Chi-square Test

Definitions

The chi-square test-statistic

χ_{c}^{2} = \sum \frac{(E_{i} - O_{i})^{2}}{E_{i}}

where the subscript “c” is the degrees of freedom. “O” is observed value and E is expected value.

Chi-square distribution

The chi-square distribution (or the chi-squared distribution) is a special case of the gamma distribution.
A chi square distribution with n degrees of freedom is equal to a gamma distribution with a = n / 2 and b = 0.5 (or β = 2).

Relationship between $χ^{2}$ and $χ^{2}$ distribution

You can compare $χ^{2}$ to the critical value from the $χ^{2}$ distribution with df degrees of freedom and the selected confidence level p.

Applications

Pearson's chi-squared test is used to assess three types of comparison

A test of goodness of fit => whether an observed frequency distribution differs from a theoretical distribution.
A test of homogeneity => compares the distribution of counts for two or more groups using the same categorical variable.
A test of independence => whether observations consisting of measures on two variables, expressed in a contingency table, are independent of each other.

How to determine df

For a test of goodness-of-fit, df = Cats − Parms, where Cats is the number of observation categories recognized by the model, and Parms is the number of parameters in the model adjusted to make the model best fit the observations: The number of categories reduced by the number of fitted parameters in the distribution.
For test of homogeneity, df = (Rows − 1)×(Cols − 1), where Rows corresponds to the number of categories (i.e. rows in the associated contingency table), and Cols corresponds the number of independent groups (i.e. columns in the associated contingency table).
For test of independence, df = (Rows − 1)×(Cols − 1), where in this case, Rows corresponds to number of categories in one variable, and Cols corresponds to number of categories in the second variable.

Rules

probability of independence
it does not interpret relationship between variables
values must be absolute, not percentage
categories must be mutually exclusive
minimal 5 observations in each cell