Randomized distributions

Foundations of Inference in R

Jo Hardin

Instructor

Logic of inference

ch1_2_v2_infer.002.png

Logic of inference

ch1_2_v2_infer.003.png

Logic of inference

ch1_2_v2_infer.004.png

Logic of inference

ch1_2_v2_infer.005.png

Logic of inference

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Logic of inference

Understanding the null distribution

Generating a distribution of the statistic from the null population gives information about whether the observed data are inconsistent with the null hypothesis

Understanding the null distribution

Original data

Location	Cola	Orange
East	28	6
West	19	7

$\hat{p}_\text{east} = 28/(28 + 6) = 0.82$

$\hat{p}_\text{west} = 19/(19 + 7) = 0.73$

Understanding the null distribution

First shuffle, same as original

Location	Cola	Orange
East	28	6
West	19	7

ch1_2_v2_infer.017.png

Understanding the null distribution

Second shuffle

Location	Cola	Orange
East	27	7
West	20	6

ch1_2_v2_infer.019.png

Understanding the null distribution

Third shuffle

Location	Cola	Orange
East	28	8
West	21	5

ch1_2_v2_infer.020.png

Understanding the null distribution

Fourth shuffle

Location	Cola	Orange
East	25	9
West	22	4

ch1_2_v2_infer.021.png

Understanding the null distribution

Fifth shuffle

Location	Cola	Orange
East	29	5
West	18	8

ch1_2_v2_infer.022.png

Understanding the null distribution

Fifth shuffle

Location	Cola	Orange
East	29	5
West	18	8

ch1_2_v2_infer.023.png

Understanding the null distribution

ch1_2_v2_infer.024.png

Understanding the null distribution

ch1_2_v2_infer.025.png

Understanding the null distribution

ch1_2_v2_infer.026.png

Understanding the null distribution

ch1_2_v2_infer.027.png

Understanding the null distribution

ch1_2_v2_infer.028.png

Understanding the null distribution

ch1_2_v2_infer.029.png

One random permutation

soda %>%
group_by(location) %>%
summarize(prop_cola = 
    mean(drink == "cola")) %>%
summarize(diff(prop_cola))

# A tibble: 1 x 1
  `diff(prop_cola)`
              <dbl>
1        -0.09276018

library(infer)
soda %>% specify(drink ~ location,
                success = "cola") %>%
hypothesize(null = "independence") %>%
generate(reps = 1, type = "permute") %>%
calculate(stat = "diff in props", 
            order = c("west","east"))

# A tibble: 1 x 2
  replicate        stat
      <int>        <dbl>
1        1    -0.02488688

Many random permutations

soda %>%
  specify(drink ~ location, success = "cola") %>%
  hypothesize(null = "independence") %>%
  generate(reps = 5, type = "permute") %>%
  calculate(stat = "diff in props", order = c("west", "east"))

# A tibble: 5 x 2
  replicate        stat
      <int>       <dbl>
1         1  0.04298643
2         2 -0.09276018
3         3  0.11085973
4         4  0.17873303
5         5 -0.16063348

Random distribution

ch1_2_v2_infer.047.png

Let's practice!

Foundations of Inference in R