4 單一迴歸變數的線性迴歸
4.1 Simple Linear Regression
簡單線性模型 Yi=β0+β1Xi+.
X為自變數(independent variable),Y為因變數(dependent variable)
β0為截距,β1為斜率
注意此圖不是由(0,0)開始
4.2 估計線性迴歸模型的係數
在實務上,母體迴歸線的截距β0和斜率β1是未知的。因此,我們必須利用資料來估計這兩個未知參數。
範例:師生比為因變數(dependent variable),平均考試成績為自變數(independent variable)
產生師生比、平均考試成績
CASchools$STR <- CASchools$students/CASchools$teachers
CASchools$score <- (CASchools$read + CASchools$math)/2
一些基本統計量:
mean() (計算所提供數字的算術平均值)
sd() (計算樣本標準差)
quantile() (傳回資料的指定樣本分位數量)
avg_STR <- mean(CASchools$STR)
avg_score <- mean(CASchools$score)sd_STR <- sd(CASchools$STR)
sd_score <- sd(CASchools$score)uantiles <- c(0.10, 0.25, 0.4, 0.5, 0.6, 0.75, 0.9)
quant_STR <- quantile(CASchools$STR, quantiles)
quant_score <- quantile(CASchools$score, quantiles)DistributionSummary <- data.frame(Average = c(avg_STR, avg_score),
StandardDeviation = c(sd_STR, sd_score),
quantile = rbind(quant_STR, quant_score))
用data.frame 方式寫出呈現
Average StandardDeviation quantile.10. quantile.25. quantile.40.
quant_STR 19.64043 1.891812 17.3486 18.58236 19.26618
quant_score 654.15655 19.053347 630.3950 640.05000 649.06999
quantile.50. quantile.60. quantile.75. quantile.90.
quant_STR 19.72321 20.0783 20.87181 21.86741
quant_score 654.45000 659.4000 666.66249 678.85999
plot(score ~ STR,
data = CASchools,
main = “Scatterplot of Test Score and STR”,
xlab = “STR (X)”,
ylab = “Test Score (Y)”)
#觀察值在師生比和考試成績的散佈圖
由散佈圖可預期在較大的班級,平均分數可能會較低。
最小平方法(OLS)
Yi=β0+β1Xi+. =bo + b1 xi + ei
最小平方法及,求得上式bo 、b1 使得誤差平方(∑e^2)最小
b1=Sxy/Sxx ,b0 = Yˉ – b1 * Xˉ (回歸線必會經過X,Y平均)
Sxy=∑(Xi−Xˉ)⋅(Yi−Yˉ),Sxx=∑(Xi−Xˉ)^2
beta_1 <- sum((STR – mean(STR)) * (score – mean(score))) / sum((STR – mean(STR))^2)
beta_0 <- mean(score) – beta_1 * mean(STR)
#beta_1 =-2.279808
#beta_0 =698.9329
linear_model <- lm(score ~ STR, data = CASchools)
Coefficients:
(Intercept) STR
698.93 -2.28
注意:lm(X~Y ,data =Data)
plot(score ~ STR,
data = CASchools,
main = “Scatterplot of Test Score and STR”,
xlab = “STR (X)”,
ylab = “Test Score (Y)”,
xlim = c(10, 30),
ylim = c(600, 720))abline(linear_model)