# Correspondence Analysis

The goal of correspondence analysis is to make biplots for correspondence tables. In a correspondence table, the row and column variables are assumed to represent unordered categories; therefore, the nominal optimal scaling level is always used. Both variables are inspected for their nominal information only. That is, the only consideration is the fact that some objects are in the same category while others are not. Nothing is assumed about the distance or order between categories of the same variable.

One specific use of correspondence analysis is the
analysis of two-way contingency tables. If a table has *r* active rows and *c* active columns, the number of dimensions in the correspondence
analysis solution is the minimum of *r* minus 1 or *c* minus 1, whichever
is less. In other words, you could perfectly represent the row categories
or the column categories of a contingency table in a space of dimensions.
Practically speaking, however, you would like to represent the row
and column categories of a two-way table in a low-dimensional space,
say two dimensions, for the reason that two-dimensional plots are
more easily comprehensible than multidimensional spatial representations.

When fewer than the maximum number of possible dimensions is used, the statistics produced in the analysis describe how well the row and column categories are represented in the low-dimensional representation. Provided that the quality of representation of the two-dimensional solution is good, you can examine plots of the row points and the column points to learn which categories of the row variable are similar, which categories of the column variable are similar, and which row and column categories are similar to each other.

**Relation to other
Categories procedures. **Simple correspondence analysis
is limited to two-way tables. If there are more than two variables
of interest, you can combine variables to create interaction variables.
For example, for the variables *region*, *job*, and *age*, you can combine *region *and *job *to create a new variable *rejob* with the 12 categories shown in the
following table. This new variable forms a two-way table with *age* (12 rows, 4 columns), which can be analyzed
in correspondence analysis.

Category code | Category definition | Category code | Category definition |
---|---|---|---|

1 | North, intern | 7 | East, intern |

2 | North, sales rep | 8 | East, sales rep |

3 | North, manager | 9 | East, manager |

4 | South, intern | 10 | West, intern |

5 | South, sales rep | 11 | West, sales rep |

6 | South, manager | 12 | West, manager |

One shortcoming of this approach is that any pair of variables
can be combined. We can combine *job *and *age*, yielding another 12-category
variable. Or we can combine *region *and *age*, which results in a new
16-category variable. Each of these interaction variables forms a
two-way table with the remaining variable. Correspondence analyses
of these three tables will not yield identical results, yet each is
a valid approach. Furthermore, if there are four or more variables,
two-way tables comparing an interaction variable with another interaction
variable can be constructed. The number of possible tables to analyze
can get quite large, even for a few variables. You can select one
of these tables to analyze, or you can analyze all of them. Alternatively,
the Multiple Correspondence Analysis procedure can be used to examine
all of the variables simultaneously without the need to construct
interaction variables.

**Relation to standard
techniques. **The Crosstabs procedure can also be used to
analyze contingency tables, with independence as a common focus in
the analyses. However, even in small tables, detecting the cause of
departures from independence may be difficult. The utility of correspondence
analysis lies in displaying such patterns for two-way tables of any
size. If there is an association between the row and column variables--that
is, if the chi-square value is significant--correspondence analysis
may help reveal the nature of the relationship.