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WS 7

*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, 19 Nov 2009 16:41:15 -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/t12586741395knzm14g5a05125.htm/, Retrieved Fri, 20 Nov 2009 00:42:32 +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/t12586741395knzm14g5a05125.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:
WS 7
 
Dataseries X:
» Textbox « » Textfile « » CSV «
286602 0 283042 0 276687 0 277915 0 277128 0 277103 0 275037 0 270150 0 267140 0 264993 0 287259 0 291186 0 292300 0 288186 0 281477 0 282656 0 280190 0 280408 0 276836 0 275216 0 274352 0 271311 0 289802 0 290726 0 292300 0 278506 0 269826 0 265861 0 269034 0 264176 0 255198 0 253353 0 246057 0 235372 0 258556 0 260993 0 254663 0 250643 0 243422 0 247105 0 248541 0 245039 0 237080 0 237085 0 225554 0 226839 0 247934 0 248333 1 246969 1 245098 1 246263 1 255765 1 264319 1 268347 1 273046 1 273963 1 267430 1 271993 1 292710 1 295881 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
nwwmb[t] = + 266613.595744681 -1219.98036006547dummy_variable[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)266613.5957446812643.821107100.84400
dummy_variable-1219.980360065475679.838858-0.21480.8306830.415342


Multiple Linear Regression - Regression Statistics
Multiple R0.0281922780235840
R-squared0.00079480454015906
Adjusted R-squared-0.0164328712436312
F-TEST (value)0.0461353319004817
F-TEST (DF numerator)1
F-TEST (DF denominator)58
p-value0.830683425738388
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation18125.1243374551
Sum Squared Residuals19054167670.3961


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
1286602266613.59574468019988.4042553195
2283042266613.59574468116428.4042553191
3276687266613.59574468110073.4042553191
4277915266613.59574468111301.4042553191
5277128266613.59574468110514.4042553191
6277103266613.59574468110489.4042553191
7275037266613.5957446818423.40425531914
8270150266613.5957446813536.40425531914
9267140266613.595744681526.404255319142
10264993266613.595744681-1620.59574468086
11287259266613.59574468120645.4042553191
12291186266613.59574468124572.4042553191
13292300266613.59574468125686.4042553191
14288186266613.59574468121572.4042553191
15281477266613.59574468114863.4042553191
16282656266613.59574468116042.4042553191
17280190266613.59574468113576.4042553191
18280408266613.59574468113794.4042553191
19276836266613.59574468110222.4042553191
20275216266613.5957446818602.40425531914
21274352266613.5957446817738.40425531914
22271311266613.5957446814697.40425531914
23289802266613.59574468123188.4042553191
24290726266613.59574468124112.4042553191
25292300266613.59574468125686.4042553191
26278506266613.59574468111892.4042553191
27269826266613.5957446813212.40425531914
28265861266613.595744681-752.595744680858
29269034266613.5957446812420.40425531914
30264176266613.595744681-2437.59574468086
31255198266613.595744681-11415.5957446809
32253353266613.595744681-13260.5957446809
33246057266613.595744681-20556.5957446809
34235372266613.595744681-31241.5957446809
35258556266613.595744681-8057.59574468086
36260993266613.595744681-5620.59574468086
37254663266613.595744681-11950.5957446809
38250643266613.595744681-15970.5957446809
39243422266613.595744681-23191.5957446809
40247105266613.595744681-19508.5957446809
41248541266613.595744681-18072.5957446809
42245039266613.595744681-21574.5957446809
43237080266613.595744681-29533.5957446809
44237085266613.595744681-29528.5957446809
45225554266613.595744681-41059.5957446809
46226839266613.595744681-39774.5957446809
47247934266613.595744681-18679.5957446809
48248333265393.615384615-17060.6153846154
49246969265393.615384615-18424.6153846154
50245098265393.615384615-20295.6153846154
51246263265393.615384615-19130.6153846154
52255765265393.615384615-9628.61538461538
53264319265393.615384615-1074.61538461538
54268347265393.6153846152953.38461538462
55273046265393.6153846157652.38461538462
56273963265393.6153846158569.38461538462
57267430265393.6153846152036.38461538462
58271993265393.6153846156599.38461538462
59292710265393.61538461527316.3846153846
60295881265393.61538461530487.3846153846


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
50.02676862029067190.05353724058134370.973231379709328
60.007243317027148910.01448663405429780.992756682972851
70.00247571338284310.00495142676568620.997524286617157
80.002254085799664230.004508171599328460.997745914200336
90.002581898083698080.005163796167396160.997418101916302
100.002866716629146240.005733433258292480.997133283370854
110.003067846746281770.006135693492563550.996932153253718
120.005001695679990270.01000339135998050.99499830432001
130.007341919815458780.01468383963091760.992658080184541
140.005954716853638540.01190943370727710.994045283146362
150.003166383739294210.006332767478588420.996833616260706
160.00176401810700230.00352803621400460.998235981892998
170.0009120135356135620.001824027071227120.999087986464386
180.0004782010951843380.0009564021903686760.999521798904816
190.0002505426529216970.0005010853058433940.999749457347078
200.0001383589710214440.0002767179420428890.999861641028979
217.9696226487747e-050.0001593924529754940.999920303773512
225.63625522299001e-050.0001127251044598000.99994363744777
230.0001147398754283320.0002294797508566640.999885260124572
240.0003181223693476020.0006362447386952040.999681877630652
250.001487127688070630.002974255376141250.99851287231193
260.001779649080751360.003559298161502730.99822035091925
270.002513538503933750.00502707700786750.997486461496066
280.004445936157693310.008891872315386620.995554063842307
290.006763805728475150.01352761145695030.993236194271525
300.01242750028651960.02485500057303910.98757249971348
310.03426131785978340.06852263571956680.965738682140217
320.07181497764461720.1436299552892340.928185022355383
330.1609109137999110.3218218275998220.839089086200089
340.3890761070858190.7781522141716390.610923892914181
350.4017262156543620.8034524313087250.598273784345638
360.4283816993166740.8567633986333480.571618300683326
370.4500076583425560.9000153166851130.549992341657444
380.4722223095673260.9444446191346530.527777690432674
390.5080017504594050.9839964990811910.491998249540596
400.5161588984797030.9676822030405940.483841101520297
410.5204483620361580.9591032759276830.479551637963842
420.5239023197565930.9521953604868140.476097680243407
430.5368748086757840.9262503826484320.463125191324216
440.5336029349987360.9327941300025290.466397065001264
450.5906877247032330.8186245505935350.409312275296767
460.6434415305556710.7131169388886590.356558469444329
470.5676389279131080.8647221441737840.432361072086892
480.5453434450626470.9093131098747060.454656554937353
490.5558521151323580.8882957697352840.444147884867642
500.6310055352916050.737988929416790.368994464708395
510.7566523862462760.4866952275074480.243347613753724
520.7949695360868680.4100609278262630.205030463913132
530.7585770435208980.4828459129582040.241422956479102
540.6825581569478250.634883686104350.317441843052175
550.5464316040470980.9071367919058040.453568395952902


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level190.372549019607843NOK
5% type I error level250.490196078431373NOK
10% type I error level270.529411764705882NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/1048951258674070.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/1048951258674070.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/1b0661258674070.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/1b0661258674070.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/20t171258674070.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/20t171258674070.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/3sk1d1258674070.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/3sk1d1258674070.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/4p6h81258674070.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/4p6h81258674070.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/5aanz1258674070.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/5aanz1258674070.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/6ud9i1258674070.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/6ud9i1258674070.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/7unzi1258674070.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/7unzi1258674070.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/8ncn51258674070.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/8ncn51258674070.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/9d11c1258674070.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/20/t12586741395knzm14g5a05125/9d11c1258674070.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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