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Multiple regression - Model 3

*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: Wed, 18 Nov 2009 08:59:02 -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/18/t1258560176gm6ac52s0lxjdvb.htm/, Retrieved Wed, 18 Nov 2009 17:03:08 +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/18/t1258560176gm6ac52s0lxjdvb.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 «
3.58 98.2 3.52 98.71 3.45 98.54 3.36 98.2 3.27 96.92 3.21 99.06 3.19 99.65 3.16 99.82 3.12 99.99 3.06 100.33 3.01 99.31 2.98 101.1 2.97 101.1 3.02 100.93 3.07 100.85 3.18 100.93 3.29 99.6 3.43 101.88 3.61 101.81 3.74 102.38 3.87 102.74 3.88 102.82 4.09 101.72 4.19 103.47 4.2 102.98 4.29 102.68 4.37 102.9 4.47 103.03 4.61 101.29 4.65 103.69 4.69 103.68 4.82 104.2 4.86 104.08 4.87 104.16 5.01 103.05 5.03 104.66 5.13 104.46 5.18 104.95 5.21 105.85 5.26 106.23 5.25 104.86 5.2 107.44 5.16 108.23 5.19 108.45 5.39 109.39 5.58 110.15 5.76 109.13 5.89 110.28 5.98 110.17 6.02 109.99 5.62 109.26 4.87 109.11 4.24 107.06 4.02 109.53 3.74 108.92 3.45 109.24 3.34 109.12 3.21 109 3.12 107.23 3.04 109.49
 
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] = -37.2929573537109 + 0.411442603756895X[t] + 0.526191233621532M1[t] + 0.587269977018786M2[t] + 0.569629309773834M3[t] + 0.501280183358935M4[t] + 1.10054171525739M5[t] + 0.149656699598763M6[t] + 0.124757345940550M7[t] + 0.0265177342483104M8[t] + 0.0251825793843536M9[t] -0.00874660861197813M10[t] + 0.620510011971564M11[t] -0.0558797256602402t + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)-37.292957353710910.678323-3.49240.0010690.000534
X0.4114426037568950.1087143.78460.0004440.000222
M10.5261912336215320.4850981.08470.2837020.141851
M20.5872699770187860.4849061.21110.2320420.116021
M30.5696293097738340.4856731.17290.2468880.123444
M40.5012801833589350.4874491.02840.3091510.154575
M51.100541715257390.5468842.01240.0500560.025028
M60.1496566995987630.4825950.31010.7578790.37894
M70.1247573459405500.4826570.25850.7971890.398595
M80.02651773424831040.4815970.05510.9563270.478164
M90.02518257938435360.4812580.05230.9584950.479248
M10-0.008746608611978130.481049-0.01820.9855720.492786
M110.6205100119715640.5077461.22210.22790.11395
t-0.05587972566024020.023798-2.34810.0232240.011612


Multiple Linear Regression - Regression Statistics
Multiple R0.704343237440981
R-squared0.496099396128842
Adjusted R-squared0.353692703730472
F-TEST (value)3.48368035078749
F-TEST (DF numerator)13
F-TEST (DF denominator)46
p-value0.000849393191327819
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation0.76025094302965
Sum Squared Residuals26.5871488333637


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
13.583.58101784317753-0.00101784317752654
23.523.79605258883056-0.276052588830565
33.453.65258695328670-0.202586953286704
43.363.38846761593422-0.0284676159342199
53.273.40520288936361-0.135202889363610
63.213.2789253200845-0.0689253200844969
73.193.44089737698261-0.250897376982613
83.163.3567232822688-0.196723282268801
93.123.36945364438328-0.249453644383277
103.063.41953521600405-0.359535216004051
113.013.57324065509532-0.56324065509532
122.983.63333317818836-0.653333178188356
132.974.10364468614965-1.13364468614965
143.024.03889846124799-1.01889846124799
153.073.93246266004224-0.862462660042245
163.183.84114921626766-0.661149216267663
173.293.83731235950920-0.547312359509204
183.433.76863675475606-0.338636754756057
193.613.65905669317462-0.0490566931746234
203.743.739459639963570.000540360036429222
213.873.830364096791860.0396359032081438
223.883.773470591435840.106529408564165
234.093.894260622226550.195739377773446
244.193.937895441169320.252104558830683
254.24.20660007328973-0.00660007328973208
264.294.088366309899680.201633690100321
274.374.1053632898210.264636710178997
284.474.034621976234260.435378023765741
294.613.862093651935480.747906348064521
304.653.842791159633160.807208840366845
314.693.757897654277140.932102345722864
324.823.817728470878241.00227152912176
334.863.711140477903211.14885952209679
344.873.654246972547191.21575302745281
355.013.770922577300341.23907742269966
365.033.756955431717141.27304456828286
375.134.144978418927050.98502158107295
385.184.351784312504950.828215687495052
395.214.648562262980960.561437737019043
405.264.680681600333440.579318399666558
415.254.660387039424710.58961296057529
425.24.715144215798630.48485578420137
435.164.959404793448130.200595206551874
445.194.895802828922160.294197171077837
455.395.225343995929450.164656004070553
465.585.448231461128120.131768538871883
475.765.601936900219380.158063099780618
485.895.398706156908010.49129384309199
495.985.823758978456040.156241021543957
506.025.754898327516810.265101672483186
515.625.381024833869090.238975166130909
524.875.19507959123042-0.325079591230416
534.244.895004059767-0.655004059766998
544.024.90450254972766-0.88450254972766
553.744.5727434821175-0.8327434821175
563.454.55028577796722-1.10028577796722
573.344.4436977849922-1.10369778499221
583.214.3045157588848-1.09451575888480
593.124.1496392451584-1.02963924515840
603.044.40310979201718-1.36310979201718


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
170.03423929329123690.06847858658247380.965760706708763
180.02764791298313190.05529582596626370.972352087016868
190.04009487144343820.08018974288687650.959905128556562
200.05096055317792460.1019211063558490.949039446822075
210.07231547935310140.1446309587062030.927684520646899
220.0951073457882250.190214691576450.904892654211775
230.1604750684866060.3209501369732130.839524931513394
240.2711386531964030.5422773063928050.728861346803597
250.3948007681408520.7896015362817040.605199231859148
260.5692727327865920.8614545344268160.430727267213408
270.8364629062126310.3270741875747380.163537093787369
280.9670560119394940.06588797612101280.0329439880605064
290.9855430645848470.02891387083030680.0144569354151534
300.990874760630190.01825047873961810.00912523936980903
310.9874518297514380.02509634049712430.0125481702485622
320.9799600444464510.04007991110709730.0200399555535486
330.9637838056930830.07243238861383440.0362161943069172
340.9459571945931360.1080856108137280.054042805406864
350.9203850509508720.1592298980982560.0796149490491282
360.9119370230447030.1761259539105930.0880629769552965
370.8801788756048960.2396422487902090.119821124395104
380.8108341437699310.3783317124601380.189165856230069
390.941704464329130.1165910713417380.0582955356708691
400.942446778792180.1151064424156410.0575532212078205
410.8922445368199760.2155109263600470.107755463180024
420.8019311472906570.3961377054186870.198068852709343
430.7557598240210960.4884803519578080.244240175978904


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level00OK
5% type I error level40.148148148148148NOK
10% type I error level90.333333333333333NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258560176gm6ac52s0lxjdvb/10hygu1258559938.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258560176gm6ac52s0lxjdvb/10hygu1258559938.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258560176gm6ac52s0lxjdvb/1dvlm1258559938.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258560176gm6ac52s0lxjdvb/1dvlm1258559938.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258560176gm6ac52s0lxjdvb/285r21258559938.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258560176gm6ac52s0lxjdvb/285r21258559938.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258560176gm6ac52s0lxjdvb/3m3up1258559938.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/18/t1258560176gm6ac52s0lxjdvb/4h29u1258559938.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/18/t1258560176gm6ac52s0lxjdvb/5qmo61258559938.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/18/t1258560176gm6ac52s0lxjdvb/6qqpz1258559938.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/18/t1258560176gm6ac52s0lxjdvb/7k89k1258559938.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/18/t1258560176gm6ac52s0lxjdvb/8sqjc1258559938.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258560176gm6ac52s0lxjdvb/8sqjc1258559938.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t1258560176gm6ac52s0lxjdvb/9iigt1258559938.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t1258560176gm6ac52s0lxjdvb/9iigt1258559938.ps (open in new window)


 
Parameters (Session):
 
Parameters (R input):
par1 = 1 ; par2 = Include Monthly Dummies ; par3 = 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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