Method

• requirements to be reasonable
  1. a simple random sample from the population / randomized experiment
  2. a sample size large enough for the ${\chi }^2$ approximation to be reasonable
• $H_0:p_1=p_{1_0},p_2=p_{2_0},\dots ,p_c=p_{c_0}$
  • null hypothesis that the true proportions for the c levels of a categorical variable are equal to hypothesized values

for one-way tables

• observations are classified according to only one categorical variable
  • each count belongs to only one cell

Test Statistic

• ${\chi }^2=\sum _{\text{all cells}}\frac{{\text{Observed-Expected}}^2}{\text{Expected}}$
  • observed and expected represent counts with the specific category
  • the test statistic ${\chi }^2$ will have (approx) a ${\chi }^2$ distribution if the null hypothesis is true
    • test using the right-tailed p-value in the distribution with the value we got
    • large values of ${\chi }^2$ give evidence against the null
  • degrees of freedom are number of cells$(c)-1$
  • each cell is one part of the proportion
    • $\hat{p}=\frac{50}{50}\to$ 2 cells, 100% divided into n groups $\to$ n cells

for two-way tables

• observations are classified according to two categorical variables
• we wish to test the null hypothesis that there is no association between row and column variables
• hypothesis could be
  • single sample
    • $H_0$: row and column variables are independent
  • multiple samples
    • $H_0$: the distribution of the response variable is the same for all of the populations

Test Statistic

• ${\chi }^2=\sum _{\text{all cells}}\frac{{\text{Observed-Expected}}^2}{\text{Expected}}$
• expected count = $\frac{\text{row total x column total}}{\text{overall total}}$
• degrees of freedom: $\text{(number of rows - 1) x (number of columns - 1)}$