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\(\phi\) and Other Contingency Tables Correlations

Source: R/xtab_corr.R
phi.Rd

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" (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 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].