Hypothesis Testing

using computer simulation. Based on examples from the infer package. Code for Quiz 13.

Load the R packages we will use.

library(tidyverse)
library(infer)
library(skimr)

• Replace all the instances of ???. These are answers on your moodle quiz.

• Run all the individual code chunks to make sure the answers in this file correspond with your quiz answers

• After you check all your code chunks run then you can knit it. It won’t knit until the ??? are replaced

• Save a plot to be your preview plot

Question: t-test

• The data this quiz uses is a subset of HR
  • Look at the variable definitions
  • Note that the variables evaluation and salary have been recoded to be represented as words instead of numbers
• Set random seed generator to 123

set.seed(123)

hr_2_tidy.csv is the name of your data subset

• Read it into and assign to hr
  • Note: col_types = “fddfff” defines the column types factor-double-double-factor-factor-factor

hr  <- read_csv("https://estanny.com/static/week13/data/hr_2_tidy.csv", 
                col_types = "fddfff") 

---

use skim to summarize the data in hr

skim(hr)

Table 1: Data summary

Name	hr
Number of rows	500
Number of columns	6
_______________________
Column type frequency:
factor	4
numeric	2
________________________
Group variables	None

Variable type: factor

skim_variable	n_missing	complete_rate	ordered	n_unique	top_counts
gender	0	1	FALSE	2	mal: 256, fem: 244
evaluation	0	1	FALSE	4	bad: 154, fai: 142, goo: 108, ver: 96
salary	0	1	FALSE	6	lev: 95, lev: 94, lev: 87, lev: 85
status	0	1	FALSE	3	fir: 194, pro: 179, ok: 127

Variable type: numeric

skim_variable	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
age	0	1	39.86	11.55	20.3	29.60	40.2	50.1	59.9	▇▇▇▇▇
hours	0	1	49.39	13.15	35.0	37.48	45.6	58.9	79.9	▇▃▂▂▂

Q: Is the mean number of hours worked per week 48?

The mean hours worked per week is: 49.4

---

specify that hours is the variable of interest

hr  %>% 
  specify(response = hours)

Response: hours (numeric)
# A tibble: 500 × 1
   hours
   <dbl>
 1  78.1
 2  35.1
 3  36.9
 4  38.5
 5  36.1
 6  78.1
 7  76  
 8  35.6
 9  35.6
10  56.8
# … with 490 more rows

---

hypothesize that the average hours worked is 48

hr  %>% 
  specify(response = hours)  %>% 
  hypothesize(null = "point", mu = 48)

Response: hours (numeric)
Null Hypothesis: point
# A tibble: 500 × 1
   hours
   <dbl>
 1  78.1
 2  35.1
 3  36.9
 4  38.5
 5  36.1
 6  78.1
 7  76  
 8  35.6
 9  35.6
10  56.8
# … with 490 more rows

---

generate 1000 replicates representing the null hypothesis

hr %>% 
  specify(response = hours)  %>% 
  hypothesize(null = "point", mu = 48)  %>% 
  generate(reps = 1000, type = "bootstrap") 

Response: hours (numeric)
Null Hypothesis: point
# A tibble: 500,000 × 2
# Groups:   replicate [1,000]
   replicate hours
       <int> <dbl>
 1         1  39.7
 2         1  44.3
 3         1  46.8
 4         1  33.7
 5         1  39.6
 6         1  39.5
 7         1  40.5
 8         1  55.8
 9         1  72.6
10         1  35.7
# … with 499,990 more rows

The output has 500,000 rows

calculate the distribution of statistics from the generated data

• Assign the output null_t_distribution
• Display null_t_distribution

null_t_distribution  <- hr  %>% 
  specify(response = age)  %>% 
  hypothesize(null = "point", mu = 48)  %>% 
  generate(reps = 1000, type = "bootstrap")  %>% 
  calculate(stat = "t")

null_t_distribution

Response: age (numeric)
Null Hypothesis: point
# A tibble: 1,000 × 2
   replicate     stat
       <int>    <dbl>
 1         1  0.144  
 2         2 -1.72   
 3         3  0.404  
 4         4 -1.11   
 5         5  0.00894
 6         6  1.46   
 7         7 -0.905  
 8         8 -0.663  
 9         9  0.291  
10        10  3.09   
# … with 990 more rows

• null_t_distribution has 1000 t-stats

---

visualize the simulated null distribution

visualize(null_t_distribution)

---

calculate the statistic from your observed data

• Assign the output observed_t_statistic

• Display observed_t_statistic

observed_t_statistic  <- hr  %>%
  specify(response = hours)  %>% 
  hypothesize(null = "point", mu = 48)  %>%
  calculate(stat = "t")

observed_t_statistic

Response: hours (numeric)
Null Hypothesis: point
# A tibble: 1 × 1
   stat
  <dbl>
1  2.37

---

get_p_value from the simulated null distribution and the observed statistic

null_t_distribution  %>% 
  get_p_value(obs_stat = observed_t_statistic, direction = "two-sided")

# A tibble: 1 × 1
  p_value
    <dbl>
1   0.014

shade_p_value on the simulated null distribution

null_t_distribution  %>% 
  visualize() +
  shade_p_value(obs_stat = observed_t_statistic, direction = "two-sided")

---

Is the p-value < 0.05? yes

Does your analysis support the null hypothesis that the true mean number of hours worked was 48? no

---

Question: 2 sample t-test

hr_2_tidy.csv is the name of your data subset

• Read it into and assign to hr_2

  • Note: col_types = “fddfff” defines the column types factor-double-double-factor-factor-factor

hr_2 <- read_csv("https://estanny.com/static/week13/data/hr_2_tidy.csv", 
                col_types = "fddfff") 

---

Q: Is the average number of hours worked the same for both genders?

---

use skim to summarize the data in hr_2 by gender

hr_2 %>% 
  group_by(gender)  %>% 
  skim()

Table 2: Data summary

Name	Piped data
Number of rows	500
Number of columns	6
_______________________
Column type frequency:
factor	3
numeric	2
________________________
Group variables	gender

Variable type: factor

skim_variable	gender	n_missing	complete_rate	ordered	n_unique	top_counts
evaluation	male	0	1	FALSE	4	bad: 79, fai: 68, goo: 61, ver: 48
evaluation	female	0	1	FALSE	4	bad: 75, fai: 74, ver: 48, goo: 47
salary	male	0	1	FALSE	6	lev: 49, lev: 48, lev: 48, lev: 44
salary	female	0	1	FALSE	6	lev: 47, lev: 46, lev: 41, lev: 39
status	male	0	1	FALSE	3	fir: 93, pro: 90, ok: 73
status	female	0	1	FALSE	3	fir: 101, pro: 89, ok: 54

Variable type: numeric

skim_variable	gender	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
age	male	0	1	38.63	11.57	20.3	28.50	37.85	49.52	59.6	▇▇▆▆▆
age	female	0	1	41.14	11.43	20.3	31.30	41.60	50.90	59.9	▆▅▇▇▇
hours	male	0	1	49.30	13.24	35.0	37.35	46.00	59.23	79.9	▇▃▂▂▂
hours	female	0	1	49.49	13.08	35.0	37.68	45.05	58.73	78.4	▇▃▃▂▂

Females worked an average of 49.5 hours per week Males worked an average of 49.3 hours per week

---

Use geom_boxplot to plot distributions of hours worked by gender

hr_2 %>% 
  ggplot(aes(x = gender, y = hours)) + 
  geom_boxplot()

---

specify the variables of interest are hours and gender

hr_2 %>% 
  specify(response = hours, explanatory = gender)

Response: hours (numeric)
Explanatory: gender (factor)
# A tibble: 500 × 2
   hours gender
   <dbl> <fct> 
 1  78.1 male  
 2  35.1 female
 3  36.9 female
 4  38.5 male  
 5  36.1 male  
 6  78.1 female
 7  76   female
 8  35.6 female
 9  35.6 male  
10  56.8 male  
# … with 490 more rows

---

hypothesize that the number of hours worked and gender are independent

hr_2  %>% 
  specify(response = hours, explanatory = gender)  %>% 
  hypothesize(null = "independence")

Response: hours (numeric)
Explanatory: gender (factor)
Null Hypothesis: independence
# A tibble: 500 × 2
   hours gender
   <dbl> <fct> 
 1  78.1 male  
 2  35.1 female
 3  36.9 female
 4  38.5 male  
 5  36.1 male  
 6  78.1 female
 7  76   female
 8  35.6 female
 9  35.6 male  
10  56.8 male  
# … with 490 more rows

---

generate 1000 replicates representing the null hypothesis

hr_2 %>% 
  specify(response = hours, explanatory = gender)  %>% 
  hypothesize(null = "independence")  %>% 
  generate(reps = 1000, type = "permute") 

Response: hours (numeric)
Explanatory: gender (factor)
Null Hypothesis: independence
# A tibble: 500,000 × 3
# Groups:   replicate [1,000]
   hours gender replicate
   <dbl> <fct>      <int>
 1  47.8 male           1
 2  60.3 female         1
 3  46.5 female         1
 4  37.2 male           1
 5  74.1 male           1
 6  35.9 female         1
 7  35.6 female         1
 8  54.5 female         1
 9  55.6 male           1
10  44.1 male           1
# … with 499,990 more rows

---

calculate the distribution of statistics from the generated data

• Assign the output null_distribution_2_sample_permute

• Display null_distribution_2_sample_permute

null_distribution_2_sample_permute  <- hr_2 %>% 
  specify(response = hours, explanatory = gender)  %>% 
  hypothesize(null = "independence")  %>% 
  generate(reps = 1000, type = "permute")  %>% 
  calculate(stat = "t", order = c("female", "male"))

null_distribution_2_sample_permute

Response: hours (numeric)
Explanatory: gender (factor)
Null Hypothesis: independence
# A tibble: 1,000 × 2
   replicate   stat
       <int>  <dbl>
 1         1  0.505
 2         2 -0.650
 3         3  0.279
 4         4  0.435
 5         5  1.73 
 6         6 -0.139
 7         7 -2.14 
 8         8  0.274
 9         9  0.766
10        10  1.52 
# … with 990 more rows

• null_t_distribution has 1000 t-stats

---

visualize the simulated null distribution

visualize(null_distribution_2_sample_permute)

---

calculate the statistic from your observed data

• Assign the output observed_t_2_sample_stat
• Display observed_t_2_sample_stat

observed_t_2_sample_stat  <- hr_2 %>%
  specify(response = hours, explanatory = gender)  %>% 
  calculate(stat = "t", order = c("female", "male"))

observed_t_2_sample_stat

Response: hours (numeric)
Explanatory: gender (factor)
# A tibble: 1 × 1
   stat
  <dbl>
1 0.160

---

get_p_value from the simulated null distribution and the observed statistic

null_t_distribution  %>% 
  get_p_value(obs_stat = observed_t_2_sample_stat, direction = "two-sided")

# A tibble: 1 × 1
  p_value
    <dbl>
1   0.924

shade_p_value on the simulated null distribution

null_t_distribution  %>% 
  visualize() +
  shade_p_value(obs_stat = observed_t_2_sample_stat, direction = "two-sided")

---

Is the p-value < 0.05? no

Does your analysis support the null hypothesis that the true mean number of hours worked by female and male employees was the same? yes

---

Question: ANOVA

hr_1_tidy.csv is the name of your data subset

• Read it into and assign to hr_anova

• Note: col_types = “fddfff” defines the column types factor-double-double-factor-factor-factor

hr_anova <- read_csv("https://estanny.com/static/week13/data/hr_1_tidy.csv", 
                col_types = "fddfff") 

### Q: Is the average number of hours worked the same for all three status (fired, ok and promoted) ?

use skim to summarize the data in hr_anova by status

hr_anova %>% 
  group_by(status)  %>% 
  skim()

Table 3: Data summary

Name	Piped data
Number of rows	500
Number of columns	6
_______________________
Column type frequency:
factor	3
numeric	2
________________________
Group variables	status

Variable type: factor

skim_variable	status	n_missing	complete_rate	ordered	n_unique	top_counts
gender	fired	0	1	FALSE	2	fem: 96, mal: 89
gender	ok	0	1	FALSE	2	fem: 77, mal: 76
gender	promoted	0	1	FALSE	2	fem: 87, mal: 75
evaluation	fired	0	1	FALSE	4	bad: 65, fai: 63, goo: 31, ver: 26
evaluation	ok	0	1	FALSE	4	bad: 69, fai: 59, goo: 15, ver: 10
evaluation	promoted	0	1	FALSE	4	ver: 63, goo: 60, fai: 20, bad: 19
salary	fired	0	1	FALSE	6	lev: 41, lev: 37, lev: 32, lev: 32
salary	ok	0	1	FALSE	6	lev: 40, lev: 37, lev: 29, lev: 23
salary	promoted	0	1	FALSE	6	lev: 37, lev: 35, lev: 29, lev: 23

Variable type: numeric

skim_variable	status	n_missing	complete_rate	mean	sd	p0	p25	p50	p75	p100	hist
age	fired	0	1	38.64	11.43	20.2	28.30	38.30	47.60	59.6	▇▇▇▅▆
age	ok	0	1	41.34	12.11	20.3	31.00	42.10	51.70	59.9	▆▆▆▆▇
age	promoted	0	1	42.13	10.98	21.0	33.40	42.95	50.98	59.9	▆▅▆▇▇
hours	fired	0	1	41.67	7.88	35.0	36.10	38.90	43.90	75.5	▇▂▁▁▁
hours	ok	0	1	48.05	11.65	35.0	37.70	45.60	56.10	78.2	▇▃▃▂▁
hours	promoted	0	1	59.27	12.90	35.0	51.12	60.10	70.15	79.7	▆▅▇▇▇

• Employees that were fired worked an average of 41.7 hours per week
• Employees that were ok worked an average of 48.0 hours per week

• Employees that were promoted worked an average of 59.3 hours per week

Use geom_boxplot to plot distributions of hours worked by status

hr_anova %>% 
  ggplot(aes(x = status, y = hours)) + 
  geom_boxplot()

---

specify the variables of interest are hours and status

hr_anova %>% 
  specify(response = hours, explanatory = status)

Response: hours (numeric)
Explanatory: status (factor)
# A tibble: 500 × 2
   hours status  
   <dbl> <fct>   
 1  36.5 fired   
 2  55.8 ok      
 3  35   fired   
 4  52   promoted
 5  35.1 ok      
 6  36.3 ok      
 7  40.1 promoted
 8  42.7 fired   
 9  66.6 promoted
10  35.5 ok      
# … with 490 more rows

---

hypothesize that the number of hours worked and status are independent

hr_anova  %>% 
  specify(response = hours, explanatory = status)  %>% 
  hypothesize(null = "independence")

Response: hours (numeric)
Explanatory: status (factor)
Null Hypothesis: independence
# A tibble: 500 × 2
   hours status  
   <dbl> <fct>   
 1  36.5 fired   
 2  55.8 ok      
 3  35   fired   
 4  52   promoted
 5  35.1 ok      
 6  36.3 ok      
 7  40.1 promoted
 8  42.7 fired   
 9  66.6 promoted
10  35.5 ok      
# … with 490 more rows

---

generate 1000 replicates representing the null hypothesis

hr_anova %>% 
  specify(response = hours, explanatory = status)  %>% 
  hypothesize(null = "independence")  %>% 
  generate(reps = 1000, type = "permute") 

Response: hours (numeric)
Explanatory: status (factor)
Null Hypothesis: independence
# A tibble: 500,000 × 3
# Groups:   replicate [1,000]
   hours status   replicate
   <dbl> <fct>        <int>
 1  40.3 fired            1
 2  40.3 ok               1
 3  37.3 fired            1
 4  50.5 promoted         1
 5  35.1 ok               1
 6  67.8 ok               1
 7  39.3 promoted         1
 8  35.7 fired            1
 9  40.2 promoted         1
10  38.4 ok               1
# … with 499,990 more rows

The output has 500,000 rows

---

calculate the distribution of statistics from the generated data

• Assign the output null_distribution_anova

• Display null_distribution_anova

null_distribution_anova  <- hr_anova %>% 
  specify(response = hours, explanatory = status)  %>% 
  hypothesize(null = "independence")  %>% 
  generate(reps = 1000, type = "permute")  %>% 
  calculate(stat = "F")

null_distribution_anova

Response: hours (numeric)
Explanatory: status (factor)
Null Hypothesis: independence
# A tibble: 1,000 × 2
   replicate   stat
       <int>  <dbl>
 1         1 0.667 
 2         2 2.78  
 3         3 1.24  
 4         4 0.330 
 5         5 2.08  
 6         6 1.95  
 7         7 0.243 
 8         8 0.312 
 9         9 0.440 
10        10 0.0281
# … with 990 more rows

• null_distribution_anova has 1000 F-stats

---

visualize the simulated null distribution

visualize(null_distribution_anova)

---

calculate the statistic from your observed data

• Assign the output observed_f_sample_stat

• Display observed_f_sample_stat

observed_f_sample_stat  <- hr_anova %>%
  specify(response = hours, explanatory = status)  %>% 
  calculate(stat = "F")

observed_f_sample_stat

Response: hours (numeric)
Explanatory: status (factor)
# A tibble: 1 × 1
   stat
  <dbl>
1  115.

---

get_p_value from the simulated null distribution and the observed statistic

null_distribution_anova  %>% 
  get_p_value(obs_stat = observed_f_sample_stat, direction = "greater")

# A tibble: 1 × 1
  p_value
    <dbl>
1       0

shade_p_value on the simulated null distribution

null_t_distribution  %>% 
  visualize() +
  shade_p_value(obs_stat = observed_f_sample_stat, direction = "greater")

Save the previous plot to preview.png and add to the yaml chunk at the top

ggsave(filename = "preview.png", 
       path = here::here("_posts", "2022-05-02-hypothesis-testing"))

---

If the p-value < 0.05? yes

Does your analysis support the null hypothesis that the true means of the number of hours worked for those that were “fired”, “ok” and “promoted” were the same? no