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Multiple_Regression

*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, 29 Dec 2009 07:22:22 -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/Dec/29/t12620965963v8429bfieh3l7t.htm/, Retrieved Tue, 29 Dec 2009 15:23:28 +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/Dec/29/t12620965963v8429bfieh3l7t.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:
Paper
 
Dataseries X:
» Textbox « » Textfile « » CSV «
2529 314 2196 318 3202 320 2718 323 2728 325 2354 327 2697 330 2651 331 2067 332 2641 334 2539 334 2294 334 2712 339 2314 345 3092 346 2677 352 2813 355 2668 358 2939 361 2617 363 2231 364 2481 365 2421 366 2408 370 2560 371 2100 371 3315 372 2801 373 2403 373 3024 374 2507 375 2980 375 2211 376 2471 376 2594 377 2452 377 2232 378 2373 379 3127 380 2802 384 2641 389 2787 390 2619 391 2806 392 2193 393 2323 394 2529 394 2412 395 2262 396 2154 397 3230 398 2295 399 2715 400 2733 400 2317 401 2730 401 1913 406 2390 407 2484 423 1960 427
 
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] = + 3255.2677926153 -1.88573019166673X[t] + e[t]


Multiple Linear Regression - Ordinary Least Squares
VariableParameterS.D.T-STAT
H0: parameter = 0
2-tail p-value1-tail p-value
(Intercept)3255.2677926153535.8495476.07500
X-1.885730191666731.443548-1.30630.1966020.098301


Multiple Linear Regression - Regression Statistics
Multiple R0.169058749276525
R-squared0.0285808607069431
Adjusted R-squared0.0118322548570629
F-TEST (value)1.70646207589555
F-TEST (DF numerator)1
F-TEST (DF denominator)58
p-value0.196601812988957
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation309.957469882705
Sum Squared Residuals5572270.7218931


Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or IndexActualsInterpolation
Forecast
Residuals
Prediction Error
125292663.14851243195-134.148512431948
221962655.60559166528-459.605591665281
332022651.83413128195550.165868718053
427182646.1769407069571.8230592930527
527282642.4054803236185.5945196763861
623542638.63401994028-284.634019940280
726972632.9768293652864.0231706347198
826512631.0910991736119.9089008263865
920672629.20536898195-562.205368981947
1026412625.4339085986115.5660914013867
1125392625.43390859861-86.4339085986133
1222942625.43390859861-331.433908598613
1327122616.0052576402895.9947423597203
1423142604.69087649028-290.690876490279
1530922602.80514629861489.194853701387
1626772591.4907651486185.5092348513878
1728132585.83357457361227.166425426388
1826682580.1763839986187.8236160013881
1929392574.51919342361364.480806576388
2026172570.7477330402846.2522669597218
2122312568.86200284861-337.862002848612
2224812566.97627265694-85.9762726569448
2324212565.09054246528-144.090542465278
2424082557.54762169861-149.547621698611
2525602555.661891506944.33810849305561
2621002555.66189150694-455.661891506944
2733152553.77616131528761.223838684722
2828012551.89043112361249.109568876389
2924032551.89043112361-148.890431123611
3030242550.00470093194473.995299068056
3125072548.11897074028-41.1189707402775
3229802548.11897074028431.881029259722
3322112546.23324054861-335.233240548611
3424712546.23324054861-75.2332405486108
3525942544.3475103569449.652489643056
3624522544.34751035694-92.347510356944
3722322542.46178016528-310.461780165277
3823732540.57604997361-167.576049973611
3931272538.69031978194588.309680218056
4028022531.14739901528270.852600984723
4126412521.71874805694119.281251943057
4227872519.83301786528267.166982134723
4326192517.94728767361101.052712326390
4428062516.06155748194289.938442518057
4521932514.17582729028-321.175827290276
4623232512.29009709861-189.290097098610
4725292512.2900970986116.7099029013903
4824122510.40436690694-98.4043669069429
4922622508.51863671528-246.518636715276
5021542506.63290652361-352.632906523609
5132302504.74717633194725.252823668057
5222952502.86144614028-207.861446140276
5327152500.97571594861214.024284051391
5427332500.97571594861232.024284051391
5523172499.08998575694-182.089985756943
5627302499.08998575694230.910014243057
5719132489.66133479861-576.661334798609
5823902487.77560460694-97.7756046069422
5924842457.6039215402726.3960784597255
6019602450.06100077361-490.061000773608


Goldfeld-Quandt test for Heteroskedasticity
p-valuesAlternative Hypothesis
breakpoint indexgreater2-sidedless
50.8083562012735150.383287597452970.191643798726485
60.8179567004105020.3640865991789950.182043299589498
70.7120789305239460.5758421389521090.287921069476054
80.591851408632060.8162971827358790.408148591367940
90.7487928361869140.5024143276261720.251207163813086
100.6680697345789130.6638605308421750.331930265421087
110.5709491245582320.8581017508835370.429050875441768
120.5431696118063430.9136607763873140.456830388193657
130.4910472609248850.982094521849770.508952739075115
140.4502642950464520.9005285900929040.549735704953548
150.6231667395088280.7536665209823430.376833260491172
160.5378660527435060.9242678945129880.462133947256494
170.4675052478571870.9350104957143750.532494752142813
180.3830419730923490.7660839461846990.61695802690765
190.3478286697551780.6956573395103560.652171330244822
200.2838564547384770.5677129094769540.716143545261523
210.3673854995353210.7347709990706410.632614500464679
220.3136774620338960.6273549240677920.686322537966104
230.2768946656338760.5537893312677520.723105334366124
240.2414456005536410.4828912011072820.758554399446359
250.1867324592223450.3734649184446890.813267540777655
260.2924062853682880.5848125707365760.707593714631712
270.6076927340934740.7846145318130530.392307265906526
280.554873886286990.890252227426020.44512611371301
290.521659710194150.956680579611700.47834028980585
300.5639170465252570.8721659069494860.436082953474743
310.5001395392450880.9997209215098240.499860460754912
320.5196376345230070.9607247309539860.480362365476993
330.5793711812690680.8412576374618640.420628818730932
340.5245908055881270.9508183888237470.475409194411873
350.4491371312631510.8982742625263030.550862868736849
360.4036083748835010.8072167497670010.5963916251165
370.4844258192820470.9688516385640950.515574180717953
380.5095183999896630.9809632000206740.490481600010337
390.5856673866392260.8286652267215490.414332613360774
400.5240654242723980.9518691514552040.475934575727602
410.4424487978779120.8848975957558230.557551202122088
420.3930191655344580.7860383310689160.606980834465542
430.316547905692960.633095811385920.68345209430704
440.2916787421265690.5833574842531390.708321257873431
450.3107683717141500.6215367434283010.68923162828585
460.272976102028060.545952204056120.72702389797194
470.2020172216269080.4040344432538160.797982778373092
480.1521245738337100.3042491476674210.84787542616629
490.1428746229816900.2857492459633810.85712537701831
500.1942759915730680.3885519831461370.805724008426931
510.5049970639891810.9900058720216380.495002936010819
520.4515264027576580.9030528055153160.548473597242342
530.3715016084404730.7430032168809460.628498391559527
540.3321019595527650.664203919105530.667898040447235
550.2131821875251700.4263643750503400.78681781247483


Meta Analysis of Goldfeld-Quandt test for Heteroskedasticity
Description# significant tests% significant testsOK/NOK
1% type I error level00OK
5% type I error level00OK
10% type I error level00OK
 
Charts produced by software:
http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/109kg11262096536.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/109kg11262096536.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/1xt5j1262096536.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/1xt5j1262096536.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/2pzg71262096536.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/2pzg71262096536.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/3ms731262096536.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/3ms731262096536.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/4v8za1262096536.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/4v8za1262096536.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/5qsnr1262096536.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/5qsnr1262096536.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/6i5131262096536.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/6i5131262096536.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/7kff01262096536.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/7kff01262096536.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/89i8r1262096536.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/89i8r1262096536.ps (open in new window)


http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/99ljd1262096536.png (open in new window)
http://www.freestatistics.org/blog/date/2009/Dec/29/t12620965963v8429bfieh3l7t/99ljd1262096536.ps (open in new window)


 
Parameters (Session):
 
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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