These functions are convenience functions to convert t, z and F test statistics to Cohen's d and partial r. These are useful in cases where the data required to compute these are not easily available or their computation is not straightforward (e.g., in liner mixed models, contrasts, etc.).
See Effect Size from Test Statistics vignette.

t_to_d(t, df_error, paired = FALSE, ci = 0.95, pooled, ...)

z_to_d(z, n, paired = FALSE, ci = 0.95, pooled, ...)

F_to_d(f, df, df_error, paired = FALSE, ci = 0.95, ...)

t_to_r(t, df_error, ci = 0.95, ...)

z_to_r(z, n, ci = 0.95, ...)

F_to_r(f, df, df_error, ci = 0.95, ...)

Arguments

t, f, z

The t, the F or the z statistics.

paired

Should the estimate account for the t-value being testing the difference between dependent means?

ci

Confidence Interval (CI) level

pooled

Deprecated. Use paired.

...

Arguments passed to or from other methods.

n

The number of observations (the sample size).

df, df_error

Degrees of freedom of numerator or of the error estimate (i.e., the residuals).

Value

A data frame with the effect size(s)(r or d), and their CIs (CI_low and CI_high).

Details

These functions use the following formulae to approximate r and d:

$$r_{partial} = t / \sqrt{t^2 + df_{error}}$$

$$r_{partial} = z / \sqrt{z^2 + N}$$

$$d = 2 * t / \sqrt{df_{error}}$$

$$d_z = t / \sqrt{df_{error}}$$

$$d = 2 * z / \sqrt{N}$$

The resulting d effect size is an approximation to Cohen's d, and assumes two equal group sizes. When possible, it is advised to directly estimate Cohen's d, with cohens_d(), emmeans::eff_size(), or similar functions.

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]

References

  • Friedman, H. (1982). Simplified determinations of statistical power, magnitude of effect and research sample sizes. Educational and Psychological Measurement, 42(2), 521-526. doi: 10.1177/001316448204200214

  • Wolf, F. M. (1986). Meta-analysis: Quantitative methods for research synthesis (Vol. 59). Sage.

  • Rosenthal, R. (1994) Parametric measures of effect size. In H. Cooper and L.V. Hedges (Eds.). The handbook of research synthesis. New York: Russell Sage Foundation.

  • Steiger, J. H. (2004). Beyond the F test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis. Psychological Methods, 9, 164-182.

  • 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.

See also

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

Examples

## t Tests
res <- t.test(1:10, y = c(7:20), var.equal = TRUE)
t_to_d(t = res$statistic, res$parameter)
#> d     |         95% CI
#> ----------------------
#> -2.19 | [-3.23, -1.12]
t_to_r(t = res$statistic, res$parameter)
#> r     |         95% CI
#> ----------------------
#> -0.74 | [-0.85, -0.49]

res <- with(sleep, t.test(extra[group == 1], extra[group == 2], paired = TRUE))
t_to_d(t = res$statistic, res$parameter, paired = TRUE)
#> d     |         95% CI
#> ----------------------
#> -1.35 | [-2.23, -0.44]
t_to_r(t = res$statistic, res$parameter)
#> r     |         95% CI
#> ----------------------
#> -0.80 | [-0.91, -0.40]
# \donttest{
## Linear Regression
model <- lm(rating ~ complaints + critical, data = attitude)
library(parameters)
(param_tab <- parameters(model))
#> Parameter   | Coefficient |    SE |         95% CI | t(27) |      p
#> -------------------------------------------------------------------
#> (Intercept) |       14.25 | 11.17 | [-8.67, 37.18] |  1.28 | 0.213 
#> complaints  |        0.75 |  0.10 | [ 0.55,  0.96] |  7.46 | < .001
#> critical    |    1.91e-03 |  0.14 | [-0.28,  0.28] |  0.01 | 0.989 

(rs <- t_to_r(param_tab$t[2:3], param_tab$df_error[2:3]))
#> r        |        95% CI
#> ------------------------
#> 0.82     | [ 0.67, 0.89]
#> 2.70e-03 | [-0.35, 0.36]

if (require(see)) plot(rs)
#> Warning: It is deprecated to specify `guide = FALSE` to remove a guide. Please use `guide = "none"` instead.


# How does this compare to actual partial correlations?
if (require("correlation")) {
  correlation::correlation(attitude[, c(1, 2, 6)], partial = TRUE)[1:2, c(2, 3, 7, 8)]
}
#> Loading required package: correlation
#> 
#> Attaching package: ‘correlation’
#> The following object is masked from ‘package:bayestestR’:
#> 
#>     simulate_simpson
#> Parameter2 |        r | t(28)
#> -----------------------------
#> complaints |     0.82 |  7.60
#> critical   | 2.70e-03 |  0.01

## Use with emmeans based contrasts (see also t_to_eta2)
if (require(emmeans)) {
  warp.lm <- lm(breaks ~ wool * tension, data = warpbreaks)


  # Also see emmeans::eff_size()
  em_tension <- emmeans(warp.lm, ~tension) #'
  diff_tension <- summary(pairs(em_tension))
  t_to_d(diff_tension$t.ratio, diff_tension$df)
}
#> NOTE: Results may be misleading due to involvement in interactions
#> d    |        95% CI
#> --------------------
#> 0.79 | [ 0.20, 1.38]
#> 1.17 | [ 0.55, 1.77]
#> 0.37 | [-0.20, 0.94]
# }