A New F Approximation for the Pillai–Bartlett Trace under H₀

Abstract

Pillai suggested two approximations for the Pillai–Bartlett trace statistic in the null case. The first one matches one moment of a β₁ random variable, and corresponds to an F random variable, and the second matches four moments in the Pearson system. Although intuitively appealing and widely used in current statistical packages, the first lacks accuracy even with moderate sample size. The second matches two moment ratios in the Pearson system and provides much greater accuracy. Two new approximations match two moments of a β₁ random variable, and hence correspond to an F random variable, yet achieve most of the accuracy of Pillai’s second approximation. The second of the two new approximations provides the best combination of logical properties and numerical accuracy.

1. INTRODUCTION

1.1 Motivation

Consider H and E, independent, b × b, central Wishart matrices, with common covariance matrix, Σ_*, and respective degrees of freedom a and ν_E. In turn T = H + E represents a b × b central Wishart matrix, with covariance Σ_* and degrees of freedom a + ν_E. Such matrices arise under the null in testing the general linear hypothesis, in the context of the general linear multivariate model (GLMM). The Pillai-Bartlett trace, V =tr(HT^−l), provides a common test statistic. See Muller, LaVange, Ramey, and Ramey (1992) for a detailed statement of the underlying problem in the context of power analysis. See Pillai (1976, 1977) for a detailed survey of distributional results for GLMM tests.

Pillai (1954, 1955) suggested approximating V by a β₁, which corresponds to an F random variable. For a = 2, b = 3, and ν_E = 24, the approximation yields around two digits of accuracy for quantiles in small samples (Pillai 1954, tab. 5.5.1). Exact probabilities exceed the corresponding approximate ones by as much as .01 in the same setting. The approach (1) matches the first moment of V with the β₁; (2) uses the intuitively appealing value of ab for the numerator degrees of freedom of the F; (3) reduces to the exact answer if s = min(a, b) = 1; and (4) provides asymptotically correct performance. However, Pillai recommended avoiding the approximation in small samples due to limited accuracy. Itô (1956) suggested a series approximation. Pillai (1957) used the Pearson system to provide an approximation that uses four moments to match two moment ratios. Davis (1970) described a method for computing exact values, based on solving a differential equation. He also examined the accuracy of the Itô (1956) and Pillai (1957) approximations in providing percentiles. Davis reported that Pillai’s Pearson curve approximation provides four digits of accuracy, except when the degrees of freedom for both H and T are small, and that Itô’s approximation does not perform as well.

A parallel problem arises for the Hotelling–Lawley trace. McKeon (1974) used the moments of U =tr(HE⁻¹) to chose {γ_m, ν_1,m, ν_2,m} to define approximating random variables of the form U_*m = γ_m · F(ν_1,m, ν_2,m), a scaled central F random variable. Matching the first three moments of U with U_*3 yields one approximation. Another form matches only the first two moments, but adds the constraint ν_1,2 = ab to define the third equation needed to uniquely determine U_*2. The constraint describes the limiting value. Both forms reduce to the exact answer if s = 1. Either has much better accuracy than previous approximations, including a one-moment method of Pillai and Samson (1959). McKeon (1974) recommended U_*2 due to slightly better average accuracy for the conditions he studied and the simplicity of the numerator degrees of freedom. The success of McKeon’s approximation encourages examining a similar strategy for the Pillai-Bartlett trace.

1.2 Moments of V

Pillai (1954, p. 95) derived the moments of V in the null case. With εV = μ₁ and ε(V − εV)² = μ₂,

$${\mu }_1=\frac{ab}{{\nu }_E+a}$$ (1.1)

and

$${\mu }_2=\frac{2ab{\nu }_E({\nu }_E+a-b)}{{({\nu }_E+a)}^2({\nu }_E+a+2)({\nu }_E+a-1)}.$$ (1.2)

2. Approximations

2.1 The Form of the Approximations

The desire to reduce to the exact result if s = min(a, b) = 1 leads to approximating V by V_*m = γ_m · β₁ (ν_1,m/2, ν_2,m/2), a β₁ random variable (Johnson and Kotz 1970, chap. 24). Observe that

$$\epsilon V_{*m}={\gamma }_m\frac{{\nu }_{1,m}}{{\nu }_{1,m}+{\nu }_{2,m}},$$ (2.1)

and

$$\epsilon {(V_{*m}-\epsilon V_{*m})}^2={\gamma }_m^2\frac{2{\nu }_{1,m}{\nu }_{2,m}}{{({\nu }_{1,m}+{\nu }_{2,m})}^2({\nu }_{1,m}+{\nu }_{2,m}+2)}.$$ (2.2)

With V ≈ V_*m = γ_m · β₁ (ν_1,m/2, ν_2,m/2), consider F_*m = F (ν_1,m, ν_2,m), a central F random variable. The fact that V_*m = ν_1,mF_*m/(ν_1,mF_*1 + ν_2,m) allows writing

$$V\approx {\gamma }_m·\frac{{\nu }_{1,m}{F}_{*m}}{{\nu }_{1,m}{F}_{*m}+{\nu }_{2,m}},$$ (2.3)

and

$$\frac{(V/{\gamma }_m)/{\nu }_{1,m}}{(1-V/{\gamma }_m)/{\nu }_{2,m}}\approx F({\nu }_{1,m},{\nu }_{2,m}).$$ (2.4)

2.2 Pillai’S Approximation

Assume ν_1,0 = ab and γ₀ = s. The Pillai (1954) approximation results from assuming γ₀ = s and ν_1,0 = ab, then solving

$$s·\frac{ab}{ab+{\nu }_{2,0}}={\mu }_1$$ (2.5)

for ν_2,0. This yields ν_2,0 = s(ν_E + a) − ab = s(ν_E + s − b).

2.3 Method 1: A New Approximation

Assume ν_1,1 = ab and γ₁ free to vary. With the assumption ν_1,1 = ab, the asymptotic value, write the equations of interest as

$${\gamma }_1\frac{ab}{ab+{\nu }_{2,1}}={\mu }_1,$$ (2.6)

and

$${\gamma }_1^2\frac{2ab{\nu }_{2,1}}{{(ab+{\nu }_{2,1})}^2(ab+{\nu }_{2,1}+2)}={\mu }_2.$$ (2.7)

Solving for ν_2,1 and γ₁ yields

$${\nu }_{2,1}=\frac{(ab+2){\nu }_E({\nu }_E+a-b)}{{\nu }_E(a+1+b)+a^2+a-2},$$ (2.8)

$${\gamma }_1=\frac{ab+{\nu }_{2,1}}{({\nu }_E+a)}.$$ (2.9)

Note that Method 1 implies Pr {V ≤ υ} = 1.0 if υ > γ₁.

2.4 Method 2: A New Approximation

Assume ν_1,2 free to vary and γ₂ = s. With the assumption γ₂ = s, the upper bound on V, write the system of equations of interest as

$$\frac{{\nu }_{1,2}}{{\nu }_{1,2}+{\nu }_{2,2}}=\frac{{\mu }_1}{s},$$ (2.10)

$$\frac{{\nu }_{1,2}{\nu }_{2,2}}{{({\nu }_{1,2}+{\nu }_{2,2})}^2({\nu }_{1,2}+{\nu }_{2,2}+1)}=\frac{{\mu }_2}{s^2}.$$ (2.11)

Solving for ν_1,2 and ν_2,2 yields

$${\nu }_{1,2}=\frac{ab}{s({\nu }_E+a)}[\frac{s({\nu }_E+s-b)({\nu }_E+a+2)({\nu }_E+a-1)}{{\nu }_E({\nu }_E+a-b)}-2]$$ (2.12)

and

$${\nu }_{2,2}=\frac{({\nu }_E+s-b)}{({\nu }_E+a)}[\frac{s({\nu }_E+s-b)({\nu }_E+a+2)({\nu }_E+a-1)}{{\nu }_E({\nu }_E+a-b)}-2].$$ (2.13)

Observe that, with

$$K=\frac{1}{s({\nu }_E+a)}[\frac{s({\nu }_E+s-b)({\nu }_E+a+2)({\nu }_E+a-1)}{{\nu }_E({\nu }_E+a-b)}-2],$$ (2.14)

ν_1,2 = Kν_1,0 and ν_2,2 = Kν_2,0. If s = a then

$$K=\frac{a{\nu }_E+(a-1)(a+2)}{a{\nu }_E}\ge 1,$$ (2.15)

while if s = b then

$$K=\frac{b({\nu }_E+a)+(b-2)}{b[({\nu }_E+a)-b]}\ge 1.$$ (2.16)

Montonicity properties of the F distribution and K ≥ 1 ensure that the probability for Method 2 will never be smaller than for Method 0, and hence the p value for a test will never be larger.

3. NUMERICAL COMPARISONS

Pillai (1954, p. 111) reported some α = .05 critical values for m = (|a − b| − l)/2 = 0, and a range of n = (ν_E − b − l)/2. Table 1 contains the critical values, as well as approximate probabilities computed with Pillai’s approximation, as well as the two new methods (Method 1 in Section 2.3 and Method 2 in Section 2.4). Both of the new methods provide substantially better accuracy than does Pillai’s approximation, with Method 1 slightly better than Method 2. Values in Table 1 greater than .05 imply that the approximation provides a conservative test in data analysis, while values less than .05 imply that the approximation provides a liberal test.

Table 1.

Test Size for Exact Critical Values α =.05, m = 0)

			Test size
s	n	V.95	Pillai	Method 1	Method 2
2	10	.451	.0567	.0503	.0508
	15	.332	.0549	.0506	.0508
	20	.264	.0526	.0494	.0495
	25	.218	.0524	.0498	.0499
	30	.186	.0519	.0497	.0497
3	10	.697	.0615	.0501	.0515

Pillai (1954) also reported some exact probabilities for certain other quantiles. Table 2 contains exact probabilities of the exact quantiles, as well as approximate quantiles from Pillai’s approximation and the two new methods. The pattern parallels that in Table 1, except for the most extreme quantile. Large quantiles create difficulty for Method 1, which uses γ₁ ≤ s and implies Pr {V ≤ υ} = 1 for υ > γ₁ (which occurs with small but nonzero probability).

Table 2.

Approximate Quantiles for Exact Probabilities (s = 2, m = 0)

		Exact quantile	Approximate quantile
n	Pr{V≤ v}	v	Pillai	Method 1	Method 2
10	.1192	.1000	.0971	.0998	.1016
10	.4534	.2000	.1972	.1999	.1992
10	.7471	.3000	.3004	.3003	.2983
10	.9084	.4000	.4067	.4006	.3997
30	.6353	.1000	.0998	.1000	.0998
30	.9661	.2000	.2022	.2001	.2003
30	.9984	.3000	3071	.2995	.3029
30	.9999	.4000	.3913	.3771	.3853

A simulation was conducted in SAS IML^® to examine performance for extreme quantiles. A total of 500,000 values of V were tabulated for a = 2, b = 3, and ν_E = 10 (and hence n = 3 and m = 0). For each replication the SAS function NORMAL created two matrices, Z_H (a × b) and Z_E (ν_E × b), of pseudo-random, i.i.d. Gaussian data having mean zero and unit variance. In turn H = Z′_H Z_H, E = Z′_E Z_E, and V = tr[H (H + E)⁻¹]. No other covariance structure needs to be considered due to the invariance of V under full rank linear transformation of the data.

Table 3 contains empirical quantiles and associated approximate probabilities. The results parallel those in Tables 1 and 2: overall, both new methods performed noticeably better than Pillai’s approximation. Method 1 performs the best for probabilities no more than .99, with Method 2 superior in the extreme right tail. As suspected, the need to map all probabilities for values greater than γ₁ into 1.0 hurts the accuracy of Method 1 near the boundary.

Table 3.

Empirical Quantiles and Approximate Probabilities (a = 2, b = 3, ν_E = 10)

Empirical quantile			Approximate probability, FV* (v)
v	F̂V(v)	±st. dev.	Pillai	Method 1	Method 2
.093	.01	± .00014	.01257	.01024	.00653
.170	.05	± .00031	.06073	.05158	.04238
.225	.10	± .00042	.11748	.10250	.09225
.336	.25	± .00061	.27669	.25375	.25050
.483	.50	± .00071	.51792	.50071	.51125
.646	.75	± .00061	.74265	.74548	.75658
.798	.90	± .00042	.87949	.89612	.89765
.890	.95	± .00031	.93064	.94921	.94599
1.071	.99	± .00014	.98118	.99289	.98824
1.284	.999	± .00004	.99755	.99993	.98894
1.453	.9999	± .00001	.99973	1.0	.99992
1.558	.99999	± .000004	.99996	1.0	.99999

Table 2 in Anderson (1984, appendix B) contains approximate critical values for a simple multiple of the Pillai–Bartlett trace, computed with the Pearson system method of Pillai. Extensive comparison of those values to ones computed with the new methods provides results consistent with the results presented here. Some related additional simulations were also conducted, and also supported the conclusions presented here.

4. CONCLUSIONS

1. Pillai’s F and the two new F approximations have logical and practical appeal because they

   1. correctly reduce to exact answer if s = 1;
   2. always match at least one moment exactly;
   3. have appropriate asymptotic behavior; and
   4. allow simple and convenient computations.

2. The new approximations provide substantially greater accuracy than the widely used Pillai F, which tends to be too conservative in small samples.

3. Method 1 provides the best average performance among the F approximations, but can sometimes be liberal.

4. The possibility of liberality, difficulties with V > γ₁, and the importance of Bonferroni corrections in multivariate data analysis, combine to substantially reduce the appeal of Method 1.

5. Pillai’s Pearson system approximation provides some additional accuracy, but lacks most of the appeal of the F approximations.

6. Method 2 (Sec. 2.4) provides the best combination of accuracy of type I error control and logical properties, and deserves to replace Pillai’s F approximation in data analysis.

7. Method 2 merits consideration in future research on power approximations.