Question

What are contingency tables? How do they work? Please give an example or two.

Answer 1

ontingency tables (also called crosstabs or two-way tables) are used in statistics to summarize the relationship between several categorical variables. A contingency table is a special type of frequency distribution table, where two variables are shown simultaneously.

For example, a researcher might be investigating the relationship between AIDS and sexual preference. The two variables would be AIDS and SEXUAL PREFERENCE. The question is “Is there a significant relationship between AIDS and sexual preference?”. A chi-square test could then be run on the table to determine if there is a relationship between the two variables.

The following contingency table shows exposure to a potential source of food-borne illness (in this case, ice-cream). From the table, you can see that 13 people in a case study ate ice cream; 17 people did not:

Contingency Table: What is it used for?

Statistics Definitions > Contingency Table

Contingency Table: Overview

Contingency tables (also called crosstabs or two-way tables) are used in statistics to summarize the relationship between several categorical variables. A contingency table is a special type of frequency distribution table, where two variables are shown simultaneously.

For example, a researcher might be investigating the relationship between AIDS and sexual preference. The two variables would be AIDS and SEXUAL PREFERENCE. The question is “Is there a significant relationship between AIDS and sexual preference?”. A chi-square test could then be run on the table to determine if there is a relationship between the two variables.

The following contingency table shows exposure to a potential source of food-borne illness (in this case, ice-cream). From the table, you can see that 13 people in a case study ate ice cream; 17 people did not:

Image: Michigan Dept. of Agriculture

In the above image, there’s an Odds Ratio calculation. For more info, see: What is the Odds Ratio?

Chi-Square Tests

A chi² test can be conducted on contingency tables to test whether or not a relationship exists between variables. These effects are defined as relationships between rows and columns. The chi² test:

Where “O” is the Observed value, “E” is the expected value and “i” is the “ith” position in the table. The following picture shows what your contingency table might look like with your data, plus the results from running a chi² test on your data. A small chi² value means that there is little relationship between the categorical variables. A large chi² value means that there is a definite correlation between the two variables. As there is some pretty strong evidence that sexual orientation is linked to a higher risk of contracting AIDS, it’s no surprise that the chi² value is rather high:

However, the note under the results states that “4 cells (66.7%) have expected count less than 5.” Generally, if this is over 25%, the result could be due to chance alone. Therefore, the results from this particular test are not statistically significant.

Cases Controls 13 32 E P Exposed O (ate) R U E R Not Exposed A E (did not eat) M Odds Ratio (OR) (ac)d 17 23 -(1317)/ (3223) = 055