Home » date » 2008 » Nov » 26 »

Seatbelt Law Question 3 b

*Unverified author*

R Software Module: rwasp_multipleregression.wasp (opens new window with default values)

Title produced by software: Multiple Regression

Date of computation: Wed, 26 Nov 2008 11:17:22 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2.htm/, Retrieved Wed, 26 Nov 2008 18:18:17 +0000

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/2008/Nov/26/t1227723488jkozw8yce1vpwx2.htm/},
    year = {2008},
}
@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 = {2008},
    note = {{ISBN} 3-900051-07-0},
    url = {http://www.R-project.org},
}

Family? (F = Feedback message, R = changed R code, M = changed R Module, P = changed Parameters, D = changed Data)

Feedback Forum:

Post a new message

Original text written by user:

IsPrivate?

No (this computation is public)

User-defined keywords:

Dataseries X:

» Textbox « » Textfile « » CSV «

101 0 98.7 1 105.1 0 98.4 1 101.7 0 102.9 0 92.2 1 94.9 1 92.8 1 98.5 1 94.3 1 87.4 1 103.4 0 101.2 0 109.6 0 111.9 0 108.9 0 105.6 0 107.8 0 97.5 1 102.4 0 105.6 0 99.8 1 96.2 1 113.1 0 107.4 0 116.8 0 112.9 0 105.3 0 109.3 0 107.9 0 101.1 0 114.7 0 116.2 0 108.4 0 113.4 0 108.7 0 112.6 0 124.2 0 114.9 0 110.5 0 121.5 0 118.1 0 111.7 0 132.7 0 119 0 116.7 0 120.1 0 113.4 0 106.6 0 116.3 0 112.6 0 111.6 0 125.1 0 110.7 0 109.6 0 114.2 0 113.4 0 116 0 109.6 0 117.8 0 115.8 0 125.3 0 113 0 120.5 0 116.6 0 111.8 0 115.2 0 118.6 0 122.4 0 116.4 0 114.5 0 119.8 0 115.8 0 127.8 0 118.8 0 119.7 0 118.6 0 120.8 0 115.9 0 109.7 0 114.8 0 116.2 0 112.2 0

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 Input	view raw input (R code)
Raw Output	view raw output of R engine
Computing time	3 seconds
R Server	'Gwilym Jenkins' @ 72.249.127.135

Multiple Linear Regression - Estimated Regression Equation

y[t] = + 100.940779220779 -9.5078787878788x[t] + 2.8386219336219M1[t] + 1.27239538239537M2[t] + 9.28963203463204M3[t] + 4.36626262626262M4[t] + 2.19778499278498M5[t] + 5.05900432900432M6[t] + 1.89277777777777M7[t] -0.287734487734494M8[t] + 3.75807359307359M9[t] + 4.24786435786436M10[t] + 2.25306637806637M11[t] + 0.195923520923521t + e[t]

Multiple Linear Regression - Ordinary Least Squares
Variable	Parameter	S.D.	T-STAT H0: parameter = 0	2-tail p-value	1-tail p-value
(Intercept)	100.940779220779	2.446356	41.2617	0	0
x	-9.5078787878788	1.974079	-4.8164	8e-06	4e-06
M1	2.8386219336219	2.673044	1.0619	0.291911	0.145956
M2	1.27239538239537	2.606274	0.4882	0.626931	0.313465
M3	9.28963203463204	2.661656	3.4902	0.00084	0.00042
M4	4.36626262626262	2.598315	1.6804	0.097333	0.048667
M5	2.19778499278498	2.651261	0.829	0.409946	0.204973
M6	5.05900432900432	2.646439	1.9116	0.060018	0.030009
M7	1.89277777777777	2.588333	0.7313	0.467053	0.233527
M8	-0.287734487734494	2.56365	-0.1122	0.910957	0.455479
M9	3.75807359307359	2.582993	1.4549	0.150158	0.075079
M10	4.24786435786436	2.58072	1.646	0.104247	0.052123
M11	2.25306637806637	2.561629	0.8795	0.382116	0.191058
t	0.195923520923521	0.026275	7.4567	0	0

Multiple Linear Regression - Regression Statistics
Multiple R	0.86328875740995
R-squared	0.745267478670416
Adjusted R-squared	0.697960010423493
F-TEST (value)	15.7536961136975
F-TEST (DF numerator)	13
F-TEST (DF denominator)	70
p-value	6.66133814775094e-16
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation	4.7921172389056
Sum Squared Residuals	1607.50713419913

Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or Index	Actuals	Interpolation Forecast	Residuals Prediction Error
1	101	103.975324675325	-2.97532467532478
2	98.7	93.0971428571429	5.60285714285715
3	105.1	110.818181818182	-5.7181818181818
4	98.4	96.5828571428571	1.81714285714287
5	101.7	104.118181818182	-2.41818181818184
6	102.9	107.175324675325	-4.27532467532465
7	92.2	94.6971428571429	-2.49714285714285
8	94.9	92.7125541125541	2.18744588744589
9	92.8	96.9542857142857	-4.15428571428571
10	98.5	97.64	0.860000000000004
11	94.3	95.8411255411255	-1.54112554112554
12	87.4	93.7839826839827	-6.38398268398268
13	103.4	106.326406926407	-2.92640692640690
14	101.2	104.956103896104	-3.7561038961039
15	109.6	113.169264069264	-3.56926406926408
16	111.9	108.441818181818	3.45818181818182
17	108.9	106.469264069264	2.43073593073594
18	105.6	109.526406926407	-3.92640692640693
19	107.8	106.556103896104	1.2438961038961
20	97.5	95.0636363636364	2.43636363636364
21	102.4	108.813246753247	-6.41324675324675
22	105.6	109.498961038961	-3.89896103896105
23	99.8	98.1922077922078	1.60779220779221
24	96.2	96.135064935065	0.064935064935066
25	113.1	108.677489177489	4.42251082251084
26	107.4	107.307186147186	0.0928138528138628
27	116.8	115.520346320346	1.27965367965368
28	112.9	110.792900432900	2.10709956709957
29	105.3	108.820346320346	-3.52034632034632
30	109.3	111.877489177489	-2.57748917748918
31	107.9	108.907186147186	-1.00718614718614
32	101.1	106.922597402597	-5.82259740259741
33	114.7	111.164329004329	3.53567099567100
34	116.2	111.850043290043	4.34995670995671
35	108.4	110.051168831169	-1.65116883116882
36	113.4	107.994025974026	5.40597402597403
37	108.7	111.028571428571	-2.32857142857141
38	112.6	109.658268398268	2.9417316017316
39	124.2	117.871428571429	6.32857142857143
40	114.9	113.143982683983	1.75601731601732
41	110.5	111.171428571429	-0.671428571428568
42	121.5	114.228571428571	7.27142857142857
43	118.1	111.258268398268	6.84173160173159
44	111.7	109.273679653680	2.42632034632035
45	132.7	113.515411255411	19.1845887445887
46	119	114.201125541126	4.79887445887446
47	116.7	112.402251082251	4.29774891774892
48	120.1	110.345108225108	9.75489177489177
49	113.4	113.379653679654	0.0203463203463458
50	106.6	112.009350649351	-5.40935064935065
51	116.3	120.222510822511	-3.92251082251083
52	112.6	115.495064935065	-2.89506493506494
53	111.6	113.522510822511	-1.92251082251082
54	125.1	116.579653679654	8.52034632034632
55	110.7	113.609350649351	-2.90935064935065
56	109.6	111.624761904762	-2.02476190476191
57	114.2	115.866493506494	-1.66649350649351
58	113.4	116.552207792208	-3.15220779220779
59	116	114.753333333333	1.24666666666667
60	109.6	112.696190476190	-3.09619047619048
61	117.8	115.730735930736	2.06926406926409
62	115.8	114.360432900433	1.4395670995671
63	125.3	122.573593073593	2.72640692640692
64	113	117.846147186147	-4.84614718614719
65	120.5	115.873593073593	4.62640692640693
66	116.6	118.930735930736	-2.33073593073594
67	111.8	115.960432900433	-4.16043290043290
68	115.2	113.975844155844	1.22415584415585
69	118.6	118.217575757576	0.382424242424234
70	122.4	118.90329004329	3.49670995670996
71	116.4	117.104415584416	-0.704415584415578
72	114.5	115.047272727273	-0.547272727272732
73	119.8	118.081818181818	1.71818181818183
74	115.8	116.711515151515	-0.911515151515152
75	127.8	124.924675324675	2.87532467532467
76	118.8	120.197229437229	-1.39722943722944
77	119.7	118.224675324675	1.47532467532468
78	118.6	121.281818181818	-2.68181818181819
79	120.8	118.311515151515	2.48848484848484
80	115.9	116.326926406926	-0.426926406926401
81	109.7	120.568658008658	-10.868658008658
82	114.8	121.254372294372	-6.4543722943723
83	116.2	119.455497835498	-3.25549783549783
84	112.2	117.398354978355	-5.19835497835498

Charts produced by software:

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/1hwpt1227723438.png (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/1hwpt1227723438.ps (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/2fl1v1227723438.png (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/2fl1v1227723438.ps (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/3zv4o1227723438.png (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/3zv4o1227723438.ps (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/4o9op1227723438.png (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/4o9op1227723438.ps (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/5sy6d1227723438.png (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/5sy6d1227723438.ps (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/634or1227723438.png (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/634or1227723438.ps (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/71ake1227723438.png (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/71ake1227723438.ps (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/8rnm71227723438.png (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/8rnm71227723438.ps (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/9gx1u1227723438.png (open in new window)

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2008/Nov/26/t1227723488jkozw8yce1vpwx2/9gx1u1227723438.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)

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))

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')

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()

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')

This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 License.

Software written by Ed van Stee & Patrick Wessa

Disclaimer

Information provided on this web site is provided "AS IS" without warranty of any kind, either express or implied, including, without limitation, warranties of merchantability, fitness for a particular purpose, and noninfringement. We use reasonable efforts to include accurate and timely information and periodically update the information, and software without notice. However, we make no warranties or representations as to the accuracy or completeness of such information (or software), and we assume no liability or responsibility for errors or omissions in the content of this web site, or any software bugs in online applications. Your use of this web site is AT YOUR OWN RISK. Under no circumstances and under no legal theory shall we be liable to you or any other person for any direct, indirect, special, incidental, exemplary, or consequential damages arising from your access to, or use of, this web site.

We may request personal information to be submitted to our servers in order to be able to:

personalize online software applications according to your needs
enforce strict security rules with respect to the data that you upload (e.g. statistical data)
manage user sessions of online applications
alert you about important changes or upgrades in resources or applications

We NEVER allow other companies to directly offer registered users information about their products and services. Banner references and hyperlinks of third parties NEVER contain any personal data of the visitor.

We do NOT sell, nor transmit by any means, personal information, nor statistical data series uploaded by you to third parties.

We carefully protect your data from loss, misuse, alteration, and destruction. However, at any time, and under any circumstance you are solely responsible for managing your passwords, and keeping them secret.

We store a unique ANONYMOUS USER ID in the form of a small 'Cookie' on your computer. This allows us to track your progress when using this website which is necessary to create state-dependent features. The cookie is used for NO OTHER PURPOSE. At any time you may opt to disallow cookies from this website - this will not affect other features of this website.

We examine cookies that are used by third-parties (banner and online ads) very closely: abuse from third-parties automatically results in termination of the advertising contract without refund. We have very good reason to believe that the cookies that are produced by third parties (banner ads) do NOT cause any privacy or security risk.

FreeStatistics.org is safe. There is no need to download any software to use the applications and services contained in this website. Hence, your system's security is not compromised by their use, and your personal data - other than data you submit in the account application form, and the user-agent information that is transmitted by your browser - is never transmitted to our servers.

As a general rule, we do not log on-line behavior of individuals (other than normal logging of webserver 'hits'). However, in cases of abuse, hacking, unauthorized access, Denial of Service attacks, illegal copying, hotlinking, non-compliance with international webstandards (such as robots.txt), or any other harmful behavior, our system engineers are empowered to log, track, identify, publish, and ban misbehaving individuals - even if this leads to ban entire blocks of IP addresses, or disclosing user's identity.

FreeStatistics.org is powered by