Three Forms of Frequency Tables for Mosaic Displays

The Three Forms

Table Form

Table form represents categorical data as an array or table object where: - Dimensions correspond to categorical variables - Elements contain cell frequencies - Dimension names provide variable and level information

This is the most compact representation and the natural output from table() and xtabs().

# HairEyeColor is already in table form
str(HairEyeColor)
#>  'table' num [1:4, 1:4, 1:2] 32 53 10 3 11 50 10 30 10 25 ...
#>  - attr(*, "dimnames")=List of 3
#>   ..$ Hair: chr [1:4] "Black" "Brown" "Red" "Blond"
#>   ..$ Eye : chr [1:4] "Brown" "Blue" "Hazel" "Green"
#>   ..$ Sex : chr [1:2] "Male" "Female"
class(HairEyeColor)
#> [1] "table"

# Examine the 3-way table structure
HairEyeColor
#> , , Sex = Male
#> 
#>        Eye
#> Hair    Brown Blue Hazel Green
#>   Black    32   11    10     3
#>   Brown    53   50    25    15
#>   Red      10   10     7     7
#>   Blond     3   30     5     8
#> 
#> , , Sex = Female
#> 
#>        Eye
#> Hair    Brown Blue Hazel Green
#>   Black    36    9     5     2
#>   Brown    66   34    29    14
#>   Red      16    7     7     7
#>   Blond     4   64     5     8

# Total observations
sum(HairEyeColor)
#> [1] 592

The HairEyeColor dataset is a 4 × 4 × 2 contingency table with 592 observations classified by hair color, eye color, and sex.

Table form can also be displayed as a heatmap-style color table, using vcdExtra::color_table(), which shades each cell background by its frequency. This gives a quick visual sense of where the data are concentrated, as a complement to the mosaic plots introduced below – useful for readers less familiar with reading mosaic displays.

color_table(HairEyeColor, shade = "freq")

		Eye				Total
		Brown	Blue	Hazel	Green	Total
Black	Male	32	11	10	3	56
Black	Female	36	9	5	2	52
Brown	Male	53	50	25	15	143
Brown	Female	66	34	29	14	143
Red	Male	10	10	7	7	34
Red	Female	16	7	7	7	37
Blond	Male	3	30	5	8	46
Blond	Female	4	64	5	8	81
Total		220	215	93	64	592

Frequency Form

Frequency form is a data frame where: - Each row represents one cell of the contingency table - Factor columns specify the cell’s coordinates - A frequency column (typically Freq) contains the count for that cell - Total observations = sum(df$Freq)

This form is convenient for modeling and is the output from as.data.frame() applied to tables.

# Convert table to frequency form
hair_freq <- as.data.frame(HairEyeColor)
head(hair_freq, 10)
#>     Hair   Eye  Sex Freq
#> 1  Black Brown Male   32
#> 2  Brown Brown Male   53
#> 3    Red Brown Male   10
#> 4  Blond Brown Male    3
#> 5  Black  Blue Male   11
#> 6  Brown  Blue Male   50
#> 7    Red  Blue Male   10
#> 8  Blond  Blue Male   30
#> 9  Black Hazel Male   10
#> 10 Brown Hazel Male   25
nrow(hair_freq)  # 32 cells in the 4 × 4 × 2 table
#> [1] 32

# Verify totals match
sum(hair_freq$Freq)
#> [1] 592

Frequency form is ideal for: - Input to modeling functions (glm(), loglm(), etc.) - Filtering specific cells or combinations - Adding derived variables - Use with ggplot2 via the weight aesthetic

Case Form

Case form represents data as: - Each row is one individual observation - Factor columns contain the variable values for that individual - No frequency column needed - Total observations = nrow(df)

This is the “raw data” format and most intuitive for those familiar with data collection.

# Convert frequency form to case form using vcdExtra::expand.dft()
hair_case <- expand.dft(hair_freq, freq = "Freq")
head(hair_case, 10)
#>     Hair   Eye  Sex
#> 1  Black Brown Male
#> 2  Black Brown Male
#> 3  Black Brown Male
#> 4  Black Brown Male
#> 5  Black Brown Male
#> 6  Black Brown Male
#> 7  Black Brown Male
#> 8  Black Brown Male
#> 9  Black Brown Male
#> 10 Black Brown Male
nrow(hair_case)  # 592 individual observations
#> [1] 592

# Structure
str(hair_case)
#> 'data.frame':    592 obs. of  3 variables:
#>  $ Hair: chr  "Black" "Black" "Black" "Black" ...
#>  $ Eye : chr  "Brown" "Brown" "Brown" "Brown" ...
#>  $ Sex : chr  "Male" "Male" "Male" "Male" ...

Case form is most natural when: - Working with individual-level data - Each observation represents one subject - No need to aggregate

Converting Between Forms

The table below summarizes conversion functions:

From → To	Table	Frequency	Case
Table	—	`as.data.frame()`	`expand.dft()`
Frequency	`xtabs(Freq ~ ...)`	—	`expand.dft()`
Case	`table()`, `xtabs()`	`count()`, `xtabs()`	—

Note: These conversions are clearly untidy and hard to remember which to use. They have now been implemented in a collection of vcdExtra::as_*() functions which take whatever you give it and return the desired form.

Examples

# Case → Frequency (count occurrences)
hair_case |>
  count(Hair, Eye, Sex, name = "Freq") |>
  head()
#>    Hair   Eye    Sex Freq
#> 1 Black  Blue Female    9
#> 2 Black  Blue   Male   11
#> 3 Black Brown Female   36
#> 4 Black Brown   Male   32
#> 5 Black Green Female    2
#> 6 Black Green   Male    3

# Frequency → Table
hair_table <- xtabs(Freq ~ Hair + Eye + Sex, data = hair_freq)
identical(hair_table, HairEyeColor)
#> [1] FALSE

# Table → Frequency (already shown)
# Frequency → Case (already shown)

Using Each Form with `geom_mosaic()`

Working with the Three-Way Table: Hair × Eye × Sex

Let’s demonstrate how to use each of the three forms with geom_mosaic() using the full HairEyeColor dataset.

# From frequency form (most common)
ggplot(data = hair_freq,
       aes(weight = Freq, x = product(Hair, Eye, Sex), fill = Eye)) +
  geom_mosaic() +
  labs(title = "Three-Way Mosaic: Hair × Eye × Sex",
       subtitle = "Frequency form with weight aesthetic")

# From case form
ggplot(data = hair_case,
       aes(x = product(Hair, Eye, Sex), fill = Eye)) +
  geom_mosaic() +
  labs(title = "Three-Way Mosaic: Hair × Eye × Sex",
       subtitle = "Case form without weight")

Both forms produce identical plots. The key difference: - Frequency form: Use weight = Freq in the aesthetic - Case form: No weight needed (each row counts as 1)

Note: Table form data must first be converted to frequency form using as.data.frame() before use with geom_mosaic().

Using Conditioning Variables

The conds aesthetic allows you to condition on one or more variables:

# Condition on Sex
ggplot(data = hair_freq,
       aes(weight = Freq,
           x = product(Hair, Eye),
           conds = product(Sex),
           fill = Eye)) +
  geom_mosaic() +
  labs(title = "Hair × Eye | Sex",
       subtitle = "Conditioned on Sex")

Default Shading vs. Residual-Based Shading

One of the most powerful features of mosaic plots is residual-based shading, which highlights deviations from a statistical model. This allows us to see not just the data, but whether patterns are statistically meaningful.

Default Shading

By default, geom_mosaic() uses the fill aesthetic to color tiles, typically showing one of the categorical variables:

ggplot(data = hair_freq,
       aes(weight = Freq, x = product(Hair, Eye), fill = Eye)) +
  geom_mosaic() +
  labs(title = "Hair × Eye Color: Default Shading",
       subtitle = "Fill shows Eye color levels")

This clearly shows the marginal and conditional distributions, but doesn’t tell us whether associations are statistically significant.

Residual-Based Shading

Residual-based shading fits a loglinear model to the data and colors tiles according to Pearson residuals:

\[ r_{ij} = \frac{\text{observed}_{ij} - \text{expected}_{ij}}{\sqrt{\text{expected}_{ij}}} \]

where “expected” frequencies come from the fitted model.

Blue tiles: Observed > Expected (positive association)
Red tiles: Observed < Expected (negative association)
Intensity: Magnitude of the residual
|r| > 2: Statistically significant at approximately α = 0.05

Independence Model for Two-Way Table

For a two-way table, we can visualize just Hair and Eye color. The most common model is independence (main effects only):

ggplot(data = hair_freq,
       aes(weight = Freq, x = product(Hair, Eye))) +
  geom_mosaic(expected = "independence") +
  scale_fill_residual() +
  labs(title = "Hair × Eye Color: Independence Model",
       subtitle = "Blue = more than expected, Red = fewer than expected")

The residual shading reveals: - Blue tiles (positive residuals): Black hair with brown eyes, blond hair with blue eyes occur more frequently than independence would predict - Red tiles (negative residuals): Black hair with blue eyes, blond hair with brown eyes occur less frequently - This indicates a significant association between hair and eye color

Adding Cell Values

We can display the actual residual values in each cell using geom_mosaic_text():

ggplot(data = hair_freq,
       aes(weight = Freq, x = product(Hair, Eye))) +
  mosaic_settings(expected = "independence") +
  geom_mosaic() +
  scale_fill_residual() +
  geom_mosaic_text(display_values = "residual",
                   format_digits = 1,
                   size = 3.5,
                   colour = "black") +
  labs(title = "Hair × Eye Independence Model with Residual Values",
       subtitle = "Numbers show Pearson residuals")

Values greater than 2 in absolute value indicate significant departures from independence.

Independence Model for Three-Way Table

For the three-way table, the independence model tests whether all three variables are mutually independent:

ggplot(data = hair_freq,
       aes(weight = Freq, x = product(Hair, Eye, Sex))) +
  geom_mosaic(expected = "independence") +
  scale_fill_residual() +
  labs(title = "Hair × Eye × Sex: Complete Independence",
       subtitle = "Model: ~ Hair + Eye + Sex (no interactions)")

Custom Models

You can specify custom loglinear models using formula syntax. For example, testing whether the Hair-Eye association differs by Sex:

# Model: Hair and Eye are associated, but independent of Sex
ggplot(data = hair_freq,
       aes(weight = Freq, x = product(Hair, Eye, Sex))) +
  geom_mosaic(expected = ~ Hair * Eye + Sex) +
  scale_fill_residual() +
  labs(title = "Model: (Hair × Eye) Independent of Sex",
       subtitle = "~ Hair * Eye + Sex")

Residuals close to zero suggest this model fits reasonably well—the Hair-Eye association is similar for males and females.

Comparing Observed and Expected

We can also display the expected counts under a model:

ggplot(data = hair_freq,
       aes(weight = Freq, x = product(Hair, Eye))) +
  mosaic_settings(expected = "independence") +
  geom_mosaic() +
  scale_fill_residual() +
  geom_mosaic_text(display_values = "expected",
                   format_digits = 1,
                   size = 3.5,
                   colour = "black") +
  labs(title = "Expected Frequencies Under Independence",
       subtitle = "Compare with observed to see associations")

And the observed counts:

ggplot(data = hair_freq,
       aes(weight = Freq, x = product(Hair, Eye))) +
  geom_mosaic(aes(fill = Eye)) +
  geom_mosaic_text(display_values = "observed",
                   format_digits = 0,
                   size = 3.5,
                   colour = "white") +
  labs(title = "Observed Frequencies",
       subtitle = "Actual counts in each cell")

Summary

Key Takeaways

Three forms of frequency data serve different purposes:
- Table form: Compact, ideal for arrays and mathematical operations
- Frequency form: Versatile, works well with modeling and ggplot2
- Case form: Intuitive, represents raw individual-level data
Converting between forms is straightforward:
- Use as.data.frame() for table → frequency
- Use expand.dft() for frequency → case
- Use xtabs() or table() for case/frequency → table
Using with geom_mosaic():
- Table/frequency forms require weight = Freq
- Case form works directly (no weight)
- All forms produce identical visualizations
Two types of shading:
- Default: Uses fill aesthetic to show variable levels
- Residual-based: Uses expected parameter to fit models and shade by Pearson residuals
Residual shading is powerful for:
- Identifying statistically significant associations
- Testing specific hypotheses about independence
- Comparing observed patterns to theoretical models

Recommendations

For visualization: Frequency form with weight aesthetic is most flexible
For modeling: Frequency form works with glm(), loglm(), etc.
For raw data: Case form is most intuitive
For residual plots: Always use scale_fill_residual() and consider adding geom_mosaic_text() with display_values = "residual"

References

Friendly, M. (1994). “Mosaic Displays for Multi-Way Contingency Tables.” Journal of the American Statistical Association, 89(425), 190-200.
Friendly, M., & Meyer, D. (2016). Discrete Data Analysis with R. Chapman and Hall/CRC.
Meyer, D., Zeileis, A., & Hornik, K. (2006). “The Strucplot Framework: Visualizing Multi-way Contingency Tables with vcd.” Journal of Statistical Software, 17(3), 1-48.
Zeileis, A., Meyer, D., & Hornik, K. (2007). “Residual-based Shadings for Visualizing (Conditional) Independence.” Journal of Computational and Graphical Statistics, 16(3), 507-525.

Session Info

sessionInfo()
#> R version 4.6.1 (2026-06-24 ucrt)
#> Platform: x86_64-w64-mingw32/x64
#> Running under: Windows 11 x64 (build 22631)
#> 
#> Matrix products: default
#>   LAPACK version 3.12.1
#> 
#> locale:
#> [1] LC_COLLATE=C                    LC_CTYPE=English_Canada.utf8   
#> [3] LC_MONETARY=English_Canada.utf8 LC_NUMERIC=C                   
#> [5] LC_TIME=English_Canada.utf8    
#> 
#> time zone: America/Toronto
#> tzcode source: internal
#> 
#> attached base packages:
#> [1] grid      stats     graphics  grDevices utils     datasets  methods  
#> [8] base     
#> 
#> other attached packages:
#> [1] dplyr_1.2.1     vcdExtra_0.9.8  gnm_1.1-5       vcd_1.4-14     
#> [5] ggmosaic2_0.5.1 ggplot2_4.0.3  
#> 
#> loaded via a namespace (and not attached):
#>  [1] gt_1.3.0            plotly_4.12.1       sass_0.4.10        
#>  [4] generics_0.1.4      tidyr_1.3.2         productplots_0.1.2 
#>  [7] xml2_1.6.0          lattice_0.23-1      digest_0.6.39      
#> [10] magrittr_2.0.5      evaluate_1.0.5      RColorBrewer_1.1-3 
#> [13] fastmap_1.2.0       Matrix_1.7-6        plyr_1.8.9         
#> [16] jsonlite_2.0.0      processx_3.9.0      ggrepel_0.9.8      
#> [19] nnet_7.3-21         chromote_0.5.1      promises_1.5.0     
#> [22] httr_1.4.9          purrr_1.2.2         viridisLite_0.4.3  
#> [25] scales_1.4.0        jquerylib_0.1.4     cli_3.6.6          
#> [28] rlang_1.3.0         withr_3.0.3         cachem_1.1.0       
#> [31] yaml_2.3.12         otel_0.2.0          tools_4.6.1        
#> [34] colorspace_2.1-3    forcats_1.0.1       ca_0.71.1          
#> [37] vctrs_0.7.3         R6_2.6.1            zoo_1.9-0          
#> [40] lifecycle_1.0.5     fs_2.1.0            relimp_1.0-5       
#> [43] htmlwidgets_1.6.4   MASS_7.3-66         pkgconfig_2.0.3    
#> [46] later_1.4.8         pillar_1.11.1       bslib_0.12.0       
#> [49] gtable_0.3.6        data.table_1.18.6.1 glue_1.8.1         
#> [52] Rcpp_1.1.2          xfun_0.60           tibble_3.3.1       
#> [55] lmtest_0.9-40       tidyselect_1.2.1    rstudioapi_0.19.0  
#> [58] knitr_1.52          dichromat_2.0-1     qvcalc_1.0.4       
#> [61] farver_2.1.2        websocket_1.4.4     htmltools_0.5.9    
#> [64] webshot2_0.1.2.9000 rmarkdown_2.32      compiler_4.6.1     
#> [67] S7_0.2.2

Three Forms of Frequency Tables for Mosaic Displays

Michael Friendly

2026-09-15

Introduction

The Three Forms

Table Form

Frequency Form

Case Form

Converting Between Forms

Examples

Using Each Form with `geom_mosaic()`

Working with the Three-Way Table: Hair × Eye × Sex

Using Conditioning Variables

Default Shading vs. Residual-Based Shading

Default Shading

Residual-Based Shading

Independence Model for Two-Way Table

Adding Cell Values

Independence Model for Three-Way Table

Custom Models

Comparing Observed and Expected

Summary

Key Takeaways

Recommendations

References

Session Info

Three Forms of Frequency Tables for Mosaic Displays

Michael Friendly

2026-09-15

Introduction

The Three Forms

Table Form

Frequency Form

Case Form

Converting Between Forms

Examples

Using Each Form with geom_mosaic()

Working with the Three-Way Table: Hair × Eye × Sex

Using Conditioning Variables

Default Shading vs. Residual-Based Shading

Default Shading

Residual-Based Shading

Independence Model for Two-Way Table

Adding Cell Values

Independence Model for Three-Way Table

Custom Models

Comparing Observed and Expected

Summary

Key Takeaways

Recommendations

References

Session Info

Using Each Form with `geom_mosaic()`