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Confidence Interval for the Square Root of the Signal-to-Noise Ratio

Source: R/ci_srsnr.R
ci_srsnr.Rd

Computes the exact confidence interval for the square root of the signal-to-noise ratio, the standard deviation of the group means relative to the error standard deviation. On this root scale the quantity is an effect size in standard deviation units, the multi-group analog of the standardized mean difference.

Usage

ci_srsnr(
  F_value = NULL,
  df_1 = NULL,
  df_2 = NULL,
  N = NULL,
  means = NULL,
  sigma_squared = NULL,
  n_per_group = NULL,
  conf_level = 0.95,
  alpha_lower = NULL,
  alpha_upper = NULL,
  ...
)

Arguments

F_value

Observed F-value from the analysis of variance. Use this argument when re-analyzing existing data.

df_1

Numerator degrees of freedom

df_2

Denominator degrees of freedom

N

Sample size

means

Numeric vector of population or hypothesized group means. Supply together with sigma_squared and n_per_group as a design-stage alternative to F_value: the function then computes the F-value implied by these design parameters and proceeds with the same noncentral F machinery.

sigma_squared

The within-group variance. Used with means.

n_per_group

A single per-group sample size, or a vector of per-group sample sizes the same length as means. Used with means.

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 2-row data.frame with columns term and value. The term values are "lower_limit" and "upper_limit", giving the square roots of the corresponding signal-to-noise-ratio confidence limits.

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).

The square root of the signal-to-noise ratio is defined as the standard deviation due to the particular factor over the standard deviation of the error (i.e., the square root of the mean square error). This function uses the confidence interval transformation principle (Steiger, 2004) to transform the confidence limits for the noncentrality parameter to the confidence limits for square root of signal-to-noise ratio. The confidence interval for noncentral F parameter can be obtained from function conf_limits_ncf in DMAR.

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

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

ci_snr, 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_pvaf(), ci_rc(), ci_reg_coef(), ci_rmsea(), ci_sc(), ci_sc_ancova(), ci_sm(), ci_smd(), ci_smd_c(), ci_snr(), ci_src(), contrast_adjusted(), plot_smd()

Author

Ken Kelley kkelley@nd.edu

Examples

## To illustrate the calculation of the confidence interval for noncentral
## F parameter,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 continued to be used in Venables (1975),  Fleishman (1980),
## and Steiger (2004). If one wants to calculate the exact confidence interval
## for square root of the signal-to-noise ratio of that example, this
## function can be used.

ci_srsnr(F_value = 11.221, df_1 = 4, df_2 = 50, N = 55)
#>  term        value
#>  lower_limit 0.541
#>  upper_limit 1.19 
#> 
#> Confidence level: 95%

ci_srsnr(F_value = 11.221, df_1 = 4, df_2 = 50, N = 55, conf_level = .90)
#>  term        value
#>  lower_limit 0.594
#>  upper_limit 1.14 
#> 
#> Confidence level: 90%

ci_srsnr(F_value = 11.221, df_1 = 4, df_2 = 50, N = 55, alpha_lower = .02, alpha_upper = .03)
#>  term        value
#>  lower_limit 0.525
#>  upper_limit 1.18 
#> 
#> Confidence level: 95%

# Design-stage call with population means + within-group variance + n.
# Useful when planning a study, before data are observed: derives the
# implied F-value internally and returns the resulting CI on the square
# root of the signal-to-noise ratio.
ci_srsnr(means = c(94, 91, 92, 83), sigma_squared = 67.375, n_per_group = 6)
#> Warning: The observed F_value is below the alpha_lower critical value of the central F-distribution, so the lower confidence limit on the square root of the signal-to-noise ratio is 0.
#>  term        value
#>  lower_limit 0    
#>  upper_limit 0.874
#> 
#> Confidence level: 95%