非參數 ANOVA 與不配對 t 檢定

R 中的假設檢定

Richie Cotton

Data Evangelist at DataCamp

非參數檢定

「非參數檢定」是不對檢定統計量假設機率分配的假設檢定。

非參數假設檢定有兩類：

模擬式。
名次式（以排序為基礎）。

t_test()

$H_{0}$: $\mu_{child} - \mu_{adult} = 0$ $H_{A}$: $\mu_{child} - \mu_{adult} > 0$

library(infer)
stack_overflow %>% 
  t_test(
    converted_comp ~ age_first_code_cut,
    order = c("child", "adult"),
    alternative = "greater"
  )

# A tibble: 1 x 6
  statistic  t_df p_value alternative lower_ci upper_ci
      <dbl> <dbl>   <dbl> <chr>          <dbl>    <dbl>
1      2.40 2083. 0.00814 greater        8438.      Inf

計算虛無分配

模擬式流程

null_distn <- stack_overflow %>% 
  specify(converted_comp ~ age_first_code_cut) %>%

  hypothesize(null = "independence") %>%

  generate(reps = 5000, type = "permute") %>%

  calculate(
    stat = "diff in means", 
    order = c("child", "adult")
  )

t 檢定（作為對照）

library(infer)
stack_overflow %>% 
  t_test(
    converted_comp ~ age_first_code_cut,
    order = c("child", "adult"),
    alternative = "greater"
  )

計算觀察到的統計量

模擬式流程

obs_stat <- stack_overflow %>% 
  specify(converted_comp ~ age_first_code_cut) %>% 
  calculate(
    stat = "diff in means", 
    order = c("child", "adult")
  )

t 檢定（作為對照）

library(infer)
stack_overflow %>% 
  t_test(
    converted_comp ~ age_first_code_cut,
    order = c("child", "adult"),
    alternative = "greater"
  )

取得 p 值

模擬式流程

get_p_value(
  null_distn, obs_stat, 
  direction = "greater"
)

# A tibble: 1 x 1
  p_value
    <dbl>
1  0.0066

t 檢定（作為對照）

library(infer)
stack_overflow %>% 
  t_test(
    converted_comp ~ age_first_code_cut,
    order = c("child", "adult"),
    alternative = "greater"
  )

# A tibble: 1 x 6
  statistic  t_df p_value alternative lower_ci upper_ci
      <dbl> <dbl>   <dbl> <chr>          <dbl>    <dbl>
1      2.40 2083. 0.00814 greater        8438.      Inf

向量的名次（排名）

x <- c(1, 15, 3, 10, 6)

rank(x)

1 5 2 4 3

「Wilcoxon-Mann-Whitney 檢定」（亦稱「Wilcoxon 名次和檢定」）大致上是對數值輸入之名次做 t 檢定。

Wilcoxon-Mann-Whitney 檢定

wilcox.test(
  converted_comp ~ age_first_code_cut,
  data = stack_overflow,
  alternative = "greater",
  correct = FALSE
)

    Wilcoxon rank sum test

data:  converted_comp by age_first_code_cut
W = 967298, p-value <2e-16
alternative hypothesis: true location shift is greater than 0

¹ 亦稱「Wilcoxon rank-sum test」與「Mann-Whitney U test」。

Kruskal-Wallis 檢定

Kruskal-Wallis 檢定之於 Wilcoxon-Mann-Whitney 檢定，正如 ANOVA 之於 t 檢定。

kruskal.test(
  converted_comp ~ job_sat,
  data = stack_overflow
)

    Kruskal-Wallis rank sum test

data:  converted_comp by job_sat
Kruskal-Wallis chi-square = 81, df = 4, p-value <2e-16

一起來練習吧！

R 中的假設檢定