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 (\(d\), \(r\), \(\eta^2_p\)â¦) 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 ânoiseâ representing the unaccounted variance in the outcome variable1.
(Partial) Percent Variance ExplainedThese measures represent the ratio of \(Signal^2 / (Signal^2 + Noise^2)\), with the ânoiseâ having all other âsignalsâ 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).
Letâs 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 ' ' 1Letâs compare the \(\eta^2_p\) (the pes column) obtained here with ones recovered from F_to_eta2():
library(effectsize)
options(es.use_symbols = TRUE) # get nice symbols when printing! (On Windows, requires R >= 4.2.0)
F_to_eta2(
f = c(40.72, 33.77, 45.31),
df = c(2, 1, 2),
df_error = c(18, 9, 18)
)> η² (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:
In Simple Effect and Contrast Analysis> Warning: package 'emmeans' was built under R version 4.4.2> Welcome to emmeans.
> Caution: You lose important information if you filter this package's results.
> See '? untidy'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(8, 51),
df = 2,
df_error = 9
)> η² (partial) | 95% CI
> ---------------------------
> 0.64 | [0.18, 1.00]
> 0.92 | [0.78, 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(-6.2, -8.2, -3.2),
df_error = 9
)> η² (partial) | 95% CI
> ---------------------------
> 0.81 | [0.54, 1.00]
> 0.88 | [0.70, 1.00]
> 0.53 | [0.11, 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 ' ' 1> η² (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)> η² (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)).
> η² (partial) | 95% CI
> ---------------------------
> 0.73 | [0.51, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].F_to_epsilon2(45.9, 1, 17)> ε² (partial) | 95% CI
> ---------------------------
> 0.71 | [0.48, 1.00]
>
> - One-sided CIs: upper bound fixed at [1.00].> ϲ (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 itâs 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).
For Slopesparameters::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,
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: 30This 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(-6.2, -8.2, -3.2),
df_error = 9
)> r | 95% CI
> ----------------------
> -0.90 | [-0.95, -0.67]
> -0.94 | [-0.97, -0.80]
> -0.73 | [-0.88, -0.23]These indices represent \(Signal/Noise\) with the âsignalâ representing the difference between two means. This is akin to Cohenâs \(d\), 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.
Between-Subject Contrastsm <- 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 Cohenâs d. It is advised to directly estimate Cohenâs 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(-6.2, -8.2, -3.3),
df_error = 9,
paired = TRUE
)> d | 95% CI
> ----------------------
> -2.07 | [-3.19, -0.91]
> -2.73 | [-4.12, -1.32]
> -1.10 | [-1.90, -0.26](Note set paired = TRUE to not over estimate the size of the effect; Rosenthal (1991); Rosnow, Rosenthal, and Rubin (2000))
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