Effect Size for Paired Contingency Tables

Cohen's g is an effect size of asymmetry (or marginal heterogeneity) for dependent (paired) contingency tables ranging between 0 (perfect symmetry) and 0.5 (perfect asymmetry) (see stats::mcnemar.test()). (Note this is not not a measure of (dis)agreement between the pairs, but of (a)symmetry.)

Usage

cohens_g(x, y = NULL, ci = 0.95, alternative = "two.sided", ...)

Arguments

x: a numeric vector or matrix. x and y can also both be factors.
y: a numeric vector; ignored if x is a matrix. If x is a factor, y should be a factor of the same length.
ci: Confidence Interval (CI) level
alternative: a character string specifying the alternative hypothesis; Controls the type of CI returned: "two.sided" (default, two-sided CI), "greater" or "less" (one-sided CI). Partial matching is allowed (e.g., "g", "l", "two"...). See One-Sided CIs in effectsize_CIs.
...: Ignored

Value

A data frame with the effect size (Cohens_g, Risk_ratio

(possibly with the prefix log_), Cohens_h) and its CIs (CI_low and CI_high).

Confidence (Compatibility) Intervals (CIs)

Confidence intervals are based on the proportion (\(P = g + 0.5\)) confidence intervals returned by stats::prop.test() (minus 0.5), which give a good close approximation.

CIs and Significance Tests

"Confidence intervals on measures of effect size convey all the information in a hypothesis test, and more." (Steiger, 2004). Confidence (compatibility) intervals and p values are complementary summaries of parameter uncertainty given the observed data. A dichotomous hypothesis test could be performed with either a CI or a p value. The 100 (1 - \(\alpha\))% confidence interval contains all of the parameter values for which p > \(\alpha\) for the current data and model. For example, a 95% confidence interval contains all of the values for which p > .05.

Note that a confidence interval including 0 does not indicate that the null (no effect) is true. Rather, it suggests that the observed data together with the model and its assumptions combined do not provided clear evidence against a parameter value of 0 (same as with any other value in the interval), with the level of this evidence defined by the chosen \(\alpha\) level (Rafi & Greenland, 2020; Schweder & Hjort, 2016; Xie & Singh, 2013). To infer no effect, additional judgments about what parameter values are "close enough" to 0 to be negligible are needed ("equivalence testing"; Bauer & Kiesser, 1996).

Plotting with `see`

The see package contains relevant plotting functions. See the plotting vignette in the see package.

References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed.). New York: Routledge.

Examples


data("screening_test")

phi(screening_test$Diagnosis, screening_test$Test1)
#> Phi (adj.) |       95% CI
#> -------------------------
#> 0.85       | [0.81, 1.00]
#> 
#> - One-sided CIs: upper bound fixed at [1.00].

phi(screening_test$Diagnosis, screening_test$Test2)
#> Phi (adj.) |       95% CI
#> -------------------------
#> 0.85       | [0.81, 1.00]
#> 
#> - One-sided CIs: upper bound fixed at [1.00].

# Both tests seem comparable - but are the tests actually different?

(tests <- table(Test1 = screening_test$Test1, Test2 = screening_test$Test2))
#>        Test2
#> Test1   "Neg" "Pos"
#>   "Neg"   794    86
#>   "Pos"   150   570

mcnemar.test(tests)
#> 
#> 	McNemar's Chi-squared test with continuity correction
#> 
#> data:  tests
#> McNemar's chi-squared = 16.818, df = 1, p-value = 4.115e-05
#> 

cohens_g(tests)
#> Cohen's g |       95% CI
#> ------------------------
#> 0.14      | [0.07, 0.19]

# Test 2 gives a negative result more than test 1!