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Multiple Linear Regression

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

Title produced by software: Multiple Regression

Date of computation: Sun, 02 Dec 2007 04:37:30 -0700

Cite this page as follows:

Statistical Computations at FreeStatistics.org, Office for Research Development and Education, URL http://www.freestatistics.org/blog/date/2007/Dec/02/t1196594920e26p5rf3axsgwsr.htm/, Retrieved Sun, 02 Dec 2007 12:28:51 +0100

User-defined keywords:

Multiple Linear Regression

Dataseries X:

» Textbox « » Textfile « » CSV «

100.6 115.9 59.7 125 96.1 112.9 58.2 121.7 110 126.3 75.3 134.3 108.2 116.8 69 124.3 106.9 112 66.1 119.1 117.2 129.7 77.5 137.8 105.2 113.6 69.3 120.5 106.3 115.7 70.2 122.7 95.9 119.5 70.2 127.2 107.5 125.8 78.2 133.2 113 129.6 85.4 136.3 111.4 128 82.4 134.9 95.5 112.8 61.2 120.9 90.3 101.6 52.2 109.4 110.8 123.9 85.3 129.6 107.1 118.8 79.9 124.7 101.4 109.1 72.2 114.6 112.9 130.6 85.7 137.4 98.5 112.4 75.5 117.9 100.1 111 69.2 117.4 93.4 116.2 77.6 122 104.4 119.8 85.3 124.8 101.8 117.2 77 123.3 107.9 127.3 89.9 132.8 91.3 107.7 60 115.1 86.6 97.5 54.3 104.2 111.4 120.1 84 125.5 98.4 110.6 69.9 116.8 102.2 111.3 75.1 116.8 103 119.8 81.7 125.5 95.8 105.5 69.9 110.9 96 108.7 68.3 114.9 95.7 128.7 77.3 136.4 106.4 119.5 77.4 125.8 112 121.1 85.3 126.5 116.2 128.4 91 134 93.9 108.8 60.6 116.1 100.5 107.5 57.6 115 112.5 125.6 93.8 130.3 101.2 102.9 78.7 106.5 107.8 107.5 80.3 111.6 114.3 120.4 89.8 125 99.6 104.3 77.5 108.3 98.6 100.6 71.7 105 93.6 121.9 83.2 127.4 99.6 112.7 86.2 116.6 113.1 124.9 100.7 128.6 110.7 123.9 100.8 127.5 88.1 102.2 57.1 108.4 93.1 104.9 62.5 110.8 107.4 109.8 79.7 114.2 99.5 98.9 80.3 101.8 105.6 107.3 92.4 109.8 108.3 112.6 91.8 115.9 99.2 104 85.8 106.9 99.3 110.6 84.2 114.6 107.1 100.8 93.1 105.4 106.9 103.8 101.2 108.1 115.4 117 100.6 118.4 99 108.4 106.7 112.7 100.1 95.5 64 98.4 96.2 96.9 67.5 99.6 96.9 103.9 101 103.9 96.2 101.1 95.5 101.5 91 100.6 97 100.8 99 104.3 103.8 104.5 99 98 95.2 98.2 107.2 99.5 86.7 99.9 110.8 97.4 93.5 97.5 111.1 105.6 102.5 105.7 104.6 117.5 112.3 117.7 94.3 107.4 105.5 107.4 90.7 97.8 75.4 98.4 88.8 91.5 70.4 92 90.9 107.7 108 107.7 90.5 100.1 100 100.2 95.5 96.6 93.3 96.7 103.1 106.8 111.1 106.8 100.6 98 101.1 98 103.1 98.6 98.1 98.6

Text written by user:

Multiple Linear Regression

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 compuational 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

1[t] = + 53.2368533361207 + 0.761056250969949`2`[t] + 0.217444956801800`3`[t] -0.423766010526483`4 `[t] -0.115859441483030t + 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)	53.2368533361207	11.310652	4.7068	1.1e-05	6e-06
`2`	0.761056250969949	0.849475	0.8959	0.373166	0.186583
`3`	0.217444956801800	0.104914	2.0726	0.041646	0.020823
`4 `	-0.423766010526483	0.769568	-0.5507	0.583507	0.291754
t	-0.115859441483030	0.072774	-1.592	0.11558	0.05779

Multiple Linear Regression - Regression Statistics
Multiple R	0.692089938519227
R-squared	0.478988482999547
Adjusted R-squared	0.451201202092857
F-TEST (value)	17.2376881569659
F-TEST (DF numerator)	4
F-TEST (DF denominator)	75
p-value	4.56730875342259e-10
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation	5.5816683947729
Sum Squared Residuals	2336.62665519050

Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or Index	Actuals	Interpolation Forecast	Residuals Prediction Error
1	100.6	101.338125987312	-0.738125987312079
2	96.1	100.011358192454	-3.91135819245363
3	110	108.472509542645	1.52749045735504
4	108.2	103.994372594361	4.20562740563907
5	106.9	101.798436028235	5.10156397176537
6	117.2	109.707720339615	7.49227966038502
7	105.2	102.886958593849	2.31304140615083
8	106.3	103.632732517366	2.66726748263360
9	95.9	104.5019397822	-8.6019397822
10	107.5	108.377698313083	-0.877698313083153
11	113	111.405781681627	1.59421831837322
12	111.4	110.013169782924	1.38683021707649
13	95.5	99.6521463898699	-4.15214638986986
14	90.3	93.9287614473618	-3.62876144736176
15	110.8	109.421811060013	1.37818893998678
16	107.1	106.326815423433	0.773184576566506
17	101.4	101.434420886486	-0.0344208864855695
18	112.9	110.954912717677	1.94508728232308
19	98.5	103.033328154429	-4.53332815442889
20	100.1	100.693969739000	-0.59396973899984
21	93.4	104.412816791274	-11.0128167912738
22	104.4	107.524541191182	-3.12454119118232
23	101.8	104.260791371512	-2.46079137151222
24	107.9	110.610862907567	-2.71086290756729
25	91.3	96.5773551250182	-5.27735512501821
26	86.6	92.0783351846101	-5.47833518461011
27	111.4	106.594246207847	4.80575379215273
28	98.4	99.8691427828248	-1.46914278282476
29	102.2	101.41673649239	0.783263507609949
30	103	105.518227607463	-2.51822760746306
31	95.8	98.1403970405352	-2.34039704053519
32	96	98.4169416291672	-2.41694162916718
33	95.7	106.36824259198	-10.6682425919799
34	106.4	103.764329848834	2.63567015116573
35	112	106.287339360269	5.71266063973116
36	116.2	109.788381725688	6.41161827431192
37	93.9	95.7309046668434	-1.83090466684338
38	100.5	94.4394798402732	6.06052015972685
39	112.5	109.486626016516	3.01337398348386
40	101.2	98.8970018808384	2.30299811916159
41	107.8	100.468706471015	7.33129352898504
42	114.3	106.557735215606	7.7422647843935
43	99.6	98.5911895406374	1.00881045936258
44	98.6	95.7966690558525	2.80333094414747
45	93.6	104.899566127457	-11.2995661274569
46	99.6	103.010996961142	-3.41099696114175
47	113.1	110.247783528800	2.8522164711996
48	110.7	109.858754943607	0.841245056393281
49	88.1	91.819561044893	-3.71956104489298
50	93.1	93.915717822495	-0.815717822494978
51	107.4	99.8282828319656	7.57171716803441
52	99.5	96.8020757595196	2.6979242404804
53	105.6	102.320044719274	3.27995528072595
54	108.3	103.522343769639	4.77765623036088
55	99.2	99.370624923742	-0.170624923742072
56	99.3	100.666826526724	-1.36682652672391
57	107.1	98.926523238115	8.17347676188495
58	106.9	101.710968471215	5.18903152878507
59	115.4	107.145794660031	8.25420533996864
60	99	104.226731956699	-5.22673195669872
61	100.1	91.0682011727952	9.0317988272048
62	96.2	92.2703586188446	3.92964138115538
63	96.9	102.944105141748	-6.04410514174764
64	96.2	100.518379360402	-4.31837936040241
65	91	100.644795436006	-9.64479543600565
66	99	103.255535590416	-4.25553559041568
67	99	99.1447210056433	-0.144721005643339
68	107.2	97.601761589905	9.5982384100951
69	110.8	98.3833481529008	12.4166518470992
70	111.1	102.990273294270	8.10972670572962
71	104.6	108.976751689670	-4.37675168966958
72	94.3	104.060388315561	-9.7603883155606
73	90.7	93.9071897597702	-3.20718975977023
74	88.8	90.621553620537	-1.82155362053702
75	90.9	104.357609455249	-13.4576094552490
76	90.5	99.8964079309286	-9.39640793092863
77	95.5	97.1431514373214	-1.64315143732141
78	103.1	104.380549280486	-1.28054928048642
79	100.6	99.1220861550829	1.47791384491711
80	103.1	98.5562659874605	4.54373401253946

Charts produced by software:

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Parameters:

par1 = 1 ; par2 = Do not include Seasonal 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.

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