Confidence Interval for the Proportion of Variance Accounted for (in the Dependent Variable by Knowing the Levels of the Factor)

Computes the exact confidence limits for the proportion of variance in the dependent variable accounted for by knowing the levels of the factor (group status in a single factor design) in a fixed effects analysis of variance, so an omnibus F-test is accompanied by an effect size with a statement of its precision.

Usage

ci_pvaf(
  F_value = NULL,
  df_1 = NULL,
  df_2 = NULL,
  N = NULL,
  conf_level = 0.95,
  alpha_lower = NULL,
  alpha_upper = NULL,
  ...
)

Arguments

F_value

Observed F-value from fixed effects analysis of variance

df_1

Numerator degrees of freedom

df_2

Denominator degrees of freedom

N

Sample size

conf_level

Confidence interval coverage (i.e., 1-Type I error rate); default is .95

alpha_lower

Type I error for the lower confidence limit

alpha_upper

Type I error for the upper confidence limit

...

Allows one to potentially include parameter values for inner functions

Value

A 4-row data.frame with columns term, value, prob_less, and prob_greater. The term values are "lower_limit" (the lower confidence limit on the proportion of variance accounted for, on the [0, 1] scale), "pvaf" (the sample proportion of variance accounted for, df_1 * F_value / (df_1 * F_value + df_2), the same value that eta squared reports, so the point estimate sits between its confidence limits), "upper_limit" (the upper confidence limit), and "actual_coverage" (the achieved coverage probability, which equals conf_level when both tail targets are met). The prob_less and prob_greater columns report the achieved tail-error probabilities at the two limits; NA on the "pvaf" and "actual_coverage" rows.

Details

The confidence level must be specified in one of following two ways: using confidence interval coverage (conf_level), or lower and upper confidence limits (alpha_lower and alpha_upper).

This function uses the confidence interval transformation principle (Steiger, 2004) to transform the confidence limits for the noncentrality parameter to the confidence limits for the population proportion of variance accounted for by knowing the group status. The confidence interval for the noncentral F parameter can be obtained from the function conf_limits_ncf, which is used within this function.

Note

This function can be used for single or factorial ANOVA designs.

References

Fleishman, A. I. (1980). Confidence intervals for correlation ratios. Educational and Psychological Measurement, 40(3), 659–670.

Kelley, K. (2007). Confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20(8), 1–24. doi:10.18637/jss.v020.i08

Kelley, K. (2008). Sample size planning for the squared multiple correlation coefficient: Accuracy in parameter estimation via narrow confidence intervals. Multivariate Behavioral Research, 43, 524–555. doi:10.1080/00273170802490632

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 3 on \(R^2\) as a model comparison effect size.)

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

See also

conf_limits_ncf

Other confidence intervals for effect sizes: ci_R2(), ci_c(), ci_c_ancova(), ci_c_ancova_bp(), ci_correlation, ci_cv(), ci_eta_squared(), ci_eta_squared_generalized(), ci_eta_squared_partial(), ci_mahalanobis(), ci_omega_squared(), ci_rc(), ci_reg_coef(), ci_rmsea(), ci_sc(), ci_sc_ancova(), ci_sm(), ci_smd(), ci_smd_c(), ci_snr(), ci_src(), ci_srsnr(), contrast_adjusted(), plot_smd()

Author

Ken Kelley kkelley@nd.edu

Examples

## Bargman (1970) gave an example in which a 5-group ANOVA with 11 subjects in each
## group is conducted and the observed F value is 11.221. This example was used
## in Venables (1975),  Fleishman (1980), and Steiger (2004). If one wants to calculate the
## exact confidence interval for the proportion of variance accounted for in that example,
## this function can be used.
ci_pvaf(F_value = 11.221, df_1 = 4, df_2 = 50, N = 55)
#>  term            value prob_less prob_greater
#>  lower_limit     0.226 0.025     0.975       
#>  pvaf            0.473 <NA>      <NA>        
#>  upper_limit     0.587 0.975     0.025       
#>  actual_coverage 0.95  <NA>      <NA>        
#> 
#> Confidence level: 95%

ci_pvaf(F_value = 11.221, df_1 = 4, df_2 = 50, N = 55, conf_level = .90)
#>  term            value prob_less prob_greater
#>  lower_limit     0.261 0.05      0.95        
#>  pvaf            0.473 <NA>      <NA>        
#>  upper_limit     0.565 0.95      0.05        
#>  actual_coverage 0.9   <NA>      <NA>        
#> 
#> Confidence level: 90%

ci_pvaf(F_value = 11.221, df_1 = 4, df_2 = 50, N = 55, alpha_lower = 0, alpha_upper = .05)
#>  term            value prob_less prob_greater
#>  lower_limit     0     0         1           
#>  pvaf            0.473 <NA>      <NA>        
#>  upper_limit     0.565 0.95      0.05        
#>  actual_coverage 0.95  <NA>      <NA>        
#> 
#> Confidence level: 95%