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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 14:10:22 -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/t12585786670nn88cokyorltzd.htm/, Retrieved Wed, 18 Nov 2009 22:11:19 +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/t12585786670nn88cokyorltzd.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 «
115.6 0 111.3 0 114.6 0 137.5 0 83.7 0 106.0 0 123.4 0 126.5 0 120.0 0 141.6 0 90.5 0 96.5 0 113.5 0 120.1 0 123.9 0 144.4 0 90.8 0 114.2 0 138.1 0 135.0 0 131.3 0 144.6 0 101.7 0 108.7 0 135.3 0 124.3 0 138.3 0 158.2 0 93.5 0 124.8 0 154.4 0 152.8 0 148.9 0 170.3 0 124.8 0 134.4 0 154.0 0 147.9 0 168.1 0 175.7 0 116.7 0 140.8 0 164.2 0 173.8 0 167.8 0 166.6 0 135.1 1 158.1 1 151.8 1 166.7 1 165.3 1 187.0 1 125.2 1 144.4 1 181.7 1 175.9 1 166.3 1 181.5 1 121.8 1 134.8 1 162.9 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] = + 89.1350491803279 -3.90549180327873X[t] + 17.4961821493625M1[t] + 17.5919981785064M2[t] + 24.4906885245902M3[t] + 41.9293788706739M4[t] -17.7319307832422M5[t] + 5.24675956284152M6[t] + 30.4854499089253M7[t] + 29.8441402550091M8[t] + 22.8228306010929M9[t] + 35.8015209471767M10[t] -10.6386903460838M11[t] + 1.08130965391621t + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)89.13504918032793.91895122.744600
X-3.905491803278733.358537-1.16290.2507590.125379
M117.49618214936254.4573493.92520.0002820.000141
M217.59199817850644.6779883.76060.0004690.000234
M324.49068852459024.673175.24074e-062e-06
M441.92937887067394.6697868.978900
M5-17.73193078324224.667838-3.79870.0004170.000208
M65.246759562841524.6673291.12410.266660.13333
M730.48544990892534.6682596.530400
M829.84414025500914.6706286.389700
M922.82283060109294.6744324.88251.3e-056e-06
M1035.80152094717674.6796697.650400
M11-10.63869034608384.646838-2.28940.026590.013295
t1.081309653916210.08195813.193500


Multiple Linear Regression - Regression Statistics
Multiple R0.967564799200113
R-squared0.936181640651156
Adjusted R-squared0.918529754022752
F-TEST (value)53.0357836733860
F-TEST (DF numerator)13
F-TEST (DF denominator)47
p-value0
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation7.34615257901844
Sum Squared Residuals2536.40001256831


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
1115.6107.7125409836067.8874590163935
2111.3108.8896666666672.41033333333336
3114.6116.869666666667-2.26966666666668
4137.5135.3896666666672.11033333333327
583.776.80966666666676.89033333333334
6106100.8696666666675.1303333333333
7123.4127.189666666667-3.7896666666667
8126.5127.629666666667-1.12966666666669
9120121.689666666667-1.68966666666666
10141.6135.7496666666675.85033333333333
1190.590.39076502732240.109234972677633
1296.5102.110765027322-5.61076502732237
13113.5120.688256830601-7.1882568306011
14120.1121.865382513661-1.76538251366123
15123.9129.845382513661-5.9453825136612
16144.4148.365382513661-3.96538251366118
1790.889.78538251366121.01461748633879
18114.2113.8453825136610.354617486338808
19138.1140.165382513661-2.06538251366119
20135140.605382513661-5.6053825136612
21131.3134.665382513661-3.3653825136612
22144.6148.725382513661-4.12538251366120
23101.7103.366480874317-1.66648087431694
24108.7115.086480874317-6.38648087431694
25135.3133.6639726775961.63602732240437
26124.3134.841098360656-10.5410983606557
27138.3142.821098360656-4.52109836065573
28158.2161.341098360656-3.14109836065574
2993.5102.761098360656-9.26109836065575
30124.8126.821098360656-2.02109836065573
31154.4153.1410983606561.25890163934428
32152.8153.581098360656-0.781098360655725
33148.9147.6410983606561.25890163934425
34170.3161.7010983606568.59890163934428
35124.8116.3421967213118.4578032786885
36134.4128.0621967213116.33780327868851
37154146.6396885245907.36031147540982
38147.9147.8168142076500.0831857923497378
39168.1155.79681420765012.3031857923497
40175.7174.3168142076501.38318579234973
41116.7115.7368142076500.963185792349718
42140.8139.7968142076501.00318579234973
43164.2166.116814207650-1.91681420765027
44173.8166.5568142076507.24318579234974
45167.8160.6168142076507.18318579234973
46166.6174.676814207650-8.07681420765028
47135.1125.4124207650279.68757923497267
48158.1137.13242076502720.9675792349727
49151.8155.709912568306-3.90991256830602
50166.7156.8870382513669.81296174863387
51165.3164.8670382513660.432961748633893
52187183.3870382513663.61296174863391
53125.2124.8070382513660.392961748633878
54144.4148.867038251366-4.46703825136611
55181.7175.1870382513666.51296174863388
56175.9175.6270382513660.272961748633879
57166.3169.687038251366-3.38703825136612
58181.5183.747038251366-2.24703825136613
59121.8138.388136612022-16.5881366120219
60134.8150.108136612022-15.3081366120219
61162.9168.685628415301-5.78562841530056


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
170.1080333452480840.2160666904961680.891966654751916
180.0415320794214930.0830641588429860.958467920578507
190.04061470338030180.08122940676060360.959385296619698
200.01605325380699800.03210650761399610.983946746193002
210.007163791345118680.01432758269023740.992836208654881
220.003879407396451030.007758814792902070.99612059260355
230.001667880429555420.003335760859110840.998332119570445
240.0009850352102477530.001970070420495510.999014964789752
250.001266569002472070.002533138004944150.998733430997528
260.001410518200754420.002821036401508840.998589481799246
270.001580956729552180.003161913459104360.998419043270448
280.001016159027314970.002032318054629940.998983840972685
290.002647699784217260.005295399568434530.997352300215783
300.001533850234470740.003067700468941470.99846614976553
310.004036451359596670.008072902719193340.995963548640403
320.01145731513562770.02291463027125540.988542684864372
330.0391291026483060.0782582052966120.960870897351694
340.06168880564456660.1233776112891330.938311194355433
350.07772387525729710.1554477505145940.922276124742703
360.1308935385831960.2617870771663920.869106461416804
370.1037624200648140.2075248401296290.896237579935186
380.1021934681323750.2043869362647510.897806531867624
390.1826544109712520.3653088219425040.817345589028748
400.1212498024794140.2424996049588280.878750197520586
410.07085753888927160.1417150777785430.929142461110728
420.04013459483317420.08026918966634830.959865405166826
430.03219480306848560.06438960613697110.967805196931514
440.01783835980533500.03567671961066990.982161640194665


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level100.357142857142857NOK
5% type I error level140.5NOK
10% type I error level190.678571428571429NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/104apj1258578617.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/104apj1258578617.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/1xjr01258578617.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/1xjr01258578617.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/2te1g1258578617.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/2te1g1258578617.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/3jm281258578617.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/3jm281258578617.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/43ynv1258578617.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/43ynv1258578617.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/5r31u1258578617.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/5r31u1258578617.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/6a3oe1258578617.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/6a3oe1258578617.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/77c5x1258578617.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/77c5x1258578617.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/8u7ei1258578617.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/8u7ei1258578617.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/9v67d1258578617.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/18/t12585786670nn88cokyorltzd/9v67d1258578617.ps (open in new window)


 
Parameters (Session):
par1 = 1 ; par2 = Include Monthly Dummies ; par3 = Linear Trend ;
 
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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