# Effect Size from Test Statistics

# Introduction

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 (\(d\)).

## (Partial) Percent Variance Explained

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 ' ' 1

Let’s 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

    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  <.0001

    F_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 estimates

    t_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].

### In Linear Mixed Models

    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

    F_to_eta2(45.9, 1, 17)

    > 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].

### Bias-Corrected Indices

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(45.9, 1, 17)

    > 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].

    F_to_omega2(45.9, 1, 17)

    > ω² (partial) |       95% CI
    > ---------------------------
    > 0.70         | [0.47, 1.00]
    >
    > - One-sided CIs: upper bound fixed at [1.00].

## Measure of Association

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).

### For Slopes

    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,
      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: 30

### In Contrast Analysis

This 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 estimates

    t_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]

## Measures of Difference

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.

### Between-Subject Contrasts

    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 estimates

    t_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: 0.95

### Within-Subject Contrasts

    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 estimates

    t_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))

# References

Albers, Casper, and Daniël Lakens. 2018. “When Power Analyses Based on Pilot Data Are Biased: Inaccurate Effect Size Estimators and Follow-up Bias.” Journal of Experimental Social Psychology 74: 187–95.
Cohen, Jacob et al. 1965. “Some Statistical Issues in Psychological Research.” Handbook of Clinical Psychology, 95–121.
Friedman, Herbert. 1982. “Simplified Determinations of Statistical Power, Magnitude of Effect and Research Sample Sizes.” Educational and Psychological Measurement 42 (2): 521–26.
Mordkoff, J Toby. 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): 228–32.
Rosenthal, Robert. 1991. “Meta-Analytic Procedures for Social Sciences.” Newbury Park, CA: Sage 10: 9781412984997.
Rosnow, Ralph L, Robert Rosenthal, and Donald B Rubin. 2000. “Contrasts and Correlations in Effect-Size Estimation.” Psychological Science 11 (6): 446–53.
Wolf, Fredric M. 1986. Meta-Analysis: Quantitative Methods for Research Synthesis. Vol. 59. Sage.

1. 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.↩︎

2. Note that these are partial percent variance explained, and so their sum can be larger than 1.↩︎

3. 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.↩︎