Errata to Non-central t distribution and the power of the t test: A rejoinder

Abstract

Non-central t distribution needed for assessing the power of the t test is described. Three approximations are compared and their merits discussed in regard to simplicity and accuracy. Power evaluation for the various forms of Student's t test ties in with the non-central t distribution, explicitly noted , where ν is the " degrees of freedom " parameter and δ the non-centrality value. The variable represents the quotient of a standard normal (z) variable displaced by a constant (δ), over the square root of a Chi-square (χ 2) variable divided by its parameter (ν) : . (1) coincides with the standard (central) t ν distribution. In the following, represents the effect size; by convention, a of 0.5 is considered a "medium" effect size. In Figure 1 are shown three instances of the t′ density envelope for ν = 10, one with δ = 0 (a standard t), and two with δ = 3 and δ = 6 : one may note that, whereas t10(δ = 0) is centered at 0 and symmetrical, the non-central t′'s are displaced toward δ, their variance is increased and they are skewed. The probability density function for is (Levy & Narula, 1974) : f (t) = (2) in which . (2) Its first statistical moments are: , (3a) , (3b) , (3c) (3d) in which denotes the moments about the mean (in particular, is the variance) and ' denotes the central moments (in particular, is the mean, also noted E(t′)). Note that, approximately, µ1′ ≈ δ × [1 + 14/(17ν)] and σ 2 ≈ ν/(ν–2) + δ 2 /(2ν–7). It can be shown for given δ that, as ν increases, E(t′) → δ, var(t′) → var (t) = ν / (ν – 2), skewness index γ1(t′) = µ3 / σ 3 → γ1(t) = 0 and kurtosis index γ2(t′) = µ4 / σ 4 – 3 → γ2(t) = 6 / (ν – 4). Of course, we also know that t → z with increasing ν, z being a standard normal variable. These relations have inspired some approximation procedures. For illustration, Table 1 shows calculated values of the moments µ1′, σ 2 , γ1 and γ2 for some combinations of ν and δ, together with the above approximations for µ1′ and σ 2 . Power calculation and approximations A standard reference for statistical distributions is the celebrated series by Johnson, Kotz and Balakrisnan (1994, 1995), of which Volume 2 devotes a chapter to the non-central t distribution. From expression 31.11' on p. 514, one obtains the following function for evaluating the distribution function of : 2 Pr{ } = , (4) where Φ(x) is the standard normal distribution function. Use of Pr{)}, (5) where t(ν[α]) is the appropriate critical value for the t test, produces the exact power value. 1 Cousineau (2007) proposed an approximation to t′(δ) as t + δ, the power being estimated through: Pr{ t ν ≥ t(ν[α]) – δ } ; (6) he illustrated this with an example involving the comparison of two groups, each with N1 = N2 = 64 elements, and therefore, a t test having v = 126 degrees of freedom. Farther on in the same direction, one could approximate as z + δ and obtain an approximate power through : Pr{ z ≥ z[α] – δ } , (7) z[α] being the appropriate critical value from the normal distribution. Finally, Johnson et al. (1995, eq. 31.25) report an approximation by Jennet and Welch (1939), from which we propose the following : 2 Pr{ z ≥ z* } , (8) where , (8a)