Association measures for contingency tables
Source:vignettes/association-measures.Rmd
association-measures.Rmdspicy provides a full suite of effect size and association measures for contingency tables, covering both nominal and ordinal variables. This vignette explains which measure to use depending on the measurement level of your variables, and how to obtain confidence intervals and p-values for chi-squared-based and rank-based statistics.
Choosing the right measure
The table below summarizes the recommended measures by variable type.
| Variable types | Recommended measure | Function |
|---|---|---|
| Nominal x Nominal | Cramer’s V | cramer_v() |
| Nominal x Nominal | Contingency Coefficient | contingency_coef() |
| Nominal x Nominal (2x2) | Phi | phi() |
| Ordinal x Ordinal | Kendall’s Tau-b | kendall_tau_b() |
| Ordinal x Ordinal (rectangular) | Kendall’s Tau-c | kendall_tau_c() |
| Ordinal x Ordinal | Goodman-Kruskal Gamma | gamma_gk() |
| Ordinal x Ordinal (asymmetric) | Somers’ D | somers_d() |
| Nominal (asymmetric, PRE) | Lambda | lambda_gk() |
| Nominal (asymmetric, PRE) | Goodman-Kruskal Tau | goodman_kruskal_tau() |
| Nominal (asymmetric, PRE) | Uncertainty Coefficient | uncertainty_coef() |
| 2x2 table | Yule’s Q | yule_q() |
PRE = Proportional Reduction in Error. These measures quantify how much knowing one variable reduces prediction error for the other.
All functions accept a contingency table (class table,
typically from xtabs() or table()).
Quick overview with assoc_measures()
assoc_measures() computes all available measures at
once:
tbl <- xtabs(~ smoking + education, data = sochealth)
assoc_measures(tbl)
#> Measure Estimate SE CI lower CI upper p
#> Cramer's V 0.136 -- 0.079 0.191 < 0.001
#> Contingency Coefficient 0.134 -- -- -- < 0.001
#> Lambda symmetric 0.000 0.000 0.000 0.000 --
#> Lambda R|C 0.000 0.000 0.000 0.000 --
#> Lambda C|R 0.000 0.000 0.000 0.000 --
#> Goodman-Kruskal's Tau R|C 0.018 0.008 0.003 0.034 0.023
#> Goodman-Kruskal's Tau C|R 0.008 0.003 0.001 0.014 0.022
#> Uncertainty Coefficient symmetric 0.011 0.005 0.002 0.021 0.021
#> Uncertainty Coefficient R|C 0.018 0.008 0.003 0.032 0.021
#> Uncertainty Coefficient C|R 0.009 0.004 0.001 0.016 0.021
#> Goodman-Kruskal Gamma -0.268 0.056 -0.378 -0.158 < 0.001
#> Kendall's Tau-b -0.126 0.027 -0.180 -0.073 < 0.001
#> Kendall's Tau-c -0.117 0.026 -0.167 -0.067 < 0.001
#> Somers' D R|C -0.091 0.020 -0.131 -0.052 < 0.001
#> Somers' D C|R -0.175 0.038 -0.249 -0.101 < 0.001This is useful for exploratory analysis. For reporting, pick the measure that matches your variable types.
Nominal variables
Cramer’s V
Cramer’s V measures the strength of association between two nominal variables. It ranges from 0 (no association) to 1 (perfect association).
Pass detail = TRUE for the confidence interval and
p-value. The p-value tests the null hypothesis of no association using
the Pearson chi-squared test.
cramer_v(tbl, detail = TRUE)
#> Estimate CI lower CI upper p
#> 0.136 0.079 0.191 < 0.001Phi coefficient
For 2x2 tables, Phi is equivalent to Cramer’s V. Unlike V, Phi can be negative when the table is 2x2, indicating the direction of association. The p-value tests H0: no association (Pearson chi-squared test).
Contingency coefficient
The contingency coefficient is an alternative to Cramer’s V. Its upper bound depends on the table dimensions, which makes it harder to compare across tables of different sizes. The p-value tests H0: no association (Pearson chi-squared test).
contingency_coef(tbl, detail = TRUE)
#> Estimate CI lower CI upper p
#> 0.134 -- -- < 0.001Ordinal variables
When both variables are ordinal (ordered factors), measures that account for the ordering are more appropriate than Cramer’s V.
Goodman-Kruskal Gamma
Gamma ranges from -1 to +1. It ignores tied pairs, which makes it sensitive to the direction of association but tends to overestimate strength when there are many ties.
tbl_ord <- xtabs(~ self_rated_health + education, data = sochealth)
gamma_gk(tbl_ord, detail = TRUE)
#> Estimate CI lower CI upper p
#> 0.310 0.238 0.383 < 0.001A positive value means that higher values on one variable tend to occur with higher values on the other. The p-value tests H0: Gamma = 0 using a Wald z-test.
Kendall’s Tau-b
Tau-b adjusts for ties and ranges from -1 to +1. It is generally preferred over Gamma for square or near-square tables. The p-value tests H0: Tau-b = 0 (Wald z-test).
kendall_tau_b(tbl_ord, detail = TRUE)
#> Estimate CI lower CI upper p
#> 0.205 0.155 0.254 < 0.001Kendall’s Tau-c
Tau-c is similar to Tau-b but adjusts for rectangular tables where the number of rows and columns differ. The p-value tests H0: Tau-c = 0 (Wald z-test).
kendall_tau_c(tbl_ord, detail = TRUE)
#> Estimate CI lower CI upper p
#> 0.200 0.151 0.248 < 0.001Somers’ D
Somers’ D is an asymmetric measure: it distinguishes between a dependent and an independent variable. By default, the row variable is treated as dependent (D(R|C)). The p-value tests H0: D = 0 (Wald z-test).
somers_d(tbl_ord, detail = TRUE)
#> Estimate CI lower CI upper p
#> 0.208 0.157 0.258 < 0.001Asymmetric (PRE) measures
These measures answer a specific question: how much does knowing the column variable reduce our error in predicting the row variable (or vice versa)?
Lambda
Lambda measures the proportional reduction in classification error. It can equal zero even when the variables are associated, if the modal category does not change across columns. The p-value tests H0: Lambda = 0 (Wald z-test).
Goodman-Kruskal Tau
Tau measures the proportional reduction in error when predicting the row variable from the column variable, using the full distribution (not just the mode). The p-value tests H0: Tau = 0 (Wald z-test).
goodman_kruskal_tau(tbl, detail = TRUE)
#> Estimate CI lower CI upper p
#> 0.017 0.008 0.026 < 0.001Uncertainty coefficient
The uncertainty coefficient (Theil’s U) is based on entropy. It measures how much knowing one variable reduces uncertainty about the other. The p-value tests H0: U = 0 (Wald z-test).
uncertainty_coef(tbl, detail = TRUE)
#> Estimate CI lower CI upper p
#> 0.028 0.016 0.040 < 0.001Yule’s Q
Yule’s Q is defined for 2x2 tables only. It ranges from -1 to +1 and is equivalent to Gamma for 2x2 tables. The p-value tests H0: Q = 0 (Wald z-test).
Automatic selection in cross_tab()
cross_tab() can automatically select an appropriate
measure via assoc_measure = "auto" (the default). When both
variables are ordered factors, it picks Kendall’s Tau-b; otherwise it
uses Cramer’s V.
# Nominal: Cramer's V
cross_tab(sochealth, smoking, education)
#> Crosstable: smoking x education (N)
#>
#> Values │ Lower secondary Upper secondary Tertiary │ Total
#> ─────────────┼───────────────────────────────────────────────────────────┼────────────
#> No │ 179 415 332 │ 926
#> Yes │ 78 112 59 │ 249
#> ─────────────┼───────────────────────────────────────────────────────────┼────────────
#> Total │ 257 527 391 │ 1175
#>
#> Chi-2(2) = 21.6, p < 0.001
#> Cramer's V = 0.14
# Ordinal: Kendall's Tau-b (automatic)
cross_tab(sochealth, self_rated_health, education)
#> Crosstable: self_rated_health x education (N)
#>
#> Values │ Lower secondary Upper secondary Tertiary │ Total
#> ────────────────┼───────────────────────────────────────────────────────────┼────────────
#> Poor │ 28 28 5 │ 61
#> Fair │ 86 118 62 │ 266
#> Good │ 102 263 193 │ 558
#> Very good │ 44 118 133 │ 295
#> ────────────────┼───────────────────────────────────────────────────────────┼────────────
#> Total │ 260 527 393 │ 1180
#>
#> Chi-2(6) = 73.2, p < 0.001
#> Kendall's Tau-b = 0.20You can override the automatic choice:
cross_tab(sochealth, self_rated_health, education, assoc_measure = "gamma")
#> Crosstable: self_rated_health x education (N)
#>
#> Values │ Lower secondary Upper secondary Tertiary │ Total
#> ────────────────┼───────────────────────────────────────────────────────────┼────────────
#> Poor │ 28 28 5 │ 61
#> Fair │ 86 118 62 │ 266
#> Good │ 102 263 193 │ 558
#> Very good │ 44 118 133 │ 295
#> ────────────────┼───────────────────────────────────────────────────────────┼────────────
#> Total │ 260 527 393 │ 1180
#>
#> Chi-2(6) = 73.2, p < 0.001
#> Goodman-Kruskal Gamma = 0.31Confidence intervals
All functions support confidence intervals via
detail = TRUE. The confidence level defaults to 95% and can
be changed with conf_level:
cramer_v(tbl, detail = TRUE, conf_level = 0.99)
#> Estimate CI lower CI upper p
#> 0.176 0.103 0.248 < 0.001To get only the estimate and p-value (no CI), pass
conf_level = NULL:
cramer_v(tbl, detail = TRUE, conf_level = NULL)
#> Estimate p
#> 0.176 < 0.001Controlling decimal places
When detail = FALSE (the default), functions return a
plain numeric scalar, so R’s own formatting rules apply. When
detail = TRUE, the result uses a custom print method that
defaults to 3 decimal places. Pass digits to change this
(the p-value always uses 3 decimal places or
< 0.001):
cramer_v(tbl, detail = TRUE, digits = 4)
#> Estimate CI lower CI upper p
#> 0.1762 0.1203 0.2309 < 0.001The same digits argument works for
assoc_measures():
assoc_measures(tbl, digits = 2)
#> Measure Estimate SE CI lower CI upper p
#> Cramer's V 0.18 -- 0.12 0.23 < 0.001
#> Contingency Coefficient 0.24 -- -- -- < 0.001
#> Lambda symmetric 0.01 0.01 0.00 0.04 0.389
#> Lambda R|C 0.00 0.00 0.00 0.00 --
#> Lambda C|R 0.02 0.03 0.00 0.07 0.386
#> Goodman-Kruskal's Tau R|C 0.02 0.00 0.01 0.03 < 0.001
#> Goodman-Kruskal's Tau C|R 0.03 0.01 0.01 0.04 < 0.001
#> Uncertainty Coefficient symmetric 0.03 0.01 0.02 0.04 < 0.001
#> Uncertainty Coefficient R|C 0.03 0.01 0.01 0.04 < 0.001
#> Uncertainty Coefficient C|R 0.03 0.01 0.02 0.04 < 0.001
#> Goodman-Kruskal Gamma 0.31 0.04 0.24 0.38 < 0.001
#> Kendall's Tau-b 0.20 0.03 0.16 0.25 < 0.001
#> Kendall's Tau-c 0.20 0.02 0.15 0.25 < 0.001
#> Somers' D R|C 0.21 0.03 0.16 0.26 < 0.001
#> Somers' D C|R 0.20 0.02 0.15 0.25 < 0.001You can also store a result and re-display it with a different precision without recalculating: