##
Sieve Diagrams

A sieve diagram shows the frequencies in a two-way contingency table in relation to expected frequencies under independence, and highlights the
**pattern of association** between the row and column variables.
The sieve diagram was proposed by Riedwyl and Schüpbach in a technical report in
1983 and later called a *parquet diagram* (Riedwyl and Schüpbach, 1994).
### Constructing the sieve diagram

In this display:

- A unit square is divided into rectangles, one for each cell in the
contingency table.
- The height of each rectangle in row
*i *is proportional to the marginal
frequency in that row (* f *_{ i+} ); the width of each rectangle in column *j* is proportional to the marginal frequency in that column (* f *_{ +j} ).
- Hence, the area of each rectangle is proportional to the
**expected frequency**,
* e *_{ ij} = (f _{ i+} f _{ +j} ) / f _{ ++}

under the hypothesis that the row and column variables are independent.
- The
**observed frequency** in each cell is shown by the number of squares drawn in each rectangle.
- Hence, the difference
between observed and expected frequency appears as the density of
shading, using color to indicate whether the deviation from
independence is
positive (blue) or negative (red).

### Examples, Demo, Software

- Examples
- This section from my short course, Categorical Data Analysis with Graphics shows several examples.
- Form-based Demo
- Enter your own data, and receive the graphic output.
- sieve.sas
- The SAS/IML program which draws sieve diagrams. There's also a Sieve diagram sample (sievedemo.sas).
- sieveplot.sas
- SAS macro for sieve diagrams.

Michael Friendly

friendly AT yorku DOT ca