More information regarding Confidence (Compatibiity) Intervals and how they are computed in effectsize.

## Confidence (Compatibility) Intervals (CIs)

Unless stated otherwise, confidence (compatibility) intervals (CIs) are estimated using the noncentrality parameter method (also called the "pivot method"). This method finds the noncentrality parameter ("ncp") of a noncentral t, F, or $$\chi^2$$ distribution that places the observed t, F, or $$\chi^2$$ test statistic at the desired probability point of the distribution. For example, if the observed t statistic is 2.0, with 50 degrees of freedom, for which cumulative noncentral t distribution is t = 2.0 the .025 quantile (answer: the noncentral t distribution with ncp = .04)? After estimating these confidence bounds on the ncp, they are converted into the effect size metric to obtain a confidence interval for the effect size (Steiger, 2004).

For additional details on estimation and troubleshooting, see effectsize_CIs.

## CIs and Significance Tests

"Confidence intervals on measures of effect size convey all the information in a hypothesis test, and more." (Steiger, 2004). Confidence (compatibility) intervals and p values are complementary summaries of parameter uncertainty given the observed data. A dichotomous hypothesis test could be performed with either a CI or a p value. The 100 (1 - $$\alpha$$)% confidence interval contains all of the parameter values for which p > $$\alpha$$ for the current data and model. For example, a 95% confidence interval contains all of the values for which p > .05.

Note that a confidence interval including 0 does not indicate that the null (no effect) is true. Rather, it suggests that the observed data together with the model and its assumptions combined do not provided clear evidence against a parameter value of 0 (same as with any other value in the interval), with the level of this evidence defined by the chosen $$\alpha$$ level (Rafi & Greenland, 2020; Schweder & Hjort, 2016; Xie & Singh, 2013). To infer no effect, additional judgments about what parameter values are "close enough" to 0 to be negligible are needed ("equivalence testing"; Bauer & Kiesser, 1996).

## Bootstrapped CIs

Some effect sizes are directionless--they do have a minimum value that would be interpreted as "no effect", but they cannot cross it. For example, a null value of Kendall's W is 0, indicating no difference between groups, but it can never have a negative value. Same goes for U2 and Overlap: the null value of $$U_2$$ is 0.5, but it can never be smaller than 0.5; am Overlap of 1 means "full overlap" (no difference), but it cannot be larger than 1.

When bootstrapping CIs for such effect sizes, the bounds of the CIs will never cross (and often will never cover) the null. Therefore, these CIs should not be used for statistical inference.

## One-Sided CIs

Typically, CIs are constructed as two-tailed intervals, with an equal proportion of the cumulative probability distribution above and below the interval. CIs can also be constructed as one-sided intervals, giving only a lower bound or upper bound. This is analogous to computing a 1-tailed p value or conducting a 1-tailed hypothesis test.

Significance tests conducted using CIs (whether a value is inside the interval) and using p values (whether p < alpha for that value) are only guaranteed to agree when both are constructed using the same number of sides/tails.

Most effect sizes are not bounded by zero (e.g., r, d, g), and as such are generally tested using 2-tailed tests and 2-sided CIs.

Some effect sizes are strictly positive--they do have a minimum value, of 0. For example, $$R^2$$, $$\eta^2$$, and other variance-accounted-for effect sizes, as well as Cramer's V and multiple R, range from 0 to 1. These typically involve F- or $$\chi^2$$-statistics and are generally tested using 1-tailed tests which test whether the estimated effect size is larger than the hypothesized null value (e.g., 0). In order for a CI to yield the same significance decision it must then by a 1-sided CI, estimating only a lower bound. This is the default CI computed by effectsize for these effect sizes, where alternative = "greater" is set.

This lower bound interval indicates the smallest effect size that is not significantly different from the observed effect size. That is, it is the minimum effect size compatible with the observed data, background model assumptions, and $$\alpha$$ level. This type of interval does not indicate a maximum effect size value; anything up to the maximum possible value of the effect size (e.g., 1) is in the interval.

One-sided CIs can also be used to test against a maximum effect size value (e.g., is $$R^2$$ significantly smaller than a perfect correlation of 1.0?) can by setting alternative = "less". This estimates a CI with only an upper bound; anything from the minimum possible value of the effect size (e.g., 0) up to this upper bound is in the interval.

We can also obtain a 2-sided interval by setting alternative = "two.sided". These intervals can be interpreted in the same way as other 2-sided intervals, such as those for r, d, or g.

An alternative approach to aligning significance tests using CIs and 1-tailed p values that can often be found in the literature is to construct a 2-sided CI at a lower confidence level (e.g., 100(1-2$$\alpha$$)% = 100 - 2*5% = 90%. This estimates the lower bound and upper bound for the above 1-sided intervals simultaneously. These intervals are commonly reported when conducting equivalence tests. For example, a 90% 2-sided interval gives the bounds for an equivalence test with $$\alpha$$ = .05. However, be aware that this interval does not give 95% coverage for the underlying effect size parameter value. For that, construct a 95% 2-sided CI.

data("hardlyworking")
fit <- lm(salary ~ n_comps, data = hardlyworking)
eta_squared(fit) # default, ci = 0.95, alternative = "greater"
#> For one-way between subjects designs, partial eta squared is equivalent to eta squared.
#> Returning eta squared.
#> # Effect Size for ANOVA
#>
#> Parameter | Eta2 |       95% CI
#> -------------------------------
#> n_comps   | 0.19 | [0.14, 1.00]
#>
#> - One-sided CIs: upper bound fixed at [1.00].
eta_squared(fit, alternative = "less") # Test is eta is smaller than some value
#> For one-way between subjects designs, partial eta squared is equivalent to eta squared.
#> Returning eta squared.
#> # Effect Size for ANOVA
#>
#> Parameter | Eta2 |       95% CI
#> -------------------------------
#> n_comps   | 0.19 | [0.00, 0.24]
#>
#> - One-sided CIs: lower bound fixed at [0.00].
eta_squared(fit, alternative = "two.sided") # 2-sided bounds for alpha = .05
#> For one-way between subjects designs, partial eta squared is equivalent to eta squared.
#> Returning eta squared.
#> # Effect Size for ANOVA
#>
#> Parameter | Eta2 |       95% CI
#> -------------------------------
#> n_comps   | 0.19 | [0.14, 0.25]
eta_squared(fit, ci = 0.9, alternative = "two.sided") # both 1-sided bounds for alpha = .05
#> For one-way between subjects designs, partial eta squared is equivalent to eta squared.
#> Returning eta squared.
#> # Effect Size for ANOVA
#>
#> Parameter | Eta2 |       90% CI
#> -------------------------------
#> n_comps   | 0.19 | [0.14, 0.24]

## CI Does Not Contain the Estimate

For very large sample sizes or effect 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:

t_to_d(80, df_error = 4555555)
#> d    |       95% CI
#> -------------------
#> 0.07 | [0.08, 0.08]

In these cases, consider an alternative optimizer, or an alternative method for computing CIs, such as the bootstrap.

Bauer, P., & Kieser, M. (1996). A unifying approach for confidence intervals and testing of equivalence and difference. Biometrika, 83(4), 934-–937. doi:10.1093/biomet/83.4.934

Rafi, Z., & Greenland, S. (2020). Semantic and cognitive tools to aid statistical science: Replace confidence and significance by compatibility and surprise. BMC Medical Research Methodology, 20(1), Article 244. doi:10.1186/s12874-020-01105-9

Schweder, T., & Hjort, N. L. (2016). Confidence, likelihood, probability: Statistical inference with confidence distributions. Cambridge University Press. doi:10.1017/CBO9781139046671

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(2), 164--182. doi:10.1037/1082-989x.9.2.164

Xie, M., & Singh, K. (2013). Confidence distribution, the frequentist distribution estimator of a parameter: A review. International Statistical Review, 81(1), 3–-39. doi:10.1111/insr.12000