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review

*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: Tue, 24 Nov 2009 02:26:23 -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/24/t1259054936kzcuudc7s16g0p1.htm/, Retrieved Tue, 24 Nov 2009 10:29:08 +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/24/t1259054936kzcuudc7s16g0p1.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 «
3,2 0 1,9 0 0 0 0,6 0 0,2 0 0,9 0 2,4 0 4,7 0 9,4 0 12,5 0 15,8 0 18,2 0 16,8 1 17,3 1 19,3 1 17,9 1 20,2 1 18,7 1 20,1 1 18,2 1 18,4 1 18,2 1 18,9 1 19,9 1 21,3 1 20 1 19,5 1 19,6 1 20,9 1 21 1 19,9 1 19,6 1 20,9 1 21,7 1 22,9 1 21,5 1 21,3 1 23,5 1 21,6 1 24,5 1 22,2 1 23,5 1 20,9 1 20,7 1 18,1 1 17,1 1 14,8 1 13,8 1 15,2 1 16 1 17,6 1 15 1 15 1 16,3 1 19,4 1 21,3 1 20,5 1 21,1 1 21,6 1 22,6 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 time5 seconds
R Server'Gwilym Jenkins' @ 72.249.127.135


Multiple Linear Regression - Estimated Regression Equation
Y[t] = + 5.81666666666664 + 13.6895833333334X[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)5.816666666666641.0432635.57551e-060
X13.68958333333341.16640311.736600


Multiple Linear Regression - Regression Statistics
Multiple R0.838867982627093
R-squared0.703699492276849
Adjusted R-squared0.698590862833346
F-TEST (value)137.747217734073
F-TEST (DF numerator)1
F-TEST (DF denominator)58
p-value0
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation3.61396904664594
Sum Squared Residuals757.524791666667


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
13.25.81666666666671-2.61666666666671
21.95.8166666666667-3.9166666666667
305.81666666666666-5.81666666666666
40.65.81666666666666-5.21666666666666
50.25.81666666666666-5.61666666666666
60.95.81666666666666-4.91666666666666
72.45.81666666666666-3.41666666666666
84.75.81666666666666-1.11666666666666
99.45.816666666666663.58333333333334
1012.55.816666666666666.68333333333334
1115.85.816666666666669.98333333333334
1218.25.8166666666666612.3833333333333
1316.819.50625-2.70625
1417.319.50625-2.20625
1519.319.50625-0.206249999999999
1617.919.50625-1.60625
1720.219.506250.69375
1818.719.50625-0.80625
1920.119.506250.593750000000002
2018.219.50625-1.30625
2118.419.50625-1.10625000000000
2218.219.50625-1.30625
2318.919.50625-0.606250000000001
2419.919.506250.393749999999999
2521.319.506251.79375
262019.506250.49375
2719.519.50625-0.00624999999999976
2819.619.506250.0937500000000017
2920.919.506251.39375
302119.506251.49375
3119.919.506250.393749999999999
3219.619.506250.0937500000000017
3320.919.506251.39375
3421.719.506252.19375
3522.919.506253.39375
3621.519.506251.99375
3721.319.506251.79375
3823.519.506253.99375
3921.619.506252.09375
4024.519.506254.99375
4122.219.506252.69375
4223.519.506253.99375
4320.919.506251.39375
4420.719.506251.19375
4518.119.50625-1.40625000000000
4617.119.50625-2.40625
4714.819.50625-4.70625
4813.819.50625-5.70625
4915.219.50625-4.30625
501619.50625-3.50625
5117.619.50625-1.90625
521519.50625-4.50625
531519.50625-4.50625
5416.319.50625-3.20625
5519.419.50625-0.106250000000001
5621.319.506251.79375
5720.519.506250.99375
5821.119.506251.59375000000000
5921.619.506252.09375
6022.619.506253.09375


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
50.1231505116097140.2463010232194280.876849488390286
60.06077206261919580.1215441252383920.939227937380804
70.04680953112788930.09361906225577860.95319046887211
80.165953326839690.331906653679380.83404667316031
90.8405656769795090.3188686460409830.159434323020491
100.9946589135756920.01068217284861680.00534108642430838
110.9999358702795950.0001282594408101056.41297204050523e-05
120.9999989574100452.0851799097921e-061.04258995489605e-06
130.9999978537390724.29252185671588e-062.14626092835794e-06
140.9999953802022739.23959545498073e-064.61979772749037e-06
150.9999891516272742.16967454527043e-051.08483727263521e-05
160.999976482427974.70351440596933e-052.35175720298467e-05
170.9999509395247229.81209505559624e-054.90604752779812e-05
180.9998938949886690.0002122100226620640.000106105011331032
190.9997851421357370.000429715728525140.00021485786426257
200.999588534624180.0008229307516405840.000411465375820292
210.99922488405890.001550231882199670.000775115941099837
220.9986199008900760.00276019821984770.00138009910992385
230.9974998670868820.005000265826237050.00250013291311852
240.9956363924673870.008727215065225920.00436360753261296
250.9935518950571710.01289620988565710.00644810494282857
260.9892968546090210.02140629078195800.0107031453909790
270.9825737050444730.03485258991105310.0174262949555265
280.9725371948437790.05492561031244210.0274628051562210
290.960576235127770.07884752974445970.0394237648722299
300.944998264727760.1100034705444800.0550017352722401
310.920346355150820.159307289698360.07965364484918
320.8876246025491180.2247507949017640.112375397450882
330.8525562124657460.2948875750685080.147443787534254
340.8218084163551140.3563831672897720.178191583644886
350.815570809684270.3688583806314610.184429190315731
360.7774070661530030.4451858676939930.222592933846996
370.731640999381930.5367180012361390.268359000618070
380.7527254675469860.4945490649060280.247274532453014
390.7150064017324490.5699871965351020.284993598267551
400.8002444569043260.3995110861913490.199755543095674
410.7952415262567260.4095169474865490.204758473743275
420.8515397505099420.2969204989801170.148460249490058
430.8281975321635480.3436049356729030.171802467836452
440.801514951848630.3969700963027390.198485048151370
450.7364371908548810.5271256182902380.263562809145119
460.6691609229567590.6616781540864820.330839077043241
470.6765180380131020.6469639239737970.323481961986898
480.7530277378049640.4939445243900720.246972262195036
490.7599132715476880.4801734569046240.240086728452312
500.7376441459154630.5247117081690730.262355854084537
510.653615238324420.6927695233511610.346384761675581
520.7282475267751110.5435049464497780.271752473224889
530.875325390994990.2493492180100210.124674609005010
540.9811583072577450.03768338548450930.0188416927422547
550.9817346285037340.03653074299253250.0182653714962663


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level140.274509803921569NOK
5% type I error level200.392156862745098NOK
10% type I error level230.450980392156863NOK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/1082mr1259054777.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/1082mr1259054777.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/1ys341259054777.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/1ys341259054777.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/26fb91259054777.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/26fb91259054777.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/3u69u1259054777.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/3u69u1259054777.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/47pev1259054777.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/47pev1259054777.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/51fz11259054777.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/51fz11259054777.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/63epa1259054777.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/63epa1259054777.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/7dzge1259054777.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/7dzge1259054777.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/8x6sl1259054777.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/8x6sl1259054777.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/97gxw1259054777.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Nov/24/t1259054936kzcuudc7s16g0p1/97gxw1259054777.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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