• Perform comparisons of multiple groups
  • Compare means/proportions
• Examine notion of variable (in)dependence
  • Test for independence of catergorical variables
• Readings
  • ISRS: ch. 3.4, 4.4

• LFS data: are average earnings the same for different levels of education?

• Test equality of #\(m\) group means \(H_0: \mu_1 = \mu_2 = \cdots = \mu_m\)

• Could test all pairs (\(\mu_1=\mu_2\), \(\mu_2=\mu_3\), etc) and reject \(H_0\) if any one is rejected
  • Problem with inflated Type I Error (multiple testing)
• Instead, test for equality of all means simultaneously
  • Use ANalysis Of VAriance (ANOVA)
  • Test statistic based on inter/intra-group variance

• Idea: compare variance (i.e. average distance) from common mean to that from individual means

library(coin)
lfs %>% 
  mutate( educ = factor(educ) ) %>% 
  kruskal_test( hrlyearn ~ educ , data = ., 
                distribution = "approx" )
## 
##  Approximative Kruskal-Wallis Test
## 
## data:  hrlyearn by educ (0, 1, 2, 3, 4, 5, 6)
## chi-squared = 5350.5, p-value < 2.2e-16

• Is unemployment rate the same for different levels of education?

• Contingency table contains frequencies of combinations of two categorical variables

lfs %>% 
  filter(lfsstat != 4 ) %>% 
  mutate( educ = factor(educ), empl = factor(lfsstat != 3)) %>% 
  xtabs( ~ empl + educ, data = .) 
##        educ
## empl        0     1     2     3     4     5     6
##   FALSE    33   251   502   200   574   683   296
##   TRUE    459  1816  6122  2237 10500 11457  5905

lfs %>% filter(lfsstat != 4 ) %>% 
  mutate( educ = factor(educ), empl = (lfsstat != 3)) %>% 
  ggplot( aes(x = educ, fill = empl)) + geom_bar()

• prop.table() for relative proportions
• addmargins() for table totals
• apply across rows/cols (margin = 1/2)

lfs %>% filter(lfsstat != 4 ) %>% 
  mutate( educ = factor(educ), empl = factor(lfsstat != 3)) %>% 
  xtabs( ~ empl + educ, data = .) %>% 
  prop.table( margin = 2 ) %>% 
  addmargins( 1 ) %>%  round(4)
##        educ
## empl         0      1      2      3      4      5      6
##   FALSE 0.0671 0.1214 0.0758 0.0821 0.0518 0.0563 0.0477
##   TRUE  0.9329 0.8786 0.9242 0.9179 0.9482 0.9437 0.9523
##   Sum   1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

lfs %>% filter(lfsstat != 4 ) %>% 
  mutate( educ = factor(educ), empl = (lfsstat != 3)) %>% 
  ggplot( aes(x = educ, fill = empl)) + 
  geom_bar(position = "fill")

library(ggmosaic)
lfs %>% filter(lfsstat != 4 ) %>% 
  mutate( educ = factor(educ), empl = (lfsstat != 3)) %>% 
  ggplot() + geom_mosaic(aes(x = product(educ), fill = empl)) + xlab("educ")

• Test equality of #\(m\) group proportions \(H_0: p_1 = p_2 = \cdots = p_m\)

• Under \(H_0\), it does not matter which group you are in
  • Proportion are independent of groups

• Equivalent to testing independence of categorical variables (prop./group)

• Chi-square (\(\chi^2\)) test statistic measures distance of observed from expected (under indep.) contingency table values
  • Sampling distribution under \(H_0\) by simulation (sample observations according to independent proportions)

library(coin)
lfs %>% filter(lfsstat != 4 ) %>% 
  mutate( educ = factor(educ), empl = factor(lfsstat != 3)) %>% 
  chisq_test( empl ~ educ, data= ., distribution = "approx")
## 
##  Approximative Pearson Chi-Squared Test
## 
## data:  empl by educ (0, 1, 2, 3, 4, 5, 6)
## chi-squared = 212.93, p-value < 2.2e-16

• Chi-square test can be extended to multi-category variables
  • E.g. Are education and marital status related?

lfs %>% 
  mutate( educ = factor(educ), marstat = factor(marstat)) %>% 
  chisq_test( marstat ~ educ, data= ., distribution = "approx")
## 
##  Approximative Pearson Chi-Squared Test
## 
## data:  marstat by educ (0, 1, 2, 3, 4, 5, 6)
## chi-squared = 10696, p-value < 2.2e-16

• coin::independence_test() provides general framework for comparing two or more groups
  • Formula-based argument Y ~ X
  • Compare Y values across levels of factor X
• \(Y\) variable type determines quantity to compare
  • Means for numeric \(Y\)
  • Proportions for factor \(Y\)
• Can use independence_test() for tests of equality of 2+ means/proportions