1. Using data set HW3_SM.txt, build a multiple linear regression model between Y
ID: 3366584 • Letter: 1
Question
1. Using data set HW3_SM.txt, build a multiple linear regression model between Y and (X1,X2,X3,X4).
(b) Construct an ANOVA table and calculate the R2 using the table.
Data set from HW3_SM.txt below
Y X1 X2 X3 X4
19.55 1.68 0.89 0.61 1.69
20.63 1.11 0.66 0.41 1.64
24.16 1.63 1.02 0.63 1.66
14.82 1.94 0.29 0.14 1.44
16.43 2.18 0.75 0.42 1.38
19.56 1.77 0.53 0.56 1.61
20.35 2.66 0.76 0.48 1.42
18.3 1.75 0.71 0.47 1.31
20.48 1.65 1.08 0.4 1.36
25.11 1.6 0.95 0.36 1.61
19.15 1.95 0.28 0.62 1.69
24.16 2.24 0.81 0.43 1.73
20.27 2.34 0.77 0.34 1.46
18.74 2.67 0.39 0.53 1.38
23.06 1.86 0.97 0.51 1.55
26.25 1.66 1.17 0.48 1.56
20.01 2.27 0.51 0.35 1.59
22.76 2.23 1.06 0.57 1.54
19.75 2.27 0.81 0.46 1.35
21.39 1.59 0.97 0.65 1.68
17.39 1.66 0.76 0.5 1.39
22.81 1.99 1.06 0.52 1.5
22.52 1.67 1.02 0.66 1.54
20.67 1.73 0.57 0.51 1.63
20.03 2.17 0.76 0.53 1.55
18 2.44 0.78 0.53 1.63
19.23 2.1 0.63 0.55 1.41
22.07 1.57 1.19 0.4 1.4
19.82 1.81 0.9 0.46 1.46
20.3 2.38 0.93 0.4 1.31
18.58 2.41 0.51 0.71 1.62
17.72 1.72 1.02 0.31 1.48
19.5 1.84 1.07 0.48 1.49
15.72 1.78 0.68 0.7 1.29
20.34 2.24 0.76 0.45 1.66
22.43 1.64 0.71 0.36 1.45
23.57 2.37 0.86 0.54 1.58
23.31 2.04 1.04 0.51 1.45
21.25 2.03 0.95 0.59 1.41
18.23 2.14 0.77 0.49 1.59
21.41 2.16 0.93 0.46 1.52
19.62 1.31 0.6 0.46 1.63
22.15 2.42 0.92 0.66 1.5
21.21 2.49 0.87 0.55 1.54
18.77 2.23 0.82 0.44 1.48
18.91 1.96 0.82 0.61 1.58
17.94 0.85 0.74 0.69 1.51
21.46 2.75 0.81 0.61 1.36
18.4 2.26 0.39 0.45 1.6
23.37 2.43 0.68 0.39 1.78
23.05 2.25 0.78 0.39 1.4
20.99 1.78 0.91 0.5 1.47
25.36 2.74 1.06 0.48 1.61
23.13 1.85 1.07 0.7 1.53
20.38 1.32 0.91 0.4 1.52
19.05 2.65 0.7 0.69 1.54
20.59 2.29 0.76 0.51 1.56
21.17 2.68 0.75 0.45 1.58
19.93 2.09 0.74 0.43 1.44
15.62 1.69 0.35 0.56 1.34
17.36 1.63 0.9 0.56 1.34
20.71 2.09 0.94 0.59 1.25
21.28 1.37 0.96 0.57 1.43
19.18 1.67 0.71 0.53 1.62
22.32 2.55 1.09 0.42 1.43
18.81 2.44 0.45 0.57 1.25
24.31 2.22 0.95 0.52 1.7
24.21 2.47 1.18 0.59 1.38
21.85 1.72 1.05 0.58 1.53
24.7 1.76 0.94 0.58 1.58
18.96 2.71 0.87 0.52 1.51
17.79 1.93 0.56 0.58 1.56
18.85 1.58 0.8 0.48 1.39
22.5 2.8 0.98 0.33 1.46
21.29 2.16 0.93 0.36 1.58
17.84 1.78 0.83 0.44 1.4
21.64 1.98 0.77 0.45 1.44
19.09 2.3 0.68 0.54 1.58
19.61 1.94 0.5 0.37 1.59
21.99 1.7 0.89 0.34 1.55
19.53 3.03 0.64 0.47 1.32
23.83 2.87 0.95 0.54 1.58
24.94 1.91 1.13 0.53 1.52
23.36 1.98 0.98 0.48 1.57
17.96 2.29 0.49 0.67 1.63
21.95 1.53 0.78 0.5 1.57
19.73 1.5 0.81 0.56 1.56
23.78 2.04 1.05 0.48 1.46
18.17 1.6 0.86 0.62 1.44
18.96 1.85 0.83 0.66 1.64
19.31 1.8 0.95 0.57 1.29
20.01 2.21 0.84 0.39 1.37
19.69 2.2 0.76 0.49 1.33
24.7 2.08 0.98 0.44 1.45
23.23 2.23 1.28 0.59 1.38
20.75 2.06 0.79 0.44 1.59
20.1 1.43 0.98 0.45 1.56
22.15 2.4 1.12 0.65 1.48
17.88 1.63 0.74 0.72 1.52
23.99 2.48 0.9 0.44 1.5
Explanation / Answer
We can easily do this in R,
CODE:
data <- read.table(text = "Y X1 X2 X3 X4
19.55 1.68 0.89 0.61 1.69
20.63 1.11 0.66 0.41 1.64
24.16 1.63 1.02 0.63 1.66
14.82 1.94 0.29 0.14 1.44
16.43 2.18 0.75 0.42 1.38
19.56 1.77 0.53 0.56 1.61
20.35 2.66 0.76 0.48 1.42
18.3 1.75 0.71 0.47 1.31
20.48 1.65 1.08 0.4 1.36
25.11 1.6 0.95 0.36 1.61
19.15 1.95 0.28 0.62 1.69
24.16 2.24 0.81 0.43 1.73
20.27 2.34 0.77 0.34 1.46
18.74 2.67 0.39 0.53 1.38
23.06 1.86 0.97 0.51 1.55
26.25 1.66 1.17 0.48 1.56
20.01 2.27 0.51 0.35 1.59
22.76 2.23 1.06 0.57 1.54
19.75 2.27 0.81 0.46 1.35
21.39 1.59 0.97 0.65 1.68
17.39 1.66 0.76 0.5 1.39
22.81 1.99 1.06 0.52 1.5
22.52 1.67 1.02 0.66 1.54
20.67 1.73 0.57 0.51 1.63
20.03 2.17 0.76 0.53 1.55
18 2.44 0.78 0.53 1.63
19.23 2.1 0.63 0.55 1.41
22.07 1.57 1.19 0.4 1.4
19.82 1.81 0.9 0.46 1.46
20.3 2.38 0.93 0.4 1.31
18.58 2.41 0.51 0.71 1.62
17.72 1.72 1.02 0.31 1.48
19.5 1.84 1.07 0.48 1.49
15.72 1.78 0.68 0.7 1.29
20.34 2.24 0.76 0.45 1.66
22.43 1.64 0.71 0.36 1.45
23.57 2.37 0.86 0.54 1.58
23.31 2.04 1.04 0.51 1.45
21.25 2.03 0.95 0.59 1.41
18.23 2.14 0.77 0.49 1.59
21.41 2.16 0.93 0.46 1.52
19.62 1.31 0.6 0.46 1.63
22.15 2.42 0.92 0.66 1.5
21.21 2.49 0.87 0.55 1.54
18.77 2.23 0.82 0.44 1.48
18.91 1.96 0.82 0.61 1.58
17.94 0.85 0.74 0.69 1.51
21.46 2.75 0.81 0.61 1.36
18.4 2.26 0.39 0.45 1.6
23.37 2.43 0.68 0.39 1.78
23.05 2.25 0.78 0.39 1.4
20.99 1.78 0.91 0.5 1.47
25.36 2.74 1.06 0.48 1.61
23.13 1.85 1.07 0.7 1.53
20.38 1.32 0.91 0.4 1.52
19.05 2.65 0.7 0.69 1.54
20.59 2.29 0.76 0.51 1.56
21.17 2.68 0.75 0.45 1.58
19.93 2.09 0.74 0.43 1.44
15.62 1.69 0.35 0.56 1.34
17.36 1.63 0.9 0.56 1.34
20.71 2.09 0.94 0.59 1.25
21.28 1.37 0.96 0.57 1.43
19.18 1.67 0.71 0.53 1.62
22.32 2.55 1.09 0.42 1.43
18.81 2.44 0.45 0.57 1.25
24.31 2.22 0.95 0.52 1.7
24.21 2.47 1.18 0.59 1.38
21.85 1.72 1.05 0.58 1.53
24.7 1.76 0.94 0.58 1.58
18.96 2.71 0.87 0.52 1.51
17.79 1.93 0.56 0.58 1.56
18.85 1.58 0.8 0.48 1.39
22.5 2.8 0.98 0.33 1.46
21.29 2.16 0.93 0.36 1.58
17.84 1.78 0.83 0.44 1.4
21.64 1.98 0.77 0.45 1.44
19.09 2.3 0.68 0.54 1.58
19.61 1.94 0.5 0.37 1.59
21.99 1.7 0.89 0.34 1.55
19.53 3.03 0.64 0.47 1.32
23.83 2.87 0.95 0.54 1.58
24.94 1.91 1.13 0.53 1.52
23.36 1.98 0.98 0.48 1.57
17.96 2.29 0.49 0.67 1.63
21.95 1.53 0.78 0.5 1.57
19.73 1.5 0.81 0.56 1.56
23.78 2.04 1.05 0.48 1.46
18.17 1.6 0.86 0.62 1.44
18.96 1.85 0.83 0.66 1.64
19.31 1.8 0.95 0.57 1.29
20.01 2.21 0.84 0.39 1.37
19.69 2.2 0.76 0.49 1.33
24.7 2.08 0.98 0.44 1.45
23.23 2.23 1.28 0.59 1.38
20.75 2.06 0.79 0.44 1.59
20.1 1.43 0.98 0.45 1.56
22.15 2.4 1.12 0.65 1.48
17.88 1.63 0.74 0.72 1.52
23.99 2.48 0.9 0.44 1.5", header = T)
reg <- lm(Y~.,data = data)
reg
summary(reg)
OUTPUT:
> data <- read.table(text = "Y X1 X2 X3 X4
+
+ 19.55 1.68 0.89 0.61 1.69
+
+ 20.63 1.11 0.66 0.41 1.64
+
+ 24.16 1.63 1.02 0.63 1.66
+
+ 14.82 1.94 0.29 0.14 1.44
+
+ 16.43 2.18 0.75 0.42 1.38
+
+ 19.56 1.77 0.53 0.56 1.61
+
+ 20.35 2.66 0.76 0.48 1.42
+
+ 18.3 1.75 0.71 0.47 1.31
+
+ 20.48 1.65 1.08 0.4 1.36
+
+ 25.11 1.6 0.95 0.36 1.61
+
+ 19.15 1.95 0.28 0.62 1.69
+
+ 24.16 2.24 0.81 0.43 1.73
+
+ 20.27 2.34 0.77 0.34 1.46
+
+ 18.74 2.67 0.39 0.53 1.38
+
+ 23.06 1.86 0.97 0.51 1.55
+
+ 26.25 1.66 1.17 0.48 1.56
+
+ 20.01 2.27 0.51 0.35 1.59
+
+ 22.76 2.23 1.06 0.57 1.54
+
+ 19.75 2.27 0.81 0.46 1.35
+
+ 21.39 1.59 0.97 0.65 1.68
+
+ 17.39 1.66 0.76 0.5 1.39
+
+ 22.81 1.99 1.06 0.52 1.5
+
+ 22.52 1.67 1.02 0.66 1.54
+
+ 20.67 1.73 0.57 0.51 1.63
+
+ 20.03 2.17 0.76 0.53 1.55
+
+ 18 2.44 0.78 0.53 1.63
+
+ 19.23 2.1 0.63 0.55 1.41
+
+ 22.07 1.57 1.19 0.4 1.4
+
+ 19.82 1.81 0.9 0.46 1.46
+
+ 20.3 2.38 0.93 0.4 1.31
+
+ 18.58 2.41 0.51 0.71 1.62
+
+ 17.72 1.72 1.02 0.31 1.48
+
+ 19.5 1.84 1.07 0.48 1.49
+
+ 15.72 1.78 0.68 0.7 1.29
+
+ 20.34 2.24 0.76 0.45 1.66
+
+ 22.43 1.64 0.71 0.36 1.45
+
+ 23.57 2.37 0.86 0.54 1.58
+
+ 23.31 2.04 1.04 0.51 1.45
+
+ 21.25 2.03 0.95 0.59 1.41
+
+ 18.23 2.14 0.77 0.49 1.59
+
+ 21.41 2.16 0.93 0.46 1.52
+
+ 19.62 1.31 0.6 0.46 1.63
+
+ 22.15 2.42 0.92 0.66 1.5
+
+ 21.21 2.49 0.87 0.55 1.54
+
+ 18.77 2.23 0.82 0.44 1.48
+
+ 18.91 1.96 0.82 0.61 1.58
+
+ 17.94 0.85 0.74 0.69 1.51
+
+ 21.46 2.75 0.81 0.61 1.36
+
+ 18.4 2.26 0.39 0.45 1.6
+
+ 23.37 2.43 0.68 0.39 1.78
+
+ 23.05 2.25 0.78 0.39 1.4
+
+ 20.99 1.78 0.91 0.5 1.47
+
+ 25.36 2.74 1.06 0.48 1.61
+
+ 23.13 1.85 1.07 0.7 1.53
+
+ 20.38 1.32 0.91 0.4 1.52
+
+ 19.05 2.65 0.7 0.69 1.54
+
+ 20.59 2.29 0.76 0.51 1.56
+
+ 21.17 2.68 0.75 0.45 1.58
+
+ 19.93 2.09 0.74 0.43 1.44
+
+ 15.62 1.69 0.35 0.56 1.34
+
+ 17.36 1.63 0.9 0.56 1.34
+
+ 20.71 2.09 0.94 0.59 1.25
+
+ 21.28 1.37 0.96 0.57 1.43
+
+ 19.18 1.67 0.71 0.53 1.62
+
+ 22.32 2.55 1.09 0.42 1.43
+
+ 18.81 2.44 0.45 0.57 1.25
+
+ 24.31 2.22 0.95 0.52 1.7
+
+ 24.21 2.47 1.18 0.59 1.38
+
+ 21.85 1.72 1.05 0.58 1.53
+
+ 24.7 1.76 0.94 0.58 1.58
+
+ 18.96 2.71 0.87 0.52 1.51
+
+ 17.79 1.93 0.56 0.58 1.56
+
+ 18.85 1.58 0.8 0.48 1.39
+
+ 22.5 2.8 0.98 0.33 1.46
+
+ 21.29 2.16 0.93 0.36 1.58
+
+ 17.84 1.78 0.83 0.44 1.4
+
+ 21.64 1.98 0.77 0.45 1.44
+
+ 19.09 2.3 0.68 0.54 1.58
+
+ 19.61 1.94 0.5 0.37 1.59
+
+ 21.99 1.7 0.89 0.34 1.55
+
+ 19.53 3.03 0.64 0.47 1.32
+
+ 23.83 2.87 0.95 0.54 1.58
+
+ 24.94 1.91 1.13 0.53 1.52
+
+ 23.36 1.98 0.98 0.48 1.57
+
+ 17.96 2.29 0.49 0.67 1.63
+
+ 21.95 1.53 0.78 0.5 1.57
+
+ 19.73 1.5 0.81 0.56 1.56
+
+ 23.78 2.04 1.05 0.48 1.46
+
+ 18.17 1.6 0.86 0.62 1.44
+
+ 18.96 1.85 0.83 0.66 1.64
+
+ 19.31 1.8 0.95 0.57 1.29
+
+ 20.01 2.21 0.84 0.39 1.37
+
+ 19.69 2.2 0.76 0.49 1.33
+
+ 24.7 2.08 0.98 0.44 1.45
+
+ 23.23 2.23 1.28 0.59 1.38
+
+ 20.75 2.06 0.79 0.44 1.59
+
+ 20.1 1.43 0.98 0.45 1.56
+
+ 22.15 2.4 1.12 0.65 1.48
+
+ 17.88 1.63 0.74 0.72 1.52
+
+ 23.99 2.48 0.9 0.44 1.5", header = T)
> reg <- lm(Y~.,data = data)
> reg
Call:
lm(formula = Y ~ ., data = data)
Coefficients:
(Intercept) X1 X2 X3 X4
1.646 1.258 7.950 -2.445 7.418
> summary(reg)
Call:
lm(formula = Y ~ ., data = data)
Residuals:
Min 1Q Median 3Q Max
-4.4194 -0.9197 -0.1146 1.0661 3.2005
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.6463 2.5773 0.639 0.52451
X1 1.2581 0.3916 3.213 0.00179 **
X2 7.9501 0.7874 10.097 < 2e-16 ***
X3 -2.4452 1.5397 -1.588 0.11558
X4 7.4176 1.4134 5.248 9.33e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 1.582 on 95 degrees of freedom
Multiple R-squared: 0.5688, Adjusted R-squared: 0.5506
F-statistic: 31.33 on 4 and 95 DF, p-value: < 2.2e-16
a) The equation is,
Y = 1.646 + 1.258 * X1 + 7.95 * X2 - 2.445 * X3 + 7.418 * X4
b) ANOVA Table:
From the ANOVA table R2 = SSModel/SSError = 313.66 / 551.44 = 0.5688
Source of Variation df SS MS Obs F Crit F(at 5%) Model 4 313.66 78.415 31.32906 2.467494 Error 95 237.78 2.502947 Total 99 551.44Related Questions
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