effectsize (version 0.4.5)

chisq_to_phi: Conversion Chi-Squared to Phi or Cramer's V

Description

Convert between Chi square \((\chi^2)\), Cramer's V, phi (\(\phi\)) and Cohen's w for contingency tables or goodness of fit.

Usage

chisq_to_phi(chisq, n, nrow, ncol, ci = 0.95, adjust = FALSE, ...)
chisq_to_cohens_w(chisq, n, nrow, ncol, ci = 0.95, adjust = FALSE, ...)
chisq_to_cramers_v(chisq, n, nrow, ncol, ci = 0.95, adjust = FALSE, ...)
phi_to_chisq(phi, n, ...)

Arguments

chisq

The Chi-squared statistic.

n

Total sample size.

nrow, ncol

The number of rows/columns in the contingency table (ignored for Phi when adjust=FALSE and CI=NULL).

ci

Confidence Interval (CI) level

adjust

Should the effect size be bias-corrected? Defaults to FALSE.

...

Arguments passed to or from other methods.

phi

The Phi statistic.

Value

A data frame with the effect size(s) between 0-1, and confidence interval(s). See cramers_v().

Confidence Intervals

Unless stated otherwise, confidence intervals are estimated using the Noncentrality parameter method; These methods searches for a the best non-central parameters (ncps) of the noncentral t-, F- or Chi-squared distribution for the desired tail-probabilities, and then convert these ncps to the corresponding effect sizes. (See full effectsize-CIs for more.)

CI Contains Zero

Keep in mind that ncp confidence intervals are inverted significance tests, and only inform us about which values are not significantly different than our sample estimate. (They do not inform us about which values are plausible, likely or compatible with our data.) Thus, when CIs contain the value 0, this should not be taken to mean that a null effect size is supported by the data; Instead this merely reflects a non-significant test statistic - i.e. the p-value is greater than alpha (Morey et al., 2016).

For positive only effect sizes (Eta squared, Cramer's V, etc.; Effect sizes associated with Chi-squared and F distributions), this applies also to cases where the lower bound of the CI is equal to 0. Even more care should be taken when the upper bound is equal to 0 - this occurs when p-value is greater than 1-alpha/2 making, the upper bound cannot be estimated, and the upper bound is arbitrarily set to 0 (Steiger, 2004). For example:

eta_squared(aov(mpg ~ factor(gear) + factor(cyl), mtcars[1:7, ]))

## # Effect Size for ANOVA (Type I)
## 
## Parameter    | Eta2 (partial) |       90% CI
## --------------------------------------------
## factor(gear) |           0.58 | [0.00, 0.84]
## factor(cyl)  |           0.46 | [0.00, 0.78]

Details

These functions use the following formulae:

$$\phi = \sqrt{\chi^2 / n}$$

$$Cramer's V = \phi / \sqrt{min(nrow,ncol)-1}$$

For adjusted versions, see Bergsma, 2013.

References

• 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.

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

See Also

Other effect size from test statistic: F_to_eta2(), t_to_d()

Examples

# NOT RUN {
contingency_table <- as.table(rbind(c(762, 327, 468), c(484, 239, 477), c(484, 239, 477)))

chisq.test(contingency_table)
#
#         Pearson's Chi-squared test
#
# data:  ctab
# X-squared = 41.234, df = 4, p-value = 2.405e-08

chisq_to_phi(41.234,
  n = sum(contingency_table),
  nrow = nrow(contingency_table),
  ncol = ncol(contingency_table)
)
chisq_to_cramers_v(41.234,
  n = sum(contingency_table),
  nrow = nrow(contingency_table),
  ncol = ncol(contingency_table)
)
# }