Types of Tables Used as Input

The CORRESP Procedure

Types of Tables Used as Input

The following example explains correspondence analysis and illustrates some capabilities of PROC CORRESP.

   data Neighbor;
      input Name $ 1-10 Age $ 12-18 Sex $ 19-25
            Height $ 26-30 Hair $ 32-37;
      datalines;
   Jones      Old    Male   Short White
   Smith      Young  Female Tall  Brown
   Kasavitz   Old    Male   Short Brown
   Ernst      Old    Female Tall  White
   Zannoria   Old    Female Short Brown
   Spangel    Young  Male   Tall  Blond
   Myers      Young  Male   Tall  Brown
   Kasinski   Old    Male   Short Blond
   Colman     Young  Female Short Blond
   Delafave   Old    Male   Tall  Brown
   Singer     Young  Male   Tall  Brown
   Igor       Old           Short
   ;

There are several types of tables, N, that can be used as input to correspondence analysis -all tables can be defined using a binary matrix, Z.

With the BINARY option, N = Z is directly analyzed. The binary matrix has one column for each category and one row for each individual or case. A binary table constructed from m categorical variables has m partitions. The following table has four partitions, one for each of the four categorical variables. Each partition has a 1 in each row, and each row contains exactly four 1s since there are four categorical variables. More generally, the binary design matrix has exactly m 1s in each row. The 1s indicate the categories to which the observation applies.

Table 24.2: Z, The Binary Coding of Neighbor Data Set

Z_Hair			Z_Height		Z_Sex		Z_Age
Blond	Brown	White	Short	Tall	Female	Male	Old	Young
0	0	1	1	0	0	1	1	0
0	1	0	0	1	1	0	0	1
0	1	0	1	0	0	1	1	0
0	0	1	0	1	1	0	1	0
0	1	0	1	0	1	0	1	0
1	0	0	0	1	0	1	0	1
0	1	0	0	1	0	1	0	1
1	0	0	1	0	0	1	1	0
1	0	0	1	0	1	0	0	1
0	1	0	0	1	0	1	1	0
0	1	0	0	1	0	1	0	1

With the MCA option, the Burt table (Z'Z) is analyzed. A Burt table is a partitioned symmetric matrix containing all pairs of crosstabulations among a set of categorical variables. Each diagonal partition is a diagonal matrix containing marginal frequencies (a crosstabulation of a variable with itself). Each off-diagonal partition is an ordinary contingency table. Each contingency table above the diagonal has a transposed counterpart below the diagonal.

Table 24.3: Z'Z, The Burt Table

	Blond	Brown	White	Short	Tall	Female	Male	Old	Young
Blond	3	0	0	2	1	1	2	1	2
Brown	0	6	0	2	4	2	4	3	3
White	0	0	2	1	1	1	1	2	0
Short	2	2	1	5	0	2	3	4	1
Tall	1	4	1	0	6	2	4	2	4
Female	1	2	1	2	2	4	0	2	2
Male	2	4	1	3	4	0	7	4	3
Old	1	3	2	4	2	2	4	6	0
Young	2	3	0	1	4	2	3	0	5

This Burt table is composed of all pairs of crosstabulations among the variables Hair, Height, Sex, and Age. It is composed of sixteen individual subtables -the number of variables squared. Both the rows and the columns have the same nine categories (in this case Blond, Brown, White, Short, Tall, Female, Male, Old, and Young). The off-diagonal partitions are crosstabulations of each variable with every other variable. Below the diagonal are the following crosstabulations (from left to right, top to bottom): Height * Hair, Sex * Hair, Sex * Height, Age * Hair, Age * Height, and Age * Sex. Each crosstabulation below the diagonal has a transposed counterpart above the diagonal. Each diagonal partition contains a crosstabulation of a variable with itself (Hair * Hair, Height * Height, Sex * Sex, and Age * Age). The diagonal elements of the diagonal partitions contain marginal frequencies of the off-diagonal partitions.

For example, the table Hair * Height has three rows for Hair and two columns for Height. The values of the Hair * Height table, summed across rows, sum to the diagonal values of the Height * Height table, as displayed in the following table.

Table 24.4: Z_Hair,Height'Z_Height, The (Hair Height) × Height Crosstabulation

	Short	Tall
Blond	2	1
Brown	2	4
White	1	1
Short	5	0
Tall	0	6

A simple crosstabulation of Hair × Height is N = Z_Hair'Z_Height. Crosstabulations such as this, involving only two variables, are the input to simple correspondence analysis.

Table 24.5: Z_Hair'Z_Height, The Hair × Height Crosstabulation

	Short	Tall
Blond	2	1
Brown	2	4
White	1	1

Tables such as the following (N = Z_Hair'Z_Height,Sex), made up of several crosstabulations, can also be analyzed in simple correspondence analysis.

Table 24.6: Z_Hair'Z_Height,Sex, The Hair × (Height Sex) Crosstabulation

	Short	Tall	Female	Male
Blond	2	1	1	2
Brown	2	4	2	4
White	1	1	1	1

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