讀取資料

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

套內數據

400位學生GPA和GRE的前六筆資料展示

head(dta)
   gpa gre
1 3.61 380
2 3.67 660
3 4.00 800
4 3.19 640
5 2.93 520
6 3.00 760

基本統計圖形

400位學生GRE(X軸)與GPA(Y軸)分數的散佈圖

plot(dta, type = "p", xlab = "GRE Score", ylab = "GPA Score",col="black")
grid()

線性模型分析

\[y_i = \beta_0 + \beta_1 x_i + \epsilon_i ,~~ \epsilon_i \sim N(0, \sigma^2) \] GPA Score＝截距參數＋斜率參數×GRE Score＋殘差（便假設為常態分配）

分析概要量表

小數點4位,去掉星星.

options(digits = 4, show.signif.stars = FALSE)
summary(m0 <- lm(gpa~gre, data =dta))

Call:
lm(formula = gpa ~ gre, data = dta)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.0867 -0.2244 -0.0002  0.2481  0.7618 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.645898   0.091310    29.0  < 2e-16
gre         0.001266   0.000152     8.3  1.6e-15

Residual standard error: 0.352 on 398 degrees of freedom
Multiple R-squared:  0.148, Adjusted R-squared:  0.146 
F-statistic: 68.9 on 1 and 398 DF,  p-value: 1.6e-15

根據這份數據, GPA分數每增加1分時，GRE會增加0.0012分(標準誤為0.0001)。 殘差標準誤\((\widehat{\sigma})\)為0.3520

方差分析表

anova(m0)
Analysis of Variance Table

Response: gpa
           Df Sum Sq Mean Sq F value  Pr(>F)
gre         1    8.5    8.53      69 1.6e-15
Residuals 398   49.3    0.12

模型擬合圖

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

殘差圖

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

##驗證殘差常態分佈

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

結束

顯示演練單元信息

sessionInfo()
R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.3

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib

locale:
[1] zh_TW.UTF-8/zh_TW.UTF-8/zh_TW.UTF-8/C/zh_TW.UTF-8/zh_TW.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

loaded via a namespace (and not attached):
 [1] compiler_3.4.3  backports_1.1.2 magrittr_1.5    rprojroot_1.3-2
 [5] tools_3.4.3     htmltools_0.3.6 yaml_2.1.17     Rcpp_0.12.15   
 [9] stringi_1.1.6   rmarkdown_1.9   knitr_1.20      stringr_1.3.0  
[13] digest_0.6.15   evaluate_0.10.1