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**|**HHS Author Manuscripts**|**PMC2762235

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- Abstract
- 1 Introduction
- 2 ANOVA and spectral decomposition
- 3 Conditions for equality
- 4 Conditions for superiority
- 5 Exact tests and confidence intervals
- 6 Applications
- 7 Simulation
- 8 Discussion
- References

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J Stat Plan Inference. Author manuscript; available in PMC 2010 December 1.

Published in final edited form as:

J Stat Plan Inference. 2009 December 1; 139(12): 3962–3973.

doi: 10.1016/j.jspi.2009.03.014PMCID: PMC2762235

NIHMSID: NIHMS129742

The mixed-effects models with two variance components are often used to analyze longitudinal data. For these models, we compare two approaches to estimating the variance components, the analysis of variance approach and the spectral decomposition approach. We establish a necessary and sufficient condition for the two approaches to yield identical estimates, and some sufficient conditions for the superiority of one approach over the other, under the mean squared error criterion. Applications of the methods to circular models and longitudinal data are discussed. Furthermore, simulation results indicate that better estimates of variance components do not necessarily imply higher power of the tests or shorter confidence intervals.

In the past two decades, the mixed-effects linear model has received considerable attention from both theoretical and practical points of view due to its extensive applications, for example, in analyzing longitudinal data arising from the health sciences, computer graphics and mechanical engineering. For a comprehensive overview of this model, see Davidian and Giltinan (1996), Diggle, *et al.* (2002) and Demidenko (2004).

Consider the mixed-effects linear model with two variance components

$${y}_{\mathit{\text{it}}}={x}_{\mathit{\text{it}}}^{\prime}\beta +{u}_{i}+{\epsilon}_{\mathit{\text{it}}},\text{}i=1,\dots ,N,\text{}t=1,\dots ,T,$$

where β is *p* × 1 vector of fixed effects, *u _{i}* is random effect,

$$y=X\beta +\mathit{\text{Zu}}+\epsilon ,$$

(1)

where *y* and *X* are of dimensions *n* × 1 and *n* × *p*, respectively, *n* = *NT*, *Z* = *I _{N}*

$$\text{Cov}(y)=V({\sigma}^{2})={\sigma}_{u}^{2}ZZ\prime +{\sigma}_{\epsilon}^{2}{I}_{n}.$$

(2)

Analysis of variance is one of the popular methods in the statistical literature to estimate the variance components ${\sigma}_{u}^{2}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{\epsilon}^{2}$. Compared to other estimates such as the maximum likelihood estimate (MLE), restricted maximum likelihood estimate (REMLE), minimum norm quadratic unbiased estimate (MINQUE), the analysis of variance estimate (ANOVAE) has simple closed forms that make it feasible to study analytically its small sample behavior and construct exact tests and confidence intervals for the variance components based on the estimates; see, e.g. Wald (1940), Searle, *et al.* (1992), and Burch and Iyer (1997). In contrast, when the parameter lies in the boundary of the parametric space, the asymptotic distributions of the likelihood-based tests, such as the likelihood ratio tests, do not follow chi-square distributions, as showed in Stram and Lee (1994) and Crainiceanu and Ruppert (2004). Moreover, solutions based on maximum likelihood (ML) estimates and restricted maximum likelihood (REML) estimates usually have very poor performance in mixed models with relatively small sample size; see Burdick and Larsen (1997).

Exact tests and confidence intervals are often required in practice. For the two-variance-component mixed-effects model (1), a new approach, referred to as the spectral decomposition, to estimating the variance components, was recently developed by Wang and Yin (2002). The resulting estimate, the spectral decomposition estimate (SDE), possesses similar advantages as the ANOVAE, having a simple closed form and allowing the construction of exact tests and confidence intervals.

The present paper serves as the first in the literature to compare analytically the small sample properties of ANOVAE and SDE under model (1). In Section 2 we introduce the ANOVAE and SDE of the variance components. A necessary and sufficient condition is derived in Section 3 under which the two methods yield identical estimates. In Section 4, a relationship between the two estimates is established, and some conditions are found under which one estimate is superior to the other in terms of mean squared error (MSE). Exact test and confidence intervals based on the two estimates are constructed in Section 5. Applications to two special models are given in Section 6. Section 7 presents some simulation results and some discussions are given in Sections 8.

Denote by *A*^{−} a generalized inverse of a matrix *A*, and write *P _{A}* =

$${\widehat{\sigma}}_{\epsilon}^{2}=\frac{1}{n-{r}_{0}}y\prime (I-{P}_{(X:Z)})y,$$

(3)

$${\widehat{\sigma}}_{u}^{2}=\{y\prime ({P}_{(X:Z)}-{P}_{X})y-({r}_{0}-\text{rk}(X)){\widehat{\sigma}}_{\epsilon}^{2}\}/\{T\text{tr}({Q}_{X}{P}_{Z})\},$$

(4)

where (*X* : *Z*) is a *n* × (*p* + *N*) matrix consisting of the column vectors of the matrices *X* and *Z*, rk(*A*) and tr(*A*) stand for the rank and trace of a matrix *A*, respectively, and *r*_{0} = rk(*X* : *Z*).

Note that the covariance matrix of *y* can be decomposed as

$$\text{Cov}(y)=\mathrm{\lambda}{P}_{Z}+{\sigma}_{\epsilon}^{2}{Q}_{Z},$$

where $\lambda =T{\sigma}_{u}^{2}+{\sigma}_{\epsilon}^{2}$. Left-multiplying model (1) by *P _{Z}* and

$$\begin{array}{c}{P}_{Z}y={P}_{Z}X\beta +{\u03f5}_{1},\text{}{\u03f5}_{1}~N(0,\lambda {P}_{Z}),\\ {Q}_{Z}y={Q}_{Z}X\beta +{\u03f5}_{2},\text{}{\u03f5}_{2}~N(0,{\sigma}_{\epsilon}^{2}{Q}_{Z}),\end{array}$$

both being singular linear models. From the unified theory of least squares (Rao, 1973), we can use

$${\tilde{\sigma}}_{\epsilon}^{2}=y\prime ({Q}_{Z}-{Q}_{Z}X{(X\prime {Q}_{Z}X)}^{-}X\prime {Q}_{Z})y/b,$$

(5)

$$\tilde{\lambda}=y\prime ({P}_{Z}-{P}_{Z}X{(X\prime {P}_{Z}X)}^{-}X\prime {P}_{Z})y/m,$$

(6)

where *m* = *N* − rk(*P _{Z}X*) and

$${\tilde{\sigma}}_{u}^{2}=(\tilde{\lambda}-{\tilde{\sigma}}_{\epsilon}^{2})/T.$$

(7)

Wang and Yin (2002) derived the two estimates ${\tilde{\sigma}}_{\u03f5}^{2}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\tilde{\sigma}}_{u}^{2}$ and termed them as the spectral decomposition estimates of ${\sigma}_{\u03f5}^{2}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{u}^{2}$, respectively. It is easy to verify (Craig’s Theorem, see Rao, 1973) that the two quadratic forms, *y*′(*P _{Z}* −

We first notice that the ANOVA and spectral decomposition approach lead to the same estimate for ${\sigma}_{\u03f5}^{2}$, as stated below.

*Under model* (1) *the ANOVAE and SDE of* ${\sigma}_{\epsilon}^{2}$ *are identical, that is*,

$${\widehat{\sigma}}_{\epsilon}^{2}={\tilde{\sigma}}_{\epsilon}^{2}.$$

It follows from (*A* : *B*) = (*A* : *Q _{A}B*) and

$${P}_{(A:B)}={P}_{A}+{Q}_{A}B{(B\prime {Q}_{A}B)}^{-}B\prime {Q}_{A},$$

(8)

and

$$\text{rk}(A:B)=\text{rk}(A)+\text{rk}({Q}_{A}B)$$

for any matrices *A* and *B* with the same number of rows. Thus

$$\begin{array}{c}{Q}_{Z}-{Q}_{Z}X{(X\prime {Q}_{Z}X)}^{-}X\prime {Q}_{Z}=I-({P}_{Z}+{Q}_{Z}X{(X\prime {Q}_{Z}X)}^{-}X\prime {Q}_{Z})=I-{P}_{(X:Z),}\hfill \\ \hfill b=\text{rk}({Q}_{Z})-\text{rk}({Q}_{Z}X)=n-{r}_{0}.\hfill \end{array}$$

Therefore ${\widehat{\sigma}}_{\epsilon}^{2}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\tilde{\sigma}}_{\epsilon}^{2}$.

However, the estimates of ${\sigma}_{u}^{2}$ from the ANOVA and spectral decomposition can be quite different. We derive below conditions under which the two estimates, ${\tilde{\sigma}}_{u}^{2}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\widehat{\sigma}}_{u}^{2}$, are identical. Write υ = *y* − *X*β,

$$\begin{array}{c}{A}_{0}=\raisebox{1ex}{$\{({P}_{(X:Z)}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{P}_{X})-\frac{({r}_{0}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}(X))(I\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{P}_{(X:Z)})}{n\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{r}_{0}}\}$}\!\left/ \!\raisebox{-1ex}{$\{T\text{tr}({Q}_{X}{P}_{Z})\}$}\right.,\hfill \\ {A}_{1}=\raisebox{1ex}{$\{\frac{{P}_{Z}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{P}_{\mathit{\text{ZX}}}{(X\prime {P}_{Z}X)}^{-}X\prime {P}_{Z}}{N\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}({P}_{Z}X)}-\frac{{Q}_{Z}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{Q}_{Z}X{(X\prime {Q}_{Z}X)}^{-}X\prime {Q}_{Z}}{n\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{r}_{0}}\}$}\!\left/ \!\raisebox{-1ex}{$T$}\right..\hfill \end{array}$$

Clearly, both ${\tilde{\sigma}}_{u}^{2}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\widehat{\sigma}}_{u}^{2}$ are unbiased estimators of ${\sigma}_{u}^{2}$, and can be reexpressed as ${\widehat{\sigma}}_{u}^{2}=\upsilon \prime {A}_{0}\upsilon ,{\tilde{\sigma}}_{u}^{2}=\upsilon \prime {A}_{1}\upsilon $, where υ ~ *N* (0, *V*(σ^{2})). Using the fact (Wang and Chow, 1994) that for any known symmetric matrix *A*, Var(υ′*A*υ) = 2tr(*AV AV*), we have

$$\begin{array}{cc}\text{Var}({\widehat{\sigma}}_{u}^{2})=\hfill & 2\{{\sigma}_{\epsilon}^{4}\frac{(n\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}(X))({r}_{0}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}(X))}{{T}^{2}{[\text{tr}({Q}_{X}{P}_{Z})]}^{2}(n\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{r}_{0})}+{\sigma}_{\epsilon}^{2}{\sigma}_{u}^{2}\frac{2}{T\text{tr}({Q}_{X}{P}_{Z})}\hfill \\ & +{\sigma}_{u}^{4}\frac{\text{tr}{({Q}_{X}{P}_{Z})}^{2}}{{[\text{tr}({Q}_{X}{P}_{Z})]}^{2}}\},\hfill \end{array}$$

(9)

and

$$\begin{array}{cc}\text{Var}({\tilde{\sigma}}_{u}^{2})=\hfill & 2\{{\sigma}_{\epsilon}^{4}\frac{(n+N\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{r}_{0}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}({P}_{Z}X))}{{T}^{2}[N-\text{rk}({P}_{Z}X)](n-{r}_{0})}+{\sigma}_{\epsilon}^{2}{\sigma}_{u}^{2}\frac{2}{T[N\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}({P}_{Z}X)]}\hfill \\ & +{\sigma}_{u}^{4}\frac{1}{N\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}({P}_{Z}X)}\}.\hfill \end{array}$$

(10)

We will need the following two lemmas to prove the main results.

(Wang and Chow, 1994). *Let P* = *P _{A}P_{B}*.

*P is an orthogonal projection matrix if and only if P*=_{A}P_{B}*P*._{B}P_{A}*If P*=_{A}P_{B}*P*(_{B}P_{A}, then P is the orthogonal projection matrix onto*A*) ∩ (*B*).

*The following statements are equivalent*.

*P*=_{A}P_{B}*P*_{B}P_{A}- (
*A*) ∩ (*B*) = (*P*)_{B}A - rk(
*P*) = dim((_{B}A*A*) ∩ (*B*)) *P*(_{B}A*A*′*P*)_{B}A^{−}*A*′*P*=_{B}*P*_{A}P_{B}

*where* dim(·) *denotes the dimension of a space*.

Note that for any vector **c** (*A*) ∩ (*B*), there exist vectors α and γ such that **c** = *A*α = *B*γ. It follows that *P _{B}P_{A}*

$$\mathcal{M}(A)\phantom{\rule{thinmathspace}{0ex}}\cap \phantom{\rule{thinmathspace}{0ex}}\mathcal{M}(B)\subseteq \mathcal{M}({P}_{B}{P}_{A})\subseteq \mathcal{M}({P}_{B}A).$$

(11)

If (i) holds, then *P _{A}*(

Conversely, if (ii) is true, then by (11) we have (*A*)∩(*B*) =(*P _{B}P_{A}*), and

That (ii) and (iii) are equivalent is obvious from (11) and the fact that dim((*P _{B}A*)) = rk(

The two lemmas lead to the following theorem.

$\text{Var}({\tilde{\sigma}}_{u}^{2})=\text{Var}({\widehat{\sigma}}_{u}^{2})$ *if and only if P _{X}P_{Z} is symmetric, i.e. P_{X}P_{Z}* =

Note that for any ${\sigma}_{u}^{2}\ge 0\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\sigma}_{\epsilon}^{2}>0,\phantom{\rule{thinmathspace}{0ex}}\text{Var}({\tilde{\sigma}}_{u}^{2})=\text{Var}({\widehat{\sigma}}_{u}^{2})$ if and only if the following three equalities all hold:

- $\frac{(n\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}(X))({r}_{0}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}(X))}{{[\text{tr}({Q}_{X}{P}_{Z})]}^{2}}=\frac{n+N\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{r}_{0}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}({P}_{Z}X)}{N-\text{rk}({P}_{Z}X)}$,
- tr(
*Q*) =_{X}P_{Z}*N*− tr(*P*) =_{X}P_{Z}*N*− rk(*P*)_{Z}X - [tr(
*Q*)]_{X}P_{Z}^{2}/[tr(*Q*)]_{X}P_{Z}^{2}= 1/(*N*− rk(*P*))._{Z}X

Write *r*_{1} = dim((*X*) ∩ (*Z*)). Then *r*_{0} = rk(*X*) + rk(*Z*) − *r*_{1}. Since rk(*Z*) = *N*, we have

$$n+N-{r}_{0}-\text{rk}({P}_{Z}X)=n-\text{rk}(X)+{r}_{1}-\text{rk}({P}_{Z}X).$$

Thus the three conditions (a)–(c) are equivalent to

$${[\text{tr}({Q}_{X}{P}_{Z})]}^{2}=\text{tr}({Q}_{X}{P}_{Z})=N-\text{rk}({P}_{Z}X)=N-{r}_{1}.$$

(12)

Note that the last equality above is equivalent to rk(*P _{Z}X*) =

We now give the main results.

${\tilde{\sigma}}_{u}^{2}={\widehat{\sigma}}_{u}^{2}$ *if and only if P _{X}P_{Z}* =

Note that *P _{X}P_{Z}* =

$${P}_{Z}X{(X\prime {P}_{Z}X)}^{-}X\prime {P}_{Z}={P}_{X}{P}_{Z},\text{}{Q}_{X}Z{(Z\prime {Q}_{X}Z)}^{-}Z\prime {Q}_{X}={Q}_{X}{P}_{Z}.$$

Using (8) we have

$${P}_{(X:Z)}-{P}_{X}={Q}_{X}{P}_{Z}={P}_{Z}-{P}_{Z}X{(X\prime {P}_{Z}X)}^{-}X\prime {P}_{Z}.$$

Thus ${\tilde{\sigma}}_{u}^{2}={\widehat{\sigma}}_{u}^{2}$.

On the other hand, if ${\tilde{\sigma}}_{u}^{2}={\widehat{\sigma}}_{u}^{2}$, then $\text{Var}({\tilde{\sigma}}_{u}^{2})=\text{Var}({\widehat{\sigma}}_{u}^{2})$. By Theorem 3.2, we have *P _{X}P_{Z}* =

Both ${\tilde{\sigma}}_{u}^{2}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\widehat{\sigma}}_{u}^{2}$ are uniformly minimum variance ubiased (UMVU) estimates of ${\sigma}_{u}^{2}$ if *P _{X}P_{Z}* =

We exemplify the main results with two popular mixed-effects models, both satisfying the identity condition *P _{X}P_{Z}* =

The one-way classification model has the form

$${y}_{\mathit{\text{ij}}}=\mu +{u}_{i}+{\epsilon}_{\mathit{\text{ij}}},\text{}i=1,\cdots ,a,\text{}j=1,\cdots ,b,$$

where μ is a fixed parameter, *u _{i}* is a random effect,

The two-way classification model is given by

$$\begin{array}{c}{y}_{\mathit{\text{ij}}}=\mu +{\alpha}_{i}+{\beta}_{j}+{\epsilon}_{\mathit{\text{ij}}},\\ i=1,2,\dots ,a,\text{}j=1,2,\dots ,b,\end{array}$$

where α_{i} is a random effect, β_{j} is a fixed effect. We assume that ${\alpha}_{i}~N(0,{\sigma}_{\alpha}^{2})\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\epsilon}_{\mathit{\text{ij}}}~N(0,{\sigma}_{\epsilon}^{2})$, and all α_{i}s and ε_{ij}s are mutually independent.

In matrix form, the model can be expressed as

$$y={X}_{\gamma}+{Z}_{\alpha}+\epsilon ,$$

where γ = (μ, β_{1}, , β_{b})′, α = (α_{1}, , ,α_{a})′, and *X* = (**1**_{a} **1**_{b} : **1**_{a} *I _{b}*),

$${P}_{X}{P}_{Z}={P}_{Z}{P}_{X}=({\overline{J}}_{a}\otimes {I}_{b})\xb7({I}_{a}\otimes {\overline{J}}_{b})={\overline{J}}_{a}\otimes {\overline{J}}_{b}.$$

An important question to be answered concerning ANOVAE and SDE is, “If *P _{X}P_{Z}* ≠

$${Q}_{X}ZZ\prime {Q}_{X}={\displaystyle \sum _{i=1}^{g}{d}_{i}{M}_{i},}$$

(13)

where *d*_{1}, , *d _{g}* are the different nonzero eigenvalues of the matrix

Using the facts that ${M}_{i}^{2}={M}_{i},{M}_{i}{M}_{j}=0(i\ne j)$,

$${P}_{(X:Z)}-{P}_{X}={Q}_{X}Z{(Z\prime {Q}_{X}Z)}^{-}Z\prime {Q}_{X}={P}_{{Q}_{X}Z}={\displaystyle \sum _{i=1}^{g}{M}_{i},}$$

we obtain

$$y\prime ({P}_{(X:Z)}-{P}_{X})y={\displaystyle \sum _{i=1}^{g}y\prime {M}_{i}y,}$$

(14)

with $y\prime {M}_{i}y~({d}_{i}{\sigma}_{u}^{2}+{\sigma}_{\epsilon}^{2}){\chi}_{{m}_{i}}^{2}$, where *m _{i}* = tr(

Since

$${Q}_{X}ZZ\prime {Q}_{X}({P}_{Z}-{P}_{Z}X{(X\prime {P}_{Z}X)}^{-}X\prime {P}_{Z})=T({P}_{Z}-{P}_{Z}X{(X\prime {P}_{Z}X)}^{-}X\prime {P}_{Z}),$$

*T* is a nonzero eigenvalue of *Q _{X}ZZ*′

$${M}_{1}={P}_{Z}-{P}_{Z}X{(X\prime {P}_{Z}X)}^{-}X\prime {P}_{Z},\text{}{d}_{1}=T,\text{}{m}_{1}=m=N-\text{rk}({P}_{Z}X),$$

and

$${\widehat{\sigma}}_{u}^{2}(i)=(y\prime {M}_{i}y/{m}_{i}-{\widehat{\sigma}}_{\epsilon}^{2})/{d}_{i},\phantom{\rule{thinmathspace}{0ex}}i=1,\cdots ,g.$$

It follows straightforwardly that ${\widehat{\sigma}}_{u}^{2}(1)={\tilde{\sigma}}_{u}^{2}$, and each ${\widehat{\sigma}}_{u}^{2}(i)$ is an unbiased estimate of ${\sigma}_{u}^{2}$. Combining (4) and (14), we obtain the following theorem.

*The ANOVAE of* ${\sigma}_{u}^{2}$ *is a weighted average of* $\{{\widehat{\sigma}}_{u}^{2}(i):i=1,\cdots ,g\}$, that is

$${\widehat{\sigma}}_{u}^{2}=T{m}_{1}{\tilde{\sigma}}_{u}^{2}/\mathrm{\Delta}+{\displaystyle \sum _{i=2}^{g}{d}_{i}{m}_{i}{\tilde{\sigma}}_{u}^{2}(i)/\mathrm{\Delta},}$$

(15)

where $\mathrm{\Delta}=T\text{tr}({Q}_{X}{P}_{Z})={\displaystyle {\sum}_{i=1}^{g}{d}_{i}{m}_{i}}$.

Thus the SDE ${\tilde{\sigma}}_{u}^{2}$ under model (1), in fact, is an unbiased estimate deduced by only one quadratic form of *y*′*M*_{1}*y*, , *y*′*M _{g}y*.

In the following, we will compare the variance of ${\widehat{\sigma}}_{u}^{2}$, the weighted average of $\{{\widehat{\sigma}}_{u}^{2}(i):i=1,\cdots ,g\}$, with that of ${\tilde{\sigma}}_{u}^{2}$ when *P _{X}P_{Z}* ≠

*The corresponding coefficients of* ${\sigma}_{u}^{2}{\sigma}_{\epsilon}^{2}$ *and* σ^{4} *in* $\text{Var}({\tilde{\sigma}}_{u}^{2})$ *are larger than their counterparts in* $\text{Var}({\tilde{\sigma}}_{u}^{2})$ *if* *P _{X}P_{Z}* ≠

Using the facts that

$$\text{rk}({P}_{Z}X)=\text{tr}({P}_{Z}X{(X\prime {P}_{Z}X)}^{-}X\prime {P}_{Z})\ge \text{tr}({P}_{Z}{P}_{X}{P}_{Z})=\text{tr}({P}_{X}{P}_{Z}),$$

and

$${[\text{tr}({Q}_{X}{P}_{Z})]}^{2}=\text{tr}({Q}_{X}{P}_{Z}{Q}_{X}{P}_{Z}{Q}_{X})\le \text{tr}({Q}_{X}{P}_{Z}{Q}_{X})=\text{tr}({Q}_{X}{P}_{Z})=N-\text{tr}({P}_{X}{P}_{Z}),$$

we have

$$\frac{{[\text{tr}({Q}_{X}{P}_{Z})]}^{2}}{{[\text{tr}({Q}_{X}{P}_{Z})]}^{2}}\le \frac{1}{\text{tr}({Q}_{X}{P}_{Z})}\le \frac{1}{N\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}({P}_{Z}X)},$$

and the equalities above hold if and only if *P _{X}P_{Z}* =

Note that

$$\begin{array}{c}{r}_{0}-\text{rk}(X)=N-\text{dim}(\mathcal{M}(X)\phantom{\rule{thinmathspace}{0ex}}\cap \phantom{\rule{thinmathspace}{0ex}}\mathcal{M}(Z)),\\ n+N-{r}_{0}-\text{rk}({P}_{Z}X)=n-\text{rk}({P}_{Z}X:{Q}_{Z}X),\end{array}$$

we obtain the ratio of the coefficient of ${\sigma}_{\epsilon}^{4}$ in $\text{Var}({\widehat{\sigma}}_{u}^{2})$ to that in $\text{Var}({\tilde{\sigma}}_{u}^{2})$:

$$\begin{array}{c}\frac{(n\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}(X))({r}_{0}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}(X))}{{[\text{tr}({Q}_{X}{P}_{Z})]}^{2}}/\frac{(n+N\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}{r}_{0}\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}({P}_{z}X))}{N\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}({P}_{Z}X)}\hfill \\ \text{}=\frac{n\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}(X)}{n\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}({P}_{Z}X:{Q}_{Z}X)}\frac{N\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{dim}(\mathcal{M}(X)\phantom{\rule{thinmathspace}{0ex}}\cap \phantom{\rule{thinmathspace}{0ex}}\mathcal{M}(Z))}{N\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{tr}({P}_{X}{P}_{Z})}\frac{N\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{rk}({P}_{Z}X)}{N\phantom{\rule{thinmathspace}{0ex}}-\phantom{\rule{thinmathspace}{0ex}}\text{tr}({P}_{X}{P}_{Z})}.\hfill \end{array}$$

(16)

Because

$$\text{rk}({P}_{Z}X)\ge \text{tr}({P}_{X}{P}_{z})\ge \text{dim}(\mathcal{M}(X)\phantom{\rule{thinmathspace}{0ex}}\cap \phantom{\rule{thinmathspace}{0ex}}\mathcal{M}(Z))\ge 0,$$

and

$$n-\text{rk}({P}_{Z}X:{Q}_{Z}X)\le n-\text{rk}(X),$$

it follows that the first two fractions in the right-hand side of (16) are larger than or equal to 1, and the third fraction is no larger than 1. Combining these with Theorem 4.2 we have

*If the right-hand side of* (16) *is less than 1, then* $\mathit{\text{Var}}({\widehat{\sigma}}_{u}^{2})<\mathit{\text{Var}}({\tilde{\sigma}}_{u}^{2})$.

The ANOVAE ${\widehat{\sigma}}_{u}^{2}$ is not uniformly superior to the SDE ${\tilde{\sigma}}_{u}^{2}$. If $({\sigma}_{e}^{2}/T)\gg {\sigma}_{u}^{2}$ and the right-hand side of (16) > 1, then $\text{Var}({\widehat{\sigma}}_{u}^{2})>\text{Var}({\tilde{\sigma}}_{u}^{2})$.

Example 6.2 in Section 6 illustrates that (16) is not always less than 1. Note that $\left(\text{Var}({\widehat{\sigma}}_{u}^{2})-\text{Var}({\tilde{\sigma}}_{u}^{2})\right)/{\sigma}_{\epsilon}^{2}$ is a quadratic function of $\tau =T{\sigma}_{u}^{2}/{\sigma}_{\epsilon}^{2}$ with the negative coefficients for the terms τ^{2} and τ, and the common term is obtained by subtracting the denominator from the numerator of (16). Thus when the right-hand side of (16) is larger than 1, $\text{Var}({\widehat{\sigma}}_{u}^{2})>\text{Var}({\tilde{\sigma}}_{u}^{2})$ for small τ and $\text{Var}({\widehat{\sigma}}_{u}^{2})>\text{Var}({\tilde{\sigma}}_{u}^{2})$ for large τ; see Example 6.2. That is, the SD estimate is preferred when the predictor varies considerably across subjects and the random effect variance and the number of repeated observations for each subject, *T*, are small.

A common interest in model (1) is testing hypotheses and constructing confidence intervals concerning the relative magnitude, $\theta ={\sigma}_{u}^{2}/{\sigma}_{\epsilon}^{2}$, of the variation due to the random effects and the random errors. To compare the two approaches under investigation, we assume that *P _{X}P_{Z}* ≠

Consider the hypothesis

$${H}_{0}:\theta ={\theta}_{0}\leftrightarrow {H}_{1}:\theta >{\theta}_{0},$$

where θ_{0} ≥ 0. Since the spectral estimates and ${\tilde{\sigma}}_{\epsilon}^{2}$ in (5) and (6) are independent, with ${m}_{1}\tilde{\lambda}/\lambda ~{\chi}_{{m}_{1}}^{2},b{\tilde{\sigma}}_{\epsilon}^{2}/{\sigma}_{\epsilon}^{2}~{\chi}_{b}^{2}$, we can construct an exact test statistic

$$F({\theta}_{0})=\frac{T\tilde{\theta}+1}{T{\theta}_{0}+1}=\frac{\tilde{\lambda}}{(T{\theta}_{0}+1){\tilde{\sigma}}_{\epsilon}^{2}}=\frac{y\prime {M}_{1}y}{(T{\theta}_{0}+1){m}_{1}{\tilde{\sigma}}_{\epsilon}^{2}},$$

(17)

which follows an F-distribution with degrees of freedom *m*_{1} and *b* under the null hypothesis *H*_{0} : θ = θ_{0}. Here, $\tilde{\theta}={\tilde{\sigma}}_{u}^{2}/{\tilde{\sigma}}_{\epsilon}^{2}$, is the spectral decomposition estimate of θ.

We reject the null hypothesis *H*_{0} if *F*(θ_{0}) > *F _{a,b}*(α). The power of the test is given by

$$P(F({\theta}_{0})>{F}_{{m}_{1},b}(\alpha ))=P\{\frac{T\theta +1}{T{\theta}_{0}+1}{F}_{{m}_{1},b}>{F}_{{m}_{1},b}(\alpha )\},$$

(18)

where ${F}_{{m}_{1},b}=b{\chi}_{{m}_{1}}^{2}/{m}_{1}{\chi}_{b}^{2}$ is an F-statistic with degrees of freedom *m*_{1} and *b*, and *F*_{m1,b}(α) is the corresponding upper 100α% quartile.

In contrast the quadratic forms *y*′(*P*_{(X:Z)}−*P _{X}*)

In the case of θ_{0} = 0, the exact test statistic based on the quadratic forms in the ANOVA method can be given by

$${F}_{A}=\frac{y\prime ({P}_{(X:Z)}-{P}_{X})y/a}{y\prime (I-{P}_{(X:Z)})y/b},$$

(19)

where *b* is defined as in (5), $a=\text{rk}({Q}_{X}Z)={\displaystyle {\sum}_{i=1}^{g}{m}_{i}}$, and *m _{i}* is defined in (14). Under

If θ_{0} > 0, *F _{A}* is intractable because of its complicated distribution. As an alternative instead, Wald’s test statistic

$$W({\theta}_{0})=\frac{\{{\displaystyle {\sum}_{i=1}^{g}y\prime {M}_{i}y/({d}_{i}{\theta}_{0}+1)}\}/a}{y\prime (I-{P}_{(X:Z)})y/b}$$

(20)

is often used in the literature, see Wald (1940). Clearly, *W*(0) = *F _{A}*.

To compare the power of the two tests based on *F*(θ_{0}) and *W*(θ_{0}), we define

$${F}_{i}({\theta}_{0})=\frac{y\prime {M}_{i}y}{({d}_{i}{\theta}_{0}+1){m}_{i}{\widehat{\sigma}}_{\epsilon}^{2}},\text{}i=2,\cdots ,g.$$

Then *W*(θ_{0}), in fact, is a weighted sum of these *F*-statistics, that is

$$W({\theta}_{0})=\frac{{m}_{1}}{a}F({\theta}_{0})+{\displaystyle \sum _{i=2}^{g}\frac{{m}_{i}}{a}{F}_{i}({\theta}_{0})}.$$

(21)

It is easy to see that *W*(θ_{0}) follows an *F*-distribution with degrees of freedom *a* and *b* under the null hypothesis *H*_{0} : θ = θ_{0}, and the power is given by

$$P\{W({\theta}_{0})>{F}_{a,b}(\alpha )\}=P\phantom{\rule{thinmathspace}{0ex}}\{\frac{{\displaystyle \sum _{i=1}^{g}({d}_{i}\theta +1){\chi}_{{m}_{i}}^{2}/({d}_{i}{\theta}_{0}+1)}}{a{\chi}_{b}^{2}/b}>{F}_{a,b}(\alpha )\}.$$

(22)

Intuitively the power of the two tests based on *F*(θ_{0}) and *W*(θ_{0}) should be close when *m*_{1}/*a* approaches to 1; see simulation results in Section 7. In this case, the former test is more appealing in practice due to its simplicity.

Note that the pivotal quantities *F*(θ) ~ *F*_{m1,b} and *W*(θ) ~ *F _{a,b}* are decreasing functions of θ. Thus two exact 100(1 − α)% confidence intervals for θ, based on pivotal quantities

$$\{\theta \in [0,+\infty ):{F}_{{m}_{1},b}(\alpha /2)\le {F}_{0}(\theta )\le {F}_{{m}_{1},b}(1-\alpha /2)\}$$

(23)

and

$$\{\theta \in [0,+\infty ):{F}_{a,b}(\alpha /2)\le W(\theta )\le {F}_{a,b}(1-\alpha /2)\}.$$

(24)

Furthermore, (23) can be simplified as

$$\{\theta \in \phantom{\rule{thinmathspace}{0ex}}[0,+\infty ):\frac{y\prime {M}_{1}y}{T{m}_{1}{\widehat{\sigma}}_{\epsilon}^{2}{F}_{{m}_{1},b}(1-\alpha /2)}-\frac{1}{T}\le \theta \le \frac{y\prime {M}_{1}y}{T{m}_{1}{\widehat{\sigma}}_{\epsilon}^{2}{F}_{{m}_{1},b}(\alpha /2)}-\frac{1}{T}\}.$$

In Section 7, we will compare the expected lengths of these confidence intervals via Monte Carlo simulation.

Section 3 describes two simple examples in which the identity condition *P _{X}P_{Z}* =

The circular feature of a machined part is one of the most basic geometric primitives, and can be described easily by its center and radius. Due to imperfections introduced in manufacturing, the desired feature may not be truly circular. In order to control the production, we need to estimate the geometric parameters (center and radius), which requires data on machined parts along their circumferences, and a corresponding statistical model. In practice the data can be obtained using a computer-controlled coordinate measuring machine, while one of the models adopted for such data is the mixed-effects model provided by Wang and Lam (1997), which takes into consideration of the variability in center location of different machined parts.

Let (*x _{ij}*,

$${\tau}_{i(j)}={\theta}_{i0}+{\theta}_{j},\text{}j=1,2,\cdots ,n,$$

α_{i} = ρ_{i}cosθ_{i0} and β_{i} = ρ_{i}sinθ_{i0}, where θ_{j} is known and θ_{i0} is fixed but unknown. Then for the data points (*x _{ij}*,

$$\{\begin{array}{c}{x}_{\mathit{\text{ij}}}=\xi +{\alpha}_{i}\phantom{\rule{thinmathspace}{0ex}}\text{cos}\phantom{\rule{thinmathspace}{0ex}}{\theta}_{j}-{\beta}_{i}\phantom{\rule{thinmathspace}{0ex}}\text{sin}\phantom{\rule{thinmathspace}{0ex}}{\theta}_{j}+{u}_{1i}+{\u03f5}_{1\mathit{\text{ij}}},\hfill \\ {y}_{\mathit{\text{ij}}}=\eta +{\alpha}_{i}\phantom{\rule{thinmathspace}{0ex}}\text{sin}\phantom{\rule{thinmathspace}{0ex}}{\theta}_{j}+{\beta}_{i}\phantom{\rule{thinmathspace}{0ex}}\text{cos}\phantom{\rule{thinmathspace}{0ex}}{\theta}_{j}+{u}_{2i}+{\u03f5}_{2\mathit{\text{ij}}},\hfill \end{array}$$

(25)

where (ξ, η)′ is the designed location of the center of the part, (*u*_{1i}, *u*_{2i})′ is the random variable in the location of the center of the *i*th machined part, ϵ_{kij} ~ *N* (0, σ^{2}), ${u}_{\mathit{\text{ki}}}~N(0,{\sigma}_{0}^{2})$, *k* = 1, 2, *i* = 1, , *m*, *j* = 1, , *n*, and all ϵ_{kij}s and *u _{ki}*s are independent.

Put *z* = (*z*_{1}, , *z _{m}*)′, ${z}_{i}=({x}_{i}^{\prime},{y}_{i}^{\prime})\prime $,

$$\mathrm{\Phi}=\left(\begin{array}{cc}{\varphi}_{1}& -{\varphi}_{2}\\ {\varphi}_{2}& {\varphi}_{1}\end{array}\right)$$

with ϕ_{1} = (cosθ_{1}, , cos θ_{n})′ , ϕ_{2} = (sin θ_{1}, , sin θ_{n})′. Then model (25) can be rewritten in matrix form as

$$z={X}_{\gamma}+{Z}_{u}+\u03f5,$$

where *X* = (**1**_{m} *I*_{2} **1**_{n} : *I _{m}* Φ),

$$\text{Cov}(z)={\sigma}^{2}{I}_{2\mathit{\text{mn}}}+{\sigma}_{0}^{2}({I}_{2m}\otimes {\mathbf{1}}_{n}{\mathbf{1}}_{n}^{\prime}).$$

It is easy to verify that *P _{X}P_{Z}* =

$$\{\begin{array}{c}\overline{c}=\frac{1}{n}{\displaystyle {\sum}_{j=1}^{n}\text{cos}\phantom{\rule{thinmathspace}{0ex}}{\theta}_{j}=0,}\hfill \\ \overline{s}=\frac{1}{n}{\displaystyle {\sum}_{j=1}^{n}\text{sin}\phantom{\rule{thinmathspace}{0ex}}{\theta}_{j}=0.}\hfill \end{array}$$

(26)

According to Theorems 3.1 and 3.2, the SDE and ANOVAE of the variance components $({\sigma}_{0}^{2},{\sigma}^{2})$ are equal if (26) holds. And the condition = = 0 can be easily satisfied by sampling *n* measurements equally spaced around the circumference of the circular feature.

Consider the popular model for longitudinal data:

$${y}_{\mathit{\text{it}}}={\beta}_{0}+{x}_{\mathit{\text{it}}}{\beta}_{1}+{u}_{i}+{\epsilon}_{\mathit{\text{it}}},\text{}i=1,2,\dots ,N,\text{}t=1,2,\dots ,T,$$

(27)

where *u _{i}* is the random effect,

Note that

$$\text{tr}({Q}_{X}{P}_{Z})=N-1-{s}_{b}(x),\text{}{[\text{tr}({Q}_{X}{P}_{Z})]}^{2}=N-1-2{s}_{b}(x)+{s}_{b}^{2}(x),$$

where

$${s}_{b}(x)=\raisebox{1ex}{$T{\displaystyle \sum _{i=1}^{N}{({\overline{x}}_{i.}-{\overline{x}}_{\u2025})}^{2}}$}\!\left/ \!\raisebox{-1ex}{$\sum _{i=1}^{N}{\displaystyle \sum _{t=1}^{T}{({x}_{\mathit{\text{it}}}-{\overline{x}}_{\u2025})}^{2}},$}\right.$$

${\overline{x}}_{i.}={\displaystyle {\sum}_{t=1}^{T}{x}_{\mathit{\text{it}}}/T}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\overline{x}}_{\u2025}={\displaystyle {\sum}_{i=1}^{N}{\displaystyle {\sum}_{t=1}^{T}{x}_{\mathit{\text{it}}}/(\mathit{\text{TN}})}}$. Thus tr(*P _{X}P_{Z}*) = [tr(

- 0 ≤
*s*(_{b}*x*) ≤ 1 *s*(_{b}*x*) = 0 if and only if_{1}. =_{2}. = =_{N}.*s*(_{b}*x*) = 1 if and only if*x*=_{i}*c*_{i}**1**_{T},*i*= 1, ,*N*

, we can show that *P _{Z}P_{X}* =

It follows from Theorem 3.3 that the necessary and sufficient condition for the identity of the SDE and ANOVAE of ${\sigma}_{u}^{2}$ under model (27) is _{1}. = _{2}. = = * _{N}*. or

In the following, we compare the the SDE and ANOVAE of ${\sigma}_{u}^{2}$ under general case 0 < *s _{b}*(

Clearly, if 0 < *s _{b}* < 1, then rk(

$$f({s}_{b})=\frac{(n-2)(N-1)(N-2)}{(n-3){(N-1-{s}_{b})}^{2}},$$

which is a continuous and increasing function of *s _{b}* in the interval (0, 1), and for any

$$\begin{array}{c}\underset{{s}_{b}\to {0}_{+}}{\text{lim}}f({s}_{b})=\frac{(n-2)(N-2)}{(n-3)(N-1)}=1-\frac{N(T-1)-1}{(n-3)(N-1)}<1,\\ \underset{{s}_{b}\to {1}_{\_}}{\text{lim}}f({s}_{b})=\frac{(n-2)(N-1)}{(n-3)(N-2)}>1.\end{array}$$

So there exists *s*_{b0} (0, 1) such that *f*(*s _{b}*) < 1 if

$${s}_{{b}_{0}}=N-1-\sqrt{\frac{(N-1)(N-2)(n-2)}{(n-3)}}.$$

(28)

From Corollary 4.1, it follows that the ANOVAE of ${\sigma}_{u}^{2}$ is superior to the SDE if 0 < *s _{b}* <

Because

$$0.5-\frac{1}{2T}<{s}_{{b}_{0}}<0.5+\frac{1}{2N-3},$$

it follows that (i) *f*(*s _{b}*) < 1 if

Combining with Theorem 4.2, we conclude that for any ${\sigma}_{\epsilon}^{2}>0,{\sigma}_{u}^{2}\ge 0$ the ANOVAE of ${\sigma}_{u}^{2}$ is superior to the SDE if *s _{b}* < 0.5 − 1/(2

Under model (27), *T* and *Ts _{b}* both are distinct nonzero eigenvalues of the matrix

We first obtain two sets of simulated values, (0.345, 0.280, 0.039) and (0.732, 0.741, 0.431), of *s _{b}* under the three sample sizes of (N,T), respectively, for

Secondly, we take random numbers from the independent χ^{2} distributions ${\chi}_{{m}_{1}}^{2},{\chi}_{1}^{2}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\chi}_{b}^{2}$ with 10000 replicates for each of the six cases above. Using formula (18) and (22), we can obtain simulated test powers of *F*(θ_{0}) and *W*(θ_{0}), see Table 1, Table 2.

Table 1 and Table 2 show that the test based on *W*(θ_{0}) is more powerful than that based on *F*(θ_{0}) for the null hypothesis *H*_{0} : θ = θ_{0} under sample size (*N, T*) = (5, 10), a scenario with relatively large number of repeated observations for each subject but small number of subjects. There is no substantial difference between the power of the two tests above under the sample size (*N, T*) = (12, 3) or (20, 4), a scenario with relatively small number of repeated observations for each subject but large number of subjects, regardless of wether the ANOVAE is superior to the SDE. As a subsequent observation, the result indicates that better estimates of variance components do not necessarily imply higher power of the tests. A similar phenomenon is observed in the expected lengths of confidence interval (24) and (23); see Figure 1 below.

Expected lengths of exact confidence intervals based on *F*(θ_{0}) (solid line) and *W*(θ_{0}) (dash line).

Thus we can adopt the simpler test statistic *F*(θ_{0}) and confidence interval (23) in practice when *T*, the number of repeated observations for each subject, is relatively small. However they are not good choices due to the low power of the test and the long length of the confidence interval when *T* is relatively large, especially, when *T* is larger than *N*, the number of subjects.

In this paper, we study the finite sample properties of ANOVAE and SDE under model (1). We establish the necessary and sufficient condition for the equality of the two estimates, and some sufficient conditions for the superiority of one estimate over the other under the mean squared error criterion. Furthermore, we consider exact tests and confidence intervals based on the two estimates, and demonstrate via simulations that better estimates of variance components do not necessarily imply higher power of the tests or shorter confidence intervals.

As comparison, the likelihood method has poor small sample size performance in mixed models (see Burdick, 1997). Moreover, when the parameter lies in the boundary of the parametric space, the likelihood-based tests such as Wald and likelihood ratio tests, do not asymptotically follow chi-square distributions as demonstrated in Stram and Lee (1994) and Crainiceanu and Ruppert (2004). For the two models considered in Section 7 with the same settings, Table 3 presents the coverage probabilities of the confidence intervals based on the maximum likelihood estimate. The results in the table clearly indicate that with relatively small sample sizes, the coverage probabilities are substantially below the nominal level of 95%.

As noticed in the literature, both ANOVAE and SDE may take negative values. This is not a problem for Genetic field in which variance component is permitted to be negative, see Burton *et al.* (1999) and Hazelton and Gurrin (2003). However, estimates of variance components are often required to be non-negative in many other cases. One remedy is to estimate the two variance components using $\text{max}\{0,{\widehat{\sigma}}_{u}^{2}\}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\text{max}\{0,{\tilde{\sigma}}_{u}^{2}\}$. However further research is needed to find conditions for the superiority among the two truncated estimates. In this regard Theorem 4.1 in this paper is a potentially useful tool for such investigation.

This research was supported by the Intramural Research Program of the National Institute of Child Health and Human Development, National Institutes of Health. Wu’s research was also partially supported by Funding Project for Academic Human Resources Development in Institutions of Higher Learning Under the Jurisdiction of Beijing Municipality PHR (IHLB) and National Natural Science Foundation of China (NSFC). The opinions expressed in the article are not necessarily of the National Institutes of Health. The authors wish to thank two referees for their valuable suggestions which considerably improved the paper.

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