某科学基金会的管理人员欲了解从事科学研究的工作人员中，高水平的数学家工资额Y与他们的研究成果（论文、著作等）的质量指标X1，从事研究工作的时间X2以及能成功获得资助的指标X3之间的关系，为此按照一定的设计方案调查了24位此类型的数学家。数据见R Code.
y <- c(33.2, 40.3, 38.7, 46.8, 41.4, 37.5, 39, 40.7, 30.1, 52.9, 38.2, 31.8,
43.3, 44.1, 42.8, 33.6, 34.2, 48, 38, 35.9, 40.4, 36.8, 45.2, 35.1)
x1 <- c(3.5, 5.3, 5.1, 5.8, 4.2, 6, 6.8, 5.5, 3.1, 7.2, 4.5, 4.9, 8, 5.6, 6.6,
3.7, 6.2, 7, 4, 4.5, 5.9, 5.6, 4.8, 3.9)
x2 <- c(9, 20, 18, 33, 31, 13, 25, 30, 5, 47, 25, 11, 23, 35, 39, 31, 7, 40,
35, 23, 33, 27, 34, 15)
x3 <- c(6.1, 6.4, 7.4, 6.7, 7.5, 5.9, 6, 4, 5.8, 8.3, 5, 6.4, 7.6, 7, 5, 4.4,
5.5, 7, 6, 3.5, 4, 4.3, 8, 5)
salary <- data.frame(y, x1, x2, x3)
lm.reg <- lm(y ~ x1 + x2 + x3, data = salary)
summary(lm.reg)
##
## Call:
## lm(formula = y ~ x1 + x2 + x3, data = salary)
##
## Residuals:
## Min 1Q Median 3Q Max
## -3.373 -1.274 0.181 1.095 3.300
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 16.5485 2.1748 7.61 2.5e-07 ***
## x1 1.3263 0.3501 3.79 0.00115 **
## x2 0.3075 0.0385 7.98 1.2e-07 ***
## x3 1.3598 0.3076 4.42 0.00026 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 1.9 on 20 degrees of freedom
## Multiple R-squared: 0.895, Adjusted R-squared: 0.879
## F-statistic: 56.8 on 3 and 20 DF, p-value: 5.79e-10
library(HH)
## Loading required package: lattice
## Loading required package: grid
## Loading required package: latticeExtra
## Loading required package: RColorBrewer
## Loading required package: multcomp
## Loading required package: mvtnorm
## Loading required package: survival
## Loading required package: splines
## Loading required package: TH.data
vif(lm.reg)
## x1 x2 x3
## 1.255 1.194 1.092
confint(lm.reg, level = 0.95)
## 2.5 % 97.5 %
## (Intercept) 12.0120 21.0850
## x1 0.5960 2.0565
## x2 0.2272 0.3879
## x3 0.7182 2.0014
par(mfrow = c(2, 2))
plot(lm.reg)
P值均小于0.05，膨胀因子较小，各回归系数显著水平高，因此建立数学家工资额与三个指标之间的回归方程为Y=16.54853+1.32625X1+0.30754X2+1.35982X3.