A correlation structure for the analysis of Gaussian and non-Gaussian responses in crossover experimental designs with repeated measures

Abstract

In this paper, we propose a family of correlation structures for crossover designs with repeated measures for both, Gaussian and non-Gaussian responses using generalized estimating equations (GEE). The structure considers two matrices: one that models between-period correlation and another one that models within-period correlation. The overall correlation matrix, which is used to build the GEE, corresponds to the Kronecker between these matrices. A procedure to estimate the parameters of the correlation matrix is proposed, its statistical properties are studied and a comparison with standard models using a single correlation matrix is carried out. A simulation study showed a superior performance of the proposed structure in terms of the quasi-likelihood criterion, efficiency, and the capacity to explain complex correlation phenomena patterns in longitudinal data from crossover designs.

Appendix A

Theorem 2

The estimator of $\pmb {R}(\pmb {\alpha })$ given by $\pmb {{\hat{\Psi }}}\otimes \pmb {R}_1(\pmb {{\hat{\alpha }}}_1)$ is asymptotically unbiased and consistent.

Proof

First, asymptotic properties of Pearson’s residuals concerning expectation, variance, and residuals are explored. Define the theoretical Pearson residuals of response of the kth of the ith experimental unit at the jth period as:

$$\begin{aligned} R_{ijk}=\frac{Y_{ijk}-\mu _{ijk}}{{\phi }\sqrt{V(\mu _{ijk})})} \end{aligned}$$

(A1)

and adapting the results of Cox and Snell (1968) it is true that:

$$\begin{aligned} E(R_{ijk})&=\sum _{l=1}^pB({\hat{\beta }}_l)E(H_l^{(ijk)})\nonumber \\&-\sum _{l,s=1}^p {\mathcal {K}}^{ls}E\left( H_l^{(ijk)}U_s^{(ijk)}+\frac{1}{2}H_{ls}^{(ijk)} \right) + O(n^{-1})\end{aligned}$$

(A2)

$$\begin{aligned} Var(R_{ijk})&=1+2\sum _{l=1}^pB(\hat{\beta _{l}})E(R_{ijk}H_l^{(ijk)})\nonumber \\&-\sum _{l,s=1}^p {\mathcal {K}}^{ls}E\left( 2R_{ijk}H_l^{(ijk)}U_s^{(ijk)}\right) -\nonumber \\&\sum _{l,s=1}^p {\mathcal {K}}^{ls}E\left( H_l^{(ijk)}H_s^{(ijk)} R_{ijk}H_{ls}^{(ijk)} \right) + O(n^{-1}) \end{aligned}$$

(A3)

where

$$\begin{aligned} H_l^{(ijk)}=\frac{\partial R_{ijk}}{\partial \beta _l}, \; H_{ls}^{(ijk)}=\frac{\partial ^2 R_{ijk}}{\partial \beta _l\partial \beta _s} \end{aligned}$$

and ${\mathcal {K}}^{ls}$ is the element at position (l, s) of the inverse of the Fisher information matrix, and according to Cordeiro and McCullagh (1991), the bias of $\hat{\pmb {\beta }}$ ($B(\hat{\pmb {\beta }})$) is given by:

$$\begin{aligned} B(\hat{\pmb {\beta }})&=-\frac{1}{2\phi }\pmb {(X^TWX)}\pmb {X}^T\pmb {D}(z_{ijk})\pmb {F}\pmb {1}\end{aligned}$$

(A4)

$$\begin{aligned} \pmb {W}&=\pmb {D}(V(Y_{ijk})^{\frac{1}{2}}) \pmb {R}(\pmb {\alpha }) \pmb {D}(V(Y_{ijk})^{\frac{1}{2}}) \pmb {D}\left( \frac{\partial \pmb {\mu }_i}{\partial \eta }\right) \nonumber \\ \pmb {F}&=\pmb {D}\left( V(\mu _{ijk})^{-1}\left( \frac{\partial \mu _{ijk}}{\partial \eta _{ijk}}\right) \left( \frac{\partial ^2 \mu _{ijk}}{\partial \eta ^2_{ijk}}\right) \right) \end{aligned}$$

(A5)

where $\pmb {1}$ is a vector of appropriate size whose entries are all equal to 1, $\pmb {D}(z_{ijk})$ is a diagonal matrix with elements on the diagonal given by the variance of the estimated linear predictors, i.e., the diagonal of the matrix

$$\begin{aligned} \pmb {z}=Var({\hat{\eta }}_{111}. \ldots , {\hat{\eta }}_{nPL}) \end{aligned}$$

(A6)

i.e., $z_{ijk}=Var({\hat{\eta }}_{ijk})$ and $\pmb {X}$ is the design matrix of the parametric effects described in Equation (3). Now, computing the expected values by taking into account the properties of the exponential family, we obtain:

$$\begin{aligned} E\left( H_l^{(ijk)}\right)&=-\sqrt{\phi V(\mu _{ijk})} \left( \frac{\partial \mu _{ijk}}{\partial \eta _{ijk}}\right) x_{l(ijk)} \end{aligned}$$

(A7)

$$\begin{aligned} E\left( H_{ls}^{(ijk)}\right)&=\left[ 2V(\mu _{ijk})^{-\frac{3}{2}} \left( \frac{\partial V(\mu _{ijk})}{\partial \mu _{ijk}}\right) \left( \frac{\partial \mu _{ijk}}{\partial \eta _{ijk}}\right) ^2 \right. \nonumber \\&\left. -2V(\mu _{ijk})^{-\frac{1}{2}}\left( \frac{\partial ^2 \mu _{ijk}}{\partial \eta _{ijk}^2}\right) \right] \times \frac{1}{2}\sqrt{\phi }x_{l(ijk)}x_{s(ijk)}\end{aligned}$$

(A8)

$$\begin{aligned} E\left( H_l^{(ijk)}U_s^{(ijk)}\right)&=-\frac{1}{2}\phi ^{\frac{1}{2}} V(\mu _{ijk})^{-\frac{3}{2}} \left( \frac{\partial V(\mu _{ijk})}{\partial \mu _{ijk}}\right) \left( \frac{\partial \mu _{ijk}}{\partial \eta _{ijk}}\right) ^2\nonumber \\&\times x_{l(ijk)}x_{s(ijk)}\end{aligned}$$

(A9)

$$\begin{aligned} E\left( 2R_{ijk} H_l^{(ijk)}U_s^{(ijk)}\right)&=-V(\mu _{ijk})^{-2} \left( \frac{\partial V(\mu _{ijk})}{\partial \mu _{ijk}}\right) ^2 \left( \frac{\partial \mu _{ijk}}{\partial \eta _{ijk}}\right) ^2\nonumber \\&\times x_{l(ijk)}x_{s(ijk)} \nonumber \\&-2\phi \left( \frac{\partial V(\mu _{ijk})}{\partial \mu _{ijk}}\right) ^{-1} \left( \frac{\partial \mu _{ijk}}{\partial \eta _{ijk}}\right) ^2 x_{l(ijk)}x_{s(ijk)}\end{aligned}$$

(A10)

$$\begin{aligned} E\left( H_l^{(ijk)}H_s^{(ijk)}\right)&= \left[ \phi + \frac{\left( \frac{\partial V(\mu _{ijk})}{\partial \mu _{ijk}}\right) }{4V(\mu _{ijk})}\right] w_{ijk} x_{l(ijk)}x_{s(ijk)}\end{aligned}$$

(A11)

$$\begin{aligned} E\left( R_{ijk}H_{ls}^{(ijk)}\right)&= \frac{1}{2}\phi V(\mu _{ijk})^{-\frac{1}{2}} \left( \frac{\partial \mu _{ijk}}{\partial \eta _{ijk}}\right) \end{aligned}$$

(A12)

where $w_{ijk}$ is the element ijk of the diagonal of the matrix $\pmb {W}$ defined in Equation (A5). From Equation (A7) and the bias of $\pmb {\beta }$ given in Equation (A4), it follows that:

$$\begin{aligned} \sum _{l=1}^p B(\hat{\beta _{l}})E\left( H_{l}^{(ijk)}\right) =-\phi ^{\frac{1}{2}} V(\mu _{ijk})^{-\frac{1}{2}} \left( \frac{\partial \mu _{ijk}}{\partial \eta _{ijk}}\right) \pmb {e}_{ijk}\pmb {X}B(\hat{\pmb {\beta }}) \end{aligned}$$

(A13)

where $\pmb {e}_{ijk}$ is a vector of zeros with a 1’s at the ijkth position. From equations (A8) and (A9), we get:

$$\begin{aligned} E\left( H_l^{(ijk)}U_s^{(ijk)}+\frac{1}{2}H_{ls}^{(ijk)}\right)&=-\frac{1}{2}\phi ^{\frac{1}{2}} V(\mu _{ijk})^{-\frac{1}{2}} \left( \frac{\partial ^2 \mu _{ijk}}{\partial \eta _{ijk}^2}\right) ^2x_{l(ijk)}x_{s(ijk)} \nonumber \\ \sum _{l,s=1}^p{\mathcal {K}}^{ls}E(H_l^{(ijk)}U_s^{(ijk)}+\frac{1}{2}H_{ls}^{(ijk)})&=-\frac{1}{2}\phi ^{\frac{1}{2}} V(\mu _{ijk})^{-\frac{1}{2}} \left( \frac{\partial ^2 \mu _{ijk}}{\partial \eta _{ijk}^2}\right) ^2x_{l(ijk)}x_{s(ijk)} \end{aligned}$$

(A14)

and therefore, from Eqs. (A13), (A14) and (A2):

$$\begin{aligned} E(R_{111}, R_{112}, \ldots , R_{nPL})=\frac{-1}{2\sqrt{\phi }}(\pmb {I}-\pmb {H})\pmb {J}\pmb {z} \end{aligned}$$

(A15)

where

$$\begin{aligned} \pmb {H}&=\pmb {W}^{\frac{1}{2}} \pmb {X(X^TWX)}^{\frac{1}{2}}\pmb {X}^T \pmb {W}^{\frac{1}{2}}\\ \pmb {J}&=\pmb {D}\left( V(Y_{ijk}) \right) \pmb {D}\left( \frac{\partial \pmb {\mu }^2_i}{\partial ^2\eta }\right) \end{aligned}$$

from Eqs. (A10), (A11) and (A12):

$$\begin{aligned} -\sum _{l,s=1}^p {\mathcal {K}}^{ls}&E\left( 2R_{ijk}H_l^{(ijk)}U_s^{(ijk)}+H_l^{(ijk)}H_s^{(ijk)} R_{ijk}H_{ls}^{(ijk)} \right) \nonumber \\&=\left[ -\phi w_{ijk}-\frac{\left( \frac{\partial V(\mu _{ijk})}{\partial \mu _{ijk}}\right) \left( \frac{\partial ^2 \mu _{ijk}}{\partial \eta _{ijk}^2}\right) }{2 V(\mu _{ijk})}-\frac{1}{2}w_{ijk}\left( \frac{\partial ^2 V(\mu _{ijk})}{\partial \mu ^2_{ijk}}\right) \right] \frac{z_{ijk}}{\phi } \end{aligned}$$

(A16)

and from equations (A4) and (A12), it follows that:

$$\begin{aligned}&2\sum _{l=1}^pB({\hat{\beta }}_l)E(R_{ijk}H_{ls}^{(ijk)})\nonumber \\&=\frac{1}{2\phi }\frac{\left( \frac{\partial V(\mu _{ijk})}{\partial \mu _{ijk}}\right) \left( \frac{\partial ^2 \mu _{ijk}}{\partial \eta _{ijk}^2}\right) }{V(\mu _{ijk})}\pmb {e}_{ijk} \pmb {Z}\pmb {D}(z_{ii})\pmb {D}\left( V(\mu _{ijk})^{-1} \left( \frac{\partial ^2 \mu _{ijk}}{\partial \eta _{ijk}^2}\right) \left( \frac{\partial \mu _{ijk}}{\partial \eta _{ijk}}\right) \right) \pmb {1} \end{aligned}$$

(A17)

Therefore, from the results (A16) and (A17), we have that:

$$\begin{aligned} \left[ Var(R_{111}), Var(R_{112}), \ldots , Var(R_{nPL})\right] =\pmb {1}+\frac{1}{2\phi }\left( \pmb {QHJ-M}\right) \pmb {z} \end{aligned}$$

(A18)

where $\pmb {1}$ is a vector of ones and

$$\begin{aligned} \pmb {Q}&=\pmb {D}\left( V(Y_{ijk}) \right) ^{\frac{1}{2}}\pmb {D}\left( \frac{\partial V(Y_{ijk})}{\partial \mu }\right) \\ \pmb {M}&=\pmb {D}\left( V(Y_{ijk}) \right) ^{\frac{1}{2}}\pmb {D}\left( \frac{\partial V(Y_{ijk})^2}{\partial ^2\mu }+2\phi \pmb {D}(\pmb {W})+\frac{\frac{\partial V(Y_{ijk})}{\partial \mu }}{V(Y_{ijk})}\right) \end{aligned}$$

By theorem 2 of Liang and Zeger (1986), it is known that the GEE estimators of $\eta _{ijk}$ is consistent and unbiased, i.e.,

$$\begin{aligned} \pmb {Z}\xrightarrow [n\rightarrow \infty ]{p} \pmb {0} \end{aligned}$$

(A19)

Thus from (A15) and (A18), we find that:

$$\begin{aligned} E(R_{ijk})&=O(n^{-1})\end{aligned}$$

(A20)

$$\begin{aligned} Var(R_{ijk})&=1+O(n^{-1}) \end{aligned}$$

(A21)

and furthermore, by Sect. 3 of Cordeiro (2004) and equations (A20) and (A21), it follows that:

$$\begin{aligned} R_{ijk}\xrightarrow [n\rightarrow \infty ]{d} N(0,1) \end{aligned}$$

(A22)

Let

$$\begin{aligned} \pmb {\Gamma }=\{\gamma _{ij}\}_{n\times n} \end{aligned}$$

(A23)

be a matrix whose first column is $\frac{\pmb {1}}{\sqrt{n}}$ and the following columns are:

$$\begin{aligned} g_{i-1}=\left( \frac{1}{\sqrt{(i-1)i}}, \ldots ,\frac{1}{\sqrt{(i-1)i}},-\frac{i-1}{\sqrt{(i-1)i}}, 0, \ldots ,0\right) , \qquad i=2, \ldots , n \end{aligned}$$

(A24)

$$\begin{aligned} \pmb {\Gamma }^T=\begin{pmatrix} \frac{\pmb {1}}{\sqrt{n}}&\vdots&\pmb {G} \end{pmatrix}^T=\begin{pmatrix} \frac{1}{\sqrt{n}} &{} \frac{1}{\sqrt{n}}&{} \frac{1}{\sqrt{n}}&{} \frac{1}{\sqrt{n}} &{} \cdots &{}\frac{1}{\sqrt{n}}\\ \frac{1}{\sqrt{2}} &{} -\frac{1}{\sqrt{2}}&{} 0&{} 0 &{}\cdots &{} 0\\ \frac{1}{\sqrt{6}} &{} \frac{1}{\sqrt{6}}&{} -\frac{2}{\sqrt{6}}&{} 0 &{} \cdots &{}0\\ \frac{1}{\sqrt{12}} &{} \frac{1}{\sqrt{12}}&{} \frac{1}{\sqrt{12}}&{} -\frac{3}{\sqrt{12}} &{} \cdots &{}0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{}\vdots \\ \frac{1}{\sqrt{(n(n-1))}} &{} \frac{1}{\sqrt{n(n-1)}}&{} \frac{1}{\sqrt{n(n-1)}}&{} \frac{1}{\sqrt{n(n-1)}} &{} \cdots &{}-\frac{n-1}{\sqrt{n(n-1)}}\\ \end{pmatrix} \end{aligned}$$

where

$$\begin{aligned} \pmb {G}^T=\begin{pmatrix} \frac{1}{\sqrt{2}} &{} -\frac{1}{\sqrt{2}}&{} 0&{} 0 &{}\cdots &{} 0\\ \frac{1}{\sqrt{6}} &{} \frac{1}{\sqrt{6}}&{} -\frac{2}{\sqrt{6}}&{} 0 &{} \cdots &{}0\\ \frac{1}{\sqrt{12}} &{} \frac{1}{\sqrt{12}}&{} \frac{1}{\sqrt{12}}&{} -\frac{3}{\sqrt{12}} &{} \cdots &{}0\\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{}\vdots \\ \frac{1}{\sqrt{(n(n-1))}} &{} \frac{1}{\sqrt{n(n-1)}}&{} \frac{1}{\sqrt{n(n-1)}}&{} \frac{1}{\sqrt{n(n-1)}} &{} \cdots &{}-\frac{n-1}{\sqrt{n(n-1)}}\\ \end{pmatrix} \end{aligned}$$

then the matrix $\pmb {\Gamma }$ is a Helmert matrix (Lancaster 1965) and therefore:

$$\begin{aligned} \pmb {\Gamma }\pmb {\Gamma }^T=\pmb {I}_n, \qquad \pmb {1}_{n-1}^T \pmb {G}=0, \qquad \pmb {G}^T\pmb {G}=\pmb {I}_{n-1}-\frac{1}{n-1}\pmb {1}_{n-1}\pmb {1}_{n-1}^T \end{aligned}$$

(A25)

If $r_{ijk}$ is defined as the estimated Pearson residual of the ith experimental unit in the jth period and the kth observation, i.e.,

$$\begin{aligned}r_{ijk}={\hat{R}}_{ijk}=\frac{Y_{ijk}-{\hat{\mu }}_{ijk}}{{\hat{\phi }}\sqrt{V({\hat{\mu }}_{ijk})})}\end{aligned}$$

and the matrix $\pmb {r}_i$ of residuals of the ith individual, where the first row has the L Pearson residuals defined in Equation (8) corresponding to the first period, and the second row of the L corresponding to the second period and so on until completing a matrix with P rows and L columns, i.e.:

$$\begin{aligned} \pmb {r}_i=\begin{pmatrix} r_{i11}&{} r_{i21}&{} \cdots &{} r_{i1L}\\ r_{i21}&{} r_{i22}&{} \cdots &{}r_{i2L}\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ r_{iP1}&{} r_{i2P}&{} \cdots &{}r_{iPL}\\ \end{pmatrix} \end{aligned}$$

(A26)

By Equation (A20) and the correlation assumption given in Equation (6), it is true that:

$$\begin{aligned}&E( \pmb {r}_i)=\pmb {0}_{P\times L} \nonumber \\&Corr\left( Vec( \pmb {r}_i)\right) =\pmb {\Psi } \otimes \pmb {R}_1(\pmb {\alpha }_1)\nonumber \\&Corr\left( Vec( \pmb {r}_i), Vec( \pmb {r}_{i'})\right) =\pmb {0}_{PL\times PL}\, \qquad i\ne i' \end{aligned}$$

(A27)

And defining $\pmb {R}$ as:

$$\begin{aligned} \pmb {R}=(\pmb {r}_1,\ldots , \pmb {r}_n)_{P\times nL} \qquad \end{aligned}$$

and since $\pmb {\Gamma }$ is orthogonal, then $\pmb {\Gamma }\otimes \pmb {I}_L$ is also orthogonal. Thus (Srivastava et al. 2008):

$$\begin{aligned} \pmb {R}(\pmb {\Gamma }\otimes \pmb {I}_L)=\begin{pmatrix} \sqrt{n}\bar{\pmb {r}}&\vdots&\pmb {R}(\pmb {G}\otimes \pmb {I}_L) \end{pmatrix} \end{aligned}$$

(A28)

and according to Equation (A27), we get:

$$\begin{aligned} \pmb {R}\left( \pmb {I}_n\otimes \pmb { \Psi }^{-1} \right) \pmb {R}^T&=(\pmb {r}_1,\ldots , \pmb {r}_n)\left( \pmb {I}_n\otimes \pmb { \Psi }^{-1} \right) (\pmb {r}_1,\ldots , \pmb {r}_n)^T \\&=n\bar{\pmb {r}}\otimes \pmb { \Psi }^{-1} \pmb {R} + \pmb {R}(\pmb {G}\otimes \pmb {I}_L)\left( \pmb {I}_n\otimes \pmb { \Psi }^{-1} \right) (\pmb {G}^T\otimes \pmb {I}_L )\pmb {R}^T\\&=n\bar{\pmb {r}}\otimes \pmb { \Psi }^{-1} \pmb {R}+\pmb {Z}(\pmb {G}^T\otimes \pmb {I}_L )\pmb {Z}^T \end{aligned}$$

where $\pmb {Z}$ is:

$$\begin{aligned}&\pmb {Z}_{P\times (n-1)L}=(\pmb {Z}_1, \ldots , \pmb {Z}_{(n-1)})= \pmb {R}(\pmb {G}\otimes \pmb {I}_L)=(\pmb {r}_1,\ldots , \pmb {r}_n)(\pmb {G}\otimes \pmb {I}_L) \end{aligned}$$

(A29)

with $\bar{\pmb {r}}$ is the matrix of the average residuals defined in Eq. (A26) for each period, that is,

$$\begin{aligned} \bar{\pmb {r}}=\frac{1}{n}\sum _{i=1}^n \pmb {r}_i =\frac{1}{n}\sum _{i=1}^n\begin{pmatrix} r_{i11}&{} r_{i21}&{} \cdots &{} r_{i1L}\\ r_{i21}&{} r_{i22}&{} \cdots &{}r_{i2L}\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ r_{iP1}&{} r_{i2P}&{} \cdots &{}r_{iPL}\\ \end{pmatrix} \end{aligned}$$

and

$$\begin{aligned}&\pmb {Z}_1=\begin{pmatrix} \frac{1}{\sqrt{2}}r_{111}-\frac{1}{\sqrt{2}}r_{211} &{} \cdots &{} \frac{1}{\sqrt{2}}r_{11L}-\frac{1}{\sqrt{2}}r_{21L}\\ \frac{1}{\sqrt{2}}r_{121}-\frac{1}{\sqrt{2}}r_{221} &{} \cdots &{} \frac{1}{\sqrt{2}}r_{12L}-\frac{1}{\sqrt{2}}r_{22L}\\ \vdots &{}\ddots &{} \vdots \\ \frac{1}{\sqrt{2}}r_{1P1}-\frac{1}{\sqrt{2}}r_{2P1} &{} \cdots &{} \frac{1}{\sqrt{2}}r_{1PL}-\frac{1}{\sqrt{2}}r_{2PL} \end{pmatrix}_{P\times L}\\&\pmb {Z}_2=\begin{pmatrix} \frac{1}{\sqrt{6}}\sum _{i=1}^2 r_{i11}-\frac{2}{\sqrt{6}}r_{311} &{} \cdots &{} \frac{1}{\sqrt{6}}\sum _{i=1}^2 r_{i1L}-\frac{2}{\sqrt{6}}r_{31L} \\ \frac{1}{\sqrt{6}}\sum _{i=1}^2 r_{i21}-\frac{2}{\sqrt{6}}r_{321} &{} \cdots &{} \frac{1}{\sqrt{6}}\sum _{i=1}^2 r_{i2L}-\frac{2}{\sqrt{6}}r_{32L} \\ \vdots &{} \vdots &{}\ddots &{} \vdots \\ \frac{1}{\sqrt{6}}\sum _{i=1}^2 r_{iP1}-\frac{2}{\sqrt{6}}r_{3P1} &{} \cdots &{} \frac{1}{\sqrt{6}}\sum _{i=1}^2 r_{iPL}-\frac{2}{\sqrt{6}}r_{3PL} \\ \end{pmatrix}_{P\times L}\\&\vdots \\&\pmb {Z}_{(n-1)}=\\&\begin{pmatrix} {\scriptscriptstyle \frac{1}{\sqrt{n(n-1)}}\sum _{i=1}^{n-1} r_{i11}-\frac{n-1}{\sqrt{n(n-1)}}r_{(n-1)11}} &{} \cdots &{} {\scriptscriptstyle \frac{1}{\sqrt{n(n-1)}}\sum _{i=1}^{n-1}r_{i1L}-\frac{n-1}{\sqrt{n(n-1)}}r_{(n-1)1L}} \\ {\scriptscriptstyle \frac{1}{\sqrt{n(n-1)}}\sum _{i=1}^{n-1} r_{i21}-\frac{n-1}{\sqrt{n(n-1)}}r_{(n-1)21}} &{} \cdots &{} {\scriptscriptstyle \frac{1}{\sqrt{n(n-1)}}\sum _{i=1}^{n-1} r_{i2L}-\frac{n-1}{\sqrt{n(n-1)}}r_{(n-1)2L}} \\ \vdots &{} \vdots &{}\ddots &{} \vdots \\ {\scriptscriptstyle \frac{1}{\sqrt{n(n-1)}}\sum _{i=1}^{n-1} r_{iP1}-\frac{n-1}{\sqrt{n(n-1)}}r_{(n-1)P1}} &{} \cdots &{} {\scriptscriptstyle \frac{1}{\sqrt{n(n-1)}}\sum _{i=1}^{n-1} r_{iPL}-\frac{n-1}{\sqrt{n(n-1)}}r_{(n-1)PL}} \\ \end{pmatrix}_{P\times L} \end{aligned}$$

Now by the properties of the Pearson residuals, we have that:

$$\begin{aligned}E(\pmb {Z}_1)&=E(\pmb {Z}_2) = \cdots =E(\pmb {Z}_{n-1}) =\pmb {0}_{P\times L} \end{aligned}$$

and by the properties given in Equation (A25) and because we assume that the experimental units are independent, that is, $Corr(r_{ijk}, r_{i'j'k'})=0$, for all $i\ne i'$ and that Equation (3) is true, then:

$$\begin{aligned}&Corr(r_{ijk}, r_{i'j'k'})=0 \qquad \forall i\ne i'\\&Corr(r_{ijk}, r_{ij'k'})=Corr(r_{i'jk}, r_{i'j'k'})=Corr(r_{i'jk}, r_{i'j'k'}), \qquad \forall i\ne i'\\ \qquad \\&Corr\left( \frac{1}{\sqrt{2}}r_{111}-\frac{1}{\sqrt{2}}r_{211}, \frac{1}{\sqrt{2}}r_{121}-\frac{1}{\sqrt{2}}r_{221}\right) \\&=\frac{1}{2}Corr(r_{111}, r_{121})+\frac{1}{2}Cov(r_{211}, r_{221})\\&=Corr(r_{111}, r_{121})=Corr(r_{111}, r_{121})\\ \qquad \\&Corr\left( \frac{1}{\sqrt{n(n-1)}}\sum _{i=1}^{n-1} r_{i11}-\frac{n-1}{\sqrt{n(n-1)}}r_{(n-1)11}, \right. \\&\left. \frac{1}{\sqrt{n(n-1)}}\sum _{i=1}^{n-1} r_{i21}-\frac{n-1}{\sqrt{n(n-1)}}r_{(n-1)21}\right) \\&=Corr(r_{111}, r_{121})=Corr(r_{111}, r_{121})\\ \end{aligned}$$

furthermore,

$$\begin{aligned} Var(Vec(\pmb {Z}_1))&=Var\left\{ \begin{pmatrix} \frac{1}{\sqrt{2}}r_{111}-\frac{1}{\sqrt{2}}r_{211} \\ \frac{1}{\sqrt{2}}r_{121}-\frac{1}{\sqrt{2}}r_{221}\\ \vdots \\ \frac{1}{\sqrt{2}}r_{1P1}-\frac{1}{\sqrt{2}}r_{2P1}\\ \frac{1}{\sqrt{2}}r_{112}-\frac{1}{\sqrt{2}}r_{212}\\ \vdots \\ \frac{1}{\sqrt{2}}r_{1PL}-\frac{1}{\sqrt{2}}r_{2PL} \end{pmatrix}_{PL\times 1} \right\} \\&=\begin{pmatrix} 1 &{} Corr(r_{111}, r_{121}) &{} \cdots &{} Corr(r_{111}, r_{1P1})\\ Corr(r_{111}, r_{121}) &{} 1 &{} \cdots &{} Corr(r_{121}, r_{1P1})\\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ Corr(r_{111}, r_{1P1}) &{} Corr(r_{121}, r_{1P1}) &{}\cdots &{}1 \end{pmatrix}_{PL\times PL}\\&=\pmb { \Psi } \otimes \pmb {R}_1(\pmb {\alpha }_1)) \end{aligned}$$

$$\begin{aligned} Var(Vec(\pmb {Z}_1))=Var(Vec(\pmb {Z}_2))=\cdots = Var(Vec(\pmb {Z}_{(n-1)}))=\pmb { \Psi } \otimes \pmb {R}_1(\pmb {\alpha }_1)) \end{aligned}$$

(A30)

By the central limit theorem, we get that:

$$\begin{aligned} Vec(\bar{\pmb {r}})&\xrightarrow [n\rightarrow \infty ]{d} N_{LP}(\pmb {0}, \pmb { \Psi } \otimes \pmb {R}_1(\pmb {\alpha }_1)) \end{aligned}$$

(A31)

By equations (A22) and (A30) it follows that:

$$\begin{aligned} Vec(\pmb {Z}_j)&\xrightarrow [n\rightarrow \infty ]{d} N_{LP}(\pmb {0},\pmb { \Psi } \otimes \pmb {R}_1(\pmb {\alpha }_1)) \end{aligned}$$

(A32)

and by equations (A25), (A31) y (A32) that:

$$\begin{aligned} Cov&(Vec(\bar{\pmb {r}}),Vec(\pmb {Z}_j) )=\pmb {0}\\ Cov&(Vec(\pmb {Z}_i) ,Vec(\pmb {Z}_j) )=\pmb {0} \end{aligned}$$

and partitioning $\pmb {Z}_1$ as follows:

$$\begin{aligned} \pmb {Z}_1&=\begin{pmatrix} \frac{1}{\sqrt{2}}r_{111}-\frac{1}{\sqrt{2}}r_{211} &{} \cdots &{} \frac{1}{\sqrt{2}}r_{11L}-\frac{1}{\sqrt{2}}r_{21L}\\ \frac{1}{\sqrt{2}}r_{121}-\frac{1}{\sqrt{2}}r_{221} &{} \cdots &{} \frac{1}{\sqrt{2}}r_{12L}-\frac{1}{\sqrt{2}}r_{22L}\\ \vdots &{}\ddots &{} \vdots \\ \frac{1}{\sqrt{2}}r_{1P1}-\frac{1}{\sqrt{2}}r_{2P1} &{} \cdots &{} \frac{1}{\sqrt{2}}r_{1PL}-\frac{1}{\sqrt{2}}r_{2PL} \end{pmatrix}_{P\times L}\\&=\begin{pmatrix} \begin{pmatrix} \frac{1}{\sqrt{2}}r_{111}-\frac{1}{\sqrt{2}}r_{211}\\ \frac{1}{\sqrt{2}}r_{121}-\frac{1}{\sqrt{2}}r_{221} \\ \vdots \\ \frac{1}{\sqrt{2}}r_{1P1}-\frac{1}{\sqrt{2}}r_{2P1} \end{pmatrix}_{P\times 1}&{} \cdots &{}\begin{pmatrix} \frac{1}{\sqrt{2}}r_{11L}-\frac{1}{\sqrt{2}}r_{21L}\\ \frac{1}{\sqrt{2}}r_{1PL}-\frac{1}{\sqrt{2}}r_{2PL}\\ \vdots \\ \frac{1}{\sqrt{2}}r_{1PL}-\frac{1}{\sqrt{2}}r_{2PL} \end{pmatrix}_{P\times 1} \end{pmatrix}\\&=(\pmb {z}_{(1)1}, \ldots , \pmb {z}_{(L)1}) \end{aligned}$$

Then, it is obtained that

$$\begin{aligned} E(\pmb {z}_{(j)1}, \pmb {z}^T_{(j)1})=\Psi _{jj}\pmb {R}_1(\pmb {\alpha }_1) \qquad E(\pmb {Z}_1, \pmb {Z}^T_1)=(trace(\pmb {\Psi }))\pmb {R}_1(\pmb {\alpha }_1) \end{aligned}$$

Similarly, it is found that:

$$\begin{aligned}&E(\pmb {z}_{(j)1}, \pmb {z}^T_{(j)1})=\Psi _{jj}\pmb {R}_1(\pmb {\alpha }_1)\nonumber \\&E(\pmb {z}_{(j)2}, \pmb {z}^T_{(j)2})=\Psi _{jj}\pmb {R}_1(\pmb {\alpha }_1)\nonumber \\ \vdots \nonumber \\&E(\pmb {z}_{(j)(n-1)}, \pmb {z}^T_{(j)(n-1)})=\Psi _{jj}\pmb {R}_1(\pmb {\alpha }_1) \end{aligned}$$

(A33)

$$\begin{aligned}&E(\pmb {Z}_1, \pmb {Z}^T_1)=(trace(\pmb {\Psi }))\pmb {R}_1(\pmb {\alpha }_1)\nonumber \\ \vdots \nonumber \\&E(\pmb {Z}_{(n-1)}, \pmb {Z}^T_{(n-q)})=(trace(\pmb {\Psi }))\pmb {R}_1(\pmb {\alpha }_1) \end{aligned}$$

(A34)

Therefore, since $\pmb {\Psi }_{jj}=1$, $\forall j=1,\ldots , P$, $trace(\pmb {\Psi })=1$ from Equation (A33), we get:

$$\begin{aligned} E\left( \pmb {R}_1(\pmb {\alpha }_1)^{\frac{1}{2}}\pmb {z}_{(j')k} \pmb {z}^T_{(j)k} \pmb {R}_1(\pmb {\alpha }_1)^{\frac{1}{2}}\right) =\pmb {\Psi }_{jj'}\pmb {I}_P, \qquad \forall k=1, \ldots , n-1 \end{aligned}$$

(A35)

$$\begin{aligned} Cov[\pmb {R}_1(\pmb {\alpha }_1)^{\frac{1}{2}} \pmb {z}_{(k)j} \pmb {z}^T_{(i)j} \pmb {R}_1(\pmb {\alpha }_1)^{\frac{1}{2}}, \pmb {R}_1(\pmb {\alpha }_1)^{\frac{1}{2}} \pmb {z}_{(k)j'} \pmb {z}^T_{(i)j'} \pmb {R}_1(\pmb {\alpha }_1)^{\frac{1}{2}} ]=\pmb {0} \end{aligned}$$

(A36)

By Theorem 2 in Liang and Zeger (1986), it is known that $\pmb {R}_1(\hat{\pmb {\alpha }}_1)$ is consistent and unbiased for $\pmb {R}_1(\pmb { \alpha }_1)$, by equations (A35), (A31) and (A32), we have that

$$\begin{aligned} {\hat{\psi }}_{jj'}=\frac{1}{n} \sum _{i=1}^n tr\left( \pmb {R}_1(\pmb {{\hat{\alpha }}}_1)^{-1}(\pmb {r}_{(j)i}-\bar{\pmb {r}}_{(j)})(\pmb {r}_{(j')i}-\bar{\pmb {r}}_{(j')})^T\right) \end{aligned}$$

(A37)

is a consistent and asymptotically unbiased estimator for $\Psi _{jj'}$, which proves the theorem. $\square$