- generalizes readily to n-way tables
- provides a method for fitting a series of sequential log-linear models to the various marginal totals of an n-way table
- is used to display the deviations (residuals) from the various log-linear models.

The new, Open Source implementation of R (www.r-project.org) now includes an object-oriented
`mosaicplot()` on which future work will build. A newly-released R package,
`vcd` extends mosaic
displays, and implements many of the graphical methods from Visualizing Categorical Data

Consider
Table 1, which
shows data on the relation between hair color and eye color among 592
subjects (students in a statistics course) collected by Snee (1974).
The Pearson * X ^{2} * for these data is 138.3 with 9 degrees
of freedom, indicating substantial departure from independence. The
question is how to understand the

Table 1: Hair-color eye-color data

Hair Color Eye Color BLACK BROWN RED BLOND | Total | Brown 68 119 26 7 | 220 Blue 20 84 17 94 | 215 Hazel 15 54 14 10 | 93 Green 5 29 14 16 | 64 --------------------------------------------+------ Total 108 286 71 127 | 592

- For such a two-way table, the mosaic display is constructed
by first dividing a unit square in porportion to the marginal
totals of one variable, say, Hair Color.
For these data, the marginal proportions are:

Marginal proportions Black Brown Red Blond 0.1824 0.4831 0.1199 0.2145

This gives the first mosaic display:- The one-way table of marginal totals can be fit to a model, in this
case, the model that all Hair colors are equally probable. This model
has expected frequencies of 592/4:
Fitted frequencies Black Brown Red Blond 148.00 148.00 148.00 148.00

- The Pearson residuals from this model, d = ( n - m ) / sqrt (m), are:
Standardized Pearson residuals Black Brown Red Blond -3.29 11.34 -6.33 -1.73

and these values are shown by color and shading as shown in the legend. The high positive value for Brown hair indicates that people with brown hair are much more frequent in the population than the Equiprobability model would predict.

- The one-way table of marginal totals can be fit to a model, in this
case, the model that all Hair colors are equally probable. This model
has expected frequencies of 592/4:
- Next, the rectangle for each Hair Color is subdivided in proportion
to the relative (conditional) frequencies of the second variable --
Eye color, giving the following conditional proportions:
Marginal proportions Brown Blue Hazel Green TOTAL Black 0.6296 0.1852 0.1389 0.0463 1.0 Brown 0.4161 0.2937 0.1888 0.1014 1.0 Red 0.3662 0.2394 0.1972 0.1972 1.0 Blond 0.0551 0.7402 0.0787 0.1260 1.0

This gives the second mosaic display:- Again, the cells are shaded in proportion to standardized
residuals from a model, here, the model that Hair Color and
Eye Color are independent in population from which this sample
was drawn.
Standardized Pearson residuals Brown Blue Hazel Green Black 4.40 -3.07 -0.48 -1.95 Brown 1.23 -1.95 1.35 -0.35 Red -0.07 -1.73 0.85 2.28 Blond -5.85 7.05 -2.23 0.61

- Thus, the two tiles shaded deep blue correspond to the two cells, (Black, Brown) and (Blond, Blue), whose residuals are greater than +4, indicating much greater frequency in those cells than would be found if Hair color and Eye Color were independent. The tile shaded deep red, (Blond, Brown) corresponds to the residual = -5.85, indicating this combination is extremely rare under the hypothesis of independence.
- The overall Pearson X
^{2}statistic is just the sum of squares of the residuals.

- Again, the cells are shaded in proportion to standardized
residuals from a model, here, the model that Hair Color and
Eye Color are independent in population from which this sample
was drawn.

This interpretation is enhanced by reordering the rows or columns of the two-way table so that the residuals have an opposite corner pattern of signs.

Here, this is achieved by reordering the Eye Colors as shown below:

Standardized Pearson residuals Brown Hazel Green Blue Black 4.40 -0.48 -1.95 -3.07 Brown 1.23 1.35 -0.35 -1.95 Red -0.07 0.85 2.28 -1.73 Blond -5.85 -2.23 0.61 7.05Thus, the mosaic shows that the association between Hair and Eye color is essentially that

- people with dark hair tend to have dark eyes,
- those with light hair tend to have light eyes
- people with red hair do not quite fit this pattern

Imagine that each cell of the two-way table for Hair and Eye color is further classified by one or more additional variables--sex and level of education, for example. Then each rectangle can be subdivided horizontally to show the proportion of males and females in that cell, and each of those horizontal portions can be subdivided vertically to show the proportions of people at each educational level in the hair-eye-sex group.

Here is the mosaic for the three-way table, with Hair and Eye color groups divided according to the proportions of Males and Females:

We see that there is no systematic association between sex and the combinations of Hair and Eye color -- except among blue-eyed blonds, where there are an overabundance of females.

For three-way tables, there are three different types of models of "independence" (with several instances each, permuting the variables A, B, and C):

Model | Log-linear model | Predicted cell probabilities | What the residuals show |
---|---|---|---|

Mutual Independence |
[A] [B] [C] | Residuals show all associations among variables | |

Joint Independence |
[A B] [C] | Residuals show associations between variable C and combinations of A and B | |

Conditional Independence |
[A C] [ B C] | No closed-form formula | Residuals show associations between A and B, holding C constant |

For higher-way tables, there are many more possibilities.

Moreover, the series of mosaic plots fitting submodels of
*Joint Independence* to
the marginal subtables have the special property that they can be viewed as partitioning the hypothesis
of Mutual Independence in the full table.

For example, for the hair-eye data, the mosaic displays for the [Hair] [Eye] marginal table and the [HairEye] [Sex] table can be viewed as representing the partition

Model dfG[Hair] [Eye] 9 146.44 [Hair, Eye] [Sex] 15 19.86 ------------------------------------------ [Hair] [Eye] [Sex] 24 155.20^{2}

This partitioning scheme extends directly to higher-way tables.

It is possible, however, for the marginal relations among variables to
differ in magnitude, or even in direction, from the relations among
those variables controlling for additional variables.
The peculiar result that a pair of variables can have a marginal
association in a different direction than their partial associations
is called *Simpson's Paradox*.

One way to determine if the marginal relations are representative
is to fit models of **Conditional Association**
and compare them with the marginal models.
For the running example, the appropriate model is the model
`[Hair, Sex] [Eye, Sex]`, which
examines the relation between Hair Color and Eye Color controlling
for Sex. The fit statistic is nearly the same as for the
unconditional marginal model:

Model dfAnd, the pattern of residuals is quite similar to that of theG[Hair] [Eye] 9 146.44 [Hair, Sex] [Eye, Sex] 15 156.68^{2}