A mosaic plot is a striking visual representation that illustrates the relationships between two or more categorical variables. Imagine a statistical canvas; it starts as a simple square, but then elegantly subdivides to reveal complex patterns within your data.
At its core, a mosaic plot appears as a large rectangle (often a square with a starting length of one unit) that is progressively partitioned into smaller, non-overlapping rectangles. The area of each final rectangle within the plot is directly proportional to the frequency or proportion of observations within that specific combination of categories.
How a Mosaic Plot Is Constructed
The unique appearance of a mosaic plot stems from its step-by-step construction, which systematically divides the total area based on the categorical variables.
- Initial Canvas: The plot begins as a square (or a rectangle) with an initial length of one, representing the entire dataset or 100% of observations.
- First Division: This square is divided first into horizontal bars. The widths of these bars are precisely proportional to the probabilities (or relative frequencies) associated with the categories of the first categorical variable. For instance, if 'Gender' is the first variable and 60% of the data is 'Female' and 40% 'Male', the square will be split into two horizontal bars, one 60% of the total width and the other 40%.
- Subsequent Divisions: Each of these initial bars is then further subdivided, but in the opposite direction. If the first division was horizontal, the next division will be vertical, with the heights of the new rectangles proportional to the probabilities of the second categorical variable within each category of the first. This process continues, alternating between horizontal and vertical divisions for each additional categorical variable.
- Area Representation: The beauty of a mosaic plot lies in how the area of each resulting small rectangle visually conveys the joint frequency or proportion of the specific combination of categories it represents. A larger rectangle signifies a higher frequency for that particular intersection of categories.
Key Visual Elements of a Mosaic Plot
| Element | Description | What it Represents |
|---|---|---|
| Overall Rectangle | The entire plotting area, typically starting as a square. | The total dataset or 100% of observations. |
| Varying Widths | The horizontal or vertical extent of the main segments for the first few variables. | The marginal probabilities or relative frequencies of the categories for the first variable(s). For instance, a wider segment for 'Category A' means 'Category A' occurs more often. |
| Varying Heights | The vertical or horizontal extent of segments for subsequent variables, often nested within the first divisions. | The conditional probabilities or relative frequencies of categories of subsequent variables given the categories of the preceding variables. |
| Area of Rectangles | The size of the smallest, terminal rectangles. | The joint frequency or proportion of the specific combination of all categorical variables represented by that rectangle. This is the primary insight into the relationships. |
| Coloring | Often, rectangles are colored. This coloring typically indicates the standardized residuals from a model of independence, or highlights specific categories of interest. | Residuals: Blue or shades of blue often indicate higher-than-expected frequencies (positive association), while red or shades of red indicate lower-than-expected frequencies (negative association) under the assumption of independence. Categories: Can also highlight specific groups for easier identification. |
Practical Insights and Use Cases
Mosaic plots are particularly useful for:
- Identifying Associations: Quickly spotting if certain combinations of categories occur more or less frequently than would be expected if the variables were independent. The presence of large, colored residuals (especially blue or red) immediately draws attention to these associations.
- Exploring Conditional Relationships: Understanding how the distribution of one variable changes across the categories of another. For example, you can see if the proportion of people who prefer a certain product varies significantly by age group.
- Visualizing Multi-Way Tables: Providing a more intuitive graphical representation of multi-dimensional contingency tables than just raw numbers, making complex relationships easier to grasp.
For example, when analyzing survey data on 'Diet Type' (Vegetarian, Vegan, Omnivore) and 'Exercise Frequency' (Daily, Weekly, Rarely), a mosaic plot could clearly show if vegetarians are more likely to exercise daily, or if omnivores are more likely to exercise rarely, by displaying the relative sizes and potential coloring of the rectangles for each combination.
Mosaic plots are a powerful tool for categorical data analysis and offer a holistic view of multivariate relationships, making them invaluable in fields like social sciences, public health, and market research.
