Contingency Tables | Cross Classification: Introduction

Contingency Tables also called cross tables or two-way frequency tables describes the relationship between several categorical (qualitative) variables. A bivariate relationship is defined by the joint distribution of the two associated random variables.

Contingency Tables

Let $X$ and $Y$ are two categorical response variables. Let variable $X$ have $I$ levels and variable $Y$ have $J$ levels. The possible combinations of classifications for both variables are $I\times J$. The response $(X, Y)$ of a subject randomly chosen from some population has a probability distribution, which can be shown in a rectangular table having $I$ rows (for categories of $X$) and $J$ columns (for categories of $Y$). The cells of this rectangular table represent the $IJ$ possible outcomes. Their probability (say $\pi_{ij}$) denotes the probability that ($X, Y$) falls in the cell in row $i$ and column $j$. When these cells contain frequency counts of outcomes, the table is called a contingency or cross-classification table and it is referred to as a $I$ by $J$ ($I \times J$) table.

The probability distribution {$\pi_{ij}$} is the joint distribution of $X$ and $Y$. The marginal distributions are the rows and columns totals obtained by summing the joint probabilities. For the row variable ($X$) the marginal probability is denoted by $\pi_{i+}$ and for column variable ($Y$) it is denoted by $\pi_{+j}$, where the subscript “+” denotes the sum over the index it replaces; that is, $\pi_{i+}=\sum_j \pi_{ij}$ and $\pi_{+j}=\sum_i \pi_{ij}$ satisfying

$l\sum_{i} \pi_{i+} =\sum_{j} \pi_{+j} = \sum_i \sum_j \pi_{ij}=1$

Note that the marginal distributions are single-variable information, and do not pertain to association linkages between the variables.

In (many) contingency tables, one variable (say, $Y$) is a response, and the other $X$) is an explanatory variable. When $X$ is fixed rather than random, the notation of a joint distribution for $X$ and $Y$ is no longer meaningful. However, for a fixed level of $X$, the variable $Y$ has a probability distribution. It is germane to study how this probability distribution of $Y$ changes as the level of $X$ changes.

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Muhammad Imdad Ullah

Currently working as Assistant Professor of Statistics in Ghazi University, Dera Ghazi Khan. Completed my Ph.D. in Statistics from the Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan. l like Applied Statistics, Mathematics, and Statistical Computing. Statistical and Mathematical software used is SAS, STATA, GRETL, EVIEWS, R, SPSS, VBA in MS-Excel. Like to use type-setting LaTeX for composing Articles, thesis, etc.

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