A contingency table, also known as a cross-classification table, describes the relationships between two or more categorical variables.
A table cross-classifying two variables is called a 2-way contingency table and forms a rectangular table with rows for the R categories of the X variable and columns for the C categories of a Y variable. Each intersection is called a cell and represents the possible outcomes. The cells contain the frequency of the joint occurrences of the X, Y outcomes. A contingency table having R rows and C columns is called an R x C table.
A variable having only two categories is called a binary variable. When both variables are binary, the resulting contingency table is a 2 x 2 table. Also, commonly known as a four-fold table because there are four cells.
A contingency table can summarize three probability distributions – joint, marginal, and conditional.
When both variables are random, you can describe the data using the joint distribution, the conditional distribution of Y given X, or the conditional distribution of X given Y.
When one variable is and explanatory variable (X, fixed) and the other a response variable (Y, random), the notion of a joint distribution is meaningless, and you should describe the data using the conditional distribution of Y given X. Likewise, if Y is a fixed variable and X random, you should describe the data using the conditional distribution of X given Y.
When the variables are matched-pairs or repeated measurements on the same sampling unit, the table is square R=C, with the same categories on both the rows and columns. For these tables, the cells may exhibit a symmetric pattern about the main diagonal of the table, or the two marginal distributions may differ in some systematic way.
