Contingency table

Builds on Partition of a set Random variable

Definition

Let $X$ and $Y$ be categorical variables with values in $\set{1, \ldots, r}$ and $\set{1, \ldots, c}$ , observed on $n$ units. The $r \times c$ contingency table of the sample is the matrix $N = (n_{ij})$ , where $n_{ij}$ is the number of units with $X = i$ and $Y = j$ . The row totals are $n_{i+} = \sum_j n_{ij}$ , the column totals are $n_{+j} = \sum_i n_{ij}$ , and $\sum_{i,j} n_{ij} = n$ . A contingency table of $m \ge 3$ variables is defined in the same way as an $m$ -way array of counts.

Remarks

The cells correspond to a partition of the sample: each unit is counted in exactly one cell. Example: among $100$ people, $10$ smoke and exercise, $20$ smoke and do not exercise, $40$ do not smoke and exercise, and $30$ do neither. The table is $\begin{pmatrix} 10 & 20 \\ 40 & 30 \end{pmatrix}$ , with row totals $30, 70$ , column totals $50, 50$ , and grand total $100$ . Non-example: in a survey where people may tick several sports, a table of persons by sport counts some people more than once. Then the cells do not partition the sample, and the entries need not add up to $n$ . To test independence, the expected counts are $E_{ij} = n_{i+} n_{+j} / n$ , and Pearson's statistic is $\sum_{i,j} (n_{ij} - E_{ij})^2 / E_{ij}$ . Under independence and for large $n$ , it is approximately chi-squared with $(r - 1)(c - 1)$ degrees of freedom.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Tag	`RZHB962EMF`
URI	`m2://RZHB962EMF`
Kind	definition
Writing	W0 stub
MSC	62H17
Defines	Contingency table
Uses	Partition of a set, Random variable

Contingency table

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here