12 2-way Chi Square Test “Independence”

12.1 Example: Violent Crime and Abuse (2x2)

tab_crime <- tibble::tribble(~abuse, ~violent, ~count,
                             "Yes Abuse", "Yes Violet", 70,
                             "Yes Abuse", "No Violet",  30,
                             "No Abuse",  "Yes Violet", 40,
                             "No Abuse",  "No Violet",  60) %>% 
  tidyr::pivot_wider(names_from = violent,
                     values_from = count) %>% 
  tibble::column_to_rownames(var = "abuse") %>% 
  as.matrix() %>% 
  as.table()
  

tab_crime

          Yes Violet No Violet
Yes Abuse         70        30
No Abuse          40        60

tab_crime %>% 
  chisq.test()

    Pearson's Chi-squared test with Yates' continuity correction

data:  .
X-squared = 16.99, df = 1, p-value = 3.758e-05

tab_crime %>% 
  chisq.test(correct = FALSE)

    Pearson's Chi-squared test

data:  .
X-squared = 18.182, df = 1, p-value = 2.008e-05

tab_crime %>% 
  effectsize::cramers_v()

# A tibble: 1 x 4
  cramers_v    CI CI_low CI_high
      <dbl> <dbl>  <dbl>   <dbl>
1     0.291  0.95  0.153   0.430

12.2 Example: Schizophrenic Treatment (Cohen textbook)

schizo_counts <- tibble::tribble(
  ~trt, ~response, ~n,
  "Psychodynamic Therapy", "Improved",  6,
  "Psychodynamic Therapy", "No Change", 6,
  "Psychodynamic Therapy", "Got Worse", 8,
  "Rogerian Therapy", "Improved",  4,
  "Rogerian Therapy", "No Change", 14,
  "Rogerian Therapy", "Got Worse", 2,
  "Group Therapy", "Improved",  8,
  "Group Therapy", "No Change", 3,
  "Group Therapy", "Got Worse", 9,
  "Token Economy", "Improved",  12,
  "Token Economy", "No Change", 5,
  "Token Economy", "Got Worse", 3)

schizo_counts

# A tibble: 12 x 3
   trt                   response      n
   <chr>                 <chr>     <dbl>
 1 Psychodynamic Therapy Improved      6
 2 Psychodynamic Therapy No Change     6
 3 Psychodynamic Therapy Got Worse     8
 4 Rogerian Therapy      Improved      4
 5 Rogerian Therapy      No Change    14
 6 Rogerian Therapy      Got Worse     2
 7 Group Therapy         Improved      8
 8 Group Therapy         No Change     3
 9 Group Therapy         Got Worse     9
10 Token Economy         Improved     12
11 Token Economy         No Change     5
12 Token Economy         Got Worse     3

df_schizo <- schizo_counts %>% 
  tidyr::uncount(weights = n) %>% 
  dplyr::mutate(id = row_number()) %>% 
  dplyr::select(id, trt, response)

psych::headTail(df_schizo)

# A tibble: 9 x 3
  id    trt                   response 
  <chr> <chr>                 <chr>    
1 1     Psychodynamic Therapy Improved 
2 2     Psychodynamic Therapy Improved 
3 3     Psychodynamic Therapy Improved 
4 4     Psychodynamic Therapy Improved 
5 ...   <NA>                  <NA>     
6 77    Token Economy         No Change
7 78    Token Economy         Got Worse
8 79    Token Economy         Got Worse
9 80    Token Economy         Got Worse

df_schizo %>% 
  furniture::tableX(trt, response)

                       response
trt                     Got Worse Improved No Change Total
  Group Therapy         9         8        3         20   
  Psychodynamic Therapy 8         6        6         20   
  Rogerian Therapy      2         4        14        20   
  Token Economy         3         12       5         20   
  Total                 22        30       28        80   

df_schizo %>% 
  dplyr::group_by("Treatment Group" = trt) %>% 
  furniture::table1("Six Months" = response)

---------------------------------------------------------------------------------
                                      Treatment Group 
              Group Therapy Psychodynamic Therapy Rogerian Therapy
              n = 20        n = 20                n = 20          
 Six Months                                                       
    Got Worse 9 (45%)       8 (40%)               2 (10%)         
    Improved  8 (40%)       6 (30%)               4 (20%)         
    No Change 3 (15%)       6 (30%)               14 (70%)        
 Token Economy
 n = 20       
              
 3 (15%)      
 12 (60%)     
 5 (25%)      
---------------------------------------------------------------------------------

df_schizo %>% 
  dplyr::group_by("Treatment Group" = trt) %>% 
  furniture::table1("Six Months" = response,
                    test = TRUE,
                    output = "markdown")

	Group Therapy	Psychodynamic Therapy	Rogerian Therapy	Token Economy	P-Value
	n = 20	n = 20	n = 20	n = 20
Six Months					0.002
Got Worse	9 (45%)	8 (40%)	2 (10%)	3 (15%)
Improved	8 (40%)	6 (30%)	4 (20%)	12 (60%)
No Change	3 (15%)	6 (30%)	14 (70%)	5 (25%)

df_schizo %>% 
  dplyr::select(response, trt) %>% 
  table() %>% 
  addmargins()

           trt
response    Group Therapy Psychodynamic Therapy Rogerian Therapy Token Economy
  Got Worse             9                     8                2             3
  Improved              8                     6                4            12
  No Change             3                     6               14             5
  Sum                  20                    20               20            20
           trt
response    Sum
  Got Worse  22
  Improved   30
  No Change  28
  Sum        80

fit_schizo <- df_schizo %>% 
  dplyr::select(response, trt) %>% 
  table() %>% 
  chisq.test()

fit_schizo

    Pearson's Chi-squared test

data:  .
X-squared = 21.394, df = 6, p-value = 0.001558

fit_schizo$resid

           trt
response    Group Therapy Psychodynamic Therapy Rogerian Therapy Token Economy
  Got Worse     1.4924050             1.0660036       -1.4924050    -1.0660036
  Improved      0.1825742            -0.5477226       -1.2780193     1.6431677
  No Change    -1.5118579            -0.3779645        2.6457513    -0.7559289

df_schizo %>% 
  dplyr::select(response, trt) %>% 
  table() %>% 
  effectsize::cramers_v()

# A tibble: 1 x 4
  cramers_v    CI CI_low CI_high
      <dbl> <dbl>  <dbl>   <dbl>
1     0.366  0.95  0.145   0.483