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## Rules of thumb on magnitudes of effect sizes

The scales of magnitude are taken from Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Lawrence Erlbaum Associates (see also here). The scales of magnitude for partial $$\omega^text{2}$$ are taken from Table 2.2 of Murphy and Myors (2004).

There is also a table of effect size magnitudes at the back of Kotrlik JW and Williams HA (2003) here. An overview of commonly used effect sizes in psychology is given by Vacha-Haase and Thompson (2004). Whitehead, Julious, Cooper and Campbell (2015) also suggest Cohen's rules of thumb for Cohen's d when comparing two independent groups with an additional suggestion of a d < 0.1 corresponding to a very small effect.

Kraemer and Thiemann (1987, p.54 and 55) use the same effect size values (which they call delta) for both intra-class correlations and Pearson correlations. This implies the below rules of thumb from Cohen (1988) for magnitudes of effect sizes for Pearson correlations could also be used for intra-class correlations. It should be noted, however, that the intra-class correlation is computed from a repeated measures ANOVA whose usual effect size (given below) is partial eta-squared. In addition, Shrout and Fleiss (1979) discuss different types of intra-class correlation coefficient and how their magnitudes can differ.

The general rules of thumb given by Cohen and Miles & Shevlin (2001) are for eta-squared, which uses the total sum of squares in the denominator, but these would arguably apply more to partial eta-squared than to eta-squared. This is because partial eta-squared in factorial ANOVA arguably more closely approximates what eta-squared would have been for the factor had it been a one-way ANOVA and it is presumably a one-way ANOVA which gave rise to Cohen's rules of thumb.

 Effect Size Use Small Medium Large Correlation inc Phi 0.1 0.3 0.5 r x c frequency tables 0.1 (Min(r-1,c-1)=1), 0.07 (Min(r-1,c-1)=2), 0.06 (Min(r-1,c-1)=3) 0.3 (Min(r-1,c-1)=1), 0.21 (Min(r-1,c-1)=2), 0.17 (Min(r-1,c-1)=3) 0.5 (Min(r-1,c-1)=1), 0.35(Min(r-1,c-1)=2), 0.29 (Min(r-1,c-1)=3) Comparing two proportions 0.2 0.5 0.8 $$\eta2$$ Anova 0.01 0.06 0.14 Anova; See Field (2013) 0.01 0.06 0.14 one-way MANOVA 0.01 0.06 0.14 Cohen's f one-way an(c)ova (regression) 0.10 0.25 0.40 $$\eta2$$ Multiple regression 0.02 0.13 0.26 $$\kappa2$$ Mediation analysis 0.01 0.09 0.25 Cohen's f Multiple Regression 0.14 0.39 0.59 Cohen's d t-tests 0.2 0.5 0.8 Cohen's $$\omega$$ chi-square 0.1 0.3 0.5 Odds Ratios 2 by 2 tables 1.5 3.5 9.0 Odds Ratios 0.55 0.65 0.75 Friedman test 0.1 0.3 0.5

Also:Haddock et al (1998) state that $$\sqrt{3/\pi}$$ multiplied by the log of the odds ratio is a standardised difference equivalent to Cohen's d.

Definitions

For two-sample t-tests Cohen's d = (difference between a pair of group means) / (averaged group standard deviation) = t Sqrt [(1/n1) + (1/n2)] (Pustejovsky (2014), p.95 and Borenstein (2009), Table 12.1)

For a one-sample t-test Cohen's d = difference between the mean and its expected value / standard deviation = t / Sqrt(n) for n subjects in each group. Cohen's d also equals t / Sqrt(n) in a paired t-test (Rosenthal, 1991) since t / Sqrt(n) = difference between two means / standard deviation of the difference and the t-test on the difference score is regarded as a special case of a one-sample t-test. Dunlap, Cortina, Vaslow and Burke (1996) suggest in their equation (3) using an alternative transformation of the paired t statistic to obtain d = t Sqrt[2(1-r)/n] for n subjects and a correlation, r, between the paired responses. They argue their estimator of d is preferred over Rosenthal's since it adjusts Cohen's d for the correlation resulting from the paired design. They do conclude, however, that for sample sizes of less than 50 the differences between the two effect size estimates for Cohen's d are 'quite small and trivial'.

Hedges and Olkin (2016) give easy to compute formulae to rescale Cohen's d to yield the proportion (and its variance) of observations in the treatment group which are higher than the control group mean. They do not, however, assess its robustness to distributional assumptions.

$$\eta2$$ = SS(effect) / [ Sum of SS(effects having the same error term as effect of interest) + SS(the error associated with these effects) ]

Cohen's f = Square Root of eta-squared / (1-eta-squared)

From here one can work out $$\eta2$$ from a F ratio in a one-way ANOVA since

$$\eta2$$ = (k-1)/(N-k) F

There is also a $$\mbox{Partial } \eta2$$ = SS(effect) / [ SS(effect) + SS(error for that effect) ]

Multivariate $$\eta^text{2}$$ = 1 - $$\Lambda1/s$$ where $$\Lambda$$ is Wilk's lambda and s is equal to the number of levels of the factor minus 1 or the number of dependent variables, whichever is the smaller (See Green et al (1997)). It may be interpreted as a partial eta-squared.

$$\kappa2$$ = ab / (Maximum value of ab) where a and b are the regression coefficients representing the independent variable to mediator effect and the mediator to outcome respectively to estimate the indirect effect of IV on outcome. See Preacher and Kelley (2011) for further details including MBESS procedure software for fitting this in R. There is also an on-line calculator for working out $$\kappa2$$ here. For further details on mediation analysis see also here. Field (2013) also refers to this measure. Wen and Fan (2015) suggest limitations in using $$\kappa2$$ and instead suggest using ab/c. where c is the sum of indirect effect (ab) and direct effect (c') using the notation in Preacher and Kelley's paper.

Suggestion : Use the square of a Pearson correlation for effect sizes for partial $$\eta2$$ (R-squared in a multiple regression) giving 0.01 (small), 0.09 (medium) and 0.25 (large) which are intuitively larger values than eta-squared. Further to this Cohen, Cohen, West and Aiken (2003) on page 95 of Applied Multiple Regression/Correlation Analysis for the behavioral Sciences third edition for looking at semi-partial effects of single predictors in a regression rather than an overall model R-squared ie looking at sqrt(change in R-squared) from a model with and without the regressor and using the Pearson correlations as a rule of thumb for effect sizes.

Cohen's $$\omega2$$ = Sum over all the groups $$(\mbox{(observed proportion - expected proportion)}2)$$ / (expected proportion)

References

Borenstein, M (2009) Effect sizes for continuous data. In H. Cooper, L. V.Hedges, & J. C. Valentine (Eds.), The handbook of research synthesis and meta-analysis (2nd ed., pp. 221–235). Sage Foundation:New York, NY.

Cohen, J (1988) Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Erlbaum.

Cohen, J, Cohen, P, West, SG and Aiken, LS (2003) Applied multiple regression/correlation analysis for the behavioral sciences. Third Edition. Routledge:New York.

Dunlap, WP, Cortina, JM, Vaslow, JB and Burke, MJ (1996). Meta-Analysis of Experiments With Matched Groups or Repeated Measures Designs. Psychological Methods 1(2) 170-177.

Field, A (2013) Discovering statistics using IBM SPSS Statistics. Fourth Edition. Sage:London.

Green, SB, Salkind, NJ & Akey, TM (1997). Using SPSS for Windows:Analyzing and understanding data. Upper Saddle River, NJ:

Haddock, CK, Rinkdskopf, D. & Shadish, C. (1998) Using odds ratios as effect sizes for meta-analysis of dichotomous data: A primer on methods and issues. Psychological Methods 3 339-353.

Hedges, LV and Olkin I (2016) Overlap Between Treatment and Control Distributions as an Effect Size Measure in Experiments. Psychological Methods 21(1) 61–68

Kotrlik, JW and Williams, HA (2003) The incorporation of effect size in information technology, learning, and performance research. Information Techology, Learning, and Performance Journal 21(1) 1-7.

Jeon, M and De Boeck, P (2017) Decision qualities of Bayes factor and p value-based hypothesis testing. Psychological Methods 22(2) 340-360.

Kraemer, HC and Thiemann, S (1987) How many subjects? Statistical power analysis in research. Sage:London. In CBSU library.

Miles, J and Shevlin, M (2001) Applying Regression and Correlation: A Guide for Students and Researchers. Sage:London.

Murphy, KR and Myors, B (2004) Statistical power analysis: A Simple and General Model for Traditional and Modern Hypothesis Tests (2nd ed.). Lawrence Erlbaum, Mahwah NJ. (Alternative rule s of thumb for effect sizes to those from Cohen are given here in Table 2.2).

Preacher, KJ and Kelley, K (2011) Effect size measures for mediation models: quantitative strategies for communicating indirect effects. Psychological Methods 16(2) 93-115.

Pustejovsky JE (2014) Converting From d to r to z When the Design Uses Extreme Groups, Dichotomization, or Experimental Control. Psychological Methods 19(1) 92-112. This reference also gives several useful formulae for variances of effect sizes such as d and also on how to convert d to a Pearson r.

Rosenthal R (1991). Meta-analytic procedures for social research. Sage:Newbury Park, CA.

Shrout, PE and Fleiss, JL (1979) Intraclass Correlations: Uses in Assessing Rater Reliability, Psychological Bulletin, 86 (2) 420-428. (A good primer showing how anova output can be used to compute ICCs).

Tabachnick, BG and Fidell, LS (2007) Using multivariate statistics. Fifth Edition. Pearson Education:London.

Vacha-Haase, T and Thompson, B (2004) How to estimate and interpret various effect ssizes. Journal of Counseling Psychology 51(4) 473-481.

Wen, Z. and Fan, X. (2015) Monotonicity of Effect Sizes: Questioning Kappa-Squared as Mediation Effect Size Measure. Psychological Methods 20(2) 193-203.

Whitehead, A. L., Julious, S. A., Cooper, C. L. and Campbell, M. J. (2015) Estimating the sample size for a pilot randomised trial to minimise the overall trial sample size for the external pilot and main trial for a continuous outcome variable. Stat Methods Med Res. (Available to read for free on-line here).

None: FAQ/effectSize (last edited 2018-12-04 11:50:31 by PeterWatson)