965-0.00193658552965142
321.35161.293492774164750.0581072258352534
331.35111.317405766071560.033694233928441
341.34191.330713655006100.0111863449938992
351.37161.337076296754160.0345237032458432
361.36221.314839980407540.0473600195924595
371.38961.302952337399600.0866476626004032
381.42271.356022110179470.0666778898205315
391.46841.359537991424300.108862008575696
401.4571.345907488304060.111092511695941
411.47181.394687968100140.0771120318998562
421.47481.404070719742090.0707292802579069
431.4371.354673689828690.0823263101713121
441.33221.36755094143171-0.035350941431705
451.27321.27804280394778-0.00484280394777827
461.34491.241871276612840.103028723387160
471.32391.264617765234060.0592822347659371
481.27851.28921997583074-0.0107199758307369
491.3051.261528726493830.0434712735061698
501.3191.246392415577570.0726075844224284
511.3651.270761188161040.0942388118389612
521.40161.353353246246300.0482467537537027
531.40881.328395485924800.080404514075203
541.42681.298210601552370.128589398447627


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
160.2691956970748480.5383913941496960.730804302925152
170.1609059528539800.3218119057079590.83909404714602
180.1313345071964590.2626690143929180.86866549280354
190.09580097915140560.1916019583028110.904199020848594
200.1693141413518030.3386282827036070.830685858648197
210.2506364970099080.5012729940198160.749363502990092
220.2846742605707100.5693485211414190.71532573942929
230.3265047186089310.6530094372178610.67349528139107
240.2873562823894510.5747125647789030.712643717610549
250.2834651945891950.566930389178390.716534805410805
260.3277990920475860.6555981840951720.672200907952414
270.4600986986566400.9201973973132790.53990130134336
280.4823193979640110.9646387959280230.517680602035989
290.6397748604922020.7204502790155950.360225139507797
300.8823482328825730.2353035342348530.117651767117427
310.9231875857532630.1536248284934750.0768124142467374
320.9453143949764460.1093712100471090.0546856050235543
330.9349990905401050.1300018189197910.0650009094598955
340.9594145106330470.08117097873390580.0405854893669529
350.934583789031040.1308324219379190.0654162109689593
360.9365220271385340.1269559457229320.063477972861466
370.933619103309270.132761793381460.06638089669073
380.8581597198198530.2836805603602950.141840280180147


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 level10.0434782608695652OK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/10lgz31258917820.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/10lgz31258917820.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/15xbq1258917820.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/15xbq1258917820.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/2095s1258917820.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/2095s1258917820.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/39i5y1258917820.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/39i5y1258917820.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/4hu2i1258917820.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/4hu2i1258917820.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/5u0b51258917820.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/5u0b51258917820.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/6cowb1258917820.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/6cowb1258917820.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/7tord1258917820.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/7tord1258917820.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/8deum1258917820.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/8deum1258917820.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/9jt151258917820.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t1258917879pt25dq99subbrnd/9jt151258917820.ps (open in new window)


 
Parameters (Session):
par1 = 2 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;
 
Parameters (R input):
par1 = 2 ; 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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