pho<-data.frame(x1 <- c(0.4,0.4,3.1,0.6,4.7,1.7,9.4,10.1,11.6,12.6,10.9,23.1,23.1,21.6,23.1,1.9,26.8,29.9), x2 <- c(52,34,19,34,24,65,44,31,29,58,37,46,50,44,56,36,58,51), x3 <- c(158,163,37,157,59,123,46,117,173,112,111,114,134,73,168,143,202,124), y <- c(64,60,71,61,54,77,81,93,93,51,76,96,77,93,95,54,168,99))
#(1)
lm.sol<-lm(y~x1+x2+x3,data=pho)
summary(lm.sol)
##
## Call:
## lm(formula = y ~ x1 + x2 + x3, data = pho)
##
## Residuals:
## Min 1Q Median 3Q Max
## -27.6 -11.2 -2.8 11.6 48.8
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 44.929 18.341 2.45 0.0281 *
## x1 1.803 0.529 3.41 0.0042 **
## x2 -0.134 0.444 -0.30 0.7677
## x3 0.167 0.114 1.46 0.1657
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 19.9 on 14 degrees of freedom
## Multiple R-squared: 0.551, Adjusted R-squared: 0.455
## F-statistic: 5.73 on 3 and 14 DF, p-value: 0.009
（1） 回归方程为 y=44.9290+1.8033x1-0.1337x2+0.1668x3
（2）回归方程显著，但有些回归系数不显著。
（3）
lm.step<-step(lm.sol)
## Start: AIC=111.2
## y ~ x1 + x2 + x3
##
## Df Sum of Sq RSS AIC
## - x2 1 36 5599 109
## <none> 5563 111
## - x3 1 850 6413 112
## - x1 1 4618 10181 120
##
## Step: AIC=109.3
## y ~ x1 + x3
##
## Df Sum of Sq RSS AIC
## <none> 5599 109
## - x3 1 833 6433 110
## - x1 1 5169 10769 119
summary(lm.step)
##
## Call:
## lm(formula = y ~ x1 + x3, data = pho)
##
## Residuals:
## Min 1Q Median 3Q Max
## -29.71 -11.32 -2.95 11.29 48.68
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 41.479 13.883 2.99 0.0092 **
## x1 1.737 0.467 3.72 0.0020 **
## x3 0.155 0.104 1.49 0.1559
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 19.3 on 15 degrees of freedom
## Multiple R-squared: 0.548, Adjusted R-squared: 0.488
## F-statistic: 9.1 on 2 and 15 DF, p-value: 0.00259
x3仍不够显著。 再用drop1函数做逐步回归。
drop1(lm.step)
## Single term deletions
##
## Model:
## y ~ x1 + x3
## Df Sum of Sq RSS AIC
## <none> 5599 109
## x1 1 5169 10769 119
## x3 1 833 6433 110
可以考虑再去掉x3.
lm.opt<-lm(y~x1,data=pho);summary(lm.opt)
##
## Call:
## lm(formula = y ~ x1, data = pho)
##
## Residuals:
## Min 1Q Median 3Q Max
## -31.49 -8.28 -1.67 5.62 59.34
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 59.259 7.420 7.99 5.7e-07 ***
## x1 1.843 0.479 3.85 0.0014 **
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 20.1 on 16 degrees of freedom
## Multiple R-squared: 0.481, Adjusted R-squared: 0.448
## F-statistic: 14.8 on 1 and 16 DF, p-value: 0.00142
皆显著。