Confidence Intervals
More information regarding Confidence Intervals and how they are computed in
effectsize.
Unless stated otherwise, confidence intervals are estimated using the
Noncentrality parameter method; These methods searches for a the best
non-central parameters (ncps) of the noncentral t-, F- or Chi-squared
distribution for the desired tail-probabilities, and then convert these
ncps to the corresponding effect sizes. (See full effectsize-CIs for
more.)
For positive only effect sizes (Eta squared, Cramer's V, etc.; Effect sizes associated with Chi-squared and F distributions), special care should be taken when interpreting CIs with a lower bound equal to 0, and even more care should be taken when the upper bound is equal to 0 (Steiger, 2004; Morey et al., 2016). For example:
eta_squared(aov(mpg ~ factor(gear) + factor(cyl), mtcars[1:7, ]))
## Parameter | Eta2 (partial) | 90% CI ## -------------------------------------------- ## factor(gear) | 0.58 | [0.00, 0.84] ## factor(cyl) | 0.46 | [0.00, 0.78]
For very large sample sizes, the width of the CI can be smaller than the tolerance of the optimizer, resulting in CIs of width 0. This can also, result in the estimated CIs excluding the point estimate. For example:
chisq_to_cramers_v(13223.73, n = 76227, nrow = 6, ncol = 1)
## Cramer's V | 95% CI ## ------------------------- ## 0.19 | [0.20, 0.20]
t_to_d(80, df_error = 4555555)
## d | 95% CI ## ------------------- ## 0.07 | [0.08, 0.08]
Morey, R. D., Hoekstra, R., Rouder, J. N., Lee, M. D., & Wagenmakers, E. J. (2016). The fallacy of placing confidence in confidence intervals. Psychonomic bulletin & review, 23(1), 103-123.
Steiger, J. H. (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, 164-182.
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