Confidence Limits for the Noncentrality Parameter of a Noncentral Chi Square Distribution

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

Usage

conf_limits_nc_chisq(
  chi_square = NULL,
  conf_level = 0.95,
  df = NULL,
  alpha_lower = NULL,
  alpha_upper = NULL,
  tol = 1e-09,
  verbose = TRUE,
  ...
)

Arguments

chi_square: The observed chi square value
conf_level: The desired degree of confidence for a symmetric interval
df: The degrees of freedom
alpha_lower: The proportion of values beyond the lower limit (cannot be used with conf_level)
alpha_upper: The proportion of values beyond the upper limit (cannot be used with conf_level)
tol: The convergence tolerance passed to uniroot
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. 0 when alpha_lower = 0 or the lower limit is unattainable; Inf when alpha_upper = 0; NA when the observed chi_square is so small that even at \(\lambda = 0\) the lower-tail probability is already at or below alpha_upper, so the upper limit is undefined (a warning is issued).
prob_less: (verbose = TRUE) The probability \(P(X \le \mathtt{chi\_square})\) that an observation from the noncentral chi square distribution centered at the row's limit falls at or below the observed chi_square.
prob_greater: (verbose = TRUE) The complementary probability \(P(X \ge \mathtt{chi\_square})\). 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 \(\lambda \ge 0\) of a noncentral chi square distribution with df degrees of freedom whose appropriate tail at the observed chi_square contains the requested probability:

the lower limit satisfies \(P(X \ge \mathtt{chi\_square}) = \mathtt{alpha\_lower}\);
the upper limit satisfies \(P(X \le \mathtt{chi\_square}) = \mathtt{alpha\_upper}\).

The two conditions run in opposite directions in \(\lambda\): the lower-tail probability \(P(X \le \mathtt{chi\_square})\) is continuous and strictly decreasing in the noncentrality parameter, so the upper-tail probability \(P(X \ge \mathtt{chi\_square})\) is continuous and strictly increasing in it. The lower limit is the \(\lambda\) at which the upper tail has grown to alpha_lower, and the upper limit is the \(\lambda\) at which the lower tail has shrunk to alpha_upper. Each is therefore the unique non-negative root of a one-dimensional equation, and both are located with uniroot on the decreasing lower-tail scale; extendInt is used to widen the search bracket if needed.

Because the noncentrality parameter is bounded below by zero, the lower limit is set to zero whenever the observed chi_square is smaller than the alpha_lower critical value of the central chi square distribution (i.e., the data is consistent with \(\lambda = 0\) at the requested confidence level). A warning is issued in that case, and the achieved probabilities reported in the output reflect the actual values at \(\lambda = 0\).

Symmetrically, when the observed chi_square is so small that even at \(\lambda = 0\) the lower-tail probability is already at or below alpha_upper, no \(\lambda \ge 0\) places as much as alpha_upper mass at or below chi_square; the upper limit is undefined and is returned as NA, with a warning.

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. (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

Author

Ken Kelley kkelley@nd.edu

Examples

# A typical call to the function.
conf_limits_nc_chisq(chi_square = 30, conf_level = .95, df = 15)
#>          term     value prob_less prob_greater
#> 1 lower_limit  1.407074     0.975        0.025
#> 2 upper_limit 38.876511     0.025        0.975

# A one-sided (upper) confidence interval.
conf_limits_nc_chisq(chi_square = 30, alpha_lower = 0, alpha_upper = .05,
                     conf_level = NULL, df = 15)
#>          term   value prob_less prob_greater
#> 1 lower_limit  0.0000      1.00         0.00
#> 2 upper_limit 34.6284      0.05         0.95