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WS 7 2

*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, 14 Nov 2009 06:20:47 -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/14/t12582048826gae5nbck4ewacy.htm/, Retrieved Sat, 14 Nov 2009 14:21:33 +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/14/t12582048826gae5nbck4ewacy.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 «
103,63 100,30 103,64 98,50 103,66 95,10 103,77 93,10 103,88 92,20 103,91 89,00 103,91 86,40 103,92 84,50 104,05 82,70 104,23 80,80 104,30 81,80 104,31 81,80 104,31 82,90 104,34 83,80 104,55 86,20 104,65 86,10 104,73 86,20 104,75 88,80 104,75 89,60 104,76 87,80 104,94 88,30 105,29 88,60 105,38 91,00 105,43 91,50 105,43 95,40 105,42 98,70 105,52 99,90 105,69 98,60 105,72 100,30 105,74 100,20 105,74 100,40 105,74 101,40 105,95 103,00 106,17 109,10 106,34 111,40 106,37 114,10 106,37 121,80 106,36 127,60 106,44 129,90 106,29 128,00 106,23 123,50 106,23 124,00 106,23 127,40 106,23 127,60 106,34 128,40 106,44 131,40 106,44 135,10 106,48 134,00 106,50 144,50 106,57 147,30 106,40 150,90 106,37 148,70 106,25 141,40 106,21 138,90 106,21 139,80 106,24 145,60 106,19 147,90 106,08 148,50 106,13 151,10 106,09 157,50
 
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] = + 102.005813794397 + 0.0322178805113383X[t] -0.268918412522903M1[t] -0.321797749647847M2[t] -0.313103563871682M3[t] -0.22477674310467M4[t] -0.146541763589954M5[t] -0.123144108113834M6[t] -0.140541763589957M7[t] -0.151805564727438M8[t] -0.0577137234751484M9[t] + 0.0380933100964853M10[t] + 0.0367703968692694M11[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)102.0058137943970.492211207.2400
X0.03221788051133830.0034739.27700
M1-0.2689184125229030.402177-0.66870.5069860.253493
M2-0.3217977496478470.401801-0.80090.4272270.213613
M3-0.3131035638716820.401655-0.77950.4395730.219787
M4-0.224776743104670.401841-0.55940.5785670.289283
M5-0.1465417635899540.402231-0.36430.7172520.358626
M6-0.1231441081138340.40235-0.30610.7609090.380455
M7-0.1405417635899570.402231-0.34940.7283460.364173
M8-0.1518055647274380.402098-0.37750.7074770.353738
M9-0.05771372347514840.401975-0.14360.8864490.443225
M100.03809331009648530.4017360.09480.9248590.46243
M110.03677039686926940.4015270.09160.9274240.463712


Multiple Linear Regression - Regression Statistics
Multiple R0.810550729538045
R-squared0.656992485154657
Adjusted R-squared0.569416098385633
F-TEST (value)7.50193641680406
F-TEST (DF numerator)12
F-TEST (DF denominator)47
p-value1.77409201107537e-07
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation0.634801017530965
Sum Squared Residuals18.9396995973424


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
1103.63104.968348797162-1.3383487971616
2103.64104.857477275116-1.21747727511623
3103.66104.756630667154-1.09663066715385
4103.77104.780521726898-1.01052172689818
5103.88104.829760613953-0.949760613952696
6103.91104.750061051793-0.840061051792532
7103.91104.648896906987-0.73889690698693
8103.92104.576419132878-0.6564191328779
9104.05104.612518789210-0.562518789209786
10104.23104.647111849810-0.41711184980987
11104.3104.678006817094-0.378006817094
12104.31104.641236420225-0.331236420224725
13104.31104.407757676264-0.0977576762642939
14104.34104.383874431600-0.0438744315995534
15104.55104.4698915306030.080108469397063
16104.65104.5549965633190.095003436681194
17104.73104.6364533308850.0935466691153425
18104.75104.7436174756900.00638252430973915
19104.75104.751994124623-0.0019941246232085
20104.76104.6827381385650.077261861434686
21104.94104.7929389200730.14706107992672
22105.29104.8984113177980.391588682201693
23105.38104.9744113177980.405588682201686
24105.43104.9537498611850.476250138815298
25105.43104.8104811826560.619518817343982
26105.42104.8639208512180.556079148781503
27105.52104.9112764936080.608723506391727
28105.69104.9577200697110.732279930289457
29105.72105.0907254460950.629274553905467
30105.74105.1109013135200.629098686480477
31105.74105.0999472341460.640052765854332
32105.74105.1209013135200.619098686480474
33105.95105.266541763590.683458236410051
34106.17105.5588778682810.611122131719253
35106.34105.6316560802300.708343919770392
36106.37105.6818739607410.68812603925905
37106.37105.6610332281550.708966771844648
38106.36105.7950175979960.564982402003824
39106.44105.8778129089480.562187091051579
40106.29105.9049257567440.385074243256118
41106.23105.8381802739580.391819726042423
42106.23105.8776868696890.352313130310634
43106.23105.9698300079520.260169992048206
44106.23105.9650097829170.264990217083420
45106.34106.0848759285780.255124071422059
46106.44106.2773366036840.162663396316404
47106.44106.3952198483480.0447801516516683
48106.48106.3230097829170.156990217083416
49106.5106.3923791157630.107620884237263
50106.57106.4297098440700.140290155930452
51106.4106.554388399687-0.154388399686519
52106.37106.571835883329-0.201835883328587
53106.25106.414880335111-0.164880335110538
54106.21106.357733289308-0.147733289308318
55106.21106.369331726292-0.159331726292400
56106.24106.544931632121-0.304931632120679
57106.19106.713124598549-0.523124598549044
58106.08106.828262360427-0.74826236042748
59106.13106.910705936530-0.780705936529747
60106.09107.080129974933-0.990129974933035


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
160.4159265781709520.8318531563419050.584073421829048
170.4588540055755720.9177080111511430.541145994424428
180.719226147672240.5615477046555210.280773852327760
190.9212586923560380.1574826152879230.0787413076439616
200.9803185861743720.03936282765125610.0196814138256281
210.9967056546383640.006588690723271060.00329434536163553
220.999292531533440.001414936933120240.00070746846656012
230.9997651561424920.0004696877150158890.000234843857507945
240.9998793945403650.0002412109192706770.000120605459635339
250.9999866929191872.66141616266778e-051.33070808133389e-05
260.9999987975260252.40494795005038e-061.20247397502519e-06
270.9999997994174834.01165034551103e-072.00582517275551e-07
280.99999988649462.27010801638524e-071.13505400819262e-07
290.9999998903530882.19293823399220e-071.09646911699610e-07
300.9999998855774272.28845145210344e-071.14422572605172e-07
310.9999999067915931.86416814764832e-079.32084073824158e-08
320.999999971441565.71168783493193e-082.85584391746596e-08
330.9999999814115913.7176817294992e-081.8588408647496e-08
340.9999999385605951.22878810943075e-076.14394054715374e-08
350.9999996619986746.76002652834084e-073.38001326417042e-07
360.9999982210009573.5579980849274e-061.7789990424637e-06
370.999995061928629.8761427595945e-064.93807137979725e-06
380.999994149107941.17017841187050e-055.85089205935252e-06
390.9999729875625555.40248748904874e-052.70124374452437e-05
400.9999486289125210.0001027421749580865.13710874790428e-05
410.9998537326242860.0002925347514287160.000146267375714358
420.9994188455090310.001162308981937530.000581154490968766
430.997517878347770.004964243304461630.00248212165223081
440.9971025215438320.005794956912335540.00289747845616777


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level240.827586206896552NOK
5% type I error level250.862068965517241NOK
10% type I error level250.862068965517241NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/14/t12582048826gae5nbck4ewacy/10q3oj1258204842.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/14/t12582048826gae5nbck4ewacy/10q3oj1258204842.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/14/t12582048826gae5nbck4ewacy/14y7t1258204842.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/14/t12582048826gae5nbck4ewacy/2thtu1258204842.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/14/t12582048826gae5nbck4ewacy/2thtu1258204842.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/14/t12582048826gae5nbck4ewacy/3jnsa1258204842.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/14/t12582048826gae5nbck4ewacy/4hljy1258204842.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/14/t12582048826gae5nbck4ewacy/5duos1258204842.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/14/t12582048826gae5nbck4ewacy/6ckft1258204842.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/14/t12582048826gae5nbck4ewacy/767kf1258204842.png (open in new window)
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http://www.freestatistics.org/blog/date/2009/Nov/14/t12582048826gae5nbck4ewacy/8odpk1258204842.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/14/t12582048826gae5nbck4ewacy/8odpk1258204842.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/14/t12582048826gae5nbck4ewacy/9xr321258204842.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/14/t12582048826gae5nbck4ewacy/9xr321258204842.ps (open in new window)


 
Parameters (Session):
par1 = 1 ; par2 = Include Monthly Dummies ; par3 = No Linear Trend ;
 
Parameters (R input):
par1 = 1 ; par2 = Include Monthly 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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