What the three measures in the pairwise association matrix mean, how to interpret their values, and why they cannot be compared with each other directly.
The Stat Analysis matrix shows how strongly every pair of variables in your dataset is associated. Statulator automatically picks the most appropriate measure of association based on whether each variable in the pair is numeric (continuous) or categorical (discrete). There are three possible pair types, each with its own measure:
| Variable pair | Measure shown | Symbol | Range |
|---|---|---|---|
| Num × Num | Pearson correlation | r | −1 to +1 |
| Num × Cat | Eta (correlation ratio) | η | 0 to 1 |
| Cat × Cat | Cramér’s V | V | 0 to 1 |
The colour and shading of each cell in the matrix encode the strength of the association. Keep in mind, however, that the three measures are not directly comparable: an r of 0.17, an η of 0.17, and a V of 0.17 mean different things, and the thresholds for “small / medium / large” differ across them. See Can I compare the three measures directly? below.
What it measures: the strength and direction of the linear relationship between two numeric variables.
Why it is used here: when both variables are numeric and roughly continuous, the natural question is “do high values of one go with high values of the other?” Pearson’s r answers this directly, on a scale that is symmetric, signed, and well understood.
| |r| | Typical label | Meaning |
|---|---|---|
| 0.0–0.1 | Trivial | Essentially no linear relationship. |
| 0.1–0.3 | Small / weak | Hint of a relationship; large samples are needed to detect it reliably. |
| 0.3–0.5 | Moderate | A noticeable trend, visible in a scatter plot. |
| 0.5–0.7 | Strong | Clear linear pattern. |
| 0.7–1.0 | Very strong | Tight linear relationship, near-deterministic at 1.0. |
The sign matters: r = −0.22 means a weak negative trend (high values of one variable tend to go with low values of the other); r = +0.22 is the same strength in the positive direction.
Caveat: Pearson’s r only measures linear association. Two variables can have a strong curved or non-monotonic relationship and still show r close to 0 — always look at the scatter plot before concluding “no relationship”.
Next step: click the cell to open the full Numeric × Numeric analysis — scatter plot, regression line, confidence interval, Spearman’s ρ, and a hypothesis test of whether r differs from zero.
What it measures: how much of the variation in a numeric variable is explained by group membership in a categorical variable — equivalently, the strength of association between a numeric response and a categorical predictor.
Why it is used here: a scatter plot does not make sense when one variable has no natural ordering. The question becomes “do the mean values of the numeric variable differ across the categories?” Eta captures that on a 0-to-1 scale and is the square root of the η² statistic from one-way ANOVA.
Cohen’s conventions, treating η² like R²:
| η² | η | Effect size |
|---|---|---|
| 0.01 | 0.10 | Small |
| 0.06 | 0.24 | Medium |
| 0.14 | 0.37 | Large |
Worked example: η = 0.17 between Age (numeric) and Treatment Group (categorical) means η² ≈ 0.029 — about 2.9% of the variation in Age is explained by group membership. This is a small effect, close to the border with medium: the group means of Age differ, but the spread within each group is much larger than the gap between them.
Direction: eta has no sign — it is always positive, because categorical variables have no “positive direction”. Open the full analysis to see which groups have higher or lower mean values.
Caveat: η assumes a one-way layout (one categorical predictor, one numeric response) and treats the categories as unordered. If your categories are actually ordinal (e.g., low / medium / high), a Spearman or Kendall rank correlation may capture more of the relationship.
Next step: click the cell to open the full Numeric × Categorical analysis — box plots, group means with confidence intervals, t-test or one-way ANOVA (with Mann-Whitney U or Kruskal-Wallis as non-parametric alternatives), and effect sizes.
What it measures: the strength of association between two categorical variables, based on the chi-square statistic of their contingency table and normalised so that V always lies between 0 and 1 regardless of table size.
Why it is used here: when both variables are categorical, the natural question is “does the distribution of one variable depend on the other?” The raw chi-square statistic answers this, but its scale depends on the sample size and the number of cells, making it impossible to compare across tables. Cramér’s V removes both dependencies.
The “small / medium / large” thresholds shift with the table size, because larger tables can hold more partial associations (Cohen 1988, adapting his w guidelines for chi-square tests):
| df* = min(r, c) − 1 | Small | Medium | Large |
|---|---|---|---|
| 1 (2 × 2 table) | 0.10 | 0.30 | 0.50 |
| 2 (3-level) | 0.07 | 0.21 | 0.35 |
| 3 (4-level) | 0.06 | 0.17 | 0.29 |
| 4 (5-level) | 0.05 | 0.15 | 0.25 |
Worked example: V = 0.13 between Outcome and Treatment Group (both categorical) is small to small-medium by every row of the table above. The two variables share modest information — knowing one shifts the distribution of the other a little, but not dramatically.
Direction: Cramér’s V has no sign — categorical variables have no direction. To see which cells contribute most to the dependence, examine the standardised residuals in the contingency table.
Caveat: Cramér’s V can be inflated when one variable has many sparsely populated levels. If cells have expected counts < 5, the chi-square statistic (and therefore V) is unreliable — switch to Fisher’s exact test.
Next step: click the cell to open the full Categorical × Categorical analysis — contingency table with observed and expected frequencies, chi-square test of independence (or Fisher’s exact test for sparse tables), and odds ratio and relative risk when both variables have two levels.
No, not directly. A V of 0.17, an η of 0.17, and an r of 0.17 do not mean the same thing. Each measure has its own scale, its own thresholds, and its own underlying assumptions. Use the matrix to rank the strongest associations within each variable-type group, not to compare values across groups.
A rough rule of thumb:
Statulator uses r, η, and V because they are the standard, most widely taught measures for each pairing of variable types, with consistent scales and well-established statistical tests behind them. Alternatives exist and may be added in future updates: