Confidence Limits for a Noncentrality Parameter From a t-distribution

Finds the noncentrality parameters of a noncentral t-distribution that bracket an observed t-value with the requested tail probabilities, giving a confidence interval on the population noncentrality parameter. Together with conf_limits_ncf and conf_limits_nc_chisq, this is one of the low-level noncentral distribution workhorses on which the ci_* confidence interval functions (e.g., ci_smd, ci_smd_c, ci_cv) are built; most analyses reach it through those functions rather than calling it directly.

Usage

conf_limits_nct(
  ncp,
  df,
  conf_level = 0.95,
  alpha_lower = NULL,
  alpha_upper = NULL,
  t_value,
  tol = 1e-09,
  verbose = TRUE,
  ...
)

Arguments

ncp: The noncentrality parameter (e.g., observed t-value) of interest
df: The degrees of freedom
conf_level: The level of confidence for a symmetric confidence interval
alpha_lower: The proportion of values beyond the lower limit of the confidence interval (cannot be used with conf_level)
alpha_upper: The proportion of values beyond the upper limit of the confidence interval (cannot be used with conf_level)
t_value: Alias for ncp
tol: The convergence tolerance passed to uniroot when locating each limit
verbose: If TRUE (the default), the returned data frame additionally reports the achieved tail probabilities at each limit; if FALSE, only term and value are returned
...: Additional arguments forwarded to uniroot

Value

A data.frame with one row per confidence limit and the columns:

term: Either "lower_limit" or "upper_limit".
value: The noncentrality parameter at that limit. -Inf when alpha_lower = 0; Inf when alpha_upper = 0.
prob_less: (verbose = TRUE) The probability \(P(T \le \mathrm{ncp})\) that a t-statistic from the noncentral t-distribution centered at the row's limit falls at or below the observed ncp. By construction this equals alpha_upper on the upper_limit row and \(1 - \mathtt{alpha\_lower}\) on the lower_limit row.
prob_greater: (verbose = TRUE) The complementary probability \(P(T \ge \mathrm{ncp})\). By construction this equals alpha_lower on the lower_limit row and \(1 - \mathtt{alpha\_upper}\) on the upper_limit row.

Details

Each confidence limit is the noncentrality parameter of a noncentral t-distribution with df degrees of freedom whose appropriate tail at the observed ncp contains the requested probability:

the lower limit satisfies \(P(T \ge \mathrm{ncp}) = \mathtt{alpha\_lower}\);
the upper limit satisfies \(P(T \le \mathrm{ncp}) = \mathtt{alpha\_upper}\).

Each tail probability is continuous and strictly monotone in the noncentrality parameter, so each limit is the unique root of a one-dimensional equation. The roots are located with uniroot starting from a bracket centered on ncp with half-width scaled by the asymptotic standard error of the noncentrality estimator; extendInt is used to widen the bracket if needed.

This function is especially useful for forming confidence intervals around standardized mean differences (Cohen's d, Glass's g, Hedges' g), standardized regression coefficients, and coefficients of variation.

Warning

As of R 4.0.0, the largest ncp that R can accurately handle is 37.62.

References

Cumming, G., & Finch, S. (2001). A primer on the understanding, use, and calculation of confidence intervals that are based on central and noncentral distributions. Educational and Psychological Measurement, 61(4), 532–574. doi:10.1177/0013164401614002

Kelley, K. (2005). The effects of nonnormal distributions on confidence intervals around the standardized mean difference: Bootstrap and parametric confidence intervals, Educational and Psychological Measurement, 65, 51–69. doi:10.1177/0013164404264850

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., & Fouladi, R. T. (1997). Noncentrality interval estimation and the evaluation of statistical methods. In L. L. Harlow, S. A. Mulaik, & J. H. Steiger (Eds.), What if there were no significance tests? (pp. 221–257). Mahwah, NJ: Lawrence Erlbaum.

Author

Ken Kelley kkelley@nd.edu

Examples

# Suppose observed t-value based on 'df'=126 is 2.83. Finding the lower
# and upper critical values for the population noncentrality parameter
# with a symmetric confidence interval with 95\% confidence is given as:
conf_limits_nct(ncp = 2.83, df = 126, conf_level = .95)
#>          term     value prob_less prob_greater
#> 1 lower_limit 0.8337503     0.975        0.025
#> 2 upper_limit 4.8153591     0.025        0.975

# Modifying the above example so that a nonsymmetric 95% confidence interval
# can be formed:
conf_limits_nct(ncp = 2.83, df = 126, alpha_lower = .01, alpha_upper = .04, conf_level = NULL)
#>          term    value prob_less prob_greater
#> 1 lower_limit 0.461692      0.99         0.01
#> 2 upper_limit 4.602743      0.04         0.96

# Modifying the above example so that a single-sided 95% confidence interval
# can be formed:
conf_limits_nct(ncp = 2.83, df = 126, alpha_lower = 0, alpha_upper = .05, conf_level = NULL)
#>          term    value prob_less prob_greater
#> 1 lower_limit     -Inf      1.00         0.00
#> 2 upper_limit 4.495225      0.05         0.95