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Confidence Interval for the Population Root Mean Square Error of Approximation

Source: R/ci_rmsea.R
ci_rmsea.Rd

Constructs a confidence interval for the population root mean square error of approximation (RMSEA), a population badness-of-fit index for structural equation models. The interval is obtained by inverting the noncentral chi square distribution of the sample fit function \(T = (N - 1) \hat F_{ML}\) under the model implied covariance structure, mapping the resulting noncentrality limits to the RMSEA metric (Steiger & Lind, 1980; Browne & Cudeck, 1993).

Usage

ci_rmsea(
  rmsea,
  df,
  N,
  conf_level = 0.95,
  alpha_lower = NULL,
  alpha_upper = NULL
)

Arguments

rmsea

Observed root mean square error of approximation

df

Degrees of freedom of the model

N

Sample size

conf_level

Desired confidence level (e.g., .90, .95, .99)

alpha_lower

The Type I error rate for the lower tail

alpha_upper

The Type I error rate for the upper tail

Value

A 3-row data.frame with columns term and value. The term values are "lower_limit" (the lower bound of the confidence interval on the population RMSEA, truncated at zero by definition), "rmsea" (the observed point estimate), and "upper_limit" (the upper bound).

Details

The RMSEA expresses the badness of model fit per degree of freedom on the noncentrality scale. Under the noncentral chi square model for the sample fit statistic, the sample \(T = (N - 1) \hat F_{ML}\) has approximate noncentral chi square distribution with \(df\) degrees of freedom and noncentrality parameter \(\lambda = (N - 1) df \cdot \mathrm{RMSEA}^2\). The CI on \(\mathrm{RMSEA}^2\) is obtained by inverting the noncentral chi square distribution at the requested confidence level (conf_limits_nc_chisq does the inversion); the bounds are then mapped back to the RMSEA scale via the square root. When the lower noncentrality limit hits zero (i.e., the data are compatible with a well-fitting model), the lower RMSEA limit is truncated at zero because RMSEA is non-negative by construction.

The 90 percent CI (rather than the usual 95 percent) is the conventional reporting choice for RMSEA (Browne & Cudeck, 1993) because the upper limit of the 90 percent CI plays a one-sided role in the test of close fit (\(H_0: \mathrm{RMSEA} \le 0.05\)). ci_rmsea defaults to conf_level = 0.95 in line with the rest of the package; pass conf_level = 0.90 when the close fit test is the intended use.

References

Browne, M. W., & Cudeck, R. (1993). Alternative ways of assessing model fit. In K. A. Bollen & J. S. Long (Eds.), Testing structural equation models (pp. 136–162). Sage.

Kelley, K., & Lai, K. (2011). Accuracy in parameter estimation for the root mean square error of approximation: Sample size planning for narrow confidence intervals. Multivariate Behavioral Research, 46, 1–32. doi:10.1080/00273171.2011.543027

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge.

Steiger, J. H., & Lind, J. C. (1980). Statistically-based tests for the number of common factors. Paper presented at the annual Spring meeting of the Psychometric Society, Iowa City, IA.

See also

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_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

# 1. A typical 95 percent CI on RMSEA.
ci_rmsea(rmsea = .055, df = 40, N = 425, conf_level = .95)
#>  term        value 
#>  lower_limit 0.037 
#>  rmsea       0.055 
#>  upper_limit 0.0727
#> 
#> Confidence level: 95%

# 2. The 90 percent CI is the conventional choice when interpretation
#    will follow the Browne and Cudeck (1993) close fit decision rule
#    (the test of H_0: RMSEA <= 0.05 vs. the upper CI limit). Here the
#    upper limit is 0.052, just above the close fit threshold of 0.05
#    that Browne and Cudeck recommend, so close fit is not established
#    even though the point estimate sits comfortably below it.
ci_rmsea(rmsea = .035, df = 40, N = 425, conf_level = .90)
#>  term        value 
#>  lower_limit 0.0147
#>  rmsea       0.035 
#>  upper_limit 0.052 
#> 
#> Confidence level: 90%

# 3. Wider model with smaller N: more uncertainty, wider CI.
ci_rmsea(rmsea = .055, df = 10, N = 100, conf_level = .90)
#> Note: The lower confidence limit of the noncentrality parameter is at its lower bound, so the lower RMSEA limit is set to 0 based on RMSEA's definition.
#>  term        value
#>  lower_limit 0    
#>  rmsea       0.055
#>  upper_limit 0.129
#> 
#> Confidence level: 90%