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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: Fri, 20 Nov 2009 07:08:59 -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/20/t1258726163o34ybi9p0vov5ha.htm/, Retrieved Fri, 20 Nov 2009 15:09:35 +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/20/t1258726163o34ybi9p0vov5ha.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 «
22 0 22 0 20 0 21 0 20 0 21 0 21 0 21 0 19 0 21 0 21 0 22 0 19 0 24 0 22 0 22 0 22 0 24 0 22 0 23 0 24 0 21 0 20 0 22 0 23 0 23 0 22 0 20 0 21 1 21 1 20 1 20 1 17 1 18 1 19 1 19 1 20 1 21 1 20 1 21 1 19 1 22 1 20 1 18 1 16 1 17 1 18 1 19 1 18 1 20 1 21 1 18 1 19 1 19 1 19 1 21 1 19 1 19 1 17 1 16 1 16 1 17 1 16 1 15 1 16 1 16 1 16 1 18 1 19 1 16 1 16 1 16 1
 
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
Y[t] = + 21.1 -3.15000000000001X[t] + 0.141666666666680M1[t] + 1.64166666666667M2[t] + 0.641666666666665M3[t] -0.0250000000000011M4[t] + 0.499999999999997M5[t] + 1.5M6[t] + 0.666666666666666M7[t] + 1.16666666666666M8[t] -1.24298986312465e-15M9[t] -0.333333333333334M10[t] -0.500000000000001M11[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)21.10.74444328.343400
X-3.150000000000010.414719-7.595500
M10.1416666666666800.9799420.14460.8855460.442773
M21.641666666666670.9799421.67530.0991740.049587
M30.6416666666666650.9799420.65480.5151410.25757
M4-0.02500000000000110.979942-0.02550.9797330.489866
M50.4999999999999970.9775010.51150.6109040.305452
M61.50.9775011.53450.1302460.065123
M70.6666666666666660.9775010.6820.4979010.24895
M81.166666666666660.9775011.19350.2374430.118722
M9-1.24298986312465e-150.977501010.5
M10-0.3333333333333340.977501-0.3410.7343110.367155
M11-0.5000000000000010.977501-0.51150.6109040.305452


Multiple Linear Regression - Regression Statistics
Multiple R0.740843854357209
R-squared0.548849616538846
Adjusted R-squared0.457090216512849
F-TEST (value)5.98139935944814
F-TEST (DF numerator)12
F-TEST (DF denominator)59
p-value1.09294272232496e-06
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation1.69308163528189
Sum Squared Residuals169.125


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
12221.24166666666660.758333333333405
22222.7416666666667-0.741666666666658
32021.7416666666667-1.74166666666667
42121.075-0.0750000000000007
52021.6-1.60000000000000
62122.6-1.6
72121.7666666666667-0.766666666666667
82122.2666666666667-1.26666666666667
91921.1-2.1
102120.76666666666670.233333333333332
112120.60.399999999999997
122221.10.899999999999997
131921.2416666666667-2.24166666666668
142422.74166666666671.25833333333333
152221.74166666666670.258333333333333
162221.0750.924999999999997
172221.60.399999999999999
182422.61.40000000000000
192221.76666666666670.233333333333331
202322.26666666666670.733333333333333
212421.12.9
222120.76666666666670.233333333333331
232020.6-0.600000000000002
242221.10.899999999999997
252321.24166666666671.75833333333332
262322.74166666666670.25833333333333
272221.74166666666670.258333333333333
282021.075-1.07500000000000
292118.452.55
302119.451.55
312018.61666666666671.38333333333333
322019.11666666666670.883333333333335
331717.95-0.949999999999998
341817.61666666666670.383333333333334
351917.451.55
361917.951.05
372018.09166666666671.90833333333332
382119.59166666666671.40833333333333
392018.59166666666671.40833333333334
402117.9253.075
411918.450.550000000000001
422219.452.55
432018.61666666666671.38333333333333
441819.1166666666667-1.11666666666666
451617.95-1.95
461717.6166666666667-0.616666666666666
471817.450.550000000000001
481917.951.05
491818.0916666666667-0.0916666666666789
502019.59166666666670.408333333333333
512118.59166666666672.40833333333334
521817.9250.0750000000000006
531918.450.550000000000001
541919.45-0.449999999999999
551918.61666666666670.383333333333334
562119.11666666666671.88333333333334
571917.951.05000000000000
581917.61666666666671.38333333333333
591717.45-0.449999999999999
601617.95-1.95
611618.0916666666667-2.09166666666668
621719.5916666666667-2.59166666666667
631618.5916666666667-2.59166666666666
641517.925-2.925
651618.45-2.45
661619.45-3.45
671618.6166666666667-2.61666666666667
681819.1166666666667-1.11666666666666
691917.951.05000000000000
701617.6166666666667-1.61666666666667
711617.45-1.45
721617.95-1.95


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
160.5705646995266030.8588706009467950.429435300473397
170.4944518886612850.988903777322570.505548111338715
180.5290396897084140.9419206205831720.470960310291586
190.4114385668635860.8228771337271730.588561433136414
200.3570358630781950.714071726156390.642964136921805
210.619909420890220.7601811582195610.380090579109781
220.5111827005970370.9776345988059260.488817299402963
230.4218057401210190.8436114802420380.578194259878981
240.327413925370550.65482785074110.67258607462945
250.3393781243199970.6787562486399940.660621875680003
260.2580756585509010.5161513171018010.7419243414491
270.1987078224443590.3974156448887190.80129217755564
280.1590394559281260.3180789118562530.840960544071874
290.1323570566735520.2647141133471040.867642943326448
300.1060900938162940.2121801876325870.893909906183706
310.07774296318300080.1554859263660020.922257036817
320.05464132161696560.1092826432339310.945358678383034
330.06567044269859620.1313408853971920.934329557301404
340.04579346422188910.09158692844377830.954206535778111
350.03337643838835810.06675287677671620.966623561611642
360.02517962204135710.05035924408271410.974820377958643
370.02179097866409650.0435819573281930.978209021335904
380.01641405178845820.03282810357691640.983585948211542
390.01085365497252920.02170730994505830.98914634502747
400.02160934728945190.04321869457890380.978390652710548
410.01504820324518100.03009640649036190.984951796754819
420.02914004784167250.05828009568334510.970859952158327
430.02532248686289180.05064497372578360.974677513137108
440.02487587234812270.04975174469624530.975124127651877
450.04361247415506720.08722494831013440.956387525844933
460.0321240760668760.0642481521337520.967875923933124
470.02258358672449140.04516717344898270.977416413275509
480.02564865792358990.05129731584717990.97435134207641
490.02094910985182460.04189821970364920.979050890148175
500.02145517621441370.04291035242882740.978544823785586
510.07841212483144670.1568242496628930.921587875168553
520.09857899464165480.1971579892833100.901421005358345
530.1215420434747290.2430840869494580.87845795652527
540.1764810686591100.3529621373182200.82351893134089
550.2500312775630790.5000625551261580.749968722436921
560.4123805183362630.8247610366725250.587619481663737


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level00OK
5% type I error level90.219512195121951NOK
10% type I error level170.414634146341463NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/10fw081258726134.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/162al1258726134.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/162al1258726134.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/20i1k1258726134.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/20i1k1258726134.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/361md1258726134.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/361md1258726134.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/4k1721258726134.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/5i8iu1258726134.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/6sy4m1258726134.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/70l4h1258726134.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/70l4h1258726134.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/8ohdk1258726134.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/8ohdk1258726134.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/9n2lh1258726134.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t1258726163o34ybi9p0vov5ha/9n2lh1258726134.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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