# Convert \(\chi^2\) to \(\phi\) and Other Correlation-like Effect Sizes

Source:`R/convert_stat_chisq.R`

`convert_chisq.Rd`

Convert between \(\chi^2\) (chi-square), \(\phi\) (phi), Cramer's \(V\), Tschuprow's \(T\), Cohen's \(w\), פ (Fei) and Pearson's \(C\) for contingency tables or goodness of fit.

## Usage

```
chisq_to_phi(
chisq,
n,
nrow = 2,
ncol = 2,
adjust = TRUE,
ci = 0.95,
alternative = "greater",
...
)
chisq_to_cohens_w(
chisq,
n,
nrow,
ncol,
p,
ci = 0.95,
alternative = "greater",
...
)
chisq_to_cramers_v(
chisq,
n,
nrow,
ncol,
adjust = TRUE,
ci = 0.95,
alternative = "greater",
...
)
chisq_to_tschuprows_t(
chisq,
n,
nrow,
ncol,
adjust = TRUE,
ci = 0.95,
alternative = "greater",
...
)
chisq_to_fei(chisq, n, nrow, ncol, p, ci = 0.95, alternative = "greater", ...)
chisq_to_pearsons_c(
chisq,
n,
nrow,
ncol,
ci = 0.95,
alternative = "greater",
...
)
phi_to_chisq(phi, n, ...)
```

## Arguments

- chisq
The \(\chi^2\) (chi-square) statistic.

- n
Total sample size.

- nrow, ncol
The number of rows/columns in the contingency table.

- 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"`

(default, two-sided CI). Partial matching is allowed (e.g.,`"g"`

,`"l"`

,`"two"`

...). See*One-Sided CIs*in effectsize_CIs.- ...
Arguments passed to or from other methods.

- p
Vector of expected values. See

`stats::chisq.test()`

.- phi
The \(\phi\) (phi) statistic.

## Value

A data frame with the effect size(s), and confidence interval(s). See
`cramers_v()`

.

## Details

These functions use the following formulas:

$$\phi = w = \sqrt{\chi^2 / n}$$ $$\textrm{Cramer's } V = \phi / \sqrt{\min(\textit{nrow}, \textit{ncol}) - 1}$$

$$\textrm{Tschuprow's } T = \phi / \sqrt[4]{(\textit{nrow} - 1) \times (\textit{ncol} - 1)}$$

$$פ = \phi / \sqrt{[1 / \min(p_E)] - 1}$$ Where \(p_E\) are the expected probabilities.

$$\textrm{Pearson's } C = \sqrt{\chi^2 / (\chi^2 + n)}$$

For versions adjusted for small-sample bias of \(\phi\), \(V\), and \(T\), see Bergsma, 2013.

## 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.

Cumming, G., & Finch, S. (2001). A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions. Educational and Psychological Measurement, 61(4), 532-574.

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

Bergsma, W. (2013). A bias-correction for Cramer's V and Tschuprow's T. Journal of the Korean Statistical Society, 42(3), 323-328.

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.

## Examples

```
data("Music_preferences")
# chisq.test(Music_preferences)
#>
#> Pearson's Chi-squared test
#>
#> data: Music_preferences
#> X-squared = 95.508, df = 6, p-value < 2.2e-16
#>
chisq_to_cohens_w(95.508,
n = sum(Music_preferences),
nrow = nrow(Music_preferences),
ncol = ncol(Music_preferences)
)
#> Cohen's w | 95% CI
#> ------------------------
#> 0.34 | [0.27, 1.41]
#>
#> - One-sided CIs: upper bound fixed at [1.41~].
data("Smoking_FASD")
# chisq.test(Smoking_FASD, p = c(0.015, 0.010, 0.975))
#>
#> Chi-squared test for given probabilities
#>
#> data: Smoking_FASD
#> X-squared = 7.8521, df = 2, p-value = 0.01972
chisq_to_fei(
7.8521,
n = sum(Smoking_FASD),
nrow = 1,
ncol = 3,
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].
```