R/convert_stat_to_r.R

In easystats/effectsize: Indices of Effect Size

# t -----------------------------------------------------------------------

#' 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.).
#' \cr
#' See [Effect Size from Test Statistics vignette.](https://easystats.github.io/effectsize/articles/from_test_statistics.html)
#'
#' @param t,f,z The t, the F or the z statistics.
#' @param df,df_error Degrees of freedom of numerator or of the error estimate
#'   (i.e., the residuals).
#' @param n The number of observations (the sample size).
#' @param paired Should the estimate account for the t-value being testing the
#'   difference between dependent means?
#' @inheritParams cohens_d
#' @param ... Arguments passed to or from other methods.
#'
#'
#' @return 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*:
#' \cr\cr
#' \deqn{r_{partial} = t / \sqrt{t^2 + df_{error}}}
#' \cr\cr
#' \deqn{r_{partial} = z / \sqrt{z^2 + N}}
#' \cr\cr
#' \deqn{d = 2 * t / \sqrt{df_{error}}}
#' \cr\cr
#' \deqn{d_z = t / \sqrt{df_{error}}}
#' \cr\cr
#' \deqn{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.
#'
#' @inheritSection effectsize_CIs Confidence (Compatibility) Intervals (CIs)
#' @inheritSection effectsize_CIs CIs and Significance Tests
#' @inheritSection print.effectsize_table Plotting with `see`
#'
#' @family effect size from test statistic
#' @seealso [cohens_d()]
#'
#' @examples
#' ## t Tests
#' res <- 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")
#'
#' @examplesIf require(correlation)
#' ## Linear Regression
#' model <- 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
#' )
#'
#' @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.
#'
#' @export
t_to_r <- function(t, df_error,
                   ci = 0.95, alternative = "two.sided",
                   ...) {
  alternative <- .match.alt(alternative)

  res <- data.frame(r = t / sqrt(t^2 + df_error))

  if (.test_ci(ci)) {
    res$CI <- ci
    ci.level <- .adjust_ci(ci, alternative)

    ts <- t(mapply(
      .get_ncp_t,
      t, df_error, ci.level
    ))

    res$CI_low <- ts[, 1] / sqrt(ts[, 1]^2 + df_error)
    res$CI_high <- ts[, 2] / sqrt(ts[, 2]^2 + df_error)

    ci_method <- list(method = "ncp", distribution = "t")
    res <- .limit_ci(res, alternative, -1, 1)
  } else {
    ci_method <- alternative <- NULL
  }

  class(res) <- c("effectsize_table", "see_effectsize_table", class(res))
  attr(res, "ci") <- ci
  attr(res, "ci_method") <- ci_method
  attr(res, "alternative") <- alternative
  return(res)
}

# z -----------------------------------------------------------------------



#' @rdname t_to_r
#' @export
z_to_r <- function(z, n,
                   ci = 0.95, alternative = "two.sided",
                   ...) {
  alternative <- .match.alt(alternative)

  res <- data.frame(r = z / sqrt(z^2 + n))

  if (.test_ci(ci)) {
    res$CI <- ci
    ci.level <- .adjust_ci(ci, alternative)

    alpha <- 1 - ci.level
    probs <- c(alpha / 2, 1 - alpha / 2)

    qs <- stats::qnorm(probs)
    zs <- cbind(qs[1] + z, qs[2] + z)

    res$CI_low <- zs[, 1] / sqrt(zs[, 1]^2 + n)
    res$CI_high <- zs[, 2] / sqrt(zs[, 2]^2 + n)

    ci_method <- list(method = "normal")
    res <- .limit_ci(res, alternative, -1, 1)
  } else {
    ci_method <- alternative <- NULL
  }

  class(res) <- c("effectsize_table", "see_effectsize_table", class(res))
  attr(res, "ci") <- ci
  attr(res, "ci_method") <- ci_method
  attr(res, "alternative") <- alternative
  return(res)
}

# F -----------------------------------------------------------------------

#' @rdname t_to_r
#' @export
F_to_r <- function(f, df, df_error,
                   ci = 0.95, alternative = "two.sided",
                   ...) {
  if (df > 1) {
    insight::format_error("Cannot convert F with more than 1 df to r.")
  }
  t_to_r(sqrt(f), df_error,
    ci = ci, alternative = alternative,
    ...
  )
}