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w7

*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 06:02:44 -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/t12588951154x0wy8rxdesgvys.htm/, Retrieved Sun, 22 Nov 2009 14:05:27 +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/t12588951154x0wy8rxdesgvys.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 «
8 0 8,1 0 7,7 0 7,5 0 7,6 0 7,8 0 7,8 0 7,8 0 7,5 0 7,5 0 7,1 0 7,5 0 7,5 0 7,6 0 7,7 0 7,7 0 7,9 0 8,1 0 8,2 0 8,2 0 8,2 0 7,9 0 7,3 0 6,9 0 6,6 0 6,7 0 6,9 0 7 0 7,1 0 7,2 0 7,1 0 6,9 0 7 0 6,8 0 6,4 0 6,7 0 6,6 0 6,4 0 6,3 0 6,2 0 6,5 0 6,8 1 6,8 1 6,4 1 6,1 1 5,8 1 6,1 1 7,2 1 7,3 1 6,9 1 6,1 1 5,8 1 6,2 1 7,1 1 7,7 1 7,9 1 7,7 1 7,4 1 7,5 1 8 1 8,1 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] = + 7.30487804878049 -0.359878048780488X[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)7.304878048780490.10090272.395500
X-0.3598780487804880.176218-2.04220.0456070.022803


Multiple Linear Regression - Regression Statistics
Multiple R0.256948656215148
R-squared0.0660226119307705
Adjusted R-squared0.0501924867092582
F-TEST (value)4.17069423058316
F-TEST (DF numerator)1
F-TEST (DF denominator)59
p-value0.0456065207076984
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation0.646090254363734
Sum Squared Residuals24.6285243902439


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
187.304878048780480.695121951219518
28.17.304878048780490.795121951219512
37.77.304878048780490.395121951219512
47.57.304878048780490.195121951219512
57.67.304878048780490.295121951219512
67.87.304878048780490.495121951219512
77.87.304878048780490.495121951219512
87.87.304878048780490.495121951219512
97.57.304878048780490.195121951219512
107.57.304878048780490.195121951219512
117.17.30487804878049-0.204878048780488
127.57.304878048780490.195121951219512
137.57.304878048780490.195121951219512
147.67.304878048780490.295121951219512
157.77.304878048780490.395121951219512
167.77.304878048780490.395121951219512
177.97.304878048780490.595121951219513
188.17.304878048780490.795121951219512
198.27.304878048780490.895121951219511
208.27.304878048780490.895121951219511
218.27.304878048780490.895121951219511
227.97.304878048780490.595121951219513
237.37.30487804878049-0.00487804878048807
246.97.30487804878049-0.404878048780487
256.67.30487804878049-0.704878048780488
266.77.30487804878049-0.604878048780488
276.97.30487804878049-0.404878048780487
2877.30487804878049-0.304878048780488
297.17.30487804878049-0.204878048780488
307.27.30487804878049-0.104878048780488
317.17.30487804878049-0.204878048780488
326.97.30487804878049-0.404878048780487
3377.30487804878049-0.304878048780488
346.87.30487804878049-0.504878048780488
356.47.30487804878049-0.904878048780488
366.77.30487804878049-0.604878048780488
376.67.30487804878049-0.704878048780488
386.47.30487804878049-0.904878048780488
396.37.30487804878049-1.00487804878049
406.27.30487804878049-1.10487804878049
416.57.30487804878049-0.804878048780488
426.86.945-0.145000000000000
436.86.945-0.145000000000000
446.46.945-0.545
456.16.945-0.845
465.86.945-1.145
476.16.945-0.845
487.26.9450.255
497.36.9450.355
506.96.945-0.0449999999999997
516.16.945-0.845
525.86.945-1.145
536.26.945-0.745
547.16.9450.155000000000000
557.76.9450.755
567.96.9450.955
577.76.9450.755
587.46.9450.455
597.56.9450.555
6086.9451.055
618.16.9451.155


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
50.1103022674752760.2206045349505530.889697732524724
60.04067791964539080.08135583929078160.95932208035461
70.01376281586577400.02752563173154800.986237184134226
80.004339764236176280.008679528472352560.995660235763824
90.002678576159428610.005357152318857220.997321423840571
100.001428888887230150.00285777777446030.99857111111277
110.005325052672164580.01065010534432920.994674947327835
120.002439586055747510.004879172111495020.997560413944252
130.001070128969454660.002140257938909310.998929871030545
140.0004117614344824670.0008235228689649330.999588238565517
150.0001591844069192980.0003183688138385960.99984081559308
166.02970242099036e-050.0001205940484198070.99993970297579
173.81328847550035e-057.6265769510007e-050.999961867115245
186.57123831294717e-050.0001314247662589430.99993428761687
190.0001778078121583320.0003556156243166640.999822192187842
200.0004196872172212660.0008393744344425320.999580312782779
210.000995012510943320.001990025021886640.999004987489057
220.0009450055981804360.001890011196360870.99905499440182
230.001190692000075530.002381384000151060.998809307999925
240.00485850614975060.00971701229950120.99514149385025
250.02707696767932080.05415393535864150.972923032320679
260.05357500663164220.1071500132632840.946424993368358
270.06113945645675090.1222789129135020.93886054354325
280.05935734843818410.1187146968763680.940642651561816
290.0527553581934420.1055107163868840.947244641806558
300.04547988513281470.09095977026562940.954520114867185
310.04066860012043870.08133720024087740.959331399879561
320.04032604539343860.08065209078687710.959673954606561
330.03738451631744360.07476903263488720.962615483682556
340.03853903911160680.07707807822321360.961460960888393
350.05747965259208310.1149593051841660.942520347407917
360.05670603478313720.1134120695662740.943293965216863
370.05789744344686850.1157948868937370.942102556553132
380.06583038756663230.1316607751332650.934169612433368
390.07573681794224160.1514736358844830.924263182057758
400.08899843808194120.1779968761638820.911001561918059
410.07826483829761960.1565296765952390.92173516170238
420.05349422452233660.1069884490446730.946505775477663
430.03527602027049490.07055204054098980.964723979729505
440.02784235358490810.05568470716981630.972157646415092
450.03111137963448050.06222275926896090.96888862036552
460.0622861708491340.1245723416982680.937713829150866
470.0848104216419360.1696208432838720.915189578358064
480.06707008166229880.1341401633245980.932929918337701
490.05109073590767630.1021814718153530.948909264092324
500.03393659448756030.06787318897512060.96606340551244
510.06276074172392020.1255214834478400.93723925827608
520.3421557543635440.6843115087270880.657844245636456
530.8799234657377010.2401530685245980.120076534262299
540.9423715365738150.1152569268523690.0576284634261846
550.8880375921716730.2239248156566540.111962407828327
560.8007275625990920.3985448748018160.199272437400908


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level160.307692307692308NOK
5% type I error level180.346153846153846NOK
10% type I error level290.557692307692308NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/10b0r71258894959.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/10b0r71258894959.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/1y3xk1258894959.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/1y3xk1258894959.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/2p8ka1258894959.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/2p8ka1258894959.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/3ireq1258894959.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/3ireq1258894959.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/4xfef1258894959.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/4xfef1258894959.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/574m51258894959.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/574m51258894959.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/6f2fw1258894959.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/6f2fw1258894959.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/7098r1258894959.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/7098r1258894959.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/8ggm51258894959.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/8ggm51258894959.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/9qjvw1258894959.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/22/t12588951154x0wy8rxdesgvys/9qjvw1258894959.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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