继续 infer 流程

R 中的假设检验

Richie Cotton

Data Evangelist at DataCamp

回顾：假设与数据集

$H_{0}$：30岁以下业余爱好者的比例与至少30岁的比例相同。

$H_{A}$：30岁以下业余爱好者的比例与至少30岁的比例不同。

alpha <- 0.1

stack_overflow_imbalanced %>% 
  count(hobbyist, age_cat, .drop = FALSE)

  hobbyist     age_cat    n
1       No At least 30    0
2       No    Under 30  191
3      Yes At least 30   15
4      Yes    Under 30 1025

回顾：工作流程

null_distn <- dataset %>% 
  specify() %>% 
  hypothesize() %>% 
  generate() %>% 
  calculate()

observed_stat <- dataset %>% 
  specify() %>% 
  calculate()

get_p_value(null_distn, observed_stat)

stack_overflow_imbalanced %>%
  specify(hobbyist ~ age_cat, success = "Yes") %>% 
  hypothesize(null = "independence")

Response: hobbyist (factor)
Explanatory: age_cat (factor)
Null Hypothesis: independence
# A tibble: 1,231 x 2
  hobbyist age_cat    
  <fct>    <fct>      
1 Yes      At least 30
2 Yes      At least 30
3 Yes      At least 30
4 Yes      Under 30   
5 Yes      At least 30
6 Yes      At least 30
7 No       Under 30   
# ... with 1,224 more rows

引出 generate()

$H_{0}$：30岁以下业余爱好者的比例与至少30岁的比例相同。

若 $H_{0}$ 为真，则：

每行的 hobbyist 值以相同概率出现在任一年龄组。
为模拟此过程，可在固定年龄组的同时置换（打乱）hobbyist 值。

stack_overflow_imbalanced

# A tibble: 1,231 x 2
  hobbyist age_cat    
  <fct>    <fct>      
1 Yes      At least 30
2 Yes      At least 30
3 Yes      At least 30
4 Yes      Under 30   
5 Yes      At least 30
6 Yes      At least 30
7 No       Under 30   
# ... with 1,224 more rows

bind_cols(
  stack_overflow_imbalanced %>% 
    select(hobbyist) %>% 
    slice_sample(prop = 1),
  stack_overflow_imbalanced %>% 
    select(age_cat)
)

# A tibble: 1,231 x 2
  hobbyist age_cat    
  <fct>    <fct>      
1 Yes      At least 30
2 Yes      At least 30
3 No       At least 30
4 No       Under 30   
5 Yes      At least 30
6 Yes      At least 30
7 Yes      Under 30   
# ... with 1,224 more rows

生成多次重复

左侧是由指定列得到的两列网格。右侧有"generate"及向右箭头。箭头右边是三份两列网格，表示重复样本。每个重复样本的右列与原数据相同，表示解释变量不变；左列不同，表示响应变量被置换。

generate()

generate() 生成符合原假设的模拟数据。

"独立性"原假设：type 设为 "permute"。
"点"原假设：type 设为 "bootstrap" 或 "simulate"。

stack_overflow_imbalanced %>%
  specify(hobbyist ~ age_cat, success = "Yes") %>% 
  hypothesize(null = "independence") %>% 
  generate(reps = 5000, type = "permute")

Response: hobbyist (factor)
Explanatory: age_cat (factor)
Null Hypothesis: independence
# A tibble: 6,155,000 x 3
# Groups:   replicate [5,000]
  hobbyist age_cat     replicate
  <fct>    <fct>           <int>
1 Yes      At least 30         1
2 Yes      At least 30         1
3 Yes      At least 30         1
4 Yes      Under 30            1
5 Yes      At least 30         1
6 Yes      At least 30         1
7 Yes      Under 30            1
# ... with 6,154,993 more rows

计算检验统计量

显示原始数据与重复样本的网格。在其下方是"calculate"，每个重复样本下有向下箭头，箭头下是一个代表检验统计量的单元格。所有重复的统计量被框出并标注为"原假设分布"。

calculate()

calculate() 计算检验统计量的分布，即"原假设分布"。

null_distn <- stack_overflow_imbalanced %>%
  specify(
    hobbyist ~ age_cat, 
    success = "Yes"
  ) %>%
  hypothesize(null = "independence") %>%
  generate(reps = 5000, type = "permute") %>%
  calculate(
    stat = "diff in props", 
    order = c("At least 30", "Under 30")
  )

# A tibble: 5,000 x 2
  replicate    stat
      <int>   <dbl>
1         1  0.0896
2         2  0.0896
3         3 -0.180 
4         4  0.157 
5         5  0.0896
6         6 -0.113 
7         7  0.0221
# ... with 4,993 more rows

¹ ?calculate 帮助页列出了所有可用的检验统计量。

可视化原假设分布

visualize(null_distn)

原假设分布的直方图。左偏，共有九个不同取值。

null_distn %>% count(stat)

# A tibble: 9 x 2
     stat     n
    <dbl> <int>
1 -0.383      2
2 -0.315     22
3 -0.248     63
4 -0.180    246
5 -0.113    641
6 -0.0454  1132
7  0.0221  1453
8  0.0896  1063
9  0.157    378

在原始数据上计算检验统计量

展示原始数据与重复样本网格，以及在 calculate 步骤得到的原假设分布单元格。本次还在原始数据下方有向下箭头，箭头下方一个被框注的单元格标注为"观测统计量"。

观测统计量：specify() %>% calculate()

obs_stat <- stack_overflow_imbalanced %>%
  specify(hobbyist ~ age_cat, success = "Yes") %>%
  # hypothesize(null = "independence") %>%
  # generate(reps = 5000, type = "permute") %>%
  calculate(
    stat = "diff in props",
    order = c("At least 30", "Under 30")
  )

# A tibble: 1 x 1
   stat
  <dbl>
1 0.157

对比原假设分布与观测统计量

visualize(null_distn) +
  geom_vline(
    aes(xintercept = stat),
    data = observed_stat, 
    color = "red"
  )

原假设分布的直方图，并在观测统计量处加一条红色竖线。该竖线位于最右侧的柱上。

获取 p 值

get_p_value(
  null_distn, obs_stat, 
  direction = "two sided"   # Not alternative = "two.sided"
)

# A tibble: 1 x 1
  p_value
    <dbl>
1   0.151

# A tibble: 1 x 6
  statistic chisq_df p_value alternative lower_ci upper_ci
      <dbl>    <dbl>   <dbl> <chr>          <dbl>    <dbl>
1      2.79        1  0.0949 two.sided    0.00718   0.0217

Passons à la pratique !

R 中的假设检验