= A guide to obtaining confidence intervals for effect sizes =

Effect sizes, specify the magnitude of a statistical comparison. However, this does not tell us how precisely it is measured.

There are [[https://thenewstatistics.com/itns/esci/|details by Geoff Cumming]], Keselman et al (2008) and  Steiger (2004) on combining these concepts by giving confidence intervals for effect sizes.
Michael Smithson [[https://www.anu.edu.au/psychology/people/smithson/details/CIstuff/CI.html|has syntax]] in SPSS and other statistical software to do the computations. There are also some [[https://www.anu.edu.au/psychology/people/smithson/details/CIstuff/Noncoht2.pdf|workshop notes]] to explain what's going on. See also Smithson (2001).

This [[https://www.danielsoper.com/statcalc/calculator.aspx?id=75|on-line calculator]] will work out CIs for f-squareds which can be converted into R-squareds using the calculator 
[[https://www.danielsoper.com/statcalc/calculator.aspx?id=76|here]] which uses the formula of R-squared = f-squared/(1+f-squared). This can be also done in R using the '''CI.Rsq''' function which uses the standard error for R-squared equal to sqrt((4*rsq*(1-rsq)^2 ^*(n-k-1)^2 ^)/((n^2 ^-1)*(n+3))) for a R-squared (rsq) with k predictors and a sample size of n. 

There is also [[http://core.ecu.edu/psyc/wuenschk/SPSS/SPSS-Programs.htm|SPSS syntax]], with an example, for obtaining a confidence interval for Cohen's d and [[FAQ/Rcis| R syntax]] for an alternative more robust nonparametric bootstrap estimate (see for example Keselman et al (2008)) for confidence intervals for Cohens' d and correlations which come from non-Normal distributions. 

Keselman et al (2008) have some [[http://www.apa.org/journals/supplemental/met_13_2_110/met_13_2_110_supp.html|SAS V9.1 code with examples]] to produce bootstrap confidence intervals for effect sizes (ie based on repeated random sampling) for a robust (winsorised) version of Cohen's d in mixed anovas (replacing the lowest and highest 20% of outcome data by their respective least extreme values. This paper is available free to CBSUers using the APA internet link.

* [[FAQ/cdse|A suggestion for directly computing the s.e. of Cohen's d for use in constructing confidence intervals]]

Bird (2022) gives formulae for the comparison of pairs of standardised differences in group means.

__References__


Bird, KD (2022, April 11). Improved Confidence Intervals for Differences Between Standardized Effect Sizes. ''Psychological Methods''. Advance online publication. http://dx.doi.org/10.1037/met0000494

Fidler, F and Thompson, B (2001) Computing correct confidence intervals for ANOVA Fixed- and random-effects effect sizes. ''Educational and Psychological Measurement'' '''61''' 575-604. CIs computed using step-by-step standard output from ANOVAs.

Keselman, HJ, Algina, J, Lix, LM, Wilcox, RR, Deering, KN (2008) A generally robust approach for testing hypotheses and setting confidence intervals for effect sizes ''Psychological Methods'' '''13(2)''' 110-129.

Smithson, M (2001) Correct confidence intervals for various regression effect sizes and parameters: the importance of noncentral distributions in computing intervals. ''Educational and Psychological Measurement'' '''61''' 605-632.

Steiger, JH (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(2)''' 164-182.