SSR(X4) = 66.775
SSR(X1|X4) = 42.2746
SSR(X2|X1,X4) = 27.85
SSR(X3|X1,X2,X4) = 0.419
ch18 <-read.table("CH06PR18.txt")
head(ch18)## V1 V2 V3 V4 V5
## 1 13.5 1 5.02 0.14 123000
## 2 12.0 14 8.19 0.27 104079
## 3 10.5 16 3.00 0.00 39998
## 4 15.0 4 10.70 0.05 57112
## 5 14.0 11 8.97 0.07 60000
## 6 10.5 15 9.45 0.24 101385head(ch18)## V1 V2 V3 V4 V5
## 1 13.5 1 5.02 0.14 123000
## 2 12.0 14 8.19 0.27 104079
## 3 10.5 16 3.00 0.00 39998
## 4 15.0 4 10.70 0.05 57112
## 5 14.0 11 8.97 0.07 60000
## 6 10.5 15 9.45 0.24 101385#i) With SSR(X4)
fit4 <- lm(V1 ~ V5, ch18)
anova(fit4) ## Analysis of Variance Table
##
## Response: V1
## Df Sum Sq Mean Sq F value Pr(>F)
## V5 1 67.775 67.775 31.723 2.628e-07 ***
## Residuals 79 168.782 2.136
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 167.775 #67.775## [1] 67.775#ii) With SSR(X1|X4) = SSE(X4) - SSE(X1,X4)
fit_x4 <- lm(V1~V5, ch18) #SSE(X4)
anova(fit_x4)## Analysis of Variance Table
##
## Response: V1
## Df Sum Sq Mean Sq F value Pr(>F)
## V5 1 67.775 67.775 31.723 2.628e-07 ***
## Residuals 79 168.782 2.136
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1fit_x1x4 <- lm(V1~V2+V5,ch18) #SSE(X1,X4)
anova(fit_x1x4)## Analysis of Variance Table
##
## Response: V1
## Df Sum Sq Mean Sq F value Pr(>F)
## V2 1 14.819 14.819 9.1365 0.003389 **
## V5 1 95.231 95.231 58.7160 4.225e-11 ***
## Residuals 78 126.508 1.622
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1168.782-126.508 #42.2746## [1] 42.274#iii) With SSR(X2|X1,X4) = SSE(X1,X4) - SSE(X1,X2,X4)
fit_x1x2x4 <- lm(V1~V2+V3+V5, ch18)
anova(fit_x1x2x4)## Analysis of Variance Table
##
## Response: V1
## Df Sum Sq Mean Sq F value Pr(>F)
## V2 1 14.819 14.819 11.566 0.001067 **
## V3 1 72.802 72.802 56.825 7.841e-11 ***
## V5 1 50.287 50.287 39.251 1.973e-08 ***
## Residuals 77 98.650 1.281
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1126.508 - 98.650 #27.85## [1] 27.858#iv) With SSR(X3|X1,X2,X4) = SSE(X1,X2,X4) - SSE(X1,X2,X3,X4)
fit_full <- lm(V1~., ch18)
anova(fit_full)## Analysis of Variance Table
##
## Response: V1
## Df Sum Sq Mean Sq F value Pr(>F)
## V2 1 14.819 14.819 11.4649 0.001125 **
## V3 1 72.802 72.802 56.3262 9.699e-11 ***
## V4 1 8.381 8.381 6.4846 0.012904 *
## V5 1 42.325 42.325 32.7464 1.976e-07 ***
## Residuals 76 98.231 1.293
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 198.650- 98.231 #0.419## [1] 0.419Hypothesis Test:
\(H_0: \beta_{3}= 0\)
\(H_A:\) \(\beta_{3} \neq 0\)
Decision Rule:
If \(F* > F\), then reject \(H_0\).
Conclusion: From the full model compared to our reduced model, we see that the F* value of 0.3248 < F = 6.9806. Thus, we fail to reject our \(H_0\)
The p-value is 0.5707886
#Method 1
anova(fit_x1x2x4, fit_full)## Analysis of Variance Table
##
## Model 1: V1 ~ V2 + V3 + V5
## Model 2: V1 ~ V2 + V3 + V4 + V5
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 77 98.650
## 2 76 98.231 1 0.41975 0.3248 0.5704n <- nrow(ch18)
p <- 5
F_test <- qf(0.99,1,n-p)
F_test## [1] 6.980578#Method 2
(0.42/1) / (98.2306/76) #(SSR(X3|X1,X2,X4)/1 / SSE(X1,X2,X3,X4)/76)## [1] 0.3249497#Method 3
n=nrow(ch18)
p=5
q=4
#SSE(X1,X2,X4)
SSE_X124 <- 98.650
#SSE(X1,X2,X3)
SSE_X1234 <- 98.231
Fstar=(SSE_X124-SSE_X1234)/(p-q)/((SSE_X1234)/(n-p))
Fstar## [1] 0.32417461-pf(Fstar,(p-q),(n-p))## [1] 0.5707886Hypothesis Test:
\(H_0: \beta_{2}} = \beta_{3} = 0\)
\(H_A:\) Either \(\beta_{2}\) or \(\beta_{3} \neq 0\)
Decision Rule:
If \(F* > F\), then reject \(H_0\).
Conclusion: From the full model compared to our reduced model, we see that the F* value of 10.93 > F = 4.8958. Thus, we reject our \(H_0\).
The p-value is 6.682e-05.
#Method 1
anova(fit_x1x4, fit_full)## Analysis of Variance Table
##
## Model 1: V1 ~ V2 + V5
## Model 2: V1 ~ V2 + V3 + V4 + V5
## Res.Df RSS Df Sum of Sq F Pr(>F)
## 1 78 126.508
## 2 76 98.231 2 28.277 10.939 6.682e-05 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#Method 2
#With SSR(X2,X3|X1,X4) = SSE(X1,X4) - SSE(X1,X2,X3,X4)
126.508 - 98.231 #28.77 ## [1] 28.277(28.277/2) / (98.2306/76) #(SSR(X3|X1,X2,X4)/1 / SSE(X1,X2,X3,X4)/76)## [1] 10.93881F_test <- qf(0.99,2,n-p)
F_test## [1] 4.89584\(R^2_{Y 1|2}\) = 0.2505
The coefficient of determination = 0.2505, is the proportion of the variation in Y explained by an independent variable X1, when the independent variable X4 is in the model.
#R^2 Y1|4
#SSR(X1|X4)/SSE(X4) = SSE(X4) - SSE(X1,X4)/SSE(X4) = 1 - SSE(X1,X4)/SSE(X4)
anova(fit_x1x4)## Analysis of Variance Table
##
## Response: V1
## Df Sum Sq Mean Sq F value Pr(>F)
## V2 1 14.819 14.819 9.1365 0.003389 **
## V5 1 95.231 95.231 58.7160 4.225e-11 ***
## Residuals 78 126.508 1.622
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1fit_x4 <- lm(V1~V5,data=ch18)
anova(fit_x4)## Analysis of Variance Table
##
## Response: V1
## Df Sum Sq Mean Sq F value Pr(>F)
## V5 1 67.775 67.775 31.723 2.628e-07 ***
## Residuals 79 168.782 2.136
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 11 - (126.508/168.782)## [1] 0.2504651The standardized regression model is \(\hat_{Y^*}\) = -5.478526e-0\(X_1^*\) + 4.236468e-01\(X_2^*\) + 4.846136e-02\(X_3^*\) + 5.027571e-01\(X_4^*\)
b2* is 9.490e-02 from the summary of the standardized model.
When \(X_2\) increases by one, \(sd(X_2)\) increases by 9.490*10^-2(sd(Y)).
Standardized coefficients: (Intercept) = -3.364857e-16, X1star = -5.478526e-01,
X2star = 4.236468e-01, X3star = 4.846136e-02, X4star = 5.027571e-01
Original coefficients for fitted reg model: X1star= 1.220059e+01
X1star = -1.420336e-0
X2star = 2.820165e-01
X3star = 6.193435e-01 X4star = 7.924302e-06
Coefficients from 6.18c - Same coefficients \(\hat{Y} = 12.2006 − 0.1420X1+ 0.2820X2 + 0.6193X3 + 0.0000079X4\)
summary(fit_full)##
## Call:
## lm(formula = V1 ~ ., data = ch18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.1872 -0.5911 -0.0910 0.5579 2.9441
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.220e+01 5.780e-01 21.110 < 2e-16 ***
## V2 -1.420e-01 2.134e-02 -6.655 3.89e-09 ***
## V3 2.820e-01 6.317e-02 4.464 2.75e-05 ***
## V4 6.193e-01 1.087e+00 0.570 0.57
## V5 7.924e-06 1.385e-06 5.722 1.98e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.137 on 76 degrees of freedom
## Multiple R-squared: 0.5847, Adjusted R-squared: 0.5629
## F-statistic: 26.76 on 4 and 76 DF, p-value: 7.272e-14n<-nrow(ch18)
v2 <- mean(ch18$V2)
sd <- sqrt(sum((ch18$V2 - v2)^2)/n-1)
(1/sqrt(n-1))*((ch18$V2 - v2)/sd)## [1] -0.117788363 0.105289238 0.139608869 -0.066308916 0.053809792
## [6] 0.122449053 -0.100628547 -0.117788363 -0.117788363 0.002330345
## [11] 0.070969607 -0.100628547 -0.100628547 0.088129422 -0.100628547
## [16] -0.117788363 -0.117788363 0.070969607 0.139608869 -0.083468732
## [21] -0.031989286 -0.083468732 -0.083468732 -0.083468732 0.105289238
## [26] -0.117788363 0.122449053 0.139608869 -0.100628547 0.139608869
## [31] -0.100628547 -0.083468732 -0.083468732 0.139608869 0.139608869
## [36] 0.088129422 -0.117788363 0.105289238 -0.083468732 0.122449053
## [41] -0.117788363 0.122449053 -0.117788363 0.122449053 -0.083468732
## [46] -0.083468732 -0.066308916 -0.117788363 -0.066308916 -0.066308916
## [51] -0.100628547 -0.100628547 -0.117788363 -0.100628547 -0.083468732
## [56] 0.122449053 0.156768684 -0.117788363 0.105289238 -0.100628547
## [61] -0.117788363 -0.134948178 0.173928499 0.139608869 0.088129422
## [66] 0.208248130 0.173928499 0.139608869 -0.117788363 -0.100628547
## [71] 0.139608869 -0.117788363 0.122449053 -0.066308916 0.208248130
## [76] -0.083468732 -0.083468732 0.173928499 0.122449053 0.053809792
## [81] 0.105289238##Standardized MR
attach(ch18)
Ystar=scale(V1)
X1star=scale(V2)
X2star=scale(V3)
X3star=scale(V4)
X4star=scale(V5)
fit4=lm(Ystar~X1star+X2star+X3star+X4star)
fit4$coefficients## (Intercept) X1star X2star X3star X4star
## -3.364857e-16 -5.478526e-01 4.236468e-01 4.846136e-02 5.027571e-01b1=sd(V1)/sd(V2)*fit4$coefficients[2]
b2=sd(V1)/sd(V3)*fit4$coefficients[3]
b3=sd(V1)/sd(V4)*fit4$coefficients[4]
b4=sd(V1)/sd(V5)*fit4$coefficients[5]
b0=mean(V1)-b1*mean(V2)-b2*mean(V3)-b3*mean(V4)-b4*mean(V5)
c(b0,b1,b2,b3,b4)## X1star X1star X2star X3star X4star
## 1.220059e+01 -1.420336e-01 2.820165e-01 6.193435e-01 7.924302e-06\(\hat{Y}\) = 14.36 - 0.1145\(X_1\) + 0.00001\(X_4\)
reg_fit <- lm(V1~ V2+V5, data=ch18)
summary(reg_fit)##
## Call:
## lm(formula = V1 ~ V2 + V5, data = ch18)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.2032 -0.4593 0.0641 0.7730 2.5083
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1.436e+01 2.771e-01 51.831 < 2e-16 ***
## V2 -1.145e-01 2.242e-02 -5.105 2.27e-06 ***
## V5 1.045e-05 1.363e-06 7.663 4.23e-11 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.274 on 78 degrees of freedom
## Multiple R-squared: 0.4652, Adjusted R-squared: 0.4515
## F-statistic: 33.93 on 2 and 78 DF, p-value: 2.506e-11In 6.18c, the model was \(\hat{Y} = 12.2006 − 0.1420X1+ 0.2820X2 + 0.6193X3 + 0.0000079X4\)
The model here is \(\hat{Y}\) = 14.36 - 0.1145\(X_1\) + 0.00001\(X_4\)
The coefficients for X1 are -0.142 and -0.1145 which are about the same, but it is a bit lower for our second, reduced model. The coefficients for X4 are 0.0000079 and 0.00001, which are about the same but it is a bit higher for our first, full model .
No. SSR(X4) = 67.775. SSR(X4|X3) = 66.859.
No SSR(X1) = 14.819 SSR(X1|X3) = 13.775
#SSR(X4)
anova(fit_x4)## Analysis of Variance Table
##
## Response: V1
## Df Sum Sq Mean Sq F value Pr(>F)
## V5 1 67.775 67.775 31.723 2.628e-07 ***
## Residuals 79 168.782 2.136
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#SSR(X4|X3) = SSE(X3) - SSE(X3,X4)
fit_x3x4 <- lm(V1~V4+V5, data=ch18)
anova(fit_x3x4)## Analysis of Variance Table
##
## Response: V1
## Df Sum Sq Mean Sq F value Pr(>F)
## V4 1 1.047 1.047 0.4842 0.4886
## V5 1 66.858 66.858 30.9213 3.626e-07 ***
## Residuals 78 168.652 2.162
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1fit_x3 <- lm(V1~V4, data=ch18)
anova(fit_x3)## Analysis of Variance Table
##
## Response: V1
## Df Sum Sq Mean Sq F value Pr(>F)
## V4 1 1.047 1.0470 0.3512 0.5551
## Residuals 79 235.511 2.9811235.511-168.652## [1] 66.859#SSR(X1)
fit_x1 <- lm(V1~V2, data=ch18)
anova(fit_x1)## Analysis of Variance Table
##
## Response: V1
## Df Sum Sq Mean Sq F value Pr(>F)
## V2 1 14.819 14.8185 5.2795 0.02422 *
## Residuals 79 221.739 2.8068
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1#SSR(X1|X3) = SSE(X3) - SSE(X1,X3)
fit_x1x3 <- lm(V1 ~ V2 + V4, data= ch18)
anova(fit_x1x3)## Analysis of Variance Table
##
## Response: V1
## Df Sum Sq Mean Sq F value Pr(>F)
## V2 1 14.819 14.8185 5.2127 0.02515 *
## V4 1 0.003 0.0027 0.0010 0.97534
## Residuals 78 221.736 2.8428
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1235.511-221.736## [1] 13.775In parts b and c, we see the first model has more significant predictors. The value of \(r_{12}\) = 0.5680, \(r_{13}\) = 0.3228, and \(r_{23}\) = 0.2538.