The t distribution characterizes how the t test statistic is distributed when the null hypothesis is assumed to be true. The noncentral t distribution instead shows how the t test statistic is distributed when the alternative hypothesis is assumed to be true (i.e. when the null hypothesis is assumed to be false). As such it is useful in calculating the power of the usual t tests.
Definition 1: The noncentral t distribution, abbreviated as T(k,δ) has the following cumulative distribution function F(t), written as Fk,δ(t) when necessary, where k = the degrees of freedom and δ = the noncentrality parameter.
when t ≥ 0, where Φ is the cumulative distribution function of the standard normal distribution, i.e.
Φ(z) = NORMSDIST(z)
and Ir(a,b) is the distribution function of the beta distribution
Since the pdf of the Poisson distribution with mean can be expressed as
and so we can express the cdf of the noncentral t distribution as
where Γ(k) is the gamma function. Since Γ(m+1)/Γ(m+3/2) can be expressed by the formula =EXP(GAMMALN(m+1)-GAMMALN(m+3/2)), the cdf of the noncentral t distribution can be expressed to any desired degree of precision in Excel as a finite sum of terms using POISSON.DIST, BETA.DIST, NORM.S.DIST and GAMMALN.
When t < 0, the noncentral t distribution is defined as
Observation: The probability density function (pdf) of the noncentral t distribution can be calculated as follows:
The mean and variance of the distribution are
Observation: The noncentral t distribution has shape similar to the central t distribution (i.e. the ordinary t distribution). The noncentrality parameter indicates how much the distribution is shifted to the right (when δ > 0) or to the left (when δ < 0). When δ = 0, the noncentral t distribution is identical to the central t distribution, and so T(k,0) = T(k).
Observation: The chart in Figure 1 shows the graphs of the noncentral t distribution with 10 degrees of freedom for δ = 0, 2, 4, 6.
Figure 1 – Noncentral t pdf by noncentrality parameter
The chart in Figure 2 shows the graphs of the noncentral t distribution with δ = 2 and the degrees of freedom = 1, 3, 5, 10.
Figure 2 – Noncentral t pdf by degrees of freedom
Real Statistics Functions: The following function is provided in the Real Statistics Resource Pack:
NT_DIST(t, df, δ, cum, iter, prec). If cum = TRUE then the value of the noncentral t distribution T(k,δ) at t is returned, while If cum = FALSE then the value of the noncentral pdf at t is returned.
NT_INV(p, df, δ, iter, iter0, prec) = the inverse of the cdf of the noncentral t distribution T(k,δ) at p, i.e. the value of t such that NT_DIST(t, df, δ, TRUE, iter, prec) = p.
NT_NCP(p, df, t, iter, iter0, prec) = the value of the noncentrality parameter δ such the cdf of the noncentral distribution T(k,δ) at t is p, i.e. NT_DIST(t, df, δ, TRUE, iter, prec) = p.
Here iter = the maximum number of terms from the infinite sum (default 1000) and prec = the maximum amount of error acceptable in the estimate of the infinite sum unless the iteration limit is reached first (default = 0.000000000001). iter0 = the number of iterations used in calculating NT_INV or NT_NCP by binary search (default 40).
Note that NT_DIST(4.5,10,4,FALSE) = .25497 and NT_DIST(4.5,10,4,TRUE) = .60368, which is consistent with the values shown in the green curve of Figure 1.