DMAR (version 1.0.0)

ci_nc_t: Confidence Interval for the Noncentrality Parameter of a Noncentral t Distribution

Description

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 ci_nc_F and ci_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. The function was conf_limits_nct() in earlier builds of DMAR and is conf.limits.nct() in MBESS; it is named into the ci_* family because a confidence interval is what it computes.

Usage

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

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.

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

Warning

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

Author

Ken Kelley [email protected]

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.

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. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_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.

See Also

stats::pt(), stats::qt(), uniroot, ci_smd, ci_smd_c, ci_nc_F, ci_nc_chisq

Other noncentral distribution confidence intervals: ci_nc_F(), ci_nc_chisq()

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:
ci_nc_t(ncp = 2.83, df = 126, conf_level = .95)

# Modifying the above example so that a nonsymmetric 95% confidence interval
# can be formed:
ci_nc_t(ncp = 2.83, df = 126, alpha_lower = .01, alpha_upper = .04, conf_level = NULL)

# Modifying the above example so that a single-sided 95% confidence interval
# can be formed:
ci_nc_t(ncp = 2.83, df = 126, alpha_lower = 0, alpha_upper = .05, conf_level = NULL)