DMAR (version 1.0.0)

moments_nc_t: Moments of the Noncentral t Distribution

Description

Returns the mean, variance, standard deviation, skewness, and excess kurtosis of a noncentral t distribution with df degrees of freedom and noncentrality parameter ncp. These are the closed-form moments surveyed by Owen (1968); they are the engine behind the bias and variance of the standardized mean difference (Cohen's d), since d is a scaled noncentral t variate. A central t (ncp = 0) is the special case with mean 0 and the familiar \(\mathit{df}/(\mathit{df}-2)\) variance. The function was moments_nct() in earlier builds of DMAR.

Usage

moments_nc_t(df, ncp = 0)

Value

A data.frame (class dmar_tbl) in

term / value layout with the mean, variance,

sd, skewness, and excess_kurtosis (any of which may be

NA when the degrees of freedom are too small), followed by the

df and ncp that produced them.

Arguments

df

Degrees of freedom, a single positive number (need not be a whole number).

ncp

Noncentrality parameter \(\delta\), a single number (may be negative, which mirrors the distribution about 0). Defaults to 0, the central t.

Author

Ken Kelley [email protected]

Details

Writing the noncentral t as \(T = (Z + \delta)/\sqrt{W/\nu}\) with \(Z \sim N(0, 1)\) and \(W \sim \chi^2_\nu\) independent, the raw moments are $$\mathrm{E}[T^k] = \mathrm{E}[(Z + \delta)^k]\, \Bigl(\tfrac{\nu}{2}\Bigr)^{k/2}\, \frac{\Gamma\!\bigl((\nu - k)/2\bigr)}{\Gamma(\nu/2)}, \qquad \nu > k,$$ computed here on the log scale for stability. The mean exists for \(\nu > 1\), the variance for \(\nu > 2\), the skewness for \(\nu > 3\), and the excess kurtosis for \(\nu > 4\); a moment whose degrees of freedom condition is not met is returned as NA. The mean is \(\delta\sqrt{\nu/2}\,\Gamma((\nu-1)/2)/\Gamma(\nu/2)\), the \(\delta\)-scaled reciprocal of the Hedges (1981) bias-correction factor that expected_smd and smd use; that is why the standardized mean difference is upward biased.

References

Owen, D. B. (1968). A survey of properties and applications of the noncentral t-distribution. Technometrics, 10(3), 445--478. tools:::Rd_expr_doi("10.1080/00401706.1968.10490590")

Hedges, L. V. (1981). Distribution theory for Glass's estimator of effect size and related estimators. Journal of Educational Statistics, 6(2), 107--128.

See Also

moments_nc_F for the noncentral F; expected_smd and var_smd for the same moments specialized to Cohen's d; dt for the density.

Other noncentral distribution moments: moments_nc_F(), moments_nc_chisq()

Examples

# A noncentral t with 20 df and noncentrality 2.5.
moments_nc_t(df = 20, ncp = 2.5)

# ncp = 0 is the central t: mean 0, variance df / (df - 2), no skew.
moments_nc_t(df = 10)

# The mean is the noncentrality times the Hedges bias factor's reciprocal,
# which is why Cohen's d (a scaled noncentral t) is upward biased.
m <- moments_nc_t(df = 18, ncp = 1.2)
m$value[m$term == "mean"]