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Relationship square
Definition
A relationship square is a plot that shows how strongly variables are associated with two selected dimensions from a factor analysis. In its common categorical-data form, each variable is represented by squared correlation-ratio values, placing its coordinates between zero and one.
What it means
Factor methods summarize a table of observations using a smaller number of dimensions. A standard display may contain many category points, making it hard to see which original variables relate most strongly to each dimension.
The relationship square represents whole variables rather than all their categories. For a categorical variable, a squared correlation ratio measures how much of a factor's variation is associated with differences between the variable's categories.
For a quantitative variable, a related display can use squared correlations with the selected factors. Squaring removes the sign, so the plot communicates strength rather than whether an association is positive or negative.
This is a specialist visualization in exploratory multivariate analysis, especially multiple correspondence analysis and mixed-data methods. It is not a diagram of social relationships or a generic chart for every kind of statistical association.
How it works
Run an appropriate factor analysis on the observation-by-variable table. Choose the dimensions to display and inspect how much of the relevant structure they summarize.
For each categorical variable, calculate its squared correlation ratio with each selected factor. This compares between-category variation in factor scores with total variation, under the analysis's weighting conventions.
Plot the two values as the variable's horizontal and vertical coordinates. This entry uses raw squared association measures; historical operator-normalized representations use different scaling, so their coordinates should not be read interchangeably. A variable near the right edge is strongly associated with the horizontal factor; one near the upper edge is strongly associated with the vertical factor.
Read the plot alongside category and observation displays. Variables with similar coordinates have similar association strengths with those two factors, but that alone does not establish that they have identical categories, are interchangeable or are strongly associated with each other in every respect.
Three practical examples
1. A survey analyst uses a relationship square to see which questions are associated with the first two dimensions of a multiple correspondence analysis. She then inspects the category plot to interpret the pattern.
2. A mixed-data study displays numerical and categorical variables using suitable squared association measures. The chart offers a compact overview while preserving the distinction between the underlying measures.
3. A statistics workshop compares a correlation circle with a relationship square. Students observe that squaring coordinates changes interpretation by removing directional sign information.
Think of it this way
Think of a two-axis scorecard showing how strongly each variable connects with two summaries of the data. A high score on one axis points to a strong connection with that summary. The analogy is limited because these are calculated association measures, not subjective ratings, and their geometry should not be read as ordinary distances between the original variables.
Worked example
Suppose a categorical variable divides factor scores into two groups: [0, 2] and [4, 6]. The overall mean is 3, and the total sum of squared deviations is 9 plus 1 plus 1 plus 9, or 20.
The group means are 1 and 5. The between-group sum of squares is 2 times (1 minus 3) squared plus 2 times (5 minus 3) squared, or 16. With equal observation weights, the squared correlation ratio is 16 divided by 20, or 0.8.
If its corresponding value for a second factor is 0.2, the variable appears at (0.8, 0.2). It is more strongly associated with the first displayed factor. This teaching calculation does not replace the weighting and conventions of a complete factor analysis.
Illustrative case study
Fictional case: Linden Research studies responses to a customer-experience survey with many categorical questions. The category map is crowded, so analysts add a relationship square to summarize which questions relate to the displayed dimensions.
They use it to prioritize detailed inspection, then return to the individual categories to understand the contrasts. Two questions with nearby square coordinates turn out to distinguish different groups, showing why the compact summary cannot replace the detailed map.
The team describes the analysis as exploratory and checks additional dimensions before drawing conclusions. This invented case illustrates interpretation rather than claiming that the plot reveals causes or validated customer segments.
Common misconceptions and mistakes
Mistake: Reading a squared value as a signed correlation. The sign has been removed, so direction must be examined elsewhere when relevant.
Mistake: Assuming nearby variable points prove the variables are equivalent. Similar strengths of association with two factors are a limited summary.
Mistake: Ignoring omitted dimensions. A variable near the origin may relate to structure outside the selected plane rather than being unimportant overall.
Business use
The relationship square can make exploratory reports easier to read when a dataset contains many variables and categories. It helps explain which variables are connected with the dimensions being discussed.
Business presentations should state the factor method, displayed dimensions and coordinate definitions. Avoid turning exploratory association into causal claims or assigning business meaning solely from a point's position. Use the plot to guide investigation and explanation alongside the other analysis outputs.
People also ask
Why is the display a square?
The plotted squared association measures lie between zero and one on each axis.
Does it show each category separately?
Its main purpose is to represent variables themselves; category plots provide the more detailed view.
Does a coordinate of zero mean a variable is useless?
It indicates no association under that displayed measure with that factor, not universal lack of relevance.
Check your understanding
A variable has coordinates (0.9, 0.1). What can be said directly from the plot?
Answer: It is more strongly associated with the first displayed factor than the second under the chosen squared measures; causation and association sign do not follow.
Related terms
- Multiple correspondence analysis
- Exploratory data analysis
- factor analysis of mixed data
- correlation ratio
- squared correlation
- factor score
- correlation circle
Learn next
Learn correlation ratio and multiple correspondence analysis. Compute between-group and total variation for a small factor-score example, then compare the variable-level summary with the separate category positions.
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Sources
Brigitte Escofier, Une représentation des variables dans l'analyse des correspondances multiples: https://www.numdam.org/article/RSA_1979__27_4_37_0.pdf. The full paper, previously unavailable, was consulted during independent review; its historical operator-normalized coordinates differ from raw squared correlation ratios. François Husson's explanation of modern FAMD variable coordinates confirms squared correlations and squared correlation ratios: https://groups.google.com/g/factominer-users/c/NHDQ8IzcOTI. This entry uses that latter convention. The arithmetic and fictional survey case are original.
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