Convert t, z, and F to Cohen's d or partial-r

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.

Usage

t_to_d(t, df_error, paired = FALSE, ci = 0.95, alternative = "two.sided", ...)

z_to_d(z, n, paired = FALSE, ci = 0.95, alternative = "two.sided", ...)

F_to_d(
  f,
  df,
  df_error,
  paired = FALSE,
  ci = 0.95,
  alternative = "two.sided",
  ...
)

t_to_r(t, df_error, ci = 0.95, alternative = "two.sided", ...)

z_to_r(z, n, ci = 0.95, alternative = "two.sided", ...)

F_to_r(f, df, df_error, ci = 0.95, alternative = "two.sided", ...)

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
alternative: a character string specifying the alternative hypothesis; Controls the type of CI returned: "two.sided" (default, two-sided CI), "greater" or "less" (one-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.
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 (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

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.

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]
t_to_r(t = res$statistic, res$parameter, alternative = "less")
#> r     |         95% CI
#> ----------------------
#> -0.74 | [-1.00, -0.54]
#> 
#> - One-sided CIs: lower bound fixed at [-1.00].

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]
t_to_r(t = res$statistic, res$parameter, alternative = "greater")
#> r     |        95% CI
#> ---------------------
#> -0.80 | [-0.90, 1.00]
#> 
#> - One-sided CIs: upper bound fixed at [1.00].

## Linear Regression
model <- lm(rating ~ complaints + critical, data = attitude)
(param_tab <- parameters::model_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 
#> 
#> Uncertainty intervals (equal-tailed) and p-values (two-tailed) computed
#>   using a Wald t-distribution approximation.

(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]

# How does this compare to actual partial correlations?
correlation::correlation(attitude,
  select = "rating",
  select2 = c("complaints", "critical"),
  partial = TRUE
)
#> # Correlation Matrix (pearson-method)
#> 
#> Parameter1 | Parameter2 |        r |        95% CI | t(28) |         p
#> ----------------------------------------------------------------------
#> rating     | complaints |     0.82 | [ 0.65, 0.91] |  7.60 | < .001***
#> rating     |   critical | 2.70e-03 | [-0.36, 0.36] |  0.01 | 0.989    
#> 
#> p-value adjustment method: Holm (1979)
#> Observations: 30