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,
adjust = TRUE,
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.

`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.- adjust
Should the effect size be corrected for small-sample bias? 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"`

(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 of`p`

is 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 (Ben-Shachar et al, 2023). 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).

## 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.

Ben-Shachar, M.S., Patil, I., Thériault, R., Wiernik, B.M., Lüdecke, D. (2023). Phi, Fei, Fo, Fum: Effect Sizes for Categorical Data That Use the Chi‑Squared Statistic. Mathematics, 11, 1982. doi:10.3390/math11091982

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 non-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 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].
```