Full infer pipeline examples using nycflights13 flights data

Chester Ismay

Updated on 2018-06-14

Source: vignettes/observed_stat_examples.Rmd

Data preparation

library(nycflights13)
library(dplyr)
library(ggplot2)
library(stringr)
library(infer)
set.seed(2017)
fli_small <- flights %>% 
  na.omit() %>%
  sample_n(size = 500) %>% 
  mutate(season = case_when(
    month %in% c(10:12, 1:3) ~ "winter",
    month %in% c(4:9) ~ "summer"
  )) %>% 
  mutate(day_hour = case_when(
    between(hour, 1, 12) ~ "morning",
    between(hour, 13, 24) ~ "not morning"
  )) %>% 
  select(arr_delay, dep_delay, season, 
         day_hour, origin, carrier)
  • Two numeric - arr_delay, dep_delay
  • Two categories
    • season ("winter", "summer"),
    • day_hour ("morning", "not morning")
  • Three categories - origin ("EWR", "JFK", "LGA")
  • Sixteen categories - carrier

Hypothesis tests

One numerical variable (mean)

Observed stat

( x_bar <- fli_small %>%
  specify(response = dep_delay) %>%
  calculate(stat = "mean") )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1  11.5
null_distn <- fli_small %>%
  specify(response = dep_delay) %>%
  hypothesize(null = "point", mu = 10) %>%
  generate(reps = 1000) %>%
  calculate(stat = "mean")
## Setting `type = "bootstrap"` in `generate()`.
visualize(null_distn) +
  shade_p_value(obs_stat = x_bar, direction = "two_sided")

null_distn %>%
  get_p_value(obs_stat = x_bar, direction = "two_sided")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1   0.356

One numerical variable (standardized mean \(t\))

Observed stat

t_bar <- fli_small %>%
  t_stat(response = dep_delay, mu = 8)
null_distn <- fli_small %>%
  specify(response = dep_delay) %>%
  hypothesize(null = "point", mu = 8) %>%
  generate(reps = 1000) %>%
  calculate(stat = "t")
## Setting `type = "bootstrap"` in `generate()`.
visualize(null_distn) +
  shade_p_value(obs_stat = t_bar, direction = "two_sided")

null_distn %>%
  get_p_value(obs_stat = t_bar, direction = "two_sided")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1   0.018

One numerical variable (median)

Observed stat

( x_tilde <- fli_small %>%
  specify(response = dep_delay) %>%
  calculate(stat = "median") )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1    -2
null_distn <- fli_small %>%
  specify(response = dep_delay) %>%
  hypothesize(null = "point", med = -1) %>% 
  generate(reps = 1000) %>% 
  calculate(stat = "median")
## Setting `type = "bootstrap"` in `generate()`.
visualize(null_distn) +
  shade_p_value(obs_stat = x_tilde, direction = "two_sided")

null_distn %>%
  get_p_value(obs_stat = x_tilde, direction = "two_sided")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1   0.018

One categorical (one proportion)

Observed stat

( p_hat <- fli_small %>%
  specify(response = day_hour, success = "morning") %>%
  calculate(stat = "prop") )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1 0.452
null_distn <- fli_small %>%
  specify(response = day_hour, success = "morning") %>%
  hypothesize(null = "point", p = .5) %>%
  generate(reps = 1000) %>%
  calculate(stat = "prop")
## Setting `type = "simulate"` in `generate()`.
visualize(null_distn) +
  shade_p_value(obs_stat = p_hat, direction = "two_sided")

null_distn %>%
  get_p_value(obs_stat = p_hat, direction = "two_sided")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1   0.036

Logical variables will be coerced to factors:

null_distn <- fli_small %>%
  mutate(day_hour_logical = (day_hour == "morning")) %>%
  specify(response = day_hour_logical, success = "TRUE") %>%
  hypothesize(null = "point", p = .5) %>%
  generate(reps = 1000) %>%
  calculate(stat = "prop")
## Setting `type = "simulate"` in `generate()`.

One categorical variable (standardized proportion \(z\))

Not yet implemented.

Two categorical (2 level) variables

Observed stat

( d_hat <- fli_small %>% 
  specify(day_hour ~ season, success = "morning") %>%
  calculate(stat = "diff in props", order = c("winter", "summer")) )
## # A tibble: 1 x 1
##      stat
##     <dbl>
## 1 0.00438
null_distn <- fli_small %>%
  specify(day_hour ~ season, success = "morning") %>%
  hypothesize(null = "independence") %>% 
  generate(reps = 1000) %>% 
  calculate(stat = "diff in props", order = c("winter", "summer"))
## Setting `type = "permute"` in `generate()`.
visualize(null_distn) +
  shade_p_value(obs_stat = d_hat, direction = "two_sided")

null_distn %>%
  get_p_value(obs_stat = d_hat, direction = "two_sided")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1   0.954

Two categorical (2 level) variables (z)

Standardized observed stat

( z_hat <- fli_small %>% 
  specify(day_hour ~ season, success = "morning") %>%
  calculate(stat = "z", order = c("winter", "summer")) )
## # A tibble: 1 x 1
##     stat
##    <dbl>
## 1 0.0985
null_distn <- fli_small %>%
  specify(day_hour ~ season, success = "morning") %>%
  hypothesize(null = "independence") %>% 
  generate(reps = 1000) %>% 
  calculate(stat = "z", order = c("winter", "summer"))
## Setting `type = "permute"` in `generate()`.
visualize(null_distn) +
  shade_p_value(obs_stat = z_hat, direction = "two_sided")

null_distn %>%
  get_p_value(obs_stat = z_hat, direction = "two_sided")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1    0.95

Note the similarities in this plot and the previous one.

One categorical (>2 level) - GoF

Observed stat

Note the need to add in the hypothesized values here to compute the observed statistic.

( Chisq_hat <- fli_small %>%
  specify(response = origin) %>%
  hypothesize(null = "point", 
              p = c("EWR" = .33, "JFK" = .33, "LGA" = .34)) %>% 
  calculate(stat = "Chisq") )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1  7.01
null_distn <- fli_small %>%
  specify(response = origin) %>%
  hypothesize(null = "point", 
              p = c("EWR" = .33, "JFK" = .33, "LGA" = .34)) %>% 
  generate(reps = 1000, type = "simulate") %>% 
  calculate(stat = "Chisq")

visualize(null_distn) +
  shade_p_value(obs_stat = Chisq_hat, direction = "greater")

null_distn %>%
  get_p_value(obs_stat = Chisq_hat, direction = "greater")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1   0.037

Two categorical (>2 level) variables

Observed stat

( Chisq_hat <- fli_small %>%
  specify(formula = day_hour ~ origin) %>% 
  calculate(stat = "Chisq") )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1 0.528
null_distn <- fli_small %>%
  specify(day_hour ~ origin) %>%
  hypothesize(null = "independence") %>% 
  generate(reps = 1000, type = "permute") %>% 
  calculate(stat = "Chisq")

visualize(null_distn) +
  shade_p_value(obs_stat = Chisq_hat, direction = "greater")

null_distn %>%
  get_p_value(obs_stat = Chisq_hat, direction = "greater")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1    0.77

One numerical variable, one categorical (2 levels) (diff in means)

Observed stat

( d_hat <- fli_small %>% 
  specify(dep_delay ~ season) %>% 
  calculate(stat = "diff in means", order = c("summer", "winter")) )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1     3
null_distn <- fli_small %>%
  specify(dep_delay ~ season) %>%
  hypothesize(null = "independence") %>%
  generate(reps = 1000, type = "permute") %>%
  calculate(stat = "diff in means", order = c("summer", "winter"))

visualize(null_distn) +
  shade_p_value(obs_stat = d_hat, direction = "two_sided")

null_distn %>%
  get_p_value(obs_stat = d_hat, direction = "two_sided")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1   0.338

One numerical variable, one categorical (2 levels) (t)

Standardized observed stat

( t_hat <- fli_small %>% 
  specify(dep_delay ~ season) %>% 
  calculate(stat = "t", order = c("summer", "winter")) )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1 0.891
null_distn <- fli_small %>%
  specify(dep_delay ~ season) %>%
  hypothesize(null = "independence") %>%
  generate(reps = 1000, type = "permute") %>%
  calculate(stat = "t", order = c("summer", "winter"))

visualize(null_distn) +
  shade_p_value(obs_stat = t_hat, direction = "two_sided")

null_distn %>%
  get_p_value(obs_stat = t_hat, direction = "two_sided")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1     0.4

Note the similarities in this plot and the previous one.

One numerical variable, one categorical (2 levels) (diff in medians)

Observed stat

( d_hat <- fli_small %>% 
  specify(dep_delay ~ season) %>% 
  calculate(stat = "diff in medians", order = c("summer", "winter")) )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1     1
null_distn <- fli_small %>%
  specify(dep_delay ~ season) %>% # alt: response = dep_delay, 
  # explanatory = season
  hypothesize(null = "independence") %>%
  generate(reps = 1000, type = "permute") %>%
  calculate(stat = "diff in medians", order = c("summer", "winter"))

visualize(null_distn) +
  shade_p_value(obs_stat = d_hat, direction = "two_sided")

null_distn %>%
  get_p_value(obs_stat = d_hat, direction = "two_sided")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1    0.64

One numerical, one categorical (>2 levels) - ANOVA

Observed stat

( F_hat <- fli_small %>% 
  specify(arr_delay ~ origin) %>%
  calculate(stat = "F") )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1 0.686
null_distn <- fli_small %>%
   specify(arr_delay ~ origin) %>%
   hypothesize(null = "independence") %>%
   generate(reps = 1000, type = "permute") %>%
   calculate(stat = "F")

visualize(null_distn) +
  shade_p_value(obs_stat = F_hat, direction = "greater")

null_distn %>%
  get_p_value(obs_stat = F_hat, direction = "greater")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1   0.529

Two numerical vars - SLR

Observed stat

( slope_hat <- fli_small %>% 
  specify(arr_delay ~ dep_delay) %>% 
  calculate(stat = "slope") )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1 0.992
null_distn <- fli_small %>%
   specify(arr_delay ~ dep_delay) %>% 
   hypothesize(null = "independence") %>%
   generate(reps = 1000, type = "permute") %>%
   calculate(stat = "slope")

visualize(null_distn) +
  shade_p_value(obs_stat = slope_hat, direction = "two_sided")

null_distn %>%
  get_p_value(obs_stat = slope_hat, direction = "two_sided")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1       0

Two numerical vars - correlation

Observed stat

( correlation_hat <- fli_small %>% 
  specify(arr_delay ~ dep_delay) %>% 
  calculate(stat = "correlation") )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1 0.895
null_distn <- fli_small %>%
   specify(arr_delay ~ dep_delay) %>% 
   hypothesize(null = "independence") %>%
   generate(reps = 1000, type = "permute") %>%
   calculate(stat = "correlation")

visualize(null_distn) +
  shade_p_value(obs_stat = correlation_hat, direction = "two_sided")

null_distn %>%
  get_p_value(obs_stat = correlation_hat, direction = "two_sided")
## # A tibble: 1 x 1
##   p_value
##     <dbl>
## 1       0

Two numerical vars - SLR (t)

Not currently implemented since \(t\) could refer to standardized slope or standardized correlation.

Confidence intervals

One numerical (one mean)

Point estimate

( x_bar <- fli_small %>% 
  specify(response = arr_delay) %>%
  calculate(stat = "mean") )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1  6.15
boot <- fli_small %>%
   specify(response = arr_delay) %>%
   generate(reps = 1000, type = "bootstrap") %>%
   calculate(stat = "mean")
( percentile_ci <- get_ci(boot) )
## # A tibble: 1 x 2
##   `2.5%` `97.5%`
##    <dbl>   <dbl>
## 1   2.61    9.60
visualize(boot) +
  shade_confidence_interval(endpoints = percentile_ci)

( standard_error_ci <- get_ci(boot, type = "se", point_estimate = x_bar) )
## # A tibble: 1 x 2
##   lower upper
##   <dbl> <dbl>
## 1  2.61  9.70
visualize(boot) +
  shade_confidence_interval(endpoints = standard_error_ci)

One numerical (one mean - standardized)

Point estimate

( t_hat <- fli_small %>% 
  specify(response = arr_delay) %>%
  calculate(stat = "t") )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1  3.30
boot <- fli_small %>%
   specify(response = arr_delay) %>%
   generate(reps = 1000, type = "bootstrap") %>%
   calculate(stat = "t")
( percentile_ci <- get_ci(boot) )
## # A tibble: 1 x 2
##   `2.5%` `97.5%`
##    <dbl>   <dbl>
## 1   1.62    4.88
visualize(boot) +
  shade_confidence_interval(endpoints = percentile_ci)

( standard_error_ci <- get_ci(boot, type = "se", point_estimate = t_hat) )
## # A tibble: 1 x 2
##   lower upper
##   <dbl> <dbl>
## 1  1.70  4.90
visualize(boot) +
  shade_confidence_interval(endpoints = standard_error_ci)

One categorical (one proportion)

Point estimate

( p_hat <- fli_small %>% 
   specify(response = day_hour, success = "morning") %>%
   calculate(stat = "prop") )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1 0.452
boot <- fli_small %>%
 specify(response = day_hour, success = "morning") %>%
 generate(reps = 1000, type = "bootstrap") %>%
 calculate(stat = "prop")
( percentile_ci <- get_ci(boot) )
## # A tibble: 1 x 2
##   `2.5%` `97.5%`
##    <dbl>   <dbl>
## 1  0.406   0.496
visualize(boot) +
  shade_confidence_interval(endpoints = percentile_ci)

( standard_error_ci <- get_ci(boot, type = "se", point_estimate = p_hat) )
## # A tibble: 1 x 2
##   lower upper
##   <dbl> <dbl>
## 1 0.408 0.496
visualize(boot) +
  shade_confidence_interval(endpoints = standard_error_ci)

One categorical variable (standardized proportion \(z\))

Not yet implemented.

One numerical variable, one categorical (2 levels) (diff in means)

Point estimate

( d_hat <- fli_small %>%
  specify(arr_delay ~ season) %>%
  calculate(stat = "diff in means", order = c("summer", "winter")) )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1  5.63
boot <- fli_small %>%
   specify(arr_delay ~ season) %>%
   generate(reps = 1000, type = "bootstrap") %>%
   calculate(stat = "diff in means", order = c("summer", "winter"))
( percentile_ci <- get_ci(boot) )
## # A tibble: 1 x 2
##   `2.5%` `97.5%`
##    <dbl>   <dbl>
## 1  -2.03    12.5
visualize(boot) +
  shade_confidence_interval(endpoints = percentile_ci)

( standard_error_ci <- get_ci(boot, type = "se", point_estimate = d_hat) )
## # A tibble: 1 x 2
##   lower upper
##   <dbl> <dbl>
## 1 -1.61  12.9
visualize(boot) +
  shade_confidence_interval(endpoints = standard_error_ci)

One numerical variable, one categorical (2 levels) (t)

Standardized point estimate

( t_hat <- fli_small %>%
  specify(arr_delay ~ season) %>%
  calculate(stat = "t", order = c("summer", "winter")) )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1  1.51
boot <- fli_small %>%
   specify(arr_delay ~ season) %>%
   generate(reps = 1000, type = "bootstrap") %>%
   calculate(stat = "t", order = c("summer", "winter"))
( percentile_ci <- get_ci(boot) )
## # A tibble: 1 x 2
##   `2.5%` `97.5%`
##    <dbl>   <dbl>
## 1 -0.359    3.74
visualize(boot) +
  shade_confidence_interval(endpoints = percentile_ci)

( standard_error_ci <- get_ci(boot, type = "se", point_estimate = t_hat) )
## # A tibble: 1 x 2
##    lower upper
##    <dbl> <dbl>
## 1 -0.578  3.60
visualize(boot) +
  shade_confidence_interval(endpoints = standard_error_ci)

Two categorical variables (diff in proportions)

Point estimate

( d_hat <- fli_small %>% 
  specify(day_hour ~ season, success = "morning") %>%
  calculate(stat = "diff in props", order = c("summer", "winter")) )
## # A tibble: 1 x 1
##       stat
##      <dbl>
## 1 -0.00438
boot <- fli_small %>%
  specify(day_hour ~ season, success = "morning") %>%
  generate(reps = 1000, type = "bootstrap") %>% 
  calculate(stat = "diff in props", order = c("summer", "winter"))
( percentile_ci <- get_ci(boot) )
## # A tibble: 1 x 2
##    `2.5%` `97.5%`
##     <dbl>   <dbl>
## 1 -0.0957  0.0818
visualize(boot) +
  shade_confidence_interval(endpoints = percentile_ci)

( standard_error_ci <- get_ci(boot, type = "se", point_estimate = d_hat) )
## # A tibble: 1 x 2
##     lower  upper
##     <dbl>  <dbl>
## 1 -0.0914 0.0826
visualize(boot) +
  shade_confidence_interval(endpoints = standard_error_ci)

Two categorical variables (z)

Standardized point estimate

( z_hat <- fli_small %>% 
  specify(day_hour ~ season, success = "morning") %>%
  calculate(stat = "z", order = c("summer", "winter")) )
## # A tibble: 1 x 1
##      stat
##     <dbl>
## 1 -0.0985
boot <- fli_small %>%
  specify(day_hour ~ season, success = "morning") %>%
  generate(reps = 1000, type = "bootstrap") %>% 
  calculate(stat = "z", order = c("summer", "winter"))
( percentile_ci <- get_ci(boot) )
## # A tibble: 1 x 2
##   `2.5%` `97.5%`
##    <dbl>   <dbl>
## 1  -1.96    1.79
visualize(boot) +
  shade_confidence_interval(endpoints = percentile_ci)

( standard_error_ci <- get_ci(boot, type = "se", point_estimate = z_hat) )
## # A tibble: 1 x 2
##   lower upper
##   <dbl> <dbl>
## 1 -2.04  1.85
visualize(boot) +
  shade_confidence_interval(endpoints = standard_error_ci)

Two numerical vars - SLR

Point estimate

( slope_hat <- fli_small %>% 
  specify(arr_delay ~ dep_delay) %>%
  calculate(stat = "slope") )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1 0.992
boot <- fli_small %>%
   specify(arr_delay ~ dep_delay) %>% 
   generate(reps = 1000, type = "bootstrap") %>%
   calculate(stat = "slope")
( percentile_ci <- get_ci(boot) )
## # A tibble: 1 x 2
##   `2.5%` `97.5%`
##    <dbl>   <dbl>
## 1  0.946    1.03
visualize(boot) +
  shade_confidence_interval(endpoints = percentile_ci)

( standard_error_ci <- get_ci(boot, type = "se", point_estimate = slope_hat) )
## # A tibble: 1 x 2
##   lower upper
##   <dbl> <dbl>
## 1 0.947  1.04
visualize(boot) +
  shade_confidence_interval(endpoints = standard_error_ci)

Two numerical vars - correlation

Point estimate

( correlation_hat <- fli_small %>% 
  specify(arr_delay ~ dep_delay) %>%
  calculate(stat = "correlation") )
## # A tibble: 1 x 1
##    stat
##   <dbl>
## 1 0.895
boot <- fli_small %>%
   specify(arr_delay ~ dep_delay) %>% 
   generate(reps = 1000, type = "bootstrap") %>%
   calculate(stat = "correlation")
( percentile_ci <- get_ci(boot) )
## # A tibble: 1 x 2
##   `2.5%` `97.5%`
##    <dbl>   <dbl>
## 1  0.827   0.933
visualize(boot) +
  shade_confidence_interval(endpoints = percentile_ci)

## # A tibble: 1 x 2
##   lower upper
##   <dbl> <dbl>
## 1 0.842 0.948
visualize(boot) +
  shade_confidence_interval(endpoints = standard_error_ci)

Two numerical vars - t

Not currently implemented since \(t\) could refer to standardized slope or standardized correlation.