Measures of Association in R: Cramer's V, Phi, Pearson r With Examples

A measure of association is a single number that tells you how strongly two variables move together. The right choice depends on whether your variables are numeric, ordinal, or nominal — and using the wrong one can silently hide a real effect or invent a fake one. This tutorial covers every major measure you will meet in practice, with runnable R examples on built-in datasets and plain-English interpretation rules.

Which measure of association fits your variables?

You want one number that says "these two variables are related — and by how much." The number you pick has to match your data: two numeric columns want a correlation coefficient, two categorical columns want a chi-square–based measure, and ordinal data has its own toolbox. Let's start with the most familiar case — two numeric columns from mtcars — so you can see the payoff before we walk through the full taxonomy.

The correlation is -0.87, which means heavier cars burn more fuel per mile — a strong, near-straight-line relationship. That's exactly the kind of signal the Pearson correlation is built to detect. But notice the ingredients: two numeric columns and a roughly linear pattern. Break either assumption and Pearson's r is no longer the right answer.

The flowchart below is the decision rule you can use for any pair of variables in any dataset.

Figure 1: Pick the right measure of association based on your variable types.

Read the flowchart left to right. Both variables numeric? Use Pearson r (or Spearman / Kendall if the relationship is curved or noisy). Both categorical? It splits again — 2×2 tables get Phi or Yule's Q, bigger nominal tables get Cramer's V, and ordinal pairs get gamma, tau-b, or Somers' D. That's the whole map. The rest of this tutorial fills in how each one works and how to compute it in R.

How do Pearson r, Spearman ρ, and Kendall τ handle numeric data?

For two numeric variables, R's cor() function gives you three options through its method argument. They answer slightly different questions, and the differences matter when your data has curves, outliers, or ties.

• Pearson r measures a linear relationship — how well a straight line fits the points.
• Spearman ρ (rho) ranks both variables first, then computes Pearson on the ranks. It captures any monotonic relationship — always rising or always falling, even if curved.
• Kendall τ (tau) counts concordant pairs (both variables move the same way) versus discordant pairs. It is the most robust to outliers but slowest to compute on large data.

Here are all three on the same mtcars pair we saw above.

All three agree on the direction (negative) and broadly agree on strength. Spearman is slightly stronger than Pearson, a hint that the true relationship between weight and mileage curves a little — the biggest cars lose mileage faster than the linear model expects. Kendall comes in lower because it counts pair-by-pair agreement rather than sizes, which always gives smaller values than the other two.

To see why Spearman can win decisively, here is a case where the relationship is perfectly monotonic but not linear.

Spearman nails the relationship at exactly 1 because the ranks line up perfectly. Pearson only reports 0.88 because the curve is not a straight line, so a linear fit leaves residuals. If you had only looked at Pearson here you would have underestimated the true association by more than 10 points.

Figure 2: Pearson, Spearman, and Kendall capture different kinds of relationships on numeric pairs.

How does the Phi coefficient measure association in 2×2 tables?

When both variables are binary — yes/no, male/female, auto/manual — you have a 2×2 contingency table and the right tool is the Phi coefficient. Phi is built directly on the chi-square statistic for the table.

$$\phi = \sqrt{\chi^2 / n}$$

Where:

• $\chi^2$ is the Pearson chi-square statistic for the 2×2 table
• $n$ is the total sample size

Phi ranges from 0 (no association) to 1 (perfect association), and for 2×2 tables you can also give it a sign based on the direction of the association. A useful mental anchor: Phi is what Pearson r becomes when you feed it two 0/1-coded variables.

Let's build a real 2×2 from mtcars: does transmission type (am: 0 auto, 1 manual) go with engine shape (vs: 0 V-shaped, 1 straight)?

A Phi of about 0.17 says the association between transmission and engine shape is weak — they are not independent, but knowing one tells you very little about the other. For context, Cohen's rule-of-thumb calls 0.10 small, 0.30 medium, and 0.50 large for 2×2 tables.

For the same 2×2 you can also compute Yule's Q, which rescales the odds ratio into the –1 to 1 range and is much more sensitive to association in small tables.

Yule's Q lands at 0.33 — about twice as strong as Phi on the same table. That is because Q ignores the marginal totals and looks only at the cross-product ratio, so it exaggerates small effects. Phi is the more conservative choice and usually the one you should report; Q is useful as a quick diagnostic.

How do you compute Cramer's V for larger contingency tables?

When your categorical variables have more than two levels — hair color, education level, department — Phi no longer applies. Its generalization is Cramer's V, which handles any k × m contingency table.

$$V = \sqrt{\chi^2 / (n \cdot \min(r-1, c-1))}$$

Where:

• $\chi^2$ is the Pearson chi-square statistic for the full table
• $n$ is the total sample size
• $r$ and $c$ are the number of rows and columns

V always lives in [0, 1]. The denominator corrects for the fact that bigger tables inflate the chi-square statistic even when the underlying association is the same. For a 2×2 table, V reduces exactly to the absolute value of Phi.

Let's use HairEyeColor, a built-in 3-dimensional contingency table of hair, eye color, and sex. We first collapse across sex to get a flat 4×4 table.

Cramer's V of 0.28 means hair color and eye color share a medium-strength association — roughly what you would expect (blonde hair goes with blue eyes more often than with brown). The bias-corrected version nudges the estimate down slightly; for samples this large the correction barely matters, but for small samples (n < 50) it can move the number noticeably.

When you want every nominal measure at once, vcd::assocstats() prints Phi, contingency coefficient, Cramer's V, and the underlying chi-square test in a single call.

The output tells you three things at once: the chi-square test is highly significant (p ≈ 4e-6), the contingency coefficient is 0.34, and Cramer's V is 0.26. Phi is reported as NA because this is a 4×4 table, not 2×2 — assocstats() is honest about which measures apply to your table shape.

Which measures fit ordinal data — gamma, tau-b, Somers' D?

When both variables have a natural order — low/medium/high, grades A–F, survey ratings — you leave information on the table if you use Cramer's V. Ordinal measures count concordant pairs (both variables move the same way) minus discordant pairs (they move in opposite directions) and scale the difference into a number between –1 and 1.

Three measures dominate this space:

• Goodman-Kruskal γ (gamma) — ignores ties entirely. Easy to interpret, but can be over-optimistic when there are lots of ties.
• Kendall τ-b (tau-b) — penalizes ties, symmetric (treats both variables the same way). The standard choice for square tables.
• Somers' D — asymmetric: it treats one variable as the "dependent" outcome and the other as the "predictor," so you get two possible values, D(Y|X) and D(X|Y).

Let's build a small 3×3 ordinal table — age group vs an income bracket — and compute all three.

All three agree on the direction (positive — older respondents tend to land in higher income brackets) but give very different magnitudes. Gamma is 0.71, the biggest because it ignores ties completely. Kendall's τ-b drops to 0.52 because it divides by a denominator that includes tied pairs, which deflates the apparent strength. Somers' D matches τ-b here because the table is symmetric; on an asymmetric table the two would diverge.

Which should you report? For a symmetric hypothesis — "are age and income associated?" — use τ-b. For a directional hypothesis — "does age predict income?" — use Somers' D and be explicit about which way round it points. Gamma is most useful as a quick first look because it has the simplest interpretation, but it can mislead when ties are common.