Correspondence Analysis Introductory Overview - Burt Table

Multiple correspondence analysis expects as input (i.e., the program will compute prior to the analysis) a so-called Burt table. The Burt table is the result of the inner product of a design or indicator matrix, and the multiple correspondence analysis results are identical to the results you would obtain for the column points from a simple correspondence analysis of the indicator or design matrix (see also Introductory Overview - MCA).

For example, suppose you had entered data concerning the Survival for different Age groups in different Locations like this:

 

SURVIVAL

AGE

LOCATION

Case No.

NO

YES

LESST50

A50TO69

OVER69

TOKYO

BOSTON

GLAMORGN

1

0

1

0

1

0

0

0

1

2

1

0

1

0

0

1

0

0

3

0

1

0

1

0

0

1

0

4

0

1

0

0

1

0

0

1

...

.

.

.

.

.

.

.

.

...

.

.

.

.

.

.

.

.

...

.

.

.

.

.

.

.

.

762

1

0

0

1

0

1

0

0

763

0

1

1

0

0

0

1

0

764

0

1

0

1

0

0

0

1

In this data arrangement, for each case a 1 was entered to indicate to which category, of a particular set of categories, a case belongs (e.g., Survival, with the categories No and Yes). For example, case 1 survived (a 0 was entered for variable No, and a 1 was entered for variable Yes), case 1 is between age 50 and 69 (a 1 was entered for variable A50to69), and was observed in Glamorgn). Overall there are 764 observations in the dataset.

If you denote the data (design or indicator matrix) shown above as matrix X, then matrix product X'X is a Burt table); shown below is an example of a Burt table that one might obtain in this manner.

The Burt table has a clearly defined structure. Overall, the data matrix is symmetrical. In the case of 3 categorical variables (as shown above), the data matrix consists 3 x 3 = 9 partitions, created by each variable being tabulated against itself, and against the categories of all other variables. Note that the sum of the diagonal elements in each diagonal partition (i.e., where the respective variables are tabulated against themselves) is constant (equal to 764 in this case).

The off-diagonal elements in each diagonal partition in this example are all 0. If the cases in the design or indicator matrix are assigned to categories via fuzzy coding (i.e., if probabilities are used to indicate likelihood of membership in a category, rather than 0/1 coding to indicate actual membership), then the off-diagonal elements of the diagonal partitions are not necessarily equal to 0. Note that complex coding schemes can easily be implemented and the respective Burt table computed via STATISTICA Visual BASIC. You can then specify as input a Burt table directly (choose the option Frequencies w/out grouping vars in the Input group box of the Table Specifications dialog). Refer to the Introductory Overview - MCA section for additional details.

See also, Exploratory Data Analysis and Data Mining Techniques.