ASSOCIATION PLOT

Name:

ASSOCIATION PLOT (LET) Type:

Graphics Command Purpose:

Generate an association plot for a two-way contingency table. Description:

\( \sum_{i=1}^{r}{\sum_{j=1}^{c}{\frac{O_{ij} - E_{ij}} {E_{ij}}}} \)

where

r	=	the number of rows in the contingency table
c	=	the number of columns in the contingency table
O_ij	=	the observed frequency of the ith row and jth column
E_ij	=	the expected frequency of the ith row and jth column
	=	\( \frac{R_i C_j}{N} \)
R_i	=	the sum of the observed frequencies for row i
C_j	=	the sum of the observed frequencies for column j
N	=	the total sample size

This test statistic can also be formulated as

\( \sum_{i=1}^{r}{\sum_{j=1}^{c}{d_{ij}^2}} \)

where

\( d_{ij}^2 = \frac{O_{ij} - E_{ij}} {\sqrt{E_{ij}}} \)

The d_ij are referred to as the standardized residuals and they show the contribution to the chi-square test statistic of each cell.

The association plot is a graphical way of showing these standardized residuals. Specifically, the cells are laid out in a two-way grid and each cell is plotted with a rectangle where

The width of the rectangle is proportional to the square root of the expected frequency.
The height of the rectangle is proportional to the standardized residual (d_ij).

For visual clarity, positive and negative standardized residuals are typically drawn in different colors.

Syntax 1:

This syntax is used for the case where you have raw data (i.e., the data has not yet been cross tabulated into a two-way table).

Syntax 2:

This syntax is used for the case where we the data have already been cross-tabulated into a two-way contingency table.

Syntax 3:

This syntax is used for the special case where you have a 2x2 table. In this case, you can enter the 4 values directly, although you do need to be careful that the parameters are entered in the order expected above.

Examples:

Note:

The example program below demonstrates how to specify the different colors for positive and negative standardized residuals. It also demonstrates how to label the rows and columns of the plot. Default:

None Synonyms:

None Related Commands:

CHI-SQUARE INDEPENDENCE TEST	= Perform a chi-square test for independence.
ODDS RATIO INDEPENDENCE TEST	= Perform a log(odds ratio) test for independence.
FISHER EXACT TEST	= Perform Fisher's exact test.
SIEVE PLOT	= Generate a sieve plot.
ROSE PLOT	= Generate a Rose plot.
BINARY TABULATION PLOT	= Generate a binary tabulation plot.
ROC CURVE	= Generate a ROC curve.
ODDS RATIO	= Compute the bias corrected odds ratio.
LOG ODDS RATIO	= Compute the bias corrected log(odds ratio).

Reference:

Visualizing Categorical Data

Cohen (1980), "On the Graphical Display of the Significant Components in a Two-Way Contingency Table," Communications in Statistics--Theory and Methods, A9:1025-1041.

Applications:

Graphical Analysis of Categorical Data Implementation Date:

2007/6 Program:

. Example from page 61 of Friendly
read matrix m
 5  29 14 16
15  54 14 10
20  84 17 94
68 119 26 7
end of data
.
label case asis
tic mark label case asis
title case asis
title offset 2
.
x3label
title Association Plot
y1label displacement 12
y1label Eye Color
x1label Hair Color
tic offset units data
xlimits 1 4
major xtic mark number 4
minor xtic mark number 0
xtic mark offset 1 1
x1tic mark label format alpha
x1tic mark label content Black Brown Red Blond
ylimits 1 4
major ytic mark number 4
minor ytic mark number 0
ytic mark offset 1 1
y1tic mark label format alpha
y1tic mark label content Green Hazel Blue Brown
y1tic mark label justification right
.
line solid
region fill color blue green
.
association plot m