Compute phi (\(\phi\)), Cramer's V, Tschuprow's T, Cohen's w,
פ (Fei), Pearson's contingency coefficient for
contingency tables or goodness-of-fit. Pair with any reported
stats::chisq.test().
Usage
phi(x, y = NULL, adjust = TRUE, ci = 0.95, alternative = "greater", ...)
cramers_v(x, y = NULL, adjust = TRUE, ci = 0.95, alternative = "greater", ...)
tschuprows_t(x, y = NULL, ci = 0.95, alternative = "greater", ...)
cohens_w(
x,
y = NULL,
p = rep(1, length(x)),
ci = 0.95,
alternative = "greater",
...
)
fei(x, p = rep(1, length(x)), ci = 0.95, alternative = "greater", ...)
pearsons_c(
x,
y = NULL,
p = rep(1, length(x)),
ci = 0.95,
alternative = "greater",
...
)Arguments
- x
a numeric vector or matrix.
xandycan also both be factors.- y
a numeric vector; ignored if
xis a matrix. Ifxis a factor,yshould be a factor of the same length.- adjust
Should the effect size be bias-corrected? Defaults to
TRUE; Advisable for small samples and large tables.- ci
Confidence Interval (CI) level
- alternative
a character string specifying the alternative hypothesis; Controls the type of CI returned:
"greater"(default) or"less"(one-sided CI), or"two.sided"(default, two-sided CI). Partial matching is allowed (e.g.,"g","l","two"...). See One-Sided CIs in effectsize_CIs.- ...
Ignored.
- p
a vector of probabilities of the same length as
x. An error is given if any entry ofpis negative.
Value
A data frame with the effect size (Cramers_v, phi (possibly with
the suffix _adjusted), Cohens_w, Fei) and its CIs (CI_low and
CI_high).
Details
phi (\(\phi\)), Cramer's V, Tschuprow's T, Cohen's w, and Pearson's
C are effect sizes for tests of independence in 2D contingency tables. For
2-by-2 tables, phi, Cramer's V, Tschuprow's T, and Cohen's w are
identical, and are equal to the simple correlation between two dichotomous
variables, ranging between 0 (no dependence) and 1 (perfect dependence).
For larger tables, Cramer's V, Tschuprow's T or Pearson's C should be
used, as they are bounded between 0-1. (Cohen's w can also be used, but
since it is not bounded at 1 (can be larger) its interpretation is more
difficult.) For square table, Cramer's V and Tschuprow's T give the same
results, but for non-square tables Tschuprow's T is more conservative:
while V will be 1 if either columns are fully dependent on rows (for each
column, there is only one non-0 cell) or rows are fully dependent on
columns, T will only be 1 if both are true.
For goodness-of-fit in 1D tables Cohen's W, פ (Fei)
or Pearson's C can be used. Cohen's w has no upper bound (can be
arbitrarily large, depending on the expected distribution). Fei is an
adjusted Cohen's w, accounting for the expected distribution, making it
bounded between 0-1. Pearson's C is also bounded between 0-1.
To summarize, for correlation-like effect sizes, we recommend:
For a 2x2 table, use
phi()For larger tables, use
cramers_v()For goodness-of-fit, use
fei()
Confidence (Compatibility) Intervals (CIs)
Unless stated otherwise, confidence (compatibility) intervals (CIs) are
estimated using the noncentrality parameter method (also called the "pivot
method"). This method finds the noncentrality parameter ("ncp") of a
noncentral t, F, or \(\chi^2\) distribution that places the observed
t, F, or \(\chi^2\) test statistic at the desired probability point of
the distribution. For example, if the observed t statistic is 2.0, with 50
degrees of freedom, for which cumulative noncentral t distribution is t =
2.0 the .025 quantile (answer: the noncentral t distribution with ncp =
.04)? After estimating these confidence bounds on the ncp, they are
converted into the effect size metric to obtain a confidence interval for the
effect size (Steiger, 2004).
For additional details on estimation and troubleshooting, see effectsize_CIs.
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).
References
Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd Ed.). New York: Routledge.
Johnston, J. E., Berry, K. J., & Mielke Jr, P. W. (2006). Measures of effect size for chi-squared and likelihood-ratio goodness-of-fit tests. Perceptual and motor skills, 103(2), 412-414.
Rosenberg, M. S. (2010). A generalized formula for converting chi-square tests to effect sizes for meta-analysis. PloS one, 5(4), e10059.
See also
chisq_to_phi() for details regarding estimation and CIs.
Other effect sizes for contingency table:
cohens_g(),
oddsratio()
Examples
## 2-by-2 tables
## -------------
data("RCT_table")
RCT_table # note groups are COLUMNS
#> Group
#> Diagnosis Treatment Control
#> Sick 71 30
#> Recovered 50 100
phi(RCT_table)
#> Phi (adj.) | 95% CI
#> -------------------------
#> 0.36 | [0.25, 1.00]
#>
#> - One-sided CIs: upper bound fixed at [1.00].
pearsons_c(RCT_table)
#> Pearson's C | 95% CI
#> --------------------------
#> 0.34 | [0.25, 1.00]
#>
#> - One-sided CIs: upper bound fixed at [1.00].
## Larger tables
## -------------
data("Music_preferences")
Music_preferences
#> Pop Rock Jazz Classic
#> Psych 150 100 165 130
#> Econ 50 65 35 10
#> Law 2 55 40 25
cramers_v(Music_preferences)
#> Cramer's V (adj.) | 95% CI
#> --------------------------------
#> 0.23 | [0.18, 1.00]
#>
#> - One-sided CIs: upper bound fixed at [1.00].
cohens_w(Music_preferences)
#> Cohen's w | 95% CI
#> ------------------------
#> 0.34 | [0.27, 1.41]
#>
#> - One-sided CIs: upper bound fixed at [1.41~].
pearsons_c(Music_preferences)
#> Pearson's C | 95% CI
#> --------------------------
#> 0.32 | [0.26, 1.00]
#>
#> - One-sided CIs: upper bound fixed at [1.00].
## Goodness of fit
## ---------------
data("Smoking_FASD")
Smoking_FASD
#> FAS PFAS TD
#> 17 11 640
fei(Smoking_FASD)
#> Fei | 95% CI
#> -------------------
#> 0.94 | [0.89, 1.00]
#>
#> - Adjusted for uniform expected probabilities.
#> - One-sided CIs: upper bound fixed at [1.00].
cohens_w(Smoking_FASD)
#> Cohen's w | 95% CI
#> ------------------------
#> 1.33 | [1.26, 1.41]
#>
#> - One-sided CIs: upper bound fixed at [1.41~].
pearsons_c(Smoking_FASD)
#> Pearson's C | 95% CI
#> --------------------------
#> 0.80 | [0.78, 1.00]
#>
#> - One-sided CIs: upper bound fixed at [1.00].
# Use custom expected values:
fei(Smoking_FASD, p = c(0.015, 0.010, 0.975))
#> Fei | 95% CI
#> -------------------
#> 0.01 | [0.00, 1.00]
#>
#> - Adjusted for non-uniform expected probabilities.
#> - One-sided CIs: upper bound fixed at [1.00].
cohens_w(Smoking_FASD, p = c(0.015, 0.010, 0.975))
#> Cohen's w | 95% CI
#> ------------------------
#> 0.11 | [0.03, 9.95]
#>
#> - One-sided CIs: upper bound fixed at [9.95~].
pearsons_c(Smoking_FASD, p = c(0.015, 0.010, 0.975))
#> Pearson's C | 95% CI
#> --------------------------
#> 0.11 | [0.03, 1.00]
#>
#> - One-sided CIs: upper bound fixed at [1.00].