Probability density function
ν > 0 degrees of freedom|
|Ex. kurtosis||see text|
The noncentral t-distribution generalizes Student's t-distribution using a noncentrality parameter. Whereas the central probability distribution describes how a test statistic t is distributed when the difference tested is null, the noncentral distribution describes how t is distributed when the null is false. This leads to its use in statistics, especially calculating statistical power. The noncentral t-distribution is also known as the singly noncentral t-distribution, and in addition to its primary use in statistical inference, is also used in robust modeling for data.
is a noncentral t-distributed random variable with ν degrees of freedom and noncentrality parameter μ ≠ 0. Note that the noncentrality parameter may be negative.
Cumulative distribution function
and Φ is the cumulative distribution function of the standard normal distribution.
Alternatively, the noncentral t-distribution CDF can be expressed as:
Although there are other forms of the cumulative distribution function, the first form presented above is very easy to evaluate through recursive computing. In statistical software R, the cumulative distribution function is implemented as pt.
Probability density function
The probability density function (pdf) for the noncentral t-distribution with ν > 0 degrees of freedom and noncentrality parameter μ can be expressed in several forms.
The confluent hypergeometric function form of the density function is
and where 1F1 is a confluent hypergeometric function.
An alternative integral form is
A third form of the density is obtained using its cumulative distribution functions, as follows.
This is the approach implemented by the dt function in R.
Moments of the noncentral t-distribution
In general, the kth raw moment of the noncentral t-distribution is
In particular, the mean and variance of the noncentral t-distribution are
The non-central t-distribution is asymmetric unless μ is zero, i.e., a central t-distribution. In addition, the asymmetry becomes smaller the larger degree of freedom. The right tail will be heavier than the left when μ > 0, and vice versa. However, the usual skewness is not generally a good measure of asymmetry for this distribution, because if the degrees of freedom is not larger than 3, the third moment does not exist at all. Even if the degrees of freedom is greater than 3, the sample estimate of the skewness is still very unstable unless the sample size is very large.
The noncentral t-distribution is always unimodal and bell shaped, but the mode is not analytically available, although for μ ≠ 0 we have
In particular, the mode always has the same sign as the noncentrality parameter μ. Moreover, the negative of the mode is exactly the mode for a noncentral t-distribution with the same number of degrees of freedom ν but noncentrality parameter −μ.
The mode is strictly increasing with μ (it always moves in the same direction as μ is adjusted in). In the limit, when μ → 0, the mode is approximated by
and when μ → ∞, the mode is approximated by
- Central t-distribution: the central t-distribution can be converted into a location/scale family. This family of distributions is used in data modeling to capture various tail behaviors. The location/scale generalization of the central t-distribution is a different distribution from the noncentral t-distribution discussed in this article. In particular, this approximation does not respect the asymmetry of the noncentral t-distribution. However, the central t-distribution can be used as an approximation to the noncentral t-distribution.
- If T is noncentral t-distributed with ν degrees of freedom and noncentrality parameter μ and F = T2, then F has a noncentral F-distribution with 1 numerator degree of freedom, ν denominator degrees of freedom, and noncentrality parameter μ2.
- If T is noncentral t-distributed with ν degrees of freedom and noncentrality parameter μ and , then Z has a normal distribution with mean μ and unit variance.
- When the denominator noncentrality parameter of a doubly noncentral t-distribution is zero, then it becomes a noncentral t-distribution.
- When μ = 0, the noncentral t-distribution becomes the central (Student's) t-distribution with the same degrees of freedom.
Occurrence and applications
Use in power analysis
Suppose we have an independent and identically distributed sample X1, ..., Xn each of which is normally distributed with mean θ and variance σ2, and we are interested in testing the null hypothesis θ = 0 vs. the alternative hypothesis θ ≠ 0. We can perform a one sample t-test using the test statistic
where is the sample mean and is the unbiased sample variance. Since the right hand side of the second equality exactly matches the characterization of a noncentral t-distribution as described above, T has a noncentral t-distribution with n−1 degrees of freedom and noncentrality parameter .
If the test procedure rejects the null hypothesis whenever , where is the upper α/2 quantile of the (central) Student's t-distribution for a pre-specified α ∈ (0, 1), then the power of this test is given by
Use in tolerance intervals
One-sided normal tolerance intervals have an exact solution in terms of the sample mean and sample variance based on the noncentral t-distribution. This enables the calculation of a statistical interval within which, with some confidence level, a specified proportion of a sampled population falls.
- Lenth, Russell V (1989). "Algorithm AS 243: Cumulative Distribution Function of the Non-central t Distribution". Journal of the Royal Statistical Society, Series C. 38 (1): 185–189. JSTOR 2347693.
- L. Scharf, Statistical Signal Processing, (Massachusetts: Addison-Wesley, 1991), p.177.
- Hogben, D; Pinkham, RS; Wilk, MB (1961). "The moments of the non-central t-distribution". Biometrika. 48 (3–4): 465–468. doi:10.1093/biomet/48.3-4.465. hdl:2027/coo.31924001119068. JSTOR 2332772.
- Hedges, Larry V. (June 1981). "Distribution Theory for Glass's Estimator of Effect size and Related Estimators". Journal of Educational Statistics. 6 (2): 107–128. doi:10.3102/2F10769986006002107.
- Tothfalusi, Laszlo; Endrenyi, Laszlo (1 March 2016). "An Exact Procedure for the Evaluation of Reference-Scaled Average Bioequivalence". The AAPS Journal. 18 (2): 476–489. doi:10.1208/s12248-016-9873-6.
- van Aubel, A; Gawronski, W (2003). "Analytic properties of noncentral distributions". Applied Mathematics and Computation. 141: 3–12. doi:10.1016/S0096-3003(02)00316-8.
- Helena Chmura Kraemer; Minja Paik (1979). "A Central t Approximation to the Noncentral t Distribution". Technometrics. 21 (3): 357–360. doi:10.1080/00401706.1979.10489781. JSTOR 1267759.
- Derek S. Young (August 2010). "tolerance: An R Package for Estimating Tolerance Intervals". Journal of Statistical Software. 36 (5): 1–39. ISSN 1548-7660. Retrieved 19 February 2013., p.23
- Eric W. Weisstein. "Noncentral Student's t-Distribution." From MathWorld—A Wolfram Web Resource
- High accuracy calculation for life or science.: Noncentral t-distribution From Casio company.