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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: Sat, 27 Nov 2010 17:21:01 +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/Nov/27/t1290878533vh4csyet83471bo.htm/, Retrieved Sat, 27 Nov 2010 18:22:16 +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/Nov/27/t1290878533vh4csyet83471bo.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 «
47.54 0 45.31 0 46.9 0 47.16 0 48.24 0 52.7 0 51.72 0 51.5 0 52.45 0 53 0 48.36 0 46.63 0 45.92 0 45.53 0 42.17 0 43.66 0 45.32 0 47.43 0 47.76 0 49.49 0 50.69 0 49.8 0 52.13 0 53.94 0 60.75 0 59.19 0 57.58 0 59.16 0 64.74 0 67.04 0 75.53 0 78.91 0 78.4 0 70.07 0 66.8 0 61.02 0 52.38 0 42.37 0 39.83 0 38.79 0 37.33 0 39.4 0 39.45 0 43.24 0 42.33 0 45.5 0 43.44 0 43.88 0 45.61 0 45.12 0 47.56 1 47.04 1 51.07 1 54.72 1 55.37 1 55.39 1 53.13 1 53.71 1 54.59 1 54.61 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 time6 seconds
R Server'RServer@AstonUniversity' @ vre.aston.ac.uk
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
Y[t] = + 51.2642 + 1.4548`X `[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)51.26421.33469238.40900
`X `1.45483.2693150.4450.6579860.328993


Multiple Linear Regression - Regression Statistics
Multiple R0.0583300545013329
R-squared0.00340239525812847
Adjusted R-squared-0.0137803220650072
F-TEST (value)0.19801264224648
F-TEST (DF numerator)1
F-TEST (DF denominator)58
p-value0.657986088726541
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation9.43770090720156
Sum Squared Residuals5166.071508


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
147.5451.2642000000001-3.72420000000009
245.3151.2642-5.95419999999999
346.951.2642-4.3642
447.1651.2642-4.1042
548.2451.2642-3.0242
652.751.26421.4358
751.7251.26420.455800000000001
851.551.26420.235800000000002
952.4551.26421.1858
105351.26421.7358
1148.3651.2642-2.9042
1246.6351.2642-4.63419999999999
1345.9251.2642-5.3442
1445.5351.2642-5.7342
1542.1751.2642-9.0942
1643.6651.2642-7.6042
1745.3251.2642-5.9442
1847.4351.2642-3.8342
1947.7651.2642-3.5042
2049.4951.2642-1.7742
2150.6951.2642-0.5742
2249.851.2642-1.4642
2352.1351.26420.865800000000004
2453.9451.26422.6758
2560.7551.26429.4858
2659.1951.26427.9258
2757.5851.26426.3158
2859.1651.26427.8958
2964.7451.264213.4758
3067.0451.264215.7758
3175.5351.264224.2658
3278.9151.264227.6458
3378.451.264227.1358
3470.0751.264218.8058
3566.851.264215.5358
3661.0251.26429.7558
3752.3851.26421.1158
3842.3751.2642-8.8942
3939.8351.2642-11.4342
4038.7951.2642-12.4742
4137.3351.2642-13.9342
4239.451.2642-11.8642
4339.4551.2642-11.8142
4443.2451.2642-8.0242
4542.3351.2642-8.9342
4645.551.2642-5.7642
4743.4451.2642-7.8242
4843.8851.2642-7.3842
4945.6151.2642-5.6542
5045.1251.2642-6.1442
5147.5652.719-5.159
5247.0452.719-5.679
5351.0752.719-1.649
5454.7252.7192.001
5555.3752.7192.651
5655.3952.7192.671
5753.1352.7190.411000000000001
5853.7152.7190.991
5954.5952.7191.871
6054.6152.7191.891


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
50.002913940528744830.005827881057489670.997086059471255
60.01212551736746740.02425103473493480.987874482632533
70.006143323908198510.0122866478163970.993856676091802
80.002475267213945970.004950534427891950.997524732786054
90.001202507769958110.002405015539916220.998797492230042
100.0006143970823107710.001228794164621540.99938560291769
110.0001801641003683690.0003603282007367380.999819835899632
127.34488299279166e-050.0001468976598558330.999926551170072
133.50670782228352e-057.01341564456703e-050.999964932921777
141.76238949664762e-053.52477899329525e-050.999982376105033
153.51391228209298e-057.02782456418595e-050.99996486087718
162.58075748922336e-055.16151497844673e-050.999974192425108
171.10583733199668e-052.21167466399335e-050.99998894162668
183.46575979911539e-066.93151959823079e-060.9999965342402
191.03910903717346e-062.07821807434693e-060.999998960890963
203.29527734096344e-076.59055468192688e-070.999999670472266
211.25338542425687e-072.50677084851374e-070.999999874661458
223.88051874549753e-087.76103749099507e-080.999999961194812
232.07336826433361e-084.14673652866721e-080.999999979266317
242.08104302735882e-084.16208605471764e-080.99999997918957
257.06431014990399e-071.4128620299808e-060.999999293568985
262.44006662829363e-064.88013325658725e-060.999997559933372
273.12901114481542e-066.25802228963085e-060.999996870988855
285.43945157941561e-061.08789031588312e-050.99999456054842
294.86301616789662e-059.72603233579325e-050.99995136983832
300.0004280028922662060.0008560057845324110.999571997107734
310.01855458477689690.03710916955379380.981445415223103
320.2704190200911980.5408380401823970.729580979908802
330.8371980285330070.3256039429339860.162801971466993
340.9798863734777060.04022725304458770.0201136265222939
350.9995703667082920.0008592665834165650.000429633291708283
360.999998346906393.30618721767932e-061.65309360883966e-06
370.9999999069279451.86144109922944e-079.30720549614718e-08
380.99999977350674.52986597253314e-072.26493298626657e-07
390.9999996103700447.79259911292286e-073.89629955646143e-07
400.9999995293991869.41201628945368e-074.70600814472684e-07
410.9999997816638824.36672236179212e-072.18336118089606e-07
420.9999997488605725.0227885596095e-072.51139427980475e-07
430.9999997790628874.41874225261978e-072.20937112630989e-07
440.9999992077894451.58442111040375e-067.92210555201875e-07
450.9999978393729234.32125415419547e-062.16062707709773e-06
460.9999919187936421.61624127161642e-058.08120635808211e-06
470.999972758814345.44823713198923e-052.72411856599461e-05
480.999909073256540.0001818534869186659.09267434593327e-05
490.9996789586693760.0006420826612478530.000321041330623927
500.998921001392790.002157997214420140.00107899860721007
510.99905354652350.00189290695299830.000946453476499152
520.9999367656780350.0001264686439292246.3234321964612e-05
530.9999805270019193.89459961628984e-051.94729980814492e-05
540.9997818325338330.0004363349323346360.000218167466167318
550.998484971103450.003030057793098980.00151502889654949


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level450.88235294117647NOK
5% type I error level490.96078431372549NOK
10% type I error level490.96078431372549NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/10jcl51290878451.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/10jcl51290878451.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/1cb6c1290878451.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/1cb6c1290878451.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/252nf1290878451.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/252nf1290878451.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/352nf1290878451.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/352nf1290878451.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/452nf1290878451.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/452nf1290878451.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/5xtmz1290878451.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/5xtmz1290878451.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/6xtmz1290878451.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/6xtmz1290878451.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/7823k1290878451.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/7823k1290878451.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/8823k1290878451.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/8823k1290878451.ps (open in new window)


http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/9jcl51290878451.png (open in new window)
http://www.freestatistics.org/blog/date/2010/Nov/27/t1290878533vh4csyet83471bo/9jcl51290878451.ps (open in new window)


 
Parameters (Session):
par1 = 1 ; par2 = Do not include Seasonal Dummies ; par3 = No Linear Trend ;
 
Parameters (R input):
par1 = 1 ; 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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Software written by Ed van Stee & Patrick Wessa


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