t_to_d function - RDocumentation (2024)


These functions are convenience functions to convert t, z and F teststatistics to Cohen's d and partial r. These are useful in cases wherethe data required to compute these are not easily available or theircomputation 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, 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", ...)


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


t, f, z

The t, the F or the z statistics.


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


Confidence Interval (CI) level


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.


The number of observations (the sample size).

df, df_error

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

Confidence (Compatibility) Intervals (CIs)

Unless stated otherwise, confidence (compatibility) intervals (CIs) areestimated using the noncentrality parameter method (also called the "pivotmethod"). This method finds the noncentrality parameter ("ncp") of anoncentral t, F, or \(\chi^2\) distribution that places the observedt, F, or \(\chi^2\) test statistic at the desired probability point ofthe distribution. For example, if the observed t statistic is 2.0, with 50degrees 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 areconverted into the effect size metric to obtain a confidence interval for theeffect 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 informationin a hypothesis test, and more." (Steiger, 2004). Confidence (compatibility)intervals and p values are complementary summaries of parameter uncertaintygiven the observed data. A dichotomous hypothesis test could be performedwith either a CI or a p value. The 100 (1 - \(\alpha\))% confidenceinterval contains all of the parameter values for which p > \(\alpha\)for the current data and model. For example, a 95% confidence intervalcontains 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 withthe model and its assumptions combined do not provided clear evidence againsta parameter value of 0 (same as with any other value in the interval), withthe level of this evidence defined by the chosen \(\alpha\) level (Rafi &Greenland, 2020; Schweder & Hjort, 2016; Xie & Singh, 2013). To infer noeffect, additional judgments about what parameter values are "close enough"to 0 to be negligible are needed ("equivalence testing"; Bauer & Kiesser,1996).

Plotting with <code>see</code>

The see package contains relevant plotting functions. See the plotting vignette in the see package.


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, andassumes two equal group sizes. When possible, it is advised to directlyestimate Cohen's d, with cohens_d(), emmeans::eff_size(), or similarfunctions.


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

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

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

  • Steiger, J. H. (2004). Beyond the F test: Effect size confidence intervalsand 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, andcalculation of confidence intervals that are based on central and noncentraldistributions. Educational and Psychological Measurement, 61(4), 532-574.

See Also


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


Run this code

## t Testsres <- t.test(1:10, y = c(7:20), var.equal = TRUE)t_to_d(t = res$statistic, res$parameter)t_to_r(t = res$statistic, res$parameter)t_to_r(t = res$statistic, res$parameter, alternative = "less")res <- with(sleep, t.test(extra[group == 1], extra[group == 2], paired = TRUE))t_to_d(t = res$statistic, res$parameter, paired = TRUE)t_to_r(t = res$statistic, res$parameter)t_to_r(t = res$statistic, res$parameter, alternative = "greater")if (FALSE) { # require(correlation)## Linear Regressionmodel <- lm(rating ~ complaints + critical, data = attitude)(param_tab <- parameters::model_parameters(model))(rs <- t_to_r(param_tab$t[2:3], param_tab$df_error[2:3]))# How does this compare to actual partial correlations?correlation::correlation(attitude, select = "rating", select2 = c("complaints", "critical"), partial = TRUE)}

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t_to_d function - RDocumentation (2024)
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