In many real world applications there are no straightforward ways of obtaining standardized effect sizes. However, it is possible to get approximations of most of the effect size indices (\(d\), \(r\), \(\eta^2_p\)…) with the use of test statistics. These conversions are based on the idea that test statistics are a function of effect size and sample size. Thus information about samples size (or more often of degrees of freedom) is used to reverse-engineer indices of effect size from test statistics. This idea and these functions also power our Effect Sizes From Test Statistics shiny app.
The measures discussed here are, in one way or another, signal to noise ratios, with the “noise” representing the unaccounted variance in the outcome variable1.
The indices are:
Percent variance explained (\(\eta^2_p\), \(\omega^2_p\), \(\epsilon^2_p\)).
Measure of association (\(r\)).
Measure of difference (\(d\)).
These measures represent the ratio of \(Signal^2 / (Signal^2 + Noise^2)\), with the “noise” having all other “signals” partial-ed out (be they of other fixed or random effects). The most popular of these indices is \(\eta^2_p\) (Eta; which is equivalent to \(R^2\)).
The conversion of the \(F\)- or \(t\)-statistic is based on Friedman (1982).
Let’s look at an example:
library(afex)
data(md_12.1)
aov_fit <- aov_car(rt ~ angle * noise + Error(id / (angle * noise)),
data = md_12.1,
anova_table = list(correction = "none", es = "pes")
)
aov_fit> Anova Table (Type 3 tests)
>
> Response: rt
> Effect df MSE F pes p.value
> 1 angle 2, 18 3560.00 40.72 *** .819 <.001
> 2 noise 1, 9 8460.00 33.77 *** .790 <.001
> 3 angle:noise 2, 18 1160.00 45.31 *** .834 <.001
> ---
> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '+' 0.1 ' ' 1Let’s compare the \(\eta^2_p\) (the
pes column) obtained here with ones recovered from
F_to_eta2():
library(effectsize)
F_to_eta2(
f = c(40.72, 33.77, 45.31),
df = c(2, 1, 2),
df_error = c(18, 9, 18)
)> Eta2 (partial) | 95% CI
> -----------------------------
> 0.82 | [0.66, 1.00]
> 0.79 | [0.49, 1.00]
> 0.83 | [0.69, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].They are identical!2 (except for the fact
that F_to_eta2() also provides confidence intervals3 :)
In this case we were able to easily obtain the effect size (thanks to
afex!), but in other cases it might not be as easy, and
using estimates based on test statistic offers a good approximation.
For example:
library(emmeans)
joint_tests(aov_fit, by = "noise")> noise = absent:
> model term df1 df2 F.ratio p.value
> angle 2 9 8.000 0.0096
>
> noise = present:
> model term df1 df2 F.ratio p.value
> angle 2 9 51.000 <.0001F_to_eta2(
f = c(5, 79),
df = 2,
df_error = 29
)> Eta2 (partial) | 95% CI
> -----------------------------
> 0.26 | [0.04, 1.00]
> 0.84 | [0.75, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].We can also use t_to_eta2() for contrast analysis:
pairs(emmeans(aov_fit, ~angle))> contrast estimate SE df t.ratio p.value
> X0 - X4 -108 17.4 9 -6.200 0.0004
> X0 - X8 -168 20.6 9 -8.200 0.0001
> X4 - X8 -60 18.4 9 -3.300 0.0244
>
> Results are averaged over the levels of: noise
> P value adjustment: tukey method for comparing a family of 3 estimatest_to_eta2(
t = c(-5.7, -8.9, -3.2),
df_error = 18
)> Eta2 (partial) | 95% CI
> -----------------------------
> 0.64 | [0.39, 1.00]
> 0.81 | [0.66, 1.00]
> 0.36 | [0.09, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].library(lmerTest)
fit_lmm <- lmer(Reaction ~ Days + (Days | Subject), sleepstudy)
anova(fit_lmm)> Type III Analysis of Variance Table with Satterthwaite's method
> Sum Sq Mean Sq NumDF DenDF F value Pr(>F)
> Days 30031 30031 1 17 45.9 3.3e-06 ***
> ---
> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1F_to_eta2(45.8, 1, 17)> Eta2 (partial) | 95% CI
> -----------------------------
> 0.73 | [0.51, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].We can also use t_to_eta2() for the slope of
Days (which in this case gives the same result).
parameters::model_parameters(fit_lmm, effects = "fixed", ci_method = "satterthwaite")> # Fixed Effects
>
> Parameter | Coefficient | SE | 95% CI | t(17.00) | p
> -----------------------------------------------------------------------
> (Intercept) | 251.41 | 6.82 | [237.01, 265.80] | 36.84 | < .001
> Days | 10.47 | 1.55 | [ 7.21, 13.73] | 6.77 | < .001>
> Uncertainty intervals (equal-tailed) and p-values (two-tailed) computed
> using a Wald t-distribution with Satterthwaite approximation.t_to_eta2(6.77, df_error = 17)> Eta2 (partial) | 95% CI
> -----------------------------
> 0.73 | [0.51, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].Alongside \(\eta^2_p\) there are also the less biased \(\omega_p^2\) (Omega) and \(\epsilon^2_p\) (Epsilon; sometimes called \(\text{Adj. }\eta^2_p\), which is equivalent to \(R^2_{adj}\); Albers and Lakens (2018), Mordkoff (2019)).
F_to_eta2(45.8, 1, 17)> Eta2 (partial) | 95% CI
> -----------------------------
> 0.73 | [0.51, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].F_to_epsilon2(45.8, 1, 17)> Epsilon2 (partial) | 95% CI
> ---------------------------------
> 0.71 | [0.48, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].F_to_omega2(45.8, 1, 17)> Omega2 (partial) | 95% CI
> -------------------------------
> 0.70 | [0.47, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].Similar to \(\eta^2_p\), \(r\) is a signal to noise ratio, and is in fact equal to \(\sqrt{\eta^2_p}\) (so it’s really a partial \(r\)). It is often used instead of \(\eta^2_p\) when discussing the strength of association (but I suspect people use it instead of \(\eta^2_p\) because it gives a bigger number, which looks better).
parameters::model_parameters(fit_lmm, effects = "fixed", ci_method = "satterthwaite")> # Fixed Effects
>
> Parameter | Coefficient | SE | 95% CI | t(17.00) | p
> -----------------------------------------------------------------------
> (Intercept) | 251.41 | 6.82 | [237.01, 265.80] | 36.84 | < .001
> Days | 10.47 | 1.55 | [ 7.21, 13.73] | 6.77 | < .001>
> Uncertainty intervals (equal-tailed) and p-values (two-tailed) computed
> using a Wald t-distribution with Satterthwaite approximation.t_to_r(6.77, df_error = 17)> r | 95% CI
> -------------------
> 0.85 | [0.67, 0.92]In a fixed-effect linear model, this returns the partial correlation. Compare:
fit_lm <- lm(rating ~ complaints + critical, data = attitude)
parameters::model_parameters(fit_lm)> 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.t_to_r(
t = c(7.46, 0.01),
df_error = 27
)> r | 95% CI
> ------------------------
> 0.82 | [ 0.67, 0.89]
> 1.92e-03 | [-0.35, 0.35]to:
correlation::correlation(attitude[, c(1, 2, 6)], partial = TRUE)[1:2, c(2, 3, 7, 8)]> Parameter2 | r | t(28)
> -----------------------------
> complaints | 0.82 | 7.60
> critical | 2.70e-03 | 0.01This measure is also sometimes used in contrast analysis, where it is called the point bi-serial correlation - \(r_{pb}\) (Cohen et al. 1965; Rosnow, Rosenthal, and Rubin 2000):
pairs(emmeans(aov_fit, ~angle))> contrast estimate SE df t.ratio p.value
> X0 - X4 -108 17.4 9 -6.200 0.0004
> X0 - X8 -168 20.6 9 -8.200 0.0001
> X4 - X8 -60 18.4 9 -3.300 0.0244
>
> Results are averaged over the levels of: noise
> P value adjustment: tukey method for comparing a family of 3 estimatest_to_r(
t = c(-5.7, -8.9, -3.2),
df_error = 18
)> r | 95% CI
> ----------------------
> -0.80 | [-0.89, -0.57]
> -0.90 | [-0.95, -0.78]
> -0.60 | [-0.79, -0.22]These indices represent \(Signal/Noise\) with the “signal” representing the difference between two means. This is akin to Cohen’s \(d\), and is a close approximation when comparing two groups of equal size (Wolf 1986; Rosnow, Rosenthal, and Rubin 2000).
These can be useful in contrast analyses.
m <- lm(breaks ~ tension, data = warpbreaks)
em_tension <- emmeans(m, ~tension)
pairs(em_tension)> contrast estimate SE df t.ratio p.value
> L - M 10.0 4 51 2.500 0.0400
> L - H 14.7 4 51 3.700 <.0001
> M - H 4.7 4 51 1.200 0.4600
>
> P value adjustment: tukey method for comparing a family of 3 estimatest_to_d(
t = c(2.53, 3.72, 1.20),
df_error = 51
)> d | 95% CI
> --------------------
> 0.71 | [ 0.14, 1.27]
> 1.04 | [ 0.45, 1.62]
> 0.34 | [-0.22, 0.89]However, these are merely approximations of a true Cohen’s
d. It is advised to directly estimate Cohen’s d,
whenever possible. For example, here with
emmeans::eff_size():
eff_size(em_tension, sigma = sigma(m), edf = df.residual(m))> contrast effect.size SE df lower.CL upper.CL
> L - M 0.84 0.34 51 0.15 1.53
> L - H 1.24 0.36 51 0.53 1.95
> M - H 0.40 0.34 51 -0.28 1.07
>
> sigma used for effect sizes: 11.88
> Confidence level used: 0.95pairs(emmeans(aov_fit, ~angle))> contrast estimate SE df t.ratio p.value
> X0 - X4 -108 17.4 9 -6.200 0.0004
> X0 - X8 -168 20.6 9 -8.200 0.0001
> X4 - X8 -60 18.4 9 -3.300 0.0244
>
> Results are averaged over the levels of: noise
> P value adjustment: tukey method for comparing a family of 3 estimatest_to_d(
t = c(-5.7, -5.9, -3.2),
df_error = 18,
paired = TRUE
)> d | 95% CI
> ----------------------
> -1.34 | [-1.97, -0.70]
> -1.39 | [-2.03, -0.74]
> -0.75 | [-1.27, -0.22](Note set paired = TRUE to not over estimate the size of
the effect; Rosenthal (1991); Rosnow, Rosenthal, and Rubin (2000))
Note that for generalized linear models (Poisson, Logistic…), where the outcome is never on an arbitrary scale, estimates themselves are indices of effect size! Thus this vignette is relevant only to general linear models.↩︎
Note that these are partial percent variance explained, and so their sum can be larger than 1.↩︎
Confidence intervals for all indices are estimated using the non-centrality parameter method; These methods search for a the best non-central parameter of the non-central \(F\)/\(t\) distribution for the desired tail-probabilities, and then convert these ncps to the corresponding effect sizes.↩︎