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Question 1: How many people are saved from death or serious injury by the seatbelt law?

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

Title produced by software: Multiple Regression

Date of computation: Thu, 15 Nov 2007 05:13:11 -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/Nov/15/t1195129421xwvo03m3ia1eb4w.htm/, Retrieved Thu, 15 Nov 2007 13:23:52 +0100

User-defined keywords:

Dataseries X:

» Textbox « » Textfile « » CSV «

1687 0 1508 0 1507 0 1385 0 1632 0 1511 0 1559 0 1630 0 1579 0 1653 0 2152 0 2148 0 1752 0 1765 0 1717 0 1558 0 1575 0 1520 0 1805 0 1800 0 1719 0 2008 0 2242 0 2478 0 2030 0 1655 0 1693 0 1623 0 1805 0 1746 0 1795 0 1926 0 1619 0 1992 0 2233 0 2192 0 2080 0 1768 0 1835 0 1569 0 1976 0 1853 0 1965 0 1689 0 1778 0 1976 0 2397 0 2654 0 2097 0 1963 0 1677 0 1941 0 2003 0 1813 0 2012 0 1912 0 2084 0 2080 0 2118 0 2150 0 1608 0 1503 0 1548 0 1382 0 1731 0 1798 0 1779 0 1887 0 2004 0 2077 0 2092 0 2051 0 1577 0 1356 0 1652 0 1382 0 1519 0 1421 0 1442 0 1543 0 1656 0 1561 0 1905 0 2199 0 1473 0 1655 0 1407 0 1395 0 1530 0 1309 0 1526 0 1327 0 1627 0 1748 0 1958 0 2274 0 1648 0 1401 0 1411 0 1403 0 1394 0 1520 0 1528 0 1643 0 1515 0 1685 0 2000 0 2215 0 1956 0 1462 0 1563 0 1459 0 1446 0 1622 0 1657 0 1638 0 1643 0 1683 0 2050 0 2262 0 1813 0 1445 0 1762 0 1461 0 1556 0 1431 0 1427 0 1554 0 1645 0 1653 0 2016 0 2207 0 1665 0 1361 0 1506 0 1360 0 1453 0 1522 0 1460 0 1552 0 1548 0 1827 0 1737 0 1941 0 1474 0 1458 0 1542 0 1404 0 1522 0 1385 0 1641 0 1510 0 1681 0 1938 0 1868 0 1726 0 1456 0 1445 0 1456 0 1365 0 1487 0 1558 0 1488 0 1684 0 1594 0 1850 0 1998 0 2079 0 1494 0 1057 1 1218 1 1168 1 1236 1 1076 1 1174 1 1139 1 1427 1 1487 1 1483 1 1513 1 1357 1 1165 1 1282 1 1110 1 1297 1 1185 1 1222 1 1284 1 1444 1 1575 1 1737 1 1763 1

Text written by user:

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

Y[t] = + 2324.06337310277 -226.385033602657X[t] -451.374973256309M1[t] -635.461053323771M2[t] -583.133697991392M3[t] -694.556342659014M4[t] -555.478987326639M5[t] -609.464131994259M6[t] -532.074276661885M7[t] -515.434421329508M8[t] -460.85706599713M9[t] -319.717210664754M10[t] -118.389855332377M11[t] -1.76485533237686t + 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)	2324.06337310277	44.029939	52.7837	0	0
X	-226.385033602657	41.037226	-5.5166	0	0
M1	-451.374973256309	53.942919	-8.3676	0	0
M2	-635.461053323771	53.941479	-11.7806	0	0
M3	-583.133697991392	53.931287	-10.8125	0	0
M4	-694.556342659014	53.922166	-12.8807	0	0
M5	-555.478987326639	53.914117	-10.303	0	0
M6	-609.464131994259	53.907141	-11.3058	0	0
M7	-532.074276661885	53.901237	-9.8713	0	0
M8	-515.434421329508	53.896405	-9.5634	0	0
M9	-460.85706599713	53.892648	-8.5514	0	0
M10	-319.717210664754	53.889963	-5.9328	0	0
M11	-118.389855332377	53.888353	-2.1969	0.029316	0.014658
t	-1.76485533237686	0.240551	-7.3367	0	0

Multiple Linear Regression - Regression Statistics
Multiple R	0.861322441473346
R-squared	0.741876348185605
Adjusted R-squared	0.723024620805902
F-TEST (value)	39.3532291891914
F-TEST (DF numerator)	13
F-TEST (DF denominator)	178
p-value	0
Multiple Linear Regression - Residual Statistics
Residual Standard Deviation	152.417759557721
Sum Squared Residuals	4135148.87028996

Multiple Linear Regression - Actuals, Interpolation, and Residuals
Time or Index	Actuals	Interpolation Forecast	Residuals Prediction Error
1	1687	1870.92354451406	-183.923544514060
2	1508	1685.07260911425	-177.072609114252
3	1507	1735.63510911425	-228.635109114254
4	1385	1622.44760911424	-237.447609114242
5	1632	1759.76010911425	-127.760109114251
6	1511	1704.01010911425	-193.010109114250
7	1559	1779.63510911425	-220.635109114248
8	1630	1794.51010911426	-164.510109114256
9	1579	1847.32260911425	-268.322609114246
10	1653	1986.69760911425	-333.697609114249
11	2152	2186.26010911425	-34.2601091142503
12	2148	2302.88510911425	-154.885109114249
13	1752	1849.74528052556	-97.7452805255622
14	1765	1663.89434512573	101.105654874272
15	1717	1714.45684512573	2.54315487427317
16	1558	1601.26934512573	-43.2693451257275
17	1575	1738.58184512573	-163.581845125727
18	1520	1682.83184512573	-162.831845125727
19	1805	1758.45684512573	46.5431548742728
20	1800	1773.33184512573	26.6681548742733
21	1719	1826.14434512573	-107.144345125727
22	2008	1965.51934512573	42.4806548742729
23	2242	2165.08184512573	76.918154874273
24	2478	2281.70684512573	196.293154874273
25	2030	1828.56701653704	201.43298346296
26	1655	1642.71608113720	12.2839188627955
27	1693	1693.27858113720	-0.278581137204542
28	1623	1580.09108113721	42.9089188627948
29	1805	1717.40358113720	87.5964188627952
30	1746	1661.65358113720	84.3464188627952
31	1795	1737.27858113721	57.721418862795
32	1926	1752.15358113720	173.846418862796
33	1619	1804.96608113721	-185.966081137205
34	1992	1944.34108113720	47.6589188627952
35	2233	2143.90358113720	89.0964188627953
36	2192	2260.52858113721	-68.5285811372049
37	2080	1807.38875254852	272.611247451482
38	1768	1621.53781714868	146.462182851318
39	1835	1672.10031714868	162.899682851318
40	1569	1558.91281714868	10.0871828513171
41	1976	1696.22531714868	279.774682851318
42	1853	1640.47531714868	212.524682851317
43	1965	1716.10031714868	248.899682851317
44	1689	1730.97531714868	-41.9753171486821
45	1778	1783.78781714868	-5.78781714868267
46	1976	1923.16281714868	52.8371828513174
47	2397	2122.72531714868	274.274682851318
48	2654	2239.35031714868	414.649682851317
49	2097	1786.21048856000	310.789511440004
50	1963	1600.35955316016	362.64044683984
51	1677	1650.92205316016	26.0779468398400
52	1941	1537.73455316016	403.265446839839
53	2003	1675.04705316016	327.95294683984
54	1813	1619.29705316016	193.702946839840
55	2012	1694.92205316016	317.07794683984
56	1912	1709.79705316016	202.20294683984
57	2084	1762.60955316016	321.39044683984
58	2080	1901.98455316016	178.015446839840
59	2118	2101.54705316016	16.4529468398398
60	2150	2218.17205316016	-68.1720531601603
61	1608	1765.03222457147	-157.032224571473
62	1503	1579.18128917164	-76.1812891716376
63	1548	1629.74378917164	-81.7437891716377
64	1382	1516.55628917164	-134.556289171638
65	1731	1653.86878917164	77.1312108283621
66	1798	1598.11878917164	199.881210828362
67	1779	1673.74378917164	105.256210828362
68	1887	1688.61878917164	198.381210828362
69	2004	1741.43128917164	262.568710828362
70	2077	1880.80628917164	196.193710828362
71	2092	2080.36878917164	11.6312108283621
72	2051	2196.99378917164	-145.993789171638
73	1577	1743.85396058295	-166.853960582951
74	1356	1558.00302518312	-202.003025183115
75	1652	1608.56552518312	43.4344748168846
76	1382	1495.37802518312	-113.378025183116
77	1519	1632.69052518312	-113.690525183116
78	1421	1576.94052518312	-155.940525183116
79	1442	1652.56552518312	-210.565525183116
80	1543	1667.44052518312	-124.440525183115
81	1656	1720.25302518312	-64.2530251831158
82	1561	1859.62802518312	-298.628025183116
83	1905	2059.19052518312	-154.190525183116
84	2199	2175.81552518312	23.1844748168843
85	1473	1722.67569659443	-249.675696594429
86	1655	1536.82476119459	118.175238805407
87	1407	1587.38726119459	-180.387261194593
88	1395	1474.19976119459	-79.1997611945937
89	1530	1611.51226119459	-81.5122611945933
90	1309	1555.76226119459	-246.762261194593
91	1526	1631.38726119459	-105.387261194593
92	1327	1646.26226119459	-319.262261194593
93	1627	1699.07476119459	-72.0747611945935
94	1748	1838.44976119459	-90.4497611945934
95	1958	2038.01226119459	-80.0122611945933
96	2274	2154.63726119459	119.362738805407
97	1648	1701.49743260591	-53.4974326059064
98	1401	1515.64649720607	-114.646497206071
99	1411	1566.20899720607	-155.208997206071
100	1403	1453.02149720607	-50.0214972060714
101	1394	1590.33399720607	-196.333997206071
102	1520	1534.58399720607	-14.5839972060711
103	1528	1610.20899720607	-82.2089972060712
104	1643	1625.08399720607	17.9160027939294
105	1515	1677.89649720607	-162.896497206071
106	1685	1817.27149720607	-132.271497206071
107	2000	2016.83399720607	-16.8339972060710
108	2215	2133.45899720607	81.5410027939288
109	1956	1680.31916861738	275.680831382616
110	1462	1494.46823321755	-32.4682332175485
111	1563	1545.03073321755	17.9692667824515
112	1459	1431.84323321755	27.1567667824508
113	1446	1569.15573321755	-123.155733217549
114	1622	1513.40573321755	108.594266782451
115	1657	1589.03073321755	67.9692667824511
116	1638	1603.90573321755	34.0942667824517
117	1643	1656.71823321755	-13.7182332175489
118	1683	1796.09323321755	-113.093233217549
119	2050	1995.65573321755	54.3442667824512
120	2262	2112.28073321755	149.719266782451
121	1813	1659.14090462886	153.859095371138
122	1445	1473.28996922903	-28.2899692290262
123	1762	1523.85246922903	238.147530770974
124	1461	1410.66496922903	50.3350307709731
125	1556	1547.97746922903	8.02253077097358
126	1431	1492.22746922903	-61.2274692290265
127	1427	1567.85246922903	-140.852469229027
128	1554	1582.72746922903	-28.7274692290261
129	1645	1635.53996922903	9.46003077097336
130	1653	1774.91496922903	-121.914969229026
131	2016	1974.47746922903	41.5225307709736
132	2207	2091.10246922903	115.897530770973
133	1665	1637.96264064034	27.0373593596605
134	1361	1452.11170524050	-91.111705240504
135	1506	1502.67420524050	3.32579475949605
136	1360	1389.48670524050	-29.4867052405046
137	1453	1526.79920524050	-73.7992052405041
138	1522	1471.04920524050	50.9507947594957
139	1460	1546.67420524050	-86.6742052405044
140	1552	1561.54920524050	-9.54920524050376
141	1548	1614.36170524050	-66.3617052405043
142	1827	1753.73670524050	73.2632947594957
143	1737	1953.29920524050	-216.299205240504
144	1941	2069.92420524050	-128.924205240504
145	1474	1616.78437665182	-142.784376651817
146	1458	1430.93344125198	27.0665587480184
147	1542	1481.49594125198	60.5040587480183
148	1404	1368.30844125198	35.6915587480177
149	1522	1505.62094125198	16.3790587480182
150	1385	1449.87094125198	-64.870941251982
151	1641	1525.49594125198	115.504058748018
152	1510	1540.37094125198	-30.3709412519815
153	1681	1593.18344125198	87.816558748018
154	1938	1732.55844125198	205.441558748018
155	1868	1932.12094125198	-64.1209412519819
156	1726	2048.74594125198	-322.745941251982
157	1456	1595.60611266329	-139.606112663295
158	1445	1409.75517726346	35.2448227365406
159	1456	1460.31767726346	-4.31767726345942
160	1365	1347.13017726346	17.8698227365400
161	1487	1484.44267726346	2.55732273654045
162	1558	1428.69267726346	129.307322736540
163	1488	1504.31767726346	-16.3176772634598
164	1684	1519.19267726346	164.807322736541
165	1594	1572.00517726346	21.9948227365402
166	1850	1711.38017726346	138.619822736540
167	1998	1910.94267726346	87.0573227365404
168	2079	2027.56767726346	51.4323227365403
169	1494	1574.42784867477	-80.4278486747727
170	1057	1162.19187967228	-105.191879672279
171	1218	1212.75437967228	5.24562032772056
172	1168	1099.56687967228	68.4331203277199
173	1236	1236.87937967228	-0.879379672279542
174	1076	1181.12937967228	-105.129379672280
175	1174	1256.75437967228	-82.7543796722797
176	1139	1271.62937967228	-132.629379672279
177	1427	1324.44187967228	102.558120327720
178	1487	1463.81687967228	23.1831203277203
179	1483	1663.37937967228	-180.379379672280
180	1513	1780.00437967228	-267.00437967228
181	1357	1326.86455108359	30.1354489164073
182	1165	1141.01361568376	23.986384316243
183	1282	1191.57611568376	90.4238843162428
184	1110	1078.38861568376	31.6113843162422
185	1297	1215.70111568376	81.2988843162427
186	1185	1159.95111568376	25.0488843162426
187	1222	1235.57611568376	-13.5761156837575
188	1284	1250.45111568376	33.548884316243
189	1444	1303.26361568376	140.736384316242
190	1575	1442.63861568376	132.361384316243
191	1737	1642.20111568376	94.7988843162427
192	1763	1758.82611568376	4.17388431624253

Charts produced by software:

http://127.0.0.1/wessadotnet/public_html/freestatisticsdotorg/blog/date/2007/Nov/15/t1195129421xwvo03m3ia1eb4w/1a2lm1195128784.png (open in new window)

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Parameters:

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

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