Chapter 3 Linear Regression, Q15
In [42]:
library(MASS)
(a)¶
In [43]:
summary(Boston)
crim zn indus chas
Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
nox rm age dis
Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
rad tax ptratio black
Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
Median : 5.000 Median :330.0 Median :19.05 Median :391.44
Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
lstat medv
Min. : 1.73 Min. : 5.00
1st Qu.: 6.95 1st Qu.:17.02
Median :11.36 Median :21.20
Mean :12.65 Mean :22.53
3rd Qu.:16.95 3rd Qu.:25.00
Max. :37.97 Max. :50.00
In [105]:
Boston2 = Boston
coef_simple = c()
Boston2$chas = as.factor(Boston$chas)
for (p in 2:length(Boston2)){
pred = Boston[,p]
print(colnames(Boston2)[p])
lm.model = lm(crim~pred,data=Boston)
coef_simple = c(coef_simple, coef(lm.model)[-1]) # for part (c)
print(summary(lm.model))
}
[1] "zn"
Call:
lm(formula = crim ~ pred, data = Boston)
Residuals:
Min 1Q Median 3Q Max
-4.429 -4.222 -2.620 1.250 84.523
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 4.45369 0.41722 10.675 < 2e-16 ***
pred -0.07393 0.01609 -4.594 5.51e-06 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 8.435 on 504 degrees of freedom
Multiple R-squared: 0.04019, Adjusted R-squared: 0.03828
F-statistic: 21.1 on 1 and 504 DF, p-value: 5.506e-06
[1] "indus"
Call:
lm(formula = crim ~ pred, data = Boston)
Residuals:
Min 1Q Median 3Q Max
-11.972 -2.698 -0.736 0.712 81.813
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.06374 0.66723 -3.093 0.00209 **
pred 0.50978 0.05102 9.991 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 7.866 on 504 degrees of freedom
Multiple R-squared: 0.1653, Adjusted R-squared: 0.1637
F-statistic: 99.82 on 1 and 504 DF, p-value: < 2.2e-16
[1] "chas"
Call:
lm(formula = crim ~ pred, data = Boston)
Residuals:
Min 1Q Median 3Q Max
-3.738 -3.661 -3.435 0.018 85.232
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.7444 0.3961 9.453 <2e-16 ***
pred -1.8928 1.5061 -1.257 0.209
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 8.597 on 504 degrees of freedom
Multiple R-squared: 0.003124, Adjusted R-squared: 0.001146
F-statistic: 1.579 on 1 and 504 DF, p-value: 0.2094
[1] "nox"
Call:
lm(formula = crim ~ pred, data = Boston)
Residuals:
Min 1Q Median 3Q Max
-12.371 -2.738 -0.974 0.559 81.728
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -13.720 1.699 -8.073 5.08e-15 ***
pred 31.249 2.999 10.419 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 7.81 on 504 degrees of freedom
Multiple R-squared: 0.1772, Adjusted R-squared: 0.1756
F-statistic: 108.6 on 1 and 504 DF, p-value: < 2.2e-16
[1] "rm"
Call:
lm(formula = crim ~ pred, data = Boston)
Residuals:
Min 1Q Median 3Q Max
-6.604 -3.952 -2.654 0.989 87.197
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 20.482 3.365 6.088 2.27e-09 ***
pred -2.684 0.532 -5.045 6.35e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 8.401 on 504 degrees of freedom
Multiple R-squared: 0.04807, Adjusted R-squared: 0.04618
F-statistic: 25.45 on 1 and 504 DF, p-value: 6.347e-07
[1] "age"
Call:
lm(formula = crim ~ pred, data = Boston)
Residuals:
Min 1Q Median 3Q Max
-6.789 -4.257 -1.230 1.527 82.849
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -3.77791 0.94398 -4.002 7.22e-05 ***
pred 0.10779 0.01274 8.463 2.85e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 8.057 on 504 degrees of freedom
Multiple R-squared: 0.1244, Adjusted R-squared: 0.1227
F-statistic: 71.62 on 1 and 504 DF, p-value: 2.855e-16
[1] "dis"
Call:
lm(formula = crim ~ pred, data = Boston)
Residuals:
Min 1Q Median 3Q Max
-6.708 -4.134 -1.527 1.516 81.674
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 9.4993 0.7304 13.006 <2e-16 ***
pred -1.5509 0.1683 -9.213 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 7.965 on 504 degrees of freedom
Multiple R-squared: 0.1441, Adjusted R-squared: 0.1425
F-statistic: 84.89 on 1 and 504 DF, p-value: < 2.2e-16
[1] "rad"
Call:
lm(formula = crim ~ pred, data = Boston)
Residuals:
Min 1Q Median 3Q Max
-10.164 -1.381 -0.141 0.660 76.433
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2.28716 0.44348 -5.157 3.61e-07 ***
pred 0.61791 0.03433 17.998 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6.718 on 504 degrees of freedom
Multiple R-squared: 0.3913, Adjusted R-squared: 0.39
F-statistic: 323.9 on 1 and 504 DF, p-value: < 2.2e-16
[1] "tax"
Call:
lm(formula = crim ~ pred, data = Boston)
Residuals:
Min 1Q Median 3Q Max
-12.513 -2.738 -0.194 1.065 77.696
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -8.528369 0.815809 -10.45 <2e-16 ***
pred 0.029742 0.001847 16.10 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6.997 on 504 degrees of freedom
Multiple R-squared: 0.3396, Adjusted R-squared: 0.3383
F-statistic: 259.2 on 1 and 504 DF, p-value: < 2.2e-16
[1] "ptratio"
Call:
lm(formula = crim ~ pred, data = Boston)
Residuals:
Min 1Q Median 3Q Max
-7.654 -3.985 -1.912 1.825 83.353
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -17.6469 3.1473 -5.607 3.40e-08 ***
pred 1.1520 0.1694 6.801 2.94e-11 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 8.24 on 504 degrees of freedom
Multiple R-squared: 0.08407, Adjusted R-squared: 0.08225
F-statistic: 46.26 on 1 and 504 DF, p-value: 2.943e-11
[1] "black"
Call:
lm(formula = crim ~ pred, data = Boston)
Residuals:
Min 1Q Median 3Q Max
-13.756 -2.299 -2.095 -1.296 86.822
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 16.553529 1.425903 11.609 <2e-16 ***
pred -0.036280 0.003873 -9.367 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 7.946 on 504 degrees of freedom
Multiple R-squared: 0.1483, Adjusted R-squared: 0.1466
F-statistic: 87.74 on 1 and 504 DF, p-value: < 2.2e-16
[1] "lstat"
Call:
lm(formula = crim ~ pred, data = Boston)
Residuals:
Min 1Q Median 3Q Max
-13.925 -2.822 -0.664 1.079 82.862
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -3.33054 0.69376 -4.801 2.09e-06 ***
pred 0.54880 0.04776 11.491 < 2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 7.664 on 504 degrees of freedom
Multiple R-squared: 0.2076, Adjusted R-squared: 0.206
F-statistic: 132 on 1 and 504 DF, p-value: < 2.2e-16
[1] "medv"
Call:
lm(formula = crim ~ pred, data = Boston)
Residuals:
Min 1Q Median 3Q Max
-9.071 -4.022 -2.343 1.298 80.957
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 11.79654 0.93419 12.63 <2e-16 ***
pred -0.36316 0.03839 -9.46 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 7.934 on 504 degrees of freedom
Multiple R-squared: 0.1508, Adjusted R-squared: 0.1491
F-statistic: 89.49 on 1 and 504 DF, p-value: < 2.2e-16
As we notice only the p-value associated with the f-statistic for the model with "chas" as predictor has a value larger than 5%. So we can conclude that the response, crim, is not associated with the predictor, "chas."
(b)¶
In [106]:
summary(Boston2)
crim zn indus chas nox
Min. : 0.00632 Min. : 0.00 Min. : 0.46 0:471 Min. :0.3850
1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1: 35 1st Qu.:0.4490
Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.5380
Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.5547
3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.6240
Max. :88.97620 Max. :100.00 Max. :27.74 Max. :0.8710
rm age dis rad
Min. :3.561 Min. : 2.90 Min. : 1.130 Min. : 1.000
1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100 1st Qu.: 4.000
Median :6.208 Median : 77.50 Median : 3.207 Median : 5.000
Mean :6.285 Mean : 68.57 Mean : 3.795 Mean : 9.549
3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188 3rd Qu.:24.000
Max. :8.780 Max. :100.00 Max. :12.127 Max. :24.000
tax ptratio black lstat
Min. :187.0 Min. :12.60 Min. : 0.32 Min. : 1.73
1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38 1st Qu.: 6.95
Median :330.0 Median :19.05 Median :391.44 Median :11.36
Mean :408.2 Mean :18.46 Mean :356.67 Mean :12.65
3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23 3rd Qu.:16.95
Max. :711.0 Max. :22.00 Max. :396.90 Max. :37.97
medv
Min. : 5.00
1st Qu.:17.02
Median :21.20
Mean :22.53
3rd Qu.:25.00
Max. :50.00
In [107]:
lm.model_all = lm(crim~.,data=Boston2)
summary(lm.model_all)
Call:
lm(formula = crim ~ ., data = Boston2)
Residuals:
Min 1Q Median 3Q Max
-9.924 -2.120 -0.353 1.019 75.051
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 17.033228 7.234903 2.354 0.018949 *
zn 0.044855 0.018734 2.394 0.017025 *
indus -0.063855 0.083407 -0.766 0.444294
chas1 -0.749134 1.180147 -0.635 0.525867
nox -10.313535 5.275536 -1.955 0.051152 .
rm 0.430131 0.612830 0.702 0.483089
age 0.001452 0.017925 0.081 0.935488
dis -0.987176 0.281817 -3.503 0.000502 ***
rad 0.588209 0.088049 6.680 6.46e-11 ***
tax -0.003780 0.005156 -0.733 0.463793
ptratio -0.271081 0.186450 -1.454 0.146611
black -0.007538 0.003673 -2.052 0.040702 *
lstat 0.126211 0.075725 1.667 0.096208 .
medv -0.198887 0.060516 -3.287 0.001087 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6.439 on 492 degrees of freedom
Multiple R-squared: 0.454, Adjusted R-squared: 0.4396
F-statistic: 31.47 on 13 and 492 DF, p-value: < 2.2e-16
For all the predictors whose associated p-value is near zero the null hypothesis that Beta-j = 0 can be rejected. We can't reject the null hypothesis for Intercept, zn, dis, rad, black, and medv.
(c)¶
In [108]:
coef_multi = coef(lm.model_all)[-1]
plot(coef_simple, coef_multi)
The coefficient for nox is -10 in multi regression model and 31 in simple linear regression model.
(d)¶
In [111]:
for (p in 2:length(Boston2)){
pred = Boston2[,p]
if (colnames(Boston)[p]!="chas"){
print(colnames(Boston)[p])
lm.model = lm(crim~poly(pred,3), data=Boston)
print(summary(lm.model))
}
}
[1] "zn"
Call:
lm(formula = crim ~ poly(pred, 3), data = Boston)
Residuals:
Min 1Q Median 3Q Max
-4.821 -4.614 -1.294 0.473 84.130
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.6135 0.3722 9.709 < 2e-16 ***
poly(pred, 3)1 -38.7498 8.3722 -4.628 4.7e-06 ***
poly(pred, 3)2 23.9398 8.3722 2.859 0.00442 **
poly(pred, 3)3 -10.0719 8.3722 -1.203 0.22954
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 8.372 on 502 degrees of freedom
Multiple R-squared: 0.05824, Adjusted R-squared: 0.05261
F-statistic: 10.35 on 3 and 502 DF, p-value: 1.281e-06
[1] "indus"
Call:
lm(formula = crim ~ poly(pred, 3), data = Boston)
Residuals:
Min 1Q Median 3Q Max
-8.278 -2.514 0.054 0.764 79.713
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.614 0.330 10.950 < 2e-16 ***
poly(pred, 3)1 78.591 7.423 10.587 < 2e-16 ***
poly(pred, 3)2 -24.395 7.423 -3.286 0.00109 **
poly(pred, 3)3 -54.130 7.423 -7.292 1.2e-12 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 7.423 on 502 degrees of freedom
Multiple R-squared: 0.2597, Adjusted R-squared: 0.2552
F-statistic: 58.69 on 3 and 502 DF, p-value: < 2.2e-16
[1] "nox"
Call:
lm(formula = crim ~ poly(pred, 3), data = Boston)
Residuals:
Min 1Q Median 3Q Max
-9.110 -2.068 -0.255 0.739 78.302
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.6135 0.3216 11.237 < 2e-16 ***
poly(pred, 3)1 81.3720 7.2336 11.249 < 2e-16 ***
poly(pred, 3)2 -28.8286 7.2336 -3.985 7.74e-05 ***
poly(pred, 3)3 -60.3619 7.2336 -8.345 6.96e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 7.234 on 502 degrees of freedom
Multiple R-squared: 0.297, Adjusted R-squared: 0.2928
F-statistic: 70.69 on 3 and 502 DF, p-value: < 2.2e-16
[1] "rm"
Call:
lm(formula = crim ~ poly(pred, 3), data = Boston)
Residuals:
Min 1Q Median 3Q Max
-18.485 -3.468 -2.221 -0.015 87.219
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.6135 0.3703 9.758 < 2e-16 ***
poly(pred, 3)1 -42.3794 8.3297 -5.088 5.13e-07 ***
poly(pred, 3)2 26.5768 8.3297 3.191 0.00151 **
poly(pred, 3)3 -5.5103 8.3297 -0.662 0.50858
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 8.33 on 502 degrees of freedom
Multiple R-squared: 0.06779, Adjusted R-squared: 0.06222
F-statistic: 12.17 on 3 and 502 DF, p-value: 1.067e-07
[1] "age"
Call:
lm(formula = crim ~ poly(pred, 3), data = Boston)
Residuals:
Min 1Q Median 3Q Max
-9.762 -2.673 -0.516 0.019 82.842
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.6135 0.3485 10.368 < 2e-16 ***
poly(pred, 3)1 68.1820 7.8397 8.697 < 2e-16 ***
poly(pred, 3)2 37.4845 7.8397 4.781 2.29e-06 ***
poly(pred, 3)3 21.3532 7.8397 2.724 0.00668 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 7.84 on 502 degrees of freedom
Multiple R-squared: 0.1742, Adjusted R-squared: 0.1693
F-statistic: 35.31 on 3 and 502 DF, p-value: < 2.2e-16
[1] "dis"
Call:
lm(formula = crim ~ poly(pred, 3), data = Boston)
Residuals:
Min 1Q Median 3Q Max
-10.757 -2.588 0.031 1.267 76.378
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.6135 0.3259 11.087 < 2e-16 ***
poly(pred, 3)1 -73.3886 7.3315 -10.010 < 2e-16 ***
poly(pred, 3)2 56.3730 7.3315 7.689 7.87e-14 ***
poly(pred, 3)3 -42.6219 7.3315 -5.814 1.09e-08 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 7.331 on 502 degrees of freedom
Multiple R-squared: 0.2778, Adjusted R-squared: 0.2735
F-statistic: 64.37 on 3 and 502 DF, p-value: < 2.2e-16
[1] "rad"
Call:
lm(formula = crim ~ poly(pred, 3), data = Boston)
Residuals:
Min 1Q Median 3Q Max
-10.381 -0.412 -0.269 0.179 76.217
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.6135 0.2971 12.164 < 2e-16 ***
poly(pred, 3)1 120.9074 6.6824 18.093 < 2e-16 ***
poly(pred, 3)2 17.4923 6.6824 2.618 0.00912 **
poly(pred, 3)3 4.6985 6.6824 0.703 0.48231
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6.682 on 502 degrees of freedom
Multiple R-squared: 0.4, Adjusted R-squared: 0.3965
F-statistic: 111.6 on 3 and 502 DF, p-value: < 2.2e-16
[1] "tax"
Call:
lm(formula = crim ~ poly(pred, 3), data = Boston)
Residuals:
Min 1Q Median 3Q Max
-13.273 -1.389 0.046 0.536 76.950
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.6135 0.3047 11.860 < 2e-16 ***
poly(pred, 3)1 112.6458 6.8537 16.436 < 2e-16 ***
poly(pred, 3)2 32.0873 6.8537 4.682 3.67e-06 ***
poly(pred, 3)3 -7.9968 6.8537 -1.167 0.244
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6.854 on 502 degrees of freedom
Multiple R-squared: 0.3689, Adjusted R-squared: 0.3651
F-statistic: 97.8 on 3 and 502 DF, p-value: < 2.2e-16
[1] "ptratio"
Call:
lm(formula = crim ~ poly(pred, 3), data = Boston)
Residuals:
Min 1Q Median 3Q Max
-6.833 -4.146 -1.655 1.408 82.697
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.614 0.361 10.008 < 2e-16 ***
poly(pred, 3)1 56.045 8.122 6.901 1.57e-11 ***
poly(pred, 3)2 24.775 8.122 3.050 0.00241 **
poly(pred, 3)3 -22.280 8.122 -2.743 0.00630 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 8.122 on 502 degrees of freedom
Multiple R-squared: 0.1138, Adjusted R-squared: 0.1085
F-statistic: 21.48 on 3 and 502 DF, p-value: 4.171e-13
[1] "black"
Call:
lm(formula = crim ~ poly(pred, 3), data = Boston)
Residuals:
Min 1Q Median 3Q Max
-13.096 -2.343 -2.128 -1.439 86.790
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.6135 0.3536 10.218 <2e-16 ***
poly(pred, 3)1 -74.4312 7.9546 -9.357 <2e-16 ***
poly(pred, 3)2 5.9264 7.9546 0.745 0.457
poly(pred, 3)3 -4.8346 7.9546 -0.608 0.544
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 7.955 on 502 degrees of freedom
Multiple R-squared: 0.1498, Adjusted R-squared: 0.1448
F-statistic: 29.49 on 3 and 502 DF, p-value: < 2.2e-16
[1] "lstat"
Call:
lm(formula = crim ~ poly(pred, 3), data = Boston)
Residuals:
Min 1Q Median 3Q Max
-15.234 -2.151 -0.486 0.066 83.353
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.6135 0.3392 10.654 <2e-16 ***
poly(pred, 3)1 88.0697 7.6294 11.543 <2e-16 ***
poly(pred, 3)2 15.8882 7.6294 2.082 0.0378 *
poly(pred, 3)3 -11.5740 7.6294 -1.517 0.1299
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 7.629 on 502 degrees of freedom
Multiple R-squared: 0.2179, Adjusted R-squared: 0.2133
F-statistic: 46.63 on 3 and 502 DF, p-value: < 2.2e-16
[1] "medv"
Call:
lm(formula = crim ~ poly(pred, 3), data = Boston)
Residuals:
Min 1Q Median 3Q Max
-24.427 -1.976 -0.437 0.439 73.655
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 3.614 0.292 12.374 < 2e-16 ***
poly(pred, 3)1 -75.058 6.569 -11.426 < 2e-16 ***
poly(pred, 3)2 88.086 6.569 13.409 < 2e-16 ***
poly(pred, 3)3 -48.033 6.569 -7.312 1.05e-12 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6.569 on 502 degrees of freedom
Multiple R-squared: 0.4202, Adjusted R-squared: 0.4167
F-statistic: 121.3 on 3 and 502 DF, p-value: < 2.2e-16
In the models in which the p-values of the quadratic or cubic terms are near zero, there is an evidence of non-linear relationship between the respecitve predictor and the response. There is nonlinearity between response and the following predictors. zn, indus, nox, rm, age, rad, tax, dis, ptratio, lastat and medv.
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