Modified-Large-Sample Confidence Intervals on Variance Components

Computes modified-large-sample (MLS) confidence intervals on the between-group and within-group variance components of a balanced one-way random-effects ANOVA, following Burdick & Graybill (1992). MLS intervals have substantially better coverage than the Satterthwaite or simple-Wald intervals when the components are far from zero, and they are the standard interval method in generalizability theory (Brennan, 2001).

Usage

variance_components_mls(
  ms_between,
  ms_within,
  df_between,
  df_within,
  n,
  conf_level = 0.95
)

Arguments

ms_between

Mean square between groups (numerator of the ANOVA F).

ms_within

Mean square within groups (denominator of the ANOVA F).

df_between

Degrees of freedom for the between-group MS (typically \(a - 1\) for \(a\) groups).

df_within

Degrees of freedom for the within-group MS (typically \(a (n - 1)\) for \(a\) groups of size \(n\)).

n

Number of observations per group (assumed balanced).

conf_level

Confidence level for the CIs. Default 0.95.

Value

A data.frame with rows for the point estimates and MLS lower / upper CIs of the between-group variance component (\(\sigma^2_b\)), the within-group component (\(\sigma^2_w\)), and the implied intraclass correlation (\(\rho = \sigma^2_b / (\sigma^2_b + \sigma^2_w)\)).

Details

Point estimates. For a one-way random-effects ANOVA on \(a\) groups of size \(n\), the method-of-moments estimators are $$\hat\sigma^2_b \;=\; \max(0,\, (\mathit{MS}_b - \mathit{MS}_w)/n), \qquad \hat\sigma^2_w \;=\; \mathit{MS}_w.$$

Modified-large-sample CIs (Burdick-Graybill 1992). The MLS interval for \(\sigma^2_b\) is $$\left[\frac{\mathit{MS}_b - \mathit{MS}_w - \sqrt{V_L}}{n},\;\; \frac{\mathit{MS}_b - \mathit{MS}_w + \sqrt{V_U}}{n}\right],$$ with $$V_L \;=\; G_1^2 \mathit{MS}_b^2 + H_2^2 \mathit{MS}_w^2 + G_{12} \mathit{MS}_b \mathit{MS}_w, \qquad V_U \;=\; H_1^2 \mathit{MS}_b^2 + G_2^2 \mathit{MS}_w^2 + H_{12} \mathit{MS}_b \mathit{MS}_w,$$ where the constants \(G_1\), \(G_2\), \(H_1\), \(H_2\) and the cross-term constants \(G_{12}\), \(H_{12}\) depend on the degrees of freedom and on \(\chi^2\) and F quantiles at the chosen confidence level (Burdick & Graybill, 1992, equations 2.4.1–2.4.5 give the explicit formulas). The lower limit is truncated at zero. For the within-group component, the standard \(\chi^2\)-based CI on \(\mathit{MS}_w\) (Searle, Casella, & McCulloch, 1992) is used.

Caveats. MLS intervals assume balanced data and homogeneous variances within groups. For unbalanced data the appropriate analog is the Burdick-Graybill MLS extension to unequal sample sizes (Burdick & Graybill, 1992, Section 2.5), which is not implemented here.

References

Brennan, R. L. (2001). Generalizability theory. Springer.

Burdick, R. K., & Graybill, F. A. (1992). Confidence intervals on variance components. Marcel Dekker.

Searle, S. R., Casella, G., & McCulloch, C. E. (1992). Variance components. Wiley.

See also

icc, var_icc, ss_aipe_icc

Other agreement and measurement: R2_mixed_effects(), content_validity_index(), gwet_ac(), icc_lmer(), krippendorff_alpha(), limits_of_agreement(), lin_ccc()

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Balanced one-way random-effects ANOVA: a = 10 groups, n = 5.
#        Hypothetical MS_b = 6.0, MS_w = 1.5.
variance_components_mls(ms_between = 6.0, ms_within = 1.5,
                        df_between = 9, df_within = 40, n = 5)
#>  term           value 
#>  sigma2_between 0.9   
#>  sigma2_b_lower 0.236 
#>  sigma2_b_upper 3.69  
#>  sigma2_within  1.5   
#>  sigma2_w_lower 1.01  
#>  sigma2_w_upper 2.46  
#>  icc            0.375 
#>  icc_lower      0.0876
#>  icc_upper      0.785 
#> 
#> Confidence level: 95%