% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/convert_tF_to_anova.R
\name{F_to_eta2}
\alias{F_to_eta2}
\alias{t_to_eta2}
\alias{F_to_epsilon2}
\alias{t_to_epsilon2}
\alias{F_to_eta2_adj}
\alias{t_to_eta2_adj}
\alias{F_to_omega2}
\alias{t_to_omega2}
\alias{F_to_f}
\alias{t_to_f}
\alias{F_to_f2}
\alias{t_to_f2}
\title{Convert test statistics (F, t) to indices of \strong{partial} variance explained (\strong{partial} Eta / Omega / Epsilon squared and Cohen's f)}
\usage{
F_to_eta2(f, df, df_error, ci = 0.9, ...)

t_to_eta2(t, df_error, ci = 0.9, ...)

F_to_epsilon2(f, df, df_error, ci = 0.9, ...)

t_to_epsilon2(t, df_error, ci = 0.9, ...)

F_to_eta2_adj(f, df, df_error, ci = 0.9, ...)

t_to_eta2_adj(t, df_error, ci = 0.9, ...)

F_to_omega2(f, df, df_error, ci = 0.9, ...)

t_to_omega2(t, df_error, ci = 0.9, ...)

F_to_f(f, df, df_error, ci = 0.9, squared = FALSE, ...)

t_to_f(t, df_error, ci = 0.9, squared = FALSE, ...)

F_to_f2(f, df, df_error, ci = 0.9, squared = TRUE, ...)

t_to_f2(t, df_error, ci = 0.9, squared = TRUE, ...)
}
\arguments{
\item{df, df_error}{Degrees of freedom of numerator or of the error estimate (i.e., the residuals).}

\item{ci}{Confidence Interval (CI) level}

\item{...}{Arguments passed to or from other methods.}

\item{t, f}{The t or the F statistics.}

\item{squared}{Return Cohen's \emph{f} or Cohen's \emph{f}-squared?}
}
\value{
A data frame with the effect size(s) between 0-1 (\code{Eta2_partial},
\code{Epsilon2_partial}, \code{Omega2_partial}, \code{Cohens_f_partial} or
\code{Cohens_f2_partial}), and their CIs (\code{CI_low} and \code{CI_high}). (Note that
for \eqn{\omega_p^2} and \eqn{\epsilon_p^2} it is possible to compute a
negative number; even though this doesn't make any practical sense, it is
recommended to report the negative number and not a 0).
}
\description{
These functions are convenience functions to convert F and t test statistics to
\strong{partial} Eta squared, (\eqn{\eta{_p}^2}), Omega squared (\eqn{\omega{_p}^2}),
Epsilon squared (\eqn{\epsilon{_p}^2}; an alias for the adjusted Eta squared) and Cohen's f.
These are useful in cases where the various Sum of Squares and Mean Squares are not
easily available or their computation is not straightforward (e.g., in liner mixed models,
contrasts, etc.). For test statistics derived from \code{lm} and \code{aov} models, these
functions give exact results. For all other cases, they return close approximations.
\cr
See \href{https://easystats.github.io/effectsize/articles/from_test_statistics.html}{Effect Size from Test Statistics vignette.}
}
\details{
These functions use the following formulae:
\cr
\deqn{\eta_p^2 = \frac{F \times df_{num}}{F \times df_{num} + df_{den}}}
\cr
\deqn{\epsilon_p^2 = \frac{(F - 1) \times df_{num}}{F \times df_{num} + df_{den}}}
\cr
\deqn{\omega_p^2 = \frac{(F - 1) \times df_{num}}{F \times df_{num} + df_{den} + 1}}
\cr
\deqn{f_p = \sqrt{\frac{\eta_p^2}{1-\eta_p^2}}}
\cr\cr
For \eqn{t}, the conversion is based on the equality of \eqn{t^2 = F} when \eqn{df_{num}=1}.
\subsection{Choosing an Un-Biased Estimate}{

Both Omega and Epsilon are unbiased estimators of the population Eta. But
which to choose? Though Omega is the more popular choice, it should be noted
that:
\enumerate{
\item The formula given above for Omega is only an approximation for complex
designs.
\item Epsilon has been found to be less biased (Carroll & Nordholm, 1975).
}
}
}
\note{
\eqn{Adj. \eta_p^2} is an alias for \eqn{\epsilon_p^2}.
}
\section{Confidence Intervals}{

Unless stated otherwise, confidence intervals are estimated using the
Noncentrality parameter method; These methods searches for a the best
non-central parameters (\code{ncp}s) of the noncentral t-, F- or Chi-squared
distribution for the desired tail-probabilities, and then convert these
\code{ncp}s to the corresponding effect sizes. (See full \link{effectsize-CIs} for
more.)
}

\section{CI Contains Zero}{

For positive only effect sizes (Eta squared, Cramer's V, etc.; Effect sizes
associated with Chi-squared and F distributions), special care should be
taken when interpreting CIs with a lower bound equal to 0, and even more care
should be taken when the \emph{upper} bound is equal to 0 (Steiger, 2004; Morey et
al., 2016). For example:\if{html}{\out{<div class="r">}}\preformatted{eta_squared(aov(mpg ~ factor(gear) + factor(cyl), mtcars[1:7, ]))
}\if{html}{\out{</div>}}\preformatted{## Parameter    | Eta2 (partial) |       90\% CI
## --------------------------------------------
## factor(gear) |           0.58 | [0.00, 0.84]
## factor(cyl)  |           0.46 | [0.00, 0.78]
}
}

\examples{
\donttest{
if (require("afex")) {
  data(md_12.1)
  aov_ez("id", "rt", md_12.1,
    within = c("angle", "noise"),
    anova_table = list(correction = "none", es = "pes")
  )
}
# compare to:
(etas <- F_to_eta2(
  f = c(40.72, 33.77, 45.31),
  df = c(2, 1, 2),
  df_error = c(18, 9, 18)
))

if (require(see)) plot(etas)


if (require("lmerTest")) { # for the df_error
  fit <- lmer(extra ~ group + (1 | ID), sleep)
  # anova(fit)
  # #> Type III Analysis of Variance Table with Satterthwaite's method
  # #>       Sum Sq Mean Sq NumDF DenDF F value   Pr(>F)
  # #> group 12.482  12.482     1     9  16.501 0.002833 **
  # #> ---
  # #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

  F_to_eta2(16.501, 1, 9)
  F_to_omega2(16.501, 1, 9)
  F_to_epsilon2(16.501, 1, 9)
  F_to_f(16.501, 1, 9)
}


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

  jt <- joint_tests(warp.lm, by = "wool")
  F_to_eta2(jt$F.ratio, jt$df1, jt$df2)
}
}
}
\references{
\itemize{
\item Albers, C., & Lakens, D. (2018). When power analyses based on pilot data are biased: Inaccurate effect size estimators and follow-up bias. Journal of experimental social psychology, 74, 187-195. \doi{10.31234/osf.io/b7z4q}
\item Carroll, R. M., & Nordholm, L. A. (1975). Sampling Characteristics of Kelley's epsilon and Hays' omega. Educational and Psychological Measurement, 35(3), 541-554.
\item 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.
\item 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}
\item Mordkoff, J. T. (2019). A Simple Method for Removing Bias From a Popular Measure of Standardized Effect Size: Adjusted Partial Eta Squared. Advances in Methods and Practices in Psychological Science, 2(3), 228-232. \doi{10.1177/2515245919855053}
\item Morey, R. D., Hoekstra, R., Rouder, J. N., Lee, M. D., & Wagenmakers, E. J. (2016). The fallacy of placing confidence in confidence intervals. Psychonomic bulletin & review, 23(1), 103-123.
\item 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.
}
}
\seealso{
\code{\link[=eta_squared]{eta_squared()}} for more details.

Other effect size from test statistic: 
\code{\link{chisq_to_phi}()},
\code{\link{t_to_d}()}
}
\concept{effect size from test statistic}
