Pragmatic, Unifying Algorithm Gives Power Probabilities for ...

O'Brien and Shieh: Power for the Multivariate Linear Hypothesis
F U ~ F(ν 1 , ν 2 (U) , 0), exactly, for s = 1 or 2; for s > 2, this is an approximation that is
“adequate for practical situations” (Seber, p. 41). Using the strategy described above,
F U /N → p f ∗ U = t{(U∗ ) –1/t – 1}/(r C r A ); where U ∗ s
= ∏k=1 (1 + φk ∗ )-1 . Thus we take
λ U = Nλ ∗ U , where λ ∗ U = t{(U ∗ ) –1/t – 1}.
Kulp and Nagarsenker (1984) provided an approximation for the noncentral
distribution of U, which quickly provides asymptotic justification for our method.
Briefly: As is commonly done (c.f. Anderson, 1984, p. 330), if we take N → ∞ and
CBA → Œ 0 under a sequence of alternatives, then their Theorem 3.1 reduces to a single
noncentral beta distribution function, which is transformable exactly to the noncentral F
prescribed here. They noted that using the noncentral beta distribution (or, equivalently,
the noncentral F) is better than using the chi-square distribution, as per Sugiura and
Fujikoshi (1969), whose method is not exact under any case, even for r A = 1.
For r A > 1, λ U > λ (M)
U . Evidence heretofore that the Muller-Peterson algorithm
systematically under approximates the power of F U comes from a study by Barton and
Cramer (1989). They used λ (M)
U to construct various s > 1 situations with nominal powers
of .80, but reported estimated powers consistently higher than this (based on 5000 trials
of each situation).
Hotelling-Lawley (F T1
, F T2
). Hotelling (1951) and Lawley (1938) proposed the
statistic T = tr[E -1 s
H] = ∑k = 1
φ k . Several F transforms have been proposed. The most
commonly used one, due to Pillai and Samson (1959), is F T1
= ν (T 1)
2 (T/s) /(r C r A ).
with ν (T 1)
2 = s(N – r X – r A – 1) + 2. McKeon (1974) proposed F T2
= ν (T 2)
2 (T/h) /(r C r A ),
with ν (T 2)
2 = 4 + ( r C r A + 2)g, where
g = (N – r X) 2 – (N – r X )(2r A + 3) + r A (r A + 3)
(N – r X )(r C + r A + 1) – (r C + 2r A + r A
2
– 1)
,
and h = (ν (T 2)
2 – 2)/(N – r X – r A – 1). For s ≥ 2, F T1
/F T2
< 1.00 (with F T1
/F T2
→ 1 as
N → ∞), but this is counterbalanced to some degree by the fact that ν (T 1)
2 > ν (T 2)
2 . We
assessed the difference between F T1
and F T2
for 108 cases formed by crossing (r A , r C ) =
{(2, 2}, (2, 3}, (3, 2}, (3, 3)}; N – r X = {31, 66, 96}; nominal percentage points for F T1
using p T1
= {.005, .010, .020, .040, .050, .075, .100, .200, .400. .600, .800, .900}. We
found that F T1
/F T2
> 0.98; 1.50 < ν (T 1)
2 /ν (T 2)
2 < 1.95; with the ratio of the resulting p
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