GPA and GRE

dta <- read.csv("https://stats.idre.ucla.edu/stat/data/binary.csv")
dta <- dta[, c("gre", "gpa")]

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基本統計圖

400位學生GRE(X軸)與GPA(Y軸)的散佈圖

plot(dta, type = "p", xlab = "GRE分數", ylab = "GPA分數")
grid()

線性模型分析

\[y_i = _0 + _1 x_i + _i ,~~ _iN(0, ^2) \] GPA＝截距參數＋斜率參數×GRE＋殘差

分析摘要表

小數點4位,去掉星星。

summary(m0 <- lm(gpa ~ gre, data = dta))
## 
## Call:
## lm(formula = gpa ~ gre, data = dta)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -1.08675 -0.22435 -0.00015  0.24809  0.76176 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 2.6458978  0.0913100  28.977  < 2e-16 ***
## gre         0.0012660  0.0001525   8.304  1.6e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3518 on 398 degrees of freedom
## Multiple R-squared:  0.1477, Adjusted R-squared:  0.1455 
## F-statistic: 68.95 on 1 and 398 DF,  p-value: 1.596e-15

根據本份資料，學生每增加1分的GRE，能夠增加0.0012分的GPA（誤差為0.0001）。殘差估計為\(\hat{\sigma}\)為0.352。

變異數分析表

模型擬合圖

plot(dta, type = "p", xlab = "GRE分數", ylab = "GPA分數")
abline(m0, lty = 2)
grid()

殘差圖

檢查殘差分配有沒有規律

plot(resid(m0) ~ fitted(m0), xlab = "Fitted values", 
     ylab = "Residuals", ylim = c(-1.5, 1.5))
abline(h = 0, lty = 2)
grid()

檢驗殘差常態分佈

qqnorm(resid(m0))
qqline(resid(m0))
grid()

the end