# Efficient designs for multivariate crossover trials

Shubham Niphadkar and Siuli Mukhopadhyay11footnotemark: 1 , Department of
Mathematics, Indian Institute of Technology Bombay, Mumbai 400 076,
IndiaCorresponding author; Department of Mathematics, Indian Institute of
Technology Bombay, Powai, Mumbai 400 076, India; Email<EMAIL_ADDRESS>
###### Abstract

This article aims to study efficient/trace optimal designs for crossover
trials, with multiple response recorded from each subject in each time period.
A multivariate fixed effect model is proposed with direct and carryover
effects corresponding to the multiple responses and the error dispersion
matrix allowing for correlations to exist between and within responses. Two
correlation structures, namely the proportional and the generalized markov
covariances are studied. The corresponding information matrices for direct
effects under the two covariances are used to determine efficient designs.
Efficiency of orthogonal array designs of Type $I$ and strength $2$ is
investigated for the two covariance forms. To motivate the multivariate
crossover designs, a gene expression data in a $3\times 3$ framework is
utilized.

Keywords— Completely symmetric; Correlated response; Markov-type covariance;
Orthogonal arrays; Proportional covariance; Trace optimal.

##  1 Introduction

In many biomedical and clinical studies, we frequently come across crossover
or repeated measure setups with $g$ responses, where $g>1$. For instance,
Grender and Johnson,  (1993) studies the effect of caffeine on the
mathematical ability of boys aged 14 and 16 years in a crossover framework. In
this study, two responses diastolic and systolic blood pressures, of each boy
are recorded in each of the two periods in a $2\times 2$ crossover design.
More recently, Leaker et al.,  (2017) considered a randomized, placebo
controlled, double-blind $3\times 3$ crossover trial, where in each period
gene expression values corresponding to multiple genes were recorded for each
subject, resulting in a multiple response crossover trial. Due to more than
one response being measured from the same subject at same and different time
periods, there is a necessity to account for both between and within response
correlations in the covariance structure. In this article, our objective is to
find trace optimal designs in a $p$-period and $t$\- treatment crossover setup
for $g$ correlated responses. Though these correlated response crossover
trials are popular in the statistical literature (see Grender and Johnson,
1993; Chinchilli and Esinhart,  1996; Tudor et al.,  2000; Pareek et al.,
2023), mostly estimation and testing of hypothesis for such trials have been
studied without any attention being paid to the actual designing of such
studies. To the best of our knowledge this is the first work in optimal
designs for such correlated response crossover experiments.

The key difficulty in finding optimal designs for such crossover models is
specifying the direct and cross covariance for the multiple responses. In the
direct covariance, correlations arising from within responses measured in
different time periods from each subject are acknowledged. This closely
resembles the covariance pattern we see in usual longitudinal/repeated
measures studies. To consider the relations between different responses
measured in the same and different time periods we use cross correlations.
Thus resembling a cokriging spatial experiment, only instead of spatial units,
here we have time periods. In this article, we propose a multivariate response
model for a $p,t\geq 3$ crossover setup with correlation within and between
responses. Two popular covariance structures from the statistical literature,
mainly proportional covariance and the markovian-type covariance (see Journel,
1999; Chiles and Delfiner,  2012; Dasgupta et al.,  2022), are utilized to
model the correlations within (direct covariance) and between (cross
covariance) responses. As in all crossover studies, subjects are assumed to be
uncorrelated. Information matrices for the proportional and markovian
structures are determined and later used to obtain optimal designs. We show
that these information matrices fail to satisfy the usual sufficient
conditions (Kiefer,  1975) and the conditions by Yeh,  (1986) for universal
optimal designs, leading us to instead search for trace optimal or efficient
crossover designs (Bate and Jones,  2006). For the proportional structure, an
orthogonal array design represented by $OA_{I}\left(n=\lambda
t\left(t-1\right),p=t,t,2\right)$, where $\lambda$ is a positive integer and
$t\geq 3$, is proved to be trace optimal/efficient. Under the more complicated
markovian structure, for $g=2$, i.e., a bivariate model, we explore high
efficiency design for various choices of within and between response
covariance matrices.

Various researchers namely, Matthews,  (1987, 1990), Kunert,  (1991), Stufken,
(1991), Kushner, 1997b ; Kushner,  (1998), Bate and Jones,  (2006) and Singh
and Kunert,  (2021) have studied efficient/highly efficient crossover designs.
While, universally optimal crossover designs have been discussed by Hedayat
and Afsarinejad,  (1978), Cheng and Wu,  (1980), Kunert,  (1984), Kunert and
Martin,  (2000), Hedayat and Yang,  (2003, 2004), to name a few. For a
detailed review of crossover designs, we would like to refer to the paper by
Bose and Dey,  (2013) and books by Senn,  (2002), Bose and Dey,  (2009) and
Kenward and Jones,  (2014). However, all of these works concentrate only on
single responses measured in each period. Very recently, Niphadkar and
Mukhopadhyay,  (2023) have determined universal optimality results for
multiple response crossover trials, where the responses are assumed to be
uncorrelated with a homoscedastic covariance matrix.

The original contributions of this article include (i) a multivariate response
crossover model with proportional and markovian covariance structures to model
correlations, (ii) information matrices for direct effects, (iii) trace
optimal design under the proportional covariance structure and (iv) highly
efficient designs for the markovian structure.

###  1.1 Motivating example: A gene expression study

We motivate our multivariate crossover model using a gene expression dataset
from Leaker et al.,  (2017). In this study, expressions from multiple genes
from the subjects are measured in a randomized, double-blind $3\times 3$
crossover setup. The main aim of the study was to understand the biomarkers of
mucosal inflammation and transcriptomics. Two doses of an oral drug were
compared with a placebo, thus resulting in a $3$ treatment trial. Each of the
subjects were assigned to any one of the $3$ treatment sequences; $ABC$, $CAB$
and $BCA$, where treatment $A$ represented the $10$ mg dose of the drug, while
$B$ and $C$ were the placebo and $25$ mg dose of the drug, respectively. In
this study cytokines, chemokines, and nasal mucosal gene expressions were
analyzed with main focus on type $2$ inflammation related to IL-$5$ and
IL-$13$. For the purpose of illustration, we considered measurements from $10$
genes in each of the periods. The information about these $10$ genes is given
in Table S.1. The dataset on gene profile is publicly available at the NCBI
Gene Expression Omnibus (Clough,  2016) with accession number $GSE67200$.
Figures 1(a) and 1(c) display the plot of average response values for each
gene, where the averaging is done over subjects. Figure 1(b) displays similar
plot with averaging done over periods. From Figures 1(a), 1(b) and 1(c) we
note that effects due to periods, subjects and treatments differ across genes.
In Figure 1(d) we plot the average response values corresponding to each gene.
From the varying height of the line segments we conclude that the mean effect
for each gene is distinct. We also studied the $10$ gene profiles in details
and noted significant correlations between multiple gene pairs.

Thus in the following section we propose a multivariate model that has varying
effects corresponding to each response variable and include correlation
between and within responses.

(a) Period effects

(b) Subject effects

(c) Treatment effects

(d) Overall mean

Figure 1: Plotting the period, subject, treatment and overall mean
corresponding to the $10$ genes.

##  2 A multivariate crossover model

We consider the class, $\varOmega_{t,n,p}$, of crossover designs with $t$
treatments, $n$ subjects and $p$ periods, where $t,p>2$. It is assumed that
measurements are recorded on $g\geq 1$ response variables from every subject
in each period. Suppose for a design $d\in\varOmega_{t,n,p}$, $Y_{dijk}$
represents the $k^{th}$ response measurement from the $j^{th}$ subject in the
$i^{th}$ period of the experiment, where $1\leq k\leq g$, $1\leq i\leq p$ and
$1\leq j\leq n$. We model $Y_{dijk}$ as:

$\displaystyle
Y_{dijk}=\mu_{k}+\alpha_{i,k}+\beta_{j,k}+\tau_{d\left(i,j\right),k}+\rho_{d\left(i-1,j\right),k}+\epsilon_{ijk},$
(2.1)

where corresponding to the $k^{th}$ response variable, $\mu_{k}$ is the
intercept effect, $\alpha_{i,k}$ is the $i^{th}$ period effect, $\beta_{j,k}$
is the $j^{th}$ subject effect, $\tau_{s,k}$ and $\rho_{s,k}$ are respectively
the direct effect and the first order carryover effect due to the $s^{th}$
treatment where $1\leq s\leq t$, and $d(i,j)$ denotes the treatment allocated
to the $j^{th}$ subject in the $i^{th}$ period. We assume that (2.1) is a
fixed effect model with a zero mean error term, $\epsilon_{ijk}$, and no
carryover effect in the first period. If $g=1$, then the above model becomes
an univariate response model similar to those found in the statistical
literature (see Bose and Dey,  2009).

Using matrix notations we rewrite the above model as:

$\displaystyle\begin{bmatrix}\bm{Y}^{{}^{\prime}}_{d1}&\ldots&\bm{Y}^{{}^{\prime}}_{dg}\end{bmatrix}^{{}^{\prime}}$
$\displaystyle=\bm{Z}_{d}\bm{\eta}+\begin{bmatrix}\bm{\epsilon}^{{}^{\prime}}_{1}&\ldots&\bm{\epsilon}^{{}^{\prime}}_{g}\end{bmatrix}^{{}^{\prime}},$
(2.2)

where
$\bm{Y}_{dk}=\left(Y_{d11k},\ldots,Y_{dp1k},\ldots,Y_{d1nk},\ldots,Y_{dpnk}\right)^{{}^{\prime}}$
and
$\bm{\epsilon}_{k}=\left(\epsilon_{11k},\ldots,\epsilon_{pnk}\right)^{{}^{\prime}},$
represent the observations and the error terms, respectively, corresponding to
the $k^{th}$ response, and $\bm{\eta}=$
$\setcounter{MaxMatrixCols}{13}\begin{bmatrix}\mu_{1}&\ldots&\mu_{g}&\bm{\alpha}^{{}^{\prime}}_{1}&\bm{\beta}^{{}^{\prime}}_{1}&\ldots&\bm{\alpha}^{{}^{\prime}}_{g}&\bm{\beta}^{{}^{\prime}}_{g}&\bm{\tau}^{{}^{\prime}}_{1}&\bm{\rho}^{{}^{\prime}}_{1}&\ldots&\bm{\tau}^{{}^{\prime}}_{g}&\bm{\rho}^{{}^{\prime}}_{g}\end{bmatrix}^{{}^{\prime}}$
is the parameter vector. The design matrix is given by,

$\displaystyle\bm{Z}_{d}$
$\displaystyle=\begin{bmatrix}\bm{I}_{g}\otimes\bm{{1}}_{np}&\bm{Z}_{1}&\bm{Z}_{2}\end{bmatrix},$
(2.3)

where $\bm{Z}_{1}=\bm{I}_{g}\otimes\bm{X}_{1}$,
$\bm{Z}_{2}=\bm{I}_{g}\otimes\bm{X}_{2}$,
$\bm{X}_{1}=\begin{bmatrix}\bm{P}&\bm{U}\end{bmatrix}$ and
$\bm{X}_{2}=\begin{bmatrix}\bm{T}_{d}&\bm{F}_{d}\end{bmatrix}$. For the
periods and subjects, we have $\bm{P}=\bm{{1}}_{n}\otimes\bm{I}_{p}$,
$\bm{U}=\bm{I}_{n}\otimes\bm{{1}}_{p}$, while $\bm{T}_{d}$ and $\bm{F}_{d}$
are the matrices for treatment and carryover effects, respectively. Note that
$\bm{F}_{d}=\left(\bm{I}_{n}\otimes\bm{\psi}\right)\bm{T}_{d}$, where
$\bm{\psi}=\begin{bmatrix}\bm{{0}}^{{}^{\prime}}_{p-1\times 1}&0\\\
\bm{I}_{p-1}&\bm{{0}}_{p-1\times 1}\end{bmatrix}$. The vector of errors is
assumed to have $E(\bm{\epsilon})=\bm{{0}}_{gnp}$, with dispersion

$\displaystyle\mathbb{D}\left(\bm{\epsilon}\right)$
$\displaystyle=\bm{\varSigma}=\begin{bmatrix}\bm{\varSigma}_{11}&\ldots&\bm{\varSigma}_{1g}\\\
\vdots&\vdots&\vdots\\\
\bm{\varSigma}_{g1}&\ldots&\bm{\varSigma}_{gg}\end{bmatrix},$ (2.4)

where
$\bm{\varSigma}_{k^{\prime}k}=\bm{\varSigma}_{kk^{\prime}}^{{}^{\prime}}$, for
$k\neq k^{\prime}=1,\ldots,g$. Here, the symmetric matrix
$\bm{\varSigma}_{kk}$ represent the variance of $\bm{\epsilon}_{k}$, while
$\bm{\varSigma}_{kk^{\prime}}$ represents the covariance between
$\bm{\epsilon}_{k}$ and $\bm{\epsilon}_{k^{\prime}}$. The matrix
$\bm{\varSigma}$ is assumed to be positive definite. Also, observations from
different subjects are taken to be uncorrelated, as is the usual practice (see
Bose and Dey,  2009).

Two different structures are considered to model the covariances between
responses, namely, the proportional structure and the generalized markov-type
structure. The generalized markov-type structure for the spatial data was
proposed recently by Dasgupta et al.,  (2022) and is a generalization of the
markov-type structure discussed in Journel,  (1999) and Chiles and Delfiner,
(2012). The proportional structure is simpler than the markovian one and is
used for the general $g>1$ case. However, in the statistical literature, the
markovian structure has been studied for only the $g=2$ case. We also restrict
our search for optimal designs for a markovian structure to $g=2$.

Proportional Structure:

Under the proportional structure the dispersion matrix $\bm{\varSigma}$ is as
follows:

$\displaystyle\bm{\varSigma}$
$\displaystyle=\bm{\varGamma}\otimes\left(\bm{I}_{n}\otimes\bm{V}\right),$

where $\bm{\varGamma}$ is assumed to be a $g\times g$ positive definite and
symmetric matrix having non-zero off diagonal elements, and $\bm{V}$ a known
positive definite and symmetric matrix. Under the proportional structure, the
within and between responses have similar covariance matrices differing by
some constants.

Generalized Markov-Type Structure:

Here, we consider $g=2$. Let $\sigma_{11}$ and $\sigma_{22}$, where
$\sigma_{11},\sigma_{22}>0$, be the respective variances of the first and
second responses and $\rho$, $0<|\rho|<1$, denotes the correlation coefficient
between the observations from the two responses measured from the same period.
Under the generalized markov-type structure the dispersion matrix
$\bm{\varSigma}$ is as follows:

$\displaystyle\bm{\varSigma}$
$\displaystyle=\begin{bmatrix}\bm{\varSigma}_{11}&\bm{\varSigma}_{12}\\\
\bm{\varSigma}_{21}&\bm{\varSigma}_{22}\end{bmatrix},$

where $\bm{\varSigma}_{11}=\bm{I}_{n}\otimes\bm{V}_{1}$,
$\bm{\varSigma}_{12}=\bm{\varSigma}_{21}=\rho\sqrt{\frac{\sigma_{22}}{\sigma_{11}}}\bm{\varSigma}_{11}$,
$\bm{\varSigma}_{22}=\rho^{2}\frac{\sigma_{22}}{\sigma_{11}}\bm{\varSigma}_{11}+\sigma_{22}(1-\rho^{2})\bm{\varSigma}_{R}$
and $\bm{\varSigma}_{R}=\bm{I}_{n}\otimes\bm{V}_{R}$. Here,
$\bm{V}_{1}=\sigma_{11}\bm{V}_{C}$, $\bm{V}_{C}$ is a known positive definite
and symmetric matrix, and $\bm{V}_{R}$ is a known correlation matrix. We see
that in this structure, direct covariances for both responses may differ,
while the cross covariances are proportional to the direct covariance within
the first response. Usually in cokriging spatial experiments the first
response is taken to be the primary response (Journel,  1999; Chiles and
Delfiner,  2012; Dasgupta et al.,  2022). In our case both responses are given
the same importance.

##  3 Search for efficient designs

We next determine a trace optimal/efficient design for the direct effects in
model (2.2). Using the idea as provided by Bate and Jones,  (2006), we say
that a design $d^{*}\in\mathcal{D}$, where $\mathcal{D}$ is a subclass of
designs, is a trace optimal/efficient design for the parameters of interest
over $\mathcal{D}$, if $d^{*}$ maximizes the trace of the corresponding
information matrix over $\mathcal{D}$.

For any design $d\in\varOmega_{t,n,p}$, let $\bm{C}_{d(s1)}$ and
$\bm{C}_{d(s2)}$ represent the information matrices of the direct effects for
the proportional and generalized markov-type structures, respectively. Lemmas
S.1 and S.3 show that the two information matrices have the following
representations:

$\displaystyle\bm{C}_{d(s1)}$
$\displaystyle=\bm{C}_{d(s1)(11)}-\bm{C}_{{d(s1)}(12)}\bm{C}^{-}_{{d(s1)}(22)}\bm{C}_{{d(s1)}(21)},$
(3.1) $\displaystyle\bm{C}_{d(s2)}$
$\displaystyle=\bm{C}_{{d(s2)}(11)(1)}-\bm{C}_{{d(s2)}(12)(1)}\bm{C}^{-}_{{d(s2)}(22)(1)}\bm{C}_{{d(s2)}(21)(1)},$
(3.2)

where
$\bm{C}_{d(s1)(11)}=\left(\bm{I}_{g}\otimes\bm{T}_{d}^{{}^{\prime}}\right)\bm{A}^{*}\left(\bm{I}_{g}\otimes\bm{T}_{d}\right)$,
$\bm{C}_{d(s1)(12)}=\bm{C}_{d(s1)(21)}^{{}^{\prime}}=\left(\bm{I}_{g}\otimes\bm{T}_{d}^{{}^{\prime}}\right)\bm{A}^{*}\left(\bm{I}_{g}\otimes\bm{F}_{d}\right)$,
$\bm{C}_{{d(s1)}(22)}=\left(\bm{I}_{g}\otimes\bm{F}^{{}^{\prime}}_{d}\right)\bm{A}^{*}\left(\bm{I}_{g}\otimes\bm{F}_{d}\right)$,
$\bm{C}_{{d(s2)}(11)(1)}=\left(\bm{I}_{2}\otimes\bm{T}_{d}^{{}^{\prime}}\right)\bm{A}^{*}\left(\bm{I}_{2}\otimes\bm{T}_{d}\right)$,
$\bm{C}_{{d(s2)}(12)(1)}=$
$\bm{C}^{{}^{\prime}}_{{d(s2)}(21)(1)}=\left(\bm{I}_{2}\otimes\bm{T}_{d}^{{}^{\prime}}\right)\bm{A}^{*}\left(\bm{I}_{2}\otimes\bm{F}_{d}\bm{H}_{t}\right)$,
$\bm{C}_{{d(s2)}(22)(1)}=\left(\bm{I}_{2}\otimes\bm{F}^{{}^{\prime}}_{d}\bm{H}_{t}\right)\bm{A}^{*}\left(\bm{I}_{2}\otimes\bm{F}_{d}\bm{H}_{t}\right)$,
and the matrix $\bm{A}^{*}$ is
$\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\bm{Z}_{1}\right)\bm{\varSigma}^{-1/2}$.
Here,
$\bm{H}_{t}=\bm{I}_{t}-\frac{1}{t}\bm{{1}}_{t}\bm{{1}}{t}^{{}^{\prime}}$. Note
that $\bm{A}^{*}$ depends on the structure of $\bm{\varSigma}$ and
$\bm{C}_{d(s1)}$ is a $gt\times gt$ matrix, while $\bm{C}_{d(s2)}$ is a
$2t\times 2t$ matrix.

###  3.1 Proportional covariance structure

Under the proportional covariance, we prove that for the $g>1$ setup, a design
represented by $OA_{I}\left(n=\lambda t\left(t-1\right),p=t,t,2\right)$, where
$\lambda$ is a positive integer and $t\geq 3$, maximizes the
$tr\left(\bm{C}_{d(s1)}\right)$ and hence is a trace optimal/efficient design.

We know from (3.1), that the information matrix for the direct effects depends
on the expression of $\bm{A}^{*}$. However, till date, there has been no work
discussing the explicit expression of $\bm{A}^{*}$ for bivariate/multivariate
responses in a crossover framework. Using Lemma S.2, the expression of the
matrix $\bm{A}^{*}$ under the proportional structure for the $g>1$ case is
obtained as

$\displaystyle\bm{A}^{*}$
$\displaystyle=\bm{\varGamma}^{-1}\otimes\left(\bm{H}_{n}\otimes\bm{V}^{*}\right),$

where
$\bm{V}^{*}=\bm{V}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{V}^{-1}\bm{J}_{p\times
p}\bm{V}^{-1}$, $\bm{J}_{p\times p}=\bm{{1}}_{p}\bm{{1}}_{p}^{{}^{\prime}}$
and $\bm{H}_{n}=\bm{I}_{n}-\frac{1}{n}\bm{{1}}_{n}\bm{{1}}_{n}^{{}^{\prime}}$.
In the next theorem, the expression of $\bm{A}^{*}$ is used to obtain the
information matrix for the direct effects under the proportional structure for
the multivariate setup.

######  Theorem 3.1.

The information matrix for the direct effects under $g>1$ case can be
expressed as

$\displaystyle\bm{C}_{d(s1)}$
$\displaystyle=\bm{\varGamma}^{-1}\otimes\bm{C}_{d(uni)},$ (3.3)

where
$\bm{C}_{d(uni)}=\bm{C}_{d(uni)(11)}-\bm{C}_{d(uni)(12)}\bm{C}_{d(uni)(22)}^{-}\bm{C}_{d(uni)(21)}$,
$\bm{C}_{d(uni)(11)}=\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{T}_{d}$,
$\bm{C}_{d(uni)(12)}=\bm{C}_{d(uni)(21)}^{{}^{\prime}}=\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{F}_{d}$,
and
$\bm{C}_{d(uni)(22)}=\bm{F}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{F}_{d}$.
Here,
$\bm{V}^{*}=\bm{V}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{V}^{-1}\bm{J}_{p\times
p}\bm{V}^{-1}$.

###### Proof.

The details of the proof are given in Appendix A.1. ∎

It should be noted $\bm{C}_{d(uni)}$ is the information matrix for the direct
effects when we have a univariate version of the model (2.1), i.e., $g=1$. To
check if $OA_{I}\left(n=\lambda t\left(t-1\right),p=t,t,2\right)$, where
$\lambda$ is a positive integer and $t\geq 3$, is universally optimal, we
explore complete symmetricity of the corresponding information matrix for
$g>1$ case.

######  Remark 3.1.

Let $d^{*}$ be a design given by $OA_{I}\left(n=\lambda
t\left(t-1\right),p=t,t,2\right)$, where $\lambda$ is a positive integer and
$t\geq 3$. Then for $g>1$, the information matrix for the direct effects
corresponding to $d^{*}$, $\bm{C}_{d^{*}(s1)}$, is not completely symmetric.

###### Proof.

The proof is given in Appendix A.1. ∎

From Remark 3.1, it is clear that the usual sufficient condition of complete
symmetricity similar to the one given by Kiefer,  (1975) for finding a
universally optimal design is not applicable here. Also, to satisfy the more
general conditions similar to those given by Yeh,  (1986), for $g>1$ case, we
need

$\displaystyle\begin{split}\frac{\gamma^{(11)}|\bm{E}|}{te_{22}}&=\cdots=\frac{\gamma^{(gg)}|\bm{E}|}{te_{22}},\text{
and}\\\
\frac{\gamma^{(kk)}|\bm{E}|}{te_{22}}&=\frac{\gamma^{(kk^{\prime})}|\bm{E}|}{te_{22}},\end{split}$

where $\gamma^{(kk^{\prime})}$ is the $(k,k^{\prime})^{th}$ element of the
matrix $\bm{\varGamma}^{-1}$. However, this condition is also seen to be
violated for the $g>1$ setup (see Proof of Remark 3.1). Note, for a
multivariate setup with no correlation between responses and a much simpler
covariance structure, Niphadkar and Mukhopadhyay,  (2023) have successfully
obtained universally optimal designs. However, this is not true when the
responses are correlated. Hence, for the multivariate response case, we resort
to identifying an efficient design in the next theorem.

######  Theorem 3.2.

Let $d^{*}\in\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$ be a design given by
$OA_{I}\left(n=\lambda t\left(t-1\right),p=t,t,2\right)$, where
$\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$ is a class of binary designs with
$p=t$, $\lambda$ is a positive integer and $t\geq 3$. Then $d^{*}$ is a trace
optimal/efficient design for the direct effects over
$\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$ for the $g>1$ case.

###### Proof.

Here, $d^{*}\in\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$ is a design given
by $OA_{I}\left(n=\lambda t\left(t-1\right),p=t,t,2\right)$, where
$\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$ is a class of binary designs with
$p=t$, $\lambda$ is a positive integer and $t\geq 3$. It has been shown by
Kunert and Martin,  (2000) that $d^{*}$ is universally optimal for the direct
effects over $\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$ in the univariate
response case. Note that universal optimality of a design over a subclass of
designs implies that the design maximizes the trace of the corresponding
information matrix over the same subclass of designs. Hence, $d^{*}$ maximizes
$tr\left(\bm{C}_{d(uni)}\right)$ over $\mathcal{D}^{(1)}_{t,n=\lambda
t(t-1),p=t}$, where $\bm{C}_{d(uni)}$ is the information matrix of the direct
effects for the univariate response case . For $g>1$, using the expression of
$\bm{C}_{d(s1)}$ from Theorem 3.1, we get

$\displaystyle tr\left(\bm{C}_{d(s1)}\right)$
$\displaystyle=tr\left(\bm{\varGamma}^{-1}\right)tr\left(\bm{C}_{d(uni)}\right).$
(3.4)

Here, $\bm{\varGamma}^{-1}$ is a positive definite matrix, and thus
$tr\left(\bm{\varGamma}^{-1}\right)>0$. So, from (3.4), we get that for $g>1$,
$d^{*}$ maximizes $tr\left(\bm{C}_{d(s1)}\right)$ over
$\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$. Thus $d^{*}$ is trace
optimal/efficient for the direct effects over $\mathcal{D}^{(1)}_{t,n=\lambda
t(t-1),p=t}$ for the $g>1$ case. ∎

###  3.2 Generalized markov-type covariance structure

For the markovian structure, we are able to show that the
$OA_{I}\left(n=\lambda t\left(t-1\right),p=t,t,2\right)$, where $\lambda$ is a
positive integer and $t\geq 3$, is a highly efficient design. To evaluate the
trace optimality/efficiency of these $OA_{I}\left(n=\lambda
t\left(t-1\right),p=t,t,2\right)$ designs, we obtain an upper bound of
$tr\left(\bm{C}_{d(s2)}\right)$, and state a necessary and sufficient
condition for a $OA_{I}\left(n=\lambda t\left(t-1\right),p=t,t,2\right)$, to
attain this upper bound of $tr\left(\bm{C}_{d(s2)}\right)$. Note, that if this
upper bound is attained then the design maximizes
$tr\left(\bm{C}_{d(s2)}\right)$ and hence is an efficient design. Also, for
various choices $\bm{V}_{1}$, $\bm{V}_{R}$ and $\rho$, we study the trace of
the information matrix for the $OA_{I}\left(n=\lambda
t\left(t-1\right),p=t,t,2\right)$ and the derived upper bound.

Proceeding as before, we see that the process of obtaining $\bm{A}^{*}$ is
more complicated as compared to the proportional structure. This is mainly
because under the Markovian structure, $\bm{\varSigma}_{22}$ is not
proportional to $\bm{\varSigma}_{11}$. By laborious calculation, (see Lemma
S.4), we obtain the expression of the matrix $\bm{A}^{*}$ under the
generalized markov-type structure as

$\displaystyle\bm{A}^{*}$
$\displaystyle=\begin{bmatrix}\bm{H}_{n}\otimes\bm{\varOmega}_{1}&-\bm{H}_{n}\otimes\bm{\varOmega}_{2}\\\
-\bm{H}_{n}\otimes\bm{\varOmega}_{2}&\bm{H}_{n}\otimes\bm{\varOmega}_{4}\end{bmatrix},$

where
$\bm{\varOmega}_{1}=\bm{V}_{1}^{*}+\frac{\bar{\rho}^{2}}{\sigma_{12}}\bm{V}_{R}^{*}$,
$\bm{\varOmega}_{2}=\frac{\bar{\rho}}{\sigma_{12}}\bm{V}_{R}^{*}$, and
$\bm{\varOmega}_{4}=\frac{1}{\sigma_{12}}\bm{V}_{R}^{*}$. Here,
$\bm{V}_{1}^{*}=\bm{V}_{1}^{-1}-$
$\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}_{1}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{V}_{1}^{-1}\bm{J}_{p\times
p}\bm{V}_{1}^{-1}$,
$\bm{V}_{R}^{*}=\bm{V}_{R}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}_{R}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{V}_{R}^{-1}\bm{J}_{p\times
p}\bm{V}_{R}^{-1}$,
$\bm{H}_{n}=\bm{I}_{n}-\frac{1}{n}\bm{{1}}_{n}\bm{{1}}_{n}^{{}^{\prime}}$,
$\bm{J}_{p\times p}=\bm{{1}}_{p}\bm{{1}}_{p}^{{}^{\prime}}$,
$\bar{\rho}=\rho\sqrt{\frac{\sigma_{22}}{\sigma_{11}}}$ and
$\sigma_{12}=\sigma_{22}\left(1-\rho^{2}\right)$. Note that further we use
$\bm{H}_{p}=\bm{I}_{p}-\frac{1}{p}\bm{{1}}_{p}\bm{{1}}_{p}^{{}^{\prime}}$.

Let $d^{*}$ be a design given by $OA_{I}\left(n=\lambda
t\left(t-1\right),p=t,t,2\right)$, where $\lambda$ is a positive integer and
$t\geq 3$. Using the above expression of $\bm{A}^{*}$, the information matrix
for the direct effects and its trace under the generalized markov-type
structure, corresponding to $d^{*}$, are obtained.

######  Theorem 3.3.

Let $d^{*}$ be a design given by $OA_{I}\left(n=\lambda
t\left(t-1\right),p=t,t,2\right)$, where $\lambda$ is a positive integer and
$t\geq 3$. Then under the generalized markov-type structure, the information
matrix for the direct effects can be expressed as

$\displaystyle\bm{{C}}_{d^{*}(s2)}$
$\displaystyle=\frac{n}{t-1}\begin{bmatrix}\bm{\varLambda}_{1}&\bm{\varLambda}_{2}\\\
\bm{\varLambda}_{2}&\bm{\varLambda}_{4}\end{bmatrix},$ (3.5)

where the matrices $\bm{\varLambda}_{1}$, $\bm{\varLambda}_{2}$ and
$\bm{\varLambda}_{4}$ are given as

$\displaystyle\begin{split}\bm{\varLambda}_{1}&=\left(tr\left(\bm{V}_{1}^{*}\right)+\frac{\bar{\rho}^{2}}{\sigma_{12}}tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{1}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)}-\frac{\bar{\rho}^{2}}{\sigma_{12}}\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right)\bm{H}_{t},\\\
\bm{\varLambda}_{2}&=-\frac{\bar{\rho}}{\sigma_{12}}\left(tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right)\bm{H}_{t},\text{
and}\\\
\bm{\varLambda}_{4}&=\frac{1}{\sigma_{12}}\left(tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right)\bm{H}_{t}.\end{split}$
(3.6)

Further, we have

$\displaystyle tr\left(\bm{{C}}_{d^{*}(s2)}\right)$
$\displaystyle=n\left(tr\left(\bm{V}_{1}^{*}\right)+\frac{1+\bar{\rho}^{2}}{\sigma_{12}}tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{1}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)}-\frac{1+\bar{\rho}^{2}}{\sigma_{12}}\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right).$
(3.7)

###### Proof.

The details of the proof are given in Appendix A.2. ∎

As in Subsection 3.1, we show next that the sufficient conditions for
universal optimality as similar to those discussed in Kiefer,  (1975) and Yeh,
(1986) are not applicable.

######  Remark 3.2.

Let $d^{*}$ be a design given by $OA_{I}\left(n=\lambda
t\left(t-1\right),p=t,t,2\right)$, where $\lambda$ is a positive integer and
$t\geq 3$. Then the information matrix for the direct effects corresponding to
a design $d^{*}$, $\bm{{C}}_{d^{*}(s2)}$, is not completely symmetric.

###### Proof.

For the proof see Appendix A.2. ∎

From Remark 3.2, it is clear that as in the proportional covariance case, here
also we cannot apply the usual sufficient condition for a universally optimal
design. For conditions similar to Yeh,  (1986), we require

$\frac{1}{t}\left(tr\left(\bm{V}_{1}^{*}\right)+\frac{\bar{\rho}^{2}}{\sigma_{12}}tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{1}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)}-\frac{\bar{\rho}^{2}}{\sigma_{12}}\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right)=\\\
\frac{1}{\sigma_{12}t}\left(tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right)=-\frac{\bar{\rho}}{\sigma_{12}t}\left(tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right).$

Since both conditions are not satisfied (see proof of Remark 3.2), we instead
focus on efficient designs. In order to check whether $d^{*}$ is an efficient
design, we obtain an upper bound of $tr\left(\bm{C}_{d(s2)}\right)$. If
$d^{*}$ attains this upper bound then we say that $d^{*}$ is an efficient
design. However, obtaining an upper bound for $tr\left(\bm{C}_{d(s2)}\right)$
is computationally challenging, so we instead consider
$\bm{\tilde{C}}_{d(s2)}$ (see Lemma S.5), the information matrix for the
direct effects with no period effects, and show that
$\bm{{C}}_{d(s2)}\leq\bm{\tilde{C}}_{d(s2)}$ with equality if $d$ is uniform
on periods (see Lemmas S.5 and S.7).

Note, $d^{*}\in\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$, where

$\displaystyle\mathcal{D}^{(1)}_{t,n=\lambda
t(t-1),p=t}=\\{d:d\in\varOmega_{t,n=\lambda t(t-1),p=t},\text{ }t\geq 3\text{
and }d\text{ is a binary design}\\}.$

Also, for $p=t$, a binary design is a design uniform on subjects, and thus
$\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$ is indeed a class of designs with
$p=t$ which are uniform on subjects.

######  Lemma 3.1.

Let $d\in\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$. Then,

$\displaystyle tr\left(\bm{{C}}_{d(s2)}\right)\leq
u\left(t,n,p,\sigma_{11},\sigma_{22},\rho,\bm{V}_{1},\bm{V}_{R}\right),$ (3.8)

where
$u\left(t,n,p,\sigma_{11},\sigma_{22},\rho,\bm{V}_{1},\bm{V}_{R}\right)\\\
=n\Bigg{(}tr\left(\bm{V}_{1}^{*}\right)+\frac{1+\bar{\rho}^{2}}{\sigma_{12}}tr\left(\bm{V}_{R}^{*}\right)-\left(tr\left(\bm{V}_{1}^{*}\bm{\psi}\right)+\frac{1+\bar{\rho}^{2}}{\sigma_{12}}tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}\times$
$\left(tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)+\frac{1+\bar{\rho}^{2}}{\sigma_{12}}tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{-1}\Bigg{)}$.
Here, $\bm{V}_{1}^{*}$, $\bm{V}_{R}^{*}$, $\bar{\rho}$ and $\sigma_{12}$ are
as defined before.

###### Proof.

Note that a design $d\in\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$, is a
binary design. In Lemma S.7, it has been shown that
$\bm{{C}}_{d(s2)}\leq\bm{\tilde{C}}_{d(s2)}$, so

$\displaystyle tr\left(\bm{{C}}_{d(s2)}\right)\leq
tr\left(\bm{\tilde{C}}_{d(s2)}\right).$ (3.9)

Using the expression of $\bm{\tilde{C}}_{d(s2)}$ from Lemma S.5 and applying
Lemma 5.1 from Kushner, 1997a , we get

$\displaystyle tr\left(\bm{\tilde{C}}_{d(s2)}\right)\leq
tr\left(\bm{\tilde{C}}_{d(s2)(11)}\right)-\left(tr\left(\bm{\tilde{C}}_{d(s2)(12)}\right)\right)^{2}\left(tr\left(\bm{\tilde{C}}_{d(s2)(22)}\right)\right)^{+},$
(3.10)

where

$\displaystyle\begin{split}tr\left(\bm{\tilde{C}}_{d(s2)(11)}\right)&=tr\left(\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{1}\right)\bm{T}_{d}\right)+tr\left(\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{4}\right)\bm{T}_{d}\right),\\\
tr\left(\bm{\tilde{C}}_{d(s2)(12)}\right)&=tr\left(\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{1}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\right)+tr\left(\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{4}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\right),\text{
and}\\\
tr\left(\bm{\tilde{C}}_{d(s2)(22)}\right)&=tr\left(\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{1}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\right)+tr\left(\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{4}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\right),\end{split}$
(3.11)

from Lemma S.5. Using Remark S.1, it is observed that
$\bm{{1}}_{p}^{{}^{\prime}}\bm{\varOmega}_{1}=\bm{{1}}_{p}^{{}^{\prime}}\bm{\varOmega}_{2}=\bm{{1}}_{p}^{{}^{\prime}}\bm{\varOmega}_{4}=\bm{{0}}_{1\times
p}$ and

$\displaystyle\begin{split}tr\left(\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{1}\right)\bm{T}_{d}\right)&=n\times
tr\left(\bm{\varOmega}_{1}\right),\\\
tr\left(\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{4}\right)\bm{T}_{d}\right)&=n\times
tr\left(\bm{\varOmega}_{4}\right),\\\
tr\left(\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{1}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\right)&=n\times
tr\left(\bm{\varOmega}_{1}\bm{\psi}\right),\\\
tr\left(\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{4}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\right)&=n\times
tr\left(\bm{\varOmega}_{4}\bm{\psi}\right),\\\
tr\left(\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{1}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\right)&=n\times
tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{1}\bm{\psi}\right),\text{
and}\\\
tr\left(\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{4}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\right)&=n\times
tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{4}\bm{\psi}\right).\end{split}$
(3.12)

Suppose we define, $c_{11}=n\left(c_{11(11)}+c_{11(22)}\right)$,
$c_{12}=n\left(c_{12(11)}+c_{12(22)}\right)$ and $c_{22}=n\big{(}c_{22(11)}+$
$c_{22(22)}\big{)}$, where $c_{11(11)}=tr\left(\bm{\varOmega}_{1}\right)$,
$c_{11(22)}=tr\left(\bm{\varOmega}_{4}\right)$,
$c_{12(11)}=tr\left(\bm{\varOmega}_{1}\bm{\psi}\right)$,
$c_{12(22)}=tr\left(\bm{\varOmega}_{4}\bm{\psi}\right)$,
$c_{22(11)}=tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{1}\bm{\psi}\right)$
and
$c_{22(22)}=tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{4}\bm{\psi}\right)$.
From Lemma S.4,
$\bm{\varOmega}_{1}=\bm{V}_{1}^{*}+\frac{\bar{\rho}^{2}}{\sigma_{12}}\bm{V}_{R}^{*}$
and $\bm{\varOmega}_{4}=\frac{1}{\sigma_{12}}\bm{V}_{R}^{*}$, where
$\bar{\rho}=\rho\sqrt{\frac{\sigma_{22}}{\sigma_{11}}}$ and
$\sigma_{12}=\sigma_{22}(1-\rho^{2})$. Using these expressions in $c_{22(11)}$
and $c_{22(22)}$, we obtain
$c_{22(11)}=tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)+\frac{\bar{\rho}^{2}}{\sigma_{12}}tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)$
and
$c_{22(22)}=\frac{1}{\sigma_{12}}tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)$.
Using Lemma S.8, $|\bar{\rho}|>0$ and $\sigma_{12}>0$ gives $c_{22(11)}$,
$c_{22(22)}>0$, implying $c_{22}=n\left(c_{22(11)}+c_{22(22)}\right)>0$.
Combining (3.9), (3.10), (3.11) and (3.12), we obtain

$\displaystyle tr\left(\bm{{C}}_{d(s2)}\right)$ $\displaystyle\leq
c_{11}-c_{12}^{2}c_{22}^{-1}.$ (3.13)

Note that since $c_{22}>0$, $c_{22}^{+}$ has been replaced by $c_{22}^{-1}$.
Using the expression of $c_{11}$, $c_{12}$ and $c_{22}$, and the expressions
of $\bm{\varOmega}_{1}$ and $\bm{\varOmega}_{4}$ in (3.13), we prove (3.8). ∎

######  Remark 3.3.

Let $d^{*}\in\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$ be a design given by
$OA_{I}\left(n=\lambda t\left(t-1\right),p=t,t,2\right)$, where $\lambda$ is a
positive integer and $t\geq 3$. Then $d^{*}$ attains
$u\left(t,n,p,\sigma_{11},\sigma_{22},\rho,\bm{V}_{1},\bm{V}_{R}\right)$ as
given in Lemma 3.1, where $n=\lambda t\left(t-1\right)$ and $p=t$, if and only
if
$tr\left(\bm{V}_{1}^{*}\bm{\psi}\right)tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)=tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)$.
Here, $\bm{V}_{1}^{*}$ and $\bm{V}_{R}^{*}$ are defined in Lemma 3.1.

###### Proof.

The proof of the above remark is given in Appendix B. ∎

From Remark 3.3, we note that if $\bm{V}_{1}$ is proportional to $\bm{V}_{R}$,
then the upper bound is attained, implying the attainment of the upper bound
for the proportional structure.

####  3.2.1 Study of design efficiency for the markovian covariance structure

In this section, for various choices for $\bm{V}_{1}$ and $\bm{V}_{R}$, we
evaluate the performance of $d^{*}$ on the basis of the upper bound of
$tr\left(\bm{C}_{d(s2)}\right)$. For the purpose of simplicity, we denote the
upper bound given in Lemma 3.1 by $u$. Different combinations of $\bm{V}_{1}$
and $\bm{V}_{R}$ are considered:

Case 1:

$\bm{V}_{1}$ is $AR(1)$ and $\bm{V}_{R}=\bm{I}_{p}$.

Case 2:

$\bm{V}_{1}$ is tridiagonal and $\bm{V}_{R}=\bm{I}_{p}$.

Case 3:

$\bm{V}_{1}=\sigma_{11}\bm{I}_{p}$, where $\sigma_{11}>0$, and $\bm{V}_{R}$ is
$AR(1)$.

Case 4:

$\bm{V}_{1}=\sigma_{11}\bm{I}_{p}$, where $\sigma_{11}>0$, and $\bm{V}_{R}$ is
tridiagonal.

For our illustrations, $p=t\in\\{3,4\\}$ is used. For application of Lemma
3.1, we are required to find $\bm{V}_{1}^{*}$ and $\bm{V}_{R}^{*}$. In the
first case and $p=t=3$,

$\displaystyle\bm{V}_{1}^{*}$
$\displaystyle=\bm{V}_{1}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}_{1}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{V}_{1}^{-1}\bm{J}_{p\times
p}\bm{V}_{1}^{-1}=\gamma_{11}^{-1}\frac{1}{3-r_{(1)}}\begin{bmatrix}2&-1-r_{(1)}&-1+r_{(1)}\\\
-1-r_{(1)}&2+2r_{(1)}&-1-r_{(1)}\\\ -1+r_{(1)}&-1-r_{(1)}&2\end{bmatrix}$

and

$\displaystyle\bm{V}_{R}^{*}$
$\displaystyle=\bm{V}_{R}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}_{R}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{V}_{R}^{-1}\bm{J}_{p\times
p}\bm{V}_{R}^{-1}=\bm{H}_{3},$

where $\bm{V}_{1}=\sigma_{11}\begin{bmatrix}1&r_{(1)}&r_{(1)}^{2}\\\
r_{(1)}&1&r_{(1)}\\\ r_{(1)}^{2}&r_{(1)}&1\end{bmatrix}$ and
$\bm{V}_{R}=\bm{I}_{p}$. Here,
$\sigma_{11}=\frac{\gamma_{11}}{1-r_{(1)}^{2}}$, $\gamma_{11}>0$ and
$0<|r_{(1)}|<1$. Note $r_{(1)}=0$ reduces the $AR(1)$ structure to the
diagonal structure, so we take $r_{(1)}\neq 0$. Also, $|r_{(1)}|<1$ is
required for the matrix $\bm{V}_{1}$ to be positive definite. To check the
necessary and sufficient condition from Remark 3.3, we evaluate
$tr\left(\bm{V}_{1}^{*}\bm{\psi}\right)=-\gamma_{11}^{-1}\frac{2+r_{(1)}}{3-r_{(1)}}$,
$tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)=-\frac{2}{3}$,
$tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)=\gamma_{11}^{-1}\frac{10+6r_{(1)}}{3\left(3-r_{(1)}\right)}$
and
$tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)=\frac{10}{9}$.
Since
$tr\left(\bm{V}_{1}^{*}\bm{\psi}\right)tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)$
and
$tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)$
are unequal, we may conclude that for $0<|r_{(1)}|<1$, $d^{*}$ does not attain
the upper bound of $tr\left(\bm{{C}}_{d(s2)}\right)$ over
$d\in\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$. We find that similar results
hold for the other cases when $p=t=3$, and for all the cases when $p=t=4$.
Thus we may state that for $p=t\in\\{3,4\\}$, and all cases of $\bm{V}_{1}$
and $\bm{V}_{R}$ considered, $d^{*}$ fails to attain the upper bound $u$.

We then evaluate the performance of $d^{*}$ for the various combinations of
$\bm{V}_{1}$ and $\bm{V}_{R}$ by the relative difference between
$tr\left(\bm{{C}}_{d^{*}(s2)}\right)$ and the upper bound $u$. The performance
of $d^{*}$ is also compared with a uniform and a balanced uniform design
having the same number of subjects as $d^{*}$. Note that a uniform and
balanced uniform design for $p=t$ are also binary designs and hence belong to
$\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$.

For any design $d^{(0)}\in\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$, where
$\lambda$ is a positive integer and $t\geq 3$, $RD_{d^{(0)}}$, the relative
difference between $tr\left(\bm{{C}}_{d^{(0)}(s2)}\right)$ and $u$ is defined
as

$\displaystyle RD_{d^{(0)}}$
$\displaystyle=1-\frac{tr\left(\bm{{C}}_{d^{(0)}(s2)}\right)}{u}.$ (3.14)

Here, $RD_{d^{(0)}}$ is a function of $t$, $n$, $p$, $\sigma_{11}$,
$\sigma_{22}$, $\rho$, $\bm{V}_{1}$ and $\bm{V}_{R}$, and its values lie
between $0$ and $1$. For $RD_{d^{(0)}}=0$, the upper bound is attained by the
design $d^{(0)}$, thus proving $d^{(0)}$ is an efficient design. Thus lower
values of $RD_{d^{(0)}}$ are preferred, as they indicate that
$tr\left(\bm{{C}}_{d^{(0)}(s2)}\right)$ is closer to the upper bound $u$, and
$d^{(0)}$ is a highly efficient design.

Let $d_{1}$ be a uniform design and $d_{2}$ be a balanced uniform design. For
illustration purpose, when $p=t=3$, we consider

$\displaystyle d_{1}=\begin{matrix}1&2&3&1&2&3\\\ 2&3&1&2&3&1\\\
3&1&2&3&1&2\end{matrix}\quad\text{and}\quad d^{*}=\begin{matrix}1&2&3&1&2&3\\\
2&3&1&3&1&2\\\ 3&1&2&2&3&1\end{matrix}.$

While for $p=t=4$, we consider

$\displaystyle
d_{1}=\setcounter{MaxMatrixCols}{12}\begin{matrix}1&4&3&2&1&4&3&2&1&4&3&2\\\
2&1&4&3&2&1&4&3&2&1&4&3\\\ 3&2&1&4&3&2&1&4&3&2&1&4\\\
4&3&2&1&4&3&2&1&4&3&2&1\end{matrix},\quad
d_{2}=\setcounter{MaxMatrixCols}{12}\begin{matrix}1&2&3&4&1&2&3&4&1&2&3&4\\\
4&1&2&3&4&1&2&3&4&1&2&3\\\ 2&3&4&1&2&3&4&1&2&3&4&1\\\
3&4&1&2&3&4&1&2&3&4&1&2\end{matrix}$

and

$\displaystyle
d^{*}=\setcounter{MaxMatrixCols}{12}\begin{matrix}1&1&1&2&2&2&3&3&3&4&4&4\\\
2&3&4&1&3&4&1&2&4&1&2&3\\\ 3&4&2&4&1&3&2&4&1&3&1&2\\\
4&2&3&3&4&1&4&1&2&2&3&1\end{matrix}.$

The balanced uniform design $d_{2}$ is constructed using the method described
by Bose and Dey,  (2009). For $p=t=3$, a balanced uniform design with the same
number of subjects as $d^{*}$ is also an orthogonal array, so it is not
considered as a candidate for comparison with $d^{*}$. A graphical procedure
is used to evaluate the designs $d^{(0)}\in\\{d_{1},d_{2},d^{*}\\}$. In our
illustrations we assume $\sigma_{11}=\sigma_{22}$. The next remark, shows that
if $\sigma_{11}=\sigma_{22}$, then for $p=t\geq 3$, the relative difference
does not depend on the value of $\sigma_{11}$.

######  Remark 3.4.

Let $d^{(0)}\in\mathcal{D}^{(1)}_{t,n=\lambda t\left(t-1\right),p=t}$ be a
design, where $\lambda$ is a positive integer and $p=t\geq 3$. Then for
$\sigma_{11}=\sigma_{22}$, $RD_{d^{(0)}}$ is independent of the value of
$\sigma_{11}$, where $RD_{d^{(0)}}$ is as given in (3.14).

###### Proof.

Details are given in Appendix B. ∎

From our choices of $\bm{V}_{1}$ and $\bm{V}_{R}$, it is clear that
$\bm{V}_{1}$ is a function of $\sigma_{11}$ and $r_{(1)}$, while $\bm{V}_{R}$
is only a function of $r_{(1)}$. In $\bm{V}_{1}$, $r_{(1)}$ is the entry just
above/below the main diagonal in the correlation matrix corresponding to
$\bm{V}_{1}$, while in $\bm{V}_{R}$, $r_{(1)}$ is the entry just above/below
the main diagonal. So, these two matrices are diagonal if and only if
$r_{(1)}=0$. Note, we have considered that when $\bm{V}_{1}$ is $AR(1)$ or
tridiagonal, $\bm{V}_{R}$ is diagonal and when $\bm{V}_{1}$ is diagonal,
$\bm{V}_{R}$ is $AR(1)$ or tridiagonal. Hence, in each case, only one of
$\bm{V}_{1}$ or $\bm{V}_{R}$ is a function of non-zero $r_{(1)}$. Thus from
Remark 3.4, we can see that for any $d^{(0)}\in\mathcal{D}^{(1)}_{t,n=\lambda
t\left(t-1\right),p=t}$, $p=t\geq 3$ and $\sigma_{11}=\sigma_{22}$, the
relative difference given as $RD_{d^{(0)}}$ is a function of $d^{(0)}$, $p$,
$n$, $\rho$ and $r_{(1)}$.

For $p=t\in\\{3,4\\}$, $n=t\left(t-1\right)$, $\sigma_{11}=\sigma_{22}$, and
$d^{(0)}\in\\{d_{1},d_{2},d^{*}\\}$, we plot
$min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ and
$max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$, corresponding to various
non-zero permissible values of $r_{(1)}$. Here, by permissible values, we mean
those values for which the matrices $\bm{V}_{1}$ and $\bm{V}_{R}$ are positive
definite. If in any case $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$
attains $0$, then this implies that design $d^{(0)}$ maximizes
$tr\left(\bm{{C}}_{d^{(0)}(s2)}\right)$ and $d^{(0)}$ is a efficient design.
If $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ does not attain $0$, we
plot $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ to get an
understanding of the worst scenario and quantify the departure of the design
under consideration from an efficient design. Lower the values of
$max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$, better is the performance
of the design in terms of efficiency. For $d^{(0)}\in\\{d_{1},d^{*}\\}$, where
$p=t=3$ and $n=6$, Figure 2 displays the plots of
$min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ and
$max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$, for all cases. While, for
$p=t=4$, $n=12$, Figure 3 illustrates the performance of the three designs.

From Figure 2, for $p=t=3$ and $n=6$, we see that
$min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d_{1}}$ never attains $0$ and hence
$d_{1}$ like $d^{*}$ fails to attain the upper bound $u$. Also, we note that
for all cases the values of $max_{0<|\rho|<1}\leavevmode\nobreak\
RD_{d_{1}}>>max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{*}}$, implying that
$d^{*}$ performs much better when compared to $d_{1}$. Considering all four
cases, the maximum value of $RD_{d^{*}}$ over $\rho\neq 0$ is at most around
$4\%$ indicating that $d^{*}$ is a highly efficient design. In the $p=t=4$ and
$n=12$ case, we observe from Figure 3 that none of the designs
$d^{(0)}\in\\{d_{1},d_{2},d^{*}\\}$, $min_{0<|\rho|<1}\leavevmode\nobreak\
RD_{d^{(0)}}$ attains $0$. In Figure 3, for all the cases, we see that
$max_{0<|\rho|<1}\leavevmode\nobreak\
RD_{d^{*}}<max_{0<|\rho|<1}\leavevmode\nobreak\
RD_{d_{2}}<max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d_{1}}$. So, $d^{*}$
performs the best among the three designs being compared, followed by $d_{2}$.
Since $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d_{1}}$ is far away from $0$,
we reject the candidature of $d_{1}$ as a highly efficient design. For some
values of $r_{(1)}$, we note $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d_{2}}$
is quite close to $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{*}}$. The value
of $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{*}}$ is found to be at most
$11\%$, while most of the values of $max_{0<|\rho|<1}\leavevmode\nobreak\
RD_{d_{2}}$ are much larger. Thus we can conclude that here $d^{*}$ is the
best design among the three designs and also is a highly efficient design.

(a) $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d^{*}\\}$ (Case $1$)

(b) $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d^{*}\\}$ (Case $1$)

(c) $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d^{*}\\}$ (Case $2$)

(d) $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d^{*}\\}$ (Case $2$)

(e) $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d^{*}\\}$ (Case $3$)

(f) $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d^{*}\\}$ (Case $3$)

(g) $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d^{*}\\}$ (Case $4$)

(h) $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d^{*}\\}$ (Case $4$)

Figure 2: Plotting $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ and
$max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d^{*}\\}$, where $p=t=3$, $n=6$ and
$\sigma_{11}=\sigma_{22}$.

(a) $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d_{2},d^{*}\\}$
(Case $1$)

(b) $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d_{2},d^{*}\\}$
(Case $1$)

(c) $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d_{2},d^{*}\\}$
(Case $2$)

(d) $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d_{2},d^{*}\\}$
(Case $2$)

(e) $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d_{2},d^{*}\\}$
(Case $3$)

(f) $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d_{2},d^{*}\\}$
(Case $3$)

(g) $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d_{2},d^{*}\\}$
(Case $4$)

(h) $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d_{2},d^{*}\\}$
(Case $4$)

Figure 3: Plotting $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ and
$max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{1},d_{2},d^{*}\\}$, where $p=t=4$, $n=12$ and
$\sigma_{11}=\sigma_{22}$.

##  4 Illustration using the gene data

We go back to use the genetic data example from Leaker et al.,  (2017) to
evaluate design efficiency under different covariance structures. In
particular, we focus on the $3\times 3$ crossover design used in the gene data
example, involving $18$ subjects and $3$ treatment sequences, namely $ABC$,
$CAB$ and $BCA$. Subjects $1-6$ receive treatment sequence $ABC$, $7-12$
receive $CAB$, while remaining receive $BCA$, and multiple gene expression
values are measured from each subject in the three time periods. Note the
study design is a uniform design with $p=t=3$. We name this design $d_{0}$,
where $d_{0}$ belongs to $\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$. Next,
we evaluate the performance of $d_{0}$ under various correlation structures
and compare it with other possible designs.

###  4.1 Proportional structure

Under the proportional structure, to investigate trace optimality/efficiency
of $d_{0}$ we consider two different structures of matrix $\bm{V}$ as follows:

  1. 1.

$\bm{V}=\left(1-r_{(1)}^{2}\right)^{-1}\begin{bmatrix}1&r_{(1)}&r_{(1)}^{2}\\\
r_{(1)}&1&r_{(1)}\\\ r_{(1)}^{2}&r_{(1)}&1\end{bmatrix}$, where
$-1<r_{(1)}<1$;

  2. 2.

$\bm{V}=\begin{bmatrix}1&r_{(1)}&0\\\ r_{(1)}&1&r_{(1)}\\\
0&r_{(1)}&1\end{bmatrix}$, where $-1/\sqrt{2}<r_{(1)}<1/\sqrt{2}$.

Structure 1 corresponds to a $AR(1)$ structure, while structure 2 corresponds
to a tridiagonal structure. The constraint on the values of $r_{(1)}$ is to
ensure positive definiteness of the matrix $\bm{V}$. From Theorem 3.2 in
Subsection 3.1, we know that under the proportional structure, a design
$d^{*}\in\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$ represented by
$OA_{I}\left(n=\lambda t\left(t-1\right),p=t,t,2\right)$, where
$\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$ is a class of binary designs with
$p=t$, $\lambda$ is a positive integer and $t\geq 3$, is trace
optimal/efficient for the direct effects over $\mathcal{D}^{(1)}_{t,n=\lambda
t(t-1),p=t}$ for the multivariate response case. For any binary design
$d\in\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$, with $t$ and $n$ same as
that of $d^{*}$, let $e$ represent the efficiency of design $d$, defined as
(Kunert,  1991)

$\displaystyle e$
$\displaystyle=\frac{tr\left(\bm{C}_{d(s1)}\right)}{tr\left(\bm{C}_{d^{*}(s1)}\right)}.$

Since $d^{*}$ maximizes $tr\left(\bm{C}_{d(s1)}\right)$, over
$\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$, $e$ lies between $0$ and $1$,
and values of $e$ close to $1$ indicate that $d$ is a highly efficient design.

Using the expression of the information matrix for the direct effects under
the proportional structure from Theorem 3.1, we get

$\displaystyle e$
$\displaystyle=\frac{tr\left(\bm{\varGamma}^{-1}\right)tr\left(\bm{C}_{d(uni)}\right)}{tr\left(\bm{\varGamma}^{-1}\right)tr\left(\bm{C}_{d^{*}(uni)}\right)}=\frac{tr\left(\bm{C}_{d(uni)}\right)}{tr\left(\bm{C}_{d^{*}(uni)}\right)},$
(4.1)

where
$\bm{C}_{d(uni)}=\bm{C}_{d(uni)(11)}-\bm{C}_{d(uni)(12)}\bm{C}_{d(uni)(22)}^{-}\bm{C}_{d(uni)(21)}$,
$\bm{C}_{d(uni)(11)}=\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{T}_{d}$,
$\bm{C}_{d(uni)(12)}=\bm{C}_{d(uni)(21)}^{{}^{\prime}}=\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{F}_{d}$,
and
$\bm{C}_{d(uni)(22)}=\bm{F}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{F}_{d}$.

From $\eqref{prop-eqn-efficiency-1}$, it is clear that for a particular design
$d\in\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$, and fixed values of $t$, $n$
and $p$, $e$ depends only on the matrix $\bm{V}$, thus only on $r_{(1)}$.
Figure 4 compares the performance of design $d_{0}$ with $d^{*}$ by plotting
values of $e$ for both structures considered for $\bm{V}$. From Figure 4, we
see that the maximum value of $e$ under both structures of $\bm{V}$ is around
$2.78\%$, indicating that $d_{0}$ performs very poorly in comparison to the
efficient design $d^{*}$.

(a) $\bm{V}$: $AR(1)$ Structure

(b) $\bm{V}$: Tridiagonal Structure

Figure 4: Efficiency of $d_{0}$.

###  4.2 Generalized markov-type structure

To check the trace optimality of design $d_{0}$ and compare its performance
with $d^{*}$ under the Markovian structure, we consider $4$ different cases
based on the structure of matrices $\bm{V}_{1}$ and $\bm{V}_{R}$. These cases
are same as considered in Subsubsection 3.2.1. Note that here $d^{*}$ is a
design represented by an orthogonal array of Type $I$ and strength $2$ with
$p=t=3$ and $n=18$. Here, we also assume that $\sigma_{11}=\sigma_{22}$.
Figure 5 displays $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d_{0}}$,
$min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{*}}$,
$max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d_{0}}$ and
$max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{*}}$ under the $4$ cases,
corresponding to various non-zero permissible values of $r_{(1)}$. Here,
$RD_{d_{0}}$ and $RD_{d^{*}}$ is as given in (3.14) with
$d^{(0)}=d_{0},d^{*}$. Note that lower values of $RD_{d^{(0)}}$ indicate that
$d^{(0)}$ is a highly efficient design. From Figure 5 and for all cases,
$min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d_{0}}$ and
$min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{*}}$ never attains $0$. So,
$d_{0}$ and $d^{*}$ fails to attain the upper bound $u$. Also,
$max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d_{0}}$ is quite higher than
$max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{*}}$ for all cases, so we may say
that $d^{*}$ performs better than $d_{0}$. Taking all the $4$ cases in
account, the maximum value of $max_{0<|\rho|<1}\leavevmode\nobreak\
RD_{d_{0}}$ and $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{*}}$ is around
$99.02\%$ and $4\%$, respectively, indicating the very poor performance of
$d_{0}$ when compared to $d^{*}$. From these comparisons we may conclude that
if the experiment for the gene study had used an orthogonal array based design
then he/she would have scored much in terms of efficiency. The poor choice of
designs may also effect the parameter estimation for the gene dataset.

(a) $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{0},d^{*}\\}$ (Case $1$)

(b) $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{0},d^{*}\\}$ (Case $1$)

(c) $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{0},d^{*}\\}$ (Case $2$)

(d) $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{0},d^{*}\\}$ (Case $2$)

(e) $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{0},d^{*}\\}$ (Case $3$)

(f) $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{0},d^{*}\\}$ (Case $3$)

(g) $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{0},d^{*}\\}$ (Case $4$)

(h) $max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{0},d^{*}\\}$ (Case $4$)

Figure 5: Plotting $min_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ and
$max_{0<|\rho|<1}\leavevmode\nobreak\ RD_{d^{(0)}}$ for
$d^{(0)}\in\\{d_{0},d^{*}\\}$, where $p=t=3$, $n=18$ and
$\sigma_{11}=\sigma_{22}$.

##  5 Conclusion and future direction

In this article, we studied the trace optimality/efficiency of an orthogonal
array of Type $I$ and strength $2$ when measurements are recorded on $g$
responses,where $g\geq 1$. To account for correlation between responses, we
considered two different types of structure for the error dispersion matrix,
namely, the proportional structure and the generalized markov-type structure.
The main conclusions of this paper are:

  1. 1.

Under the proportional structure, for $g>1$ and $p=t\geq 3$, a design given as
an orthogonal array of type $I$ and strength $2$ is trace optimal/efficient
for the direct effects over the class of binary designs.

  2. 2.

Under the generalized markov-type structure, for the $4$ different cases
depending on the choices of matrices $\bm{V}_{1}$ and $\bm{V}_{R}$, when
$p=t=3$ and $n=6$, and $p=t=4$ and $n=12$, an orthogonal array of type $I$ and
strength $2$ is highly efficient for the direct effects over the class of
binary designs.

Although in this article, we have only considered universal optimality and
trace optimality criteria, other criteria can also be investigated. In the
future, for multivariate crossover designs, we plan to investigate $A$-, $D$-,
and $E$-optimal design, under the proportional and markovian structures of the
dispersion matrix.

## Statements and Declarations

### Funding

The authors declare that no financial support has been received from any
funding agency, for this work.

### Competing Interests

The authors declare that there are no conflicts of interest regarding the
publication of this article.

### Ethical Conduct

This research does not involve human or animal participants.

### Authors’ Contributions

All authors contributed to the study conception and design. Material
preparation, analysis and investigation were performed by Shubham Niphadkar.
The first draft of the manuscript was written by Shubham Niphadkar and all
authors commented on previous versions of the manuscript. All authors read and
approved the final manuscript.

##  Appendix A Information matrix

### A.1 Proportional structure

###### Proof of Theorem 3.1.

From Bose and Dey,  (2009), we know that for the $g=1$ case with the
dispersion matrix of error terms as a positive definite and symmetric matrix
$\bm{I}_{n}\otimes\bm{V}$, the information matrix for the direct effects is
given as

$\displaystyle\bm{C}_{d(uni)}$
$\displaystyle=\bm{C}_{d(uni)(11)}-\bm{C}_{d(uni)(12)}\bm{C}_{d(uni)(22)}^{-}\bm{C}_{d(uni)(21)},$
(A.1)

where
$\bm{C}_{d(uni)(11)}=\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{T}_{d}$,
$\bm{C}_{d(uni)(12)}=\bm{C}_{d(uni)(21)}^{{}^{\prime}}=\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{F}_{d}$
and
$\bm{C}_{d(uni)(22)}=\bm{F}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{F}_{d}$.
Here, the matrix
$\bm{V}^{*}=\bm{V}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{V}^{-1}\bm{J}_{p\times
p}\bm{V}^{-1}$.

Using the expression of $\bm{A}^{*}$ as given in Lemma S.2 and the expression
of the information matrix for the direct effects as given in Lemma S.1, we get
that under the proportional structure for $g>1$,

$\bm{C}_{d(s1)}=\bm{\varGamma}^{-1}\otimes\\\
\left[\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{T}_{d}-\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{F}_{d}\left(\bm{F}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{F}_{d}\right)^{-}\bm{F}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{T}_{d}\right],$
(A.2)

where $\bm{\varGamma}$ is a $g\times g$ positive definite and symmetric matrix
with non-zero off-diagonal elements. Thus from (A.1) and (A.2), we can prove
(3.3). ∎

###### Proof of Remark 3.1.

Here, $d^{*}$ is a design given by $OA_{I}\left(n=\lambda
t\left(t-1\right),p=t,t,2\right)$, where $\lambda$ is a positive integer and
$t\geq 3$. From Theorem 3.1, we get that for $g>1$, the information matrix for
the direct effects corresponding to $d^{*}$ is given as

$\displaystyle\bm{C}_{d^{*}(s1)}$
$\displaystyle=\bm{\varGamma}^{-1}\otimes\bm{C}_{d^{*}(uni)},$ (A.3)

where
$\bm{C}_{d^{*}(uni)}=\bm{C}_{d^{*}(uni)(11)}-\bm{C}_{d^{*}(uni)(12)}\bm{C}_{d^{*}(uni)(22)}^{-}\bm{C}_{d^{*}(uni)(21)}$,
and $\bm{\varGamma}$ is a $g\times g$ positive definite and symmetric matrix
with non-zero off-diagonal elements. Here,
$\bm{C}_{d^{*}(uni)(11)}=\bm{T}_{d^{*}}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{T}_{d^{*}}$,
$\bm{C}_{d^{*}(uni)(12)}=\bm{C}_{d^{*}(uni)(21)}^{{}^{\prime}}=\bm{T}_{d^{*}}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{F}_{d^{*}}$,
$\bm{C}_{d^{*}(uni)(22)}=\bm{F}_{d^{*}}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{V}^{*}\right)\bm{F}_{d^{*}}$
and the matrix
$\bm{V}^{*}=\bm{V}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{V}^{-1}\bm{J}_{p\times
p}\bm{V}^{-1}$.

Since $\bm{F}_{d^{*}}=\left(\bm{I}_{n}\otimes\bm{\psi}\right)\bm{T}_{d^{*}}$,
where $\bm{\psi}=\begin{bmatrix}\bm{{0}}^{{}^{\prime}}_{p-1\times 1}&0\\\
\bm{I}_{p-1}&\bm{{0}}_{p-1\times 1}\end{bmatrix}$, and $\bm{V}$ is a known
positive definite and symmetric matrix, from Martin and Eccleston,  (1998), we
know

$\displaystyle\bm{C}_{d^{*}(uni)}$
$\displaystyle=\left(|\bm{E}|/e_{22}\right)\bm{H}_{t},$ (A.4)

where $\bm{E}=\frac{n}{t-1}\begin{bmatrix}e_{11}&e_{12}\\\
e_{12}&e_{22}\end{bmatrix}$,
$e_{11}=tr\left(\bm{T}^{{}^{\prime}}_{d^{*}1}\bm{V}^{*}\bm{T}_{d^{*}1}\right)$,
$e_{12}=tr\left(\bm{T}^{{}^{\prime}}_{d^{*}1}\bm{V}^{*}\bm{\psi}\bm{T}_{d^{*}1}\right)$,
$e_{22}=$
$tr\left(\bm{T}^{{}^{\prime}}_{d^{*}1}\bm{\psi}^{{}^{\prime}}\bm{V}^{*}\bm{\psi}\bm{T}_{d^{*}1}\right)-\frac{\left(\bm{V}^{*}\right)_{1,1}}{t}$
and $|\bm{E}|\neq 0$. Here, $\left(\bm{V}^{*}\right)_{1,1}$ is the element
corresponding to first row and first column of the matrix $\bm{V}^{*}$, and
$\bm{T}_{d^{*}1}$ is the treatment incidence matrix for the first subject
corresponding to $d^{*}$. For $g>1$, let us denote

$\displaystyle\bm{\varGamma}^{-1}=\begin{bmatrix}\gamma^{(11)}&\ldots&\gamma^{(1g)}\\\
\vdots&\vdots&\vdots\\\ \gamma^{(g1)}&\ldots&\gamma^{(gg)}\end{bmatrix}.$

Since $\bm{\varGamma}$ is a $g\times g$ positive definite, symmetric and non-
diagonal matrix, the same holds true for $\bm{\varGamma}^{-1}$. So, from (A.3)
and (A.4), we get that $\bm{C}_{d^{*}(s1)}$ is completely symmetric for $g>1$
case if and only if

$\displaystyle\begin{split}\frac{\gamma^{(11)}|\bm{E}|}{te_{22}}&=\cdots=\frac{\gamma^{(gg)}|\bm{E}|}{te_{22}},\\\
\frac{\gamma^{(kk)}|\bm{E}|}{te_{22}}&=\frac{\gamma^{(kk^{\prime})}|\bm{E}|}{te_{22}},\text{
and}\\\
\frac{\gamma^{(kk^{\prime})}|\bm{E}|}{te_{22}}&=\frac{\gamma^{(kk^{\prime})}|\bm{E}|\left(t-1\right)}{te_{22}},\end{split}$
(A.5)

where $\gamma^{(kk^{\prime})}$ is the $(k,k^{\prime})^{th}$ element of the
matrix $\bm{\varGamma}^{-1}$. Here, due to positive definiteness, symmetricity
and non-diagonality of $\bm{\varGamma}^{-1}$, $\gamma^{(kk)}>0$ and for
atleast one pair $(k,k^{\prime})$,
$\gamma^{(kk^{\prime})}=\gamma^{(k^{\prime}k)}\neq 0$. Thus (A.5) does not
hold true. Hence, $\bm{C}_{d^{*}(s1)}$ is not a completely symmetric matrix
for $g>1$ case. ∎

### A.2 Generalized markov-type structure

###### Proof of Theorem 3.3.

Here $d^{*}\in\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$ be a design given by
$OA_{I}\big{(}n=\lambda t\left(t-1\right),p=t,$
$t,2\big{)}$, where $\lambda$ is a positive integer and $t\geq 3$, and thus
$d^{*}$ is a design uniform on periods. Since from Remark S.1, we get that
$\bm{{1}}_{p}^{{}^{\prime}}\bm{\varOmega}_{1}=\bm{{1}}_{p}^{{}^{\prime}}\bm{\varOmega}_{2}=\bm{{1}}_{p}^{{}^{\prime}}\bm{\varOmega}_{4}=\bm{{0}}_{1\times
p}$,
$\bm{{1}}_{p}^{{}^{\prime}}\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{x}\bm{\psi}\bm{{1}}_{p}$
is the entry corresponding to the first row and first column of the matrix
$\bm{\varOmega}_{x}$, for $x=1,2,4$. Here, $\bm{\varOmega}_{1}$,
$\bm{\varOmega}_{2}$ and $\bm{\varOmega}_{4}$ are as given in Lemma S.4. Using
the expression of $\bm{{C}}_{d^{*}(s2)}$ as given in Lemma S.3 and the
expression of $\bm{A}^{*}$ in Lemma S.4, we get

$\displaystyle\bm{{C}}_{d^{*}(s2)}$
$\displaystyle=\bm{{C}}_{d^{*}(s2)(11)}-\bm{{C}}_{d^{*}(s2)(12)}\bm{{C}}_{d^{*}(s2)(22)}^{-}\bm{{C}}_{d^{*}(s2)(21)},$
(A.6)

where

$\displaystyle\begin{split}\bm{C}_{{d^{*}(s2)}(11)}&=\begin{bmatrix}\bm{T}_{d^{*}}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{1}\right)\bm{T}_{d^{*}}&-\bm{T}_{d^{*}}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{2}\right)\bm{T}_{d^{*}}\\\
-\bm{T}_{d^{*}}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{2}\right)\bm{T}_{d^{*}}&\bm{T}_{d^{*}}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{4}\right)\bm{T}_{d^{*}}\end{bmatrix},\\\
\bm{C}_{{d^{*}(s2)}(12)}&=\bm{C}^{{}^{\prime}}_{{d^{*}(s2)}(21)}=\begin{bmatrix}\bm{T}_{d^{*}}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{1}\bm{\psi}\right)\bm{T}_{d^{*}}&-\bm{T}_{d^{*}}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d^{*}}\\\
-\bm{T}_{d^{*}}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d^{*}}&\bm{T}_{d^{*}}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{4}\bm{\psi}\right)\bm{T}_{d^{*}}\end{bmatrix},\text{
and}\\\
\bm{C}_{{d^{*}(s2)}(22)}&=\begin{bmatrix}\bm{T}_{d^{*}}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{1}\bm{\psi}\right)\bm{T}_{d^{*}}&-\bm{T}_{d^{*}}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d^{*}}\\\
-\bm{T}_{d^{*}}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d^{*}}&\bm{T}_{d^{*}}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{4}\bm{\psi}\right)\bm{T}_{d^{*}}\end{bmatrix}.\end{split}$

Thus from the above equation and following Lemma 3, Lemma 4 and Lemma 5 from
Martin and Eccleston,  (1998), we get

$\displaystyle\begin{split}\bm{{C}}_{d^{*}(s2)(11)}&=\frac{n}{t-1}\begin{bmatrix}c_{11(11)}&-c_{11(12)}\\\
-c_{11(12)}&c_{11(22)}\end{bmatrix}\otimes\bm{H}_{t},\\\
\bm{{C}}_{d^{*}(s2)(12)}&=\bm{\tilde{C}}_{d^{*}(s2)(21)}=\frac{n}{t-1}\begin{bmatrix}c_{12(11)}&-c_{12(12)}\\\
-c_{12(12)}&c_{12(22)}\end{bmatrix}\otimes\bm{H}_{t},\text{ and}\\\
\bm{{C}}_{d^{*}(s2)(22)}&=\frac{n}{t-1}\begin{bmatrix}c_{22(11)}&-c_{22(12)}\\\
-c_{22(12)}&c_{22(22)}\end{bmatrix}\otimes\bm{H}_{t},\end{split}$ (A.7)

where $c_{11(11)}=tr\left(\bm{\varOmega}_{1}\right)$,
$c_{11(12)}=tr\left(\bm{\varOmega}_{2}\right)$,
$c_{11(22)}=tr\left(\bm{\varOmega}_{4}\right)$,
$c_{12(11)}=tr\left(\bm{\varOmega}_{1}\bm{\psi}\right)$,
$c_{12(12)}=tr\left(\bm{\varOmega}_{2}\bm{\psi}\right)$,
$c_{12(22)}=tr\left(\bm{\varOmega}_{4}\bm{\psi}\right)$,
$c_{22(11)}=tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{1}\bm{\psi}\right)$,
$c_{22(12)}=tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{2}\bm{\psi}\right)$,
$c_{22(22)}=tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{4}\bm{\psi}\right)$.
From Lemma S.4, we get

$\displaystyle\begin{split}\bm{\varOmega}_{1}&=\bm{V}_{1}^{*}+\frac{\bar{\rho}^{2}}{\sigma_{12}}\bm{V}_{R}^{*}=\bm{V}_{1}^{*}+\bar{\rho}^{2}\bm{\varOmega}_{4},\\\
\bm{\varOmega}_{2}&=\bar{\rho}\bm{\varOmega}_{4},\text{ and}\\\
\bm{\varOmega}_{4}&=\frac{1}{\sigma_{12}}\bm{V}_{R}^{*},\end{split}$ (A.8)

where $\bar{\rho}=\sqrt{\frac{\sigma_{22}}{\sigma_{11}}}$ and
$\sigma_{12}=\sigma_{22}(1-\rho^{2})$. Thus using the expressions of
$c_{11(11)}$, $c_{11(12)}$, $c_{11(22)}$, $c_{12(11)}$, $c_{12(12)}$,
$c_{12(22)}$, $c_{22(11)}$, $c_{22(12)}$ and $c_{22(22)}$, we get

$\displaystyle\begin{split}c_{11(11)}&=tr\left(\bm{V}_{1}^{*}\right)+\bar{\rho}^{2}c_{11(22)},\\\
c_{11(12)}&=\bar{\rho}\times c_{11(22)},\\\
c_{11(22)}&=\frac{1}{\sigma_{12}}tr\left(\bm{V}_{R}^{*}\right),\\\
c_{12(11)}&=tr\left(\bm{V}_{1}^{*}\bm{\psi}\right)+\bar{\rho}^{2}c_{12(22)},\\\
c_{12(12)}&=\bar{\rho}\times c_{12(22)},\\\
c_{12(22)}&=\frac{1}{\sigma_{12}}tr\left(\bm{V}_{R}^{*}\bm{\psi}\right),\\\
c_{22(11)}&=tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)+\bar{\rho}^{2}c_{22(22)},\\\
c_{22(12)}&=\bar{\rho}\times c_{22(22)},\text{ and}\\\
c_{22(22)}&=\frac{1}{\sigma_{12}}tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right).\end{split}$
(A.9)

Using the above equation, we get that
$c_{22(11)}c_{22(22)}-c^{2}_{22(12)}=\frac{1}{\sigma_{12}}tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)\times$
$tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)$.
Since $\sigma_{12}>0$, applying Lemma S.8, we get that
$c_{22(11)}c_{22(22)}-c^{2}_{22(12)}>0$. So, from (A.7), we get

$\displaystyle\bm{{C}}_{d^{*}(s2)(22)}^{-}$
$\displaystyle=\frac{t-1}{n\left(c_{22(11)}c_{22(22)}-c^{2}_{22(12)}\right)}\begin{bmatrix}c_{22(22)}&c_{22(12)}\\\
c_{22(12)}&c_{22(11)}\end{bmatrix}\otimes\bm{I}_{t}.$ (A.10)

From (A.6), (A.7), (A.9) and (A.10), by calculation we get that
$\bm{{C}}_{d^{*}(s2)}=$
$\frac{n}{t-1}\begin{bmatrix}\bm{\varLambda}_{1}&\bm{\varLambda}_{2}\\\
\bm{\varLambda}_{2}&\bm{\varLambda}_{4}\end{bmatrix}$, where

$\displaystyle\begin{split}\bm{\varLambda}_{1}&=\left(tr\left(\bm{V}_{1}^{*}\right)+\frac{\bar{\rho}^{2}}{\sigma_{12}}tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{1}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)}-\frac{\bar{\rho}^{2}}{\sigma_{12}}\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right)\bm{H}_{t},\\\
\bm{\varLambda}_{2}&=-\frac{\bar{\rho}}{\sigma_{12}}\left(tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right)\bm{H}_{t},\text{
and}\\\
\bm{\varLambda}_{4}&=\frac{1}{\sigma_{12}}\left(tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right)\bm{H}_{t}.\end{split}$

From the above equation, we get

$\displaystyle tr\left(\bm{{C}}_{d^{*}(s2)}\right)$
$\displaystyle=n\left(tr\left(\bm{V}_{1}^{*}\right)+\frac{1+\bar{\rho}^{2}}{\sigma_{12}}tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{1}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)}-\frac{1+\bar{\rho}^{2}}{\sigma_{12}}\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right).$

∎

###### Proof of Remark 3.2.

Here, $d^{*}$ is a design given by $OA_{I}\left(n=\lambda
t\left(t-1\right),p=t,t,2\right)$, where $\lambda$ is a positive integer and
$t\geq 3$. From Theorem 3.3, we get that the information matrix for the direct
effects corresponding to $d^{*}$ is expressed as

$\displaystyle\bm{{C}}_{d^{*}(s2)}$
$\displaystyle=\frac{n}{t-1}\begin{bmatrix}\bm{\varLambda}_{1}&\bm{\varLambda}_{2}\\\
\bm{\varLambda}_{2}&\bm{\varLambda}_{4}\end{bmatrix},$ (A.11)

where

$\displaystyle\begin{split}\bm{\varLambda}_{1}&=\left(tr\left(\bm{V}_{1}^{*}\right)+\frac{\bar{\rho}^{2}}{\sigma_{12}}tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{1}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)}-\frac{\bar{\rho}^{2}}{\sigma_{12}}\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right)\bm{H}_{t},\\\
\bm{\varLambda}_{2}&=-\frac{\bar{\rho}}{\sigma_{12}}\left(tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right)\bm{H}_{t},\text{
and}\\\
\bm{\varLambda}_{4}&=\frac{1}{\sigma_{12}}\left(tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right)\bm{H}_{t}.\end{split}$

From the above equation, we get that $\bm{{C}}_{d^{*}(s2)}$ is completely
symmetric if and only if

$\frac{1}{t}\left(tr\left(\bm{V}_{1}^{*}\right)+\frac{\bar{\rho}^{2}}{\sigma_{12}}tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{1}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)}-\frac{\bar{\rho}^{2}}{\sigma_{12}}\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right)=\\\
\frac{1}{\sigma_{12}t}\left(tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right)=-\frac{\bar{\rho}}{\sigma_{12}t}\left(tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right)$
(A.12)

and

$\displaystyle\frac{\bar{\rho}}{\sigma_{12}t}\left(tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right)$
$\displaystyle=-\frac{\bar{\rho}\left(t-1\right)}{\sigma_{12}t}\left(tr\left(\bm{V}_{R}^{*}\right)-\frac{\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}}{tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}\right).$
(A.13)

From Lemma S.8, we get that
$tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)>0$.
If we consider the univariate response case with the dispersion matrix of the
vector error terms as $\sigma^{2}\left(\bm{I}_{n}\otimes\bm{V}_{R}\right)$,
then from Lemma 6 of Martin and Eccleston,  (1998), we get that
$tr\left(\bm{V}_{R}^{*}\right)tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)-\left(tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}\neq
0$. Since $\sigma_{12}>0$ and $|\bar{\rho}|>0$, (A.12) and (A.13) does not
hold true. Hence, $\bm{{C}}_{d^{*}(s2)}$ is not a completely symmetric matrix.
∎

##  Appendix B Efficiency under the generalized markov-type structure

###### Proof of Remark 3.3.

Here, $d^{*}\in\mathcal{D}^{(1)}_{t,n=\lambda t(t-1),p=t}$ is a design given
by $OA_{I}\big{(}n=\lambda t\left(t-1\right),p=t,$
$t,2\big{)}$, where $\lambda$ is a positive integer and $t\geq 3$. Using the
expression of $tr\left(\bm{{C}}_{d^{*}(s2)}\right)$ as given in Theorem 3.3
and the expression of
$u\left(t,n,p,\sigma_{11},\sigma_{22},\rho,\bm{V}_{1},\bm{V}_{R}\right)$ as
given in Lemma 3.1, where $p=t$ and $n=\lambda t\left(t-1\right)$, by
calculation we get

$u\left(t,n,p,\sigma_{11},\sigma_{22},\rho,\bm{V}_{1},\bm{V}_{R}\right)-tr\left(\bm{{C}}_{d^{*}(s2)}\right)\\\
=n\left(1+\bar{\rho}^{2}\right)\frac{\left(tr\left(\bm{V}_{1}^{*}\bm{\psi}\right)tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)-tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)\right)^{2}}{\left(\sigma_{12}tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)+\left(1+\bar{\rho}^{2}\right)tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)\right)tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right)tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)}.$
(B.1)

Thus from (B.1), we get that $d^{*}$ attains
$u\left(t,n,p,\sigma_{11},\sigma_{22},\rho,\bm{V}_{1},\bm{V}_{R}\right)$, if
and only if

$\displaystyle
tr\left(\bm{V}_{1}^{*}\bm{\psi}\right)tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)=tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{1}^{*}\bm{\psi}\right).$

∎

###### Proof of Remark 3.4.

Here $d^{(0)}\in\mathcal{D}^{(1)}_{t,n=\lambda t\left(t-1\right),p=t}$, where
$\lambda$ is a positive integer, $p=t\geq 3$ and $\sigma_{11}=\sigma_{22}$.
So, $\bar{\rho}=\rho\sqrt{\frac{\sigma_{22}}{\sigma_{11}}}=\rho$ and
$\sigma_{12}=\sigma_{22}\left(1-\bar{\rho}^{2}\right)=\sigma_{11}\left(1-\rho^{2}\right)$.
Since $\bm{V}_{1}=\sigma_{11}\bm{V}_{C}$, from Lemma 3.1, we get

$u=u\left(t,n,p,\sigma_{11},\sigma_{22}=\sigma_{11},\rho,\bm{V}_{1},\bm{V}_{R}\right)=\frac{1}{\sigma_{11}}tr\left(\bm{V}_{C}^{*}\right)+\frac{1+\rho^{2}}{\sigma_{11}(1-\rho^{2})}tr\left(\bm{V}_{R}^{*}\right)\\\
-\frac{1}{\sigma_{11}}\left(tr\left(\bm{V}_{C}^{*}\bm{\psi}\right)+\frac{1+\rho^{2}}{1-\rho^{2}}tr\left(\bm{V}_{R}^{*}\bm{\psi}\right)\right)^{2}/\left(tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{C}^{*}\bm{\psi}\right)+\frac{1+\rho^{2}}{1-\rho^{2}}tr\left(\bm{H}_{p}\bm{\psi}^{{}^{\prime}}\bm{V}_{R}^{*}\bm{\psi}\right)\right),$
(B.2)

where
$\bm{V}_{C}^{*}=\bm{V}_{C}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}_{C}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{V}_{C}^{-1}\bm{J}_{p\times
p}\bm{V}_{C}^{-1}$ and
$\bm{V}_{R}^{*}=\bm{V}_{R}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}_{R}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{V}_{R}^{-1}\bm{J}_{p\times
p}\bm{V}_{R}^{-1}$, and using the expressions of $\bm{\varOmega}_{1}$,
$\bm{\varOmega}_{2}$ and $\bm{\varOmega}_{4}$ as given in Lemma S.4, we get

$\displaystyle\begin{split}\bm{\varOmega}_{1}&=\frac{1}{\sigma_{11}}\bm{V}^{*}_{C}+\frac{\rho^{2}}{\sigma_{11}\left(1-\rho^{2}\right)}\bm{V}_{R}^{*},\\\
\bm{\varOmega}_{2}&=\frac{\rho}{\sigma_{11}\left(1-\rho^{2}\right)}\bm{V}_{R}^{*}\\\
\bm{\varOmega}_{4}&=\frac{1}{\sigma_{11}\left(1-\rho^{2}\right)}\bm{V}_{R}^{*}\end{split}$
(B.3)

Let $\bm{B}^{*}=\sigma_{11}\bm{A}^{*}$. Hence, using the expression of
$\bm{C}_{d(s2)}$ from Lemma S.3, we get

$\displaystyle\bm{C}_{d(s2)}$
$\displaystyle=\bm{C}_{d(s2)(11)}-\bm{C}_{{d(s2)}(12)}\bm{C}^{-}_{{d(s2)}(22)}\bm{C}_{{d(s2)}(21)},$
(B.4)

where
$\bm{C}_{d(s2)(11)}=\frac{1}{\sigma_{11}}\left(\bm{I}_{2}\otimes\bm{T}^{{}^{\prime}}_{d}\right)\bm{B}^{*}\left(\bm{I}_{2}\otimes\bm{T}_{d}\right)$,
$\bm{C}_{d(s2)(12)}=\bm{C}^{{}^{\prime}}_{d(s2)(21)}=\frac{1}{\sigma_{11}}\left(\bm{I}_{2}\otimes\bm{T}^{{}^{\prime}}_{d}\right)\bm{B}^{*}\times$
$\left(\bm{I}_{2}\otimes\bm{F}_{d}\right)$ and
$\bm{C}_{d(s2)(22)}=\frac{1}{\sigma_{11}}\left(\bm{I}_{2}\otimes\bm{F}^{{}^{\prime}}_{d}\right)\bm{B}^{*}\left(\bm{I}_{2}\otimes\bm{F}_{d}\right)$.

It should be noted that since the matrices $\bm{V}^{*}_{C}$ and
$\bm{V}_{R}^{*}$ are independent of $\sigma_{11}$, from (B.3) and using the
expression of $\bm{A}^{*}$ in Lemma S.4, we get that the matrix $\bm{B}^{*}$
is independent of $\sigma_{11}$. Thus from (3.14), (B.2) and (B.4), we get
that $RD_{d^{(0)}}$ is independent of the value of $\sigma_{11}$. ∎

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## Supplementary Material

### S.1 Table

Gene Marker |  Characteristics
---|---
PNO1 (X100134584_TGI_AT) |  Facilitates ribosomal RNA maturation and processing, pivotal for ribosome biogenesis and protein synthesis control.
BPI (X100126967_TGI_AT) |  Defends against bacterial infections by neutralizing bacterial endotoxins within the innate immune system.
UNC13A (X100308708_TGI_AT) |  Regulates neurotransmitter release, contributing to synaptic plasticity and efficient neuronal communication.
MTA3 (X100154937_TGI_AT) |  Exhibits chromatin-binding and histone deacetylase activities, influencing transcriptional regulation and chromatin remodeling.
DYRK1A (X100140918_TGI_AT) |  Multifaceted kinase with roles in circadian rhythm, neuronal development, and protein phosphorylation-mediated cellular processes.
EPN3 (X100310234_TGI_AT) |  Involved in lipid and protein binding, localized in vital cellular compartments, including clathrin-coated pits and plasma membranes.
ZNF365 (X100300789_TGI_AT) |  Participates in DNA binding and transcriptional regulation, impacting gene expression and diverse biological contexts.
OR5AS1 (X100125346_TGI_AT) |  Essential component of the olfactory receptor family, contributing to chemosensory perception and the sense of smell.
EHMT1 (X100129988_TGI_AT) |  Epigenetic regulator governing histone methylation, chromatin modification, and gene expression, with implications in embryonic development and disorders.
LTBP2 (X100129645_TGI_AT) |  Crucial in the extracellular matrix, regulating transforming growth factor-beta (TGF-$\beta$) activity, influencing tissue homeostasis, and cellular behavior.
Table S.1: Characteristics of the $10$ gene markers. This information is
obtained from GEO’s annotation SOFT table.

### S.2 Results for the proportional covariance structure

######  Lemma S.1.

Under the proportional structure, $forg>1$, the information matrix for the
direct effects can be expressed as

$\displaystyle\bm{C}_{d(s1)}$
$\displaystyle=\bm{C}_{d(s1)(11)}-\bm{C}_{{d(s1)}(12)}\bm{C}^{-}_{{d(s1)}(22)}\bm{C}_{{d(s1)}(21)},$
(S.1)

where
$\bm{C}_{d(s1)(11)}=\left(\bm{I}_{g}\otimes\bm{T}^{{}^{\prime}}_{d}\right)\bm{A}^{*}\left(\bm{I}_{g}\otimes\bm{T}_{d}\right)$,
$\bm{C}_{d(s1)(12)}=\bm{C}^{{}^{\prime}}_{d(s1)(21)}=\left(\bm{I}_{g}\otimes\bm{T}^{{}^{\prime}}_{d}\right)\bm{A}^{*}\left(\bm{I}_{g}\otimes\bm{F}_{d}\right)$
and
$\bm{C}_{d(s1)(22)}=\left(\bm{I}_{g}\otimes\bm{F}^{{}^{\prime}}_{d}\right)\bm{A}^{*}\left(\bm{I}_{g}\otimes\bm{F}_{d}\right)$.
Here,
$\bm{A}^{*}=\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\bm{Z}_{1}\right)\bm{\varSigma}^{-1/2}$.

###### Proof.

Let
$\bm{\eta}^{(1)}=\begin{bmatrix}\bm{\tau}^{{}^{\prime}}_{1}&\ldots&\bm{\tau}^{{}^{\prime}}_{g}\end{bmatrix}^{{}^{\prime}}$
and
$\bm{\eta}^{(2)}=\setcounter{MaxMatrixCols}{11}\begin{bmatrix}\mu_{1}&\ldots&\mu_{g}&\bm{\alpha}^{{}^{\prime}}_{1}&\bm{\beta}^{{}^{\prime}}_{1}&\ldots&\bm{\alpha}^{{}^{\prime}}_{g}&\bm{\beta}^{{}^{\prime}}_{g}&\bm{\rho}^{{}^{\prime}}_{1}&\ldots&\bm{\rho}^{{}^{\prime}}_{g}\end{bmatrix}^{{}^{\prime}}$.
So, $\bm{\eta}^{(1)}$ denotes the vector of direct effects. Then by
rearranging the parameters, the model (2.2) can be equivalently expressed as

$\displaystyle\begin{bmatrix}\bm{Y}^{{}^{\prime}}_{d1}&\ldots&\bm{Y}^{{}^{\prime}}_{dg}\end{bmatrix}^{{}^{\prime}}$
$\displaystyle=\left(\bm{I}_{g}\otimes\bm{T}_{d}\right)\bm{\eta}^{(1)}+\begin{bmatrix}\bm{I}_{g}\otimes\bm{{1}}_{np}&\bm{Z}_{1}&\bm{I}_{g}\otimes\bm{F}_{d}\end{bmatrix}\bm{\eta}^{(2)}+\begin{bmatrix}\bm{\epsilon}^{{}^{\prime}}_{1}&\ldots&\bm{\epsilon}^{{}^{\prime}}_{g}\end{bmatrix}^{{}^{\prime}}.$
(S.2)

Premultiplying (S.2) by $\bm{\varSigma}^{-1/2}$, we get the model as

$\begin{bmatrix}\bm{Y}^{{}^{\prime}}_{d1(new)}&\ldots&\bm{Y}^{{}^{\prime}}_{dg(new)}\end{bmatrix}^{{}^{\prime}}=\bm{\varSigma}^{-1/2}\left(\bm{I}_{g}\otimes\bm{T}_{d}\right)\bm{\eta}^{(1)}+\bm{\varSigma}^{-1/2}\begin{bmatrix}\bm{I}_{g}\otimes\bm{{1}}_{np}&\bm{Z}_{1}&\bm{I}_{g}\otimes\bm{F}_{d}\end{bmatrix}\bm{\eta}^{(2)}+\\\
\begin{bmatrix}\bm{\epsilon}^{{}^{\prime}}_{1(new)}&\ldots&\bm{\epsilon}^{{}^{\prime}}_{g(new)}\end{bmatrix}^{{}^{\prime}},$
(S.3)

where $\bm{Y}_{dk(new)}=\bm{\varSigma}^{-1/2}\bm{Y}_{dk}$ and
$\bm{\epsilon}_{k(new)}=\bm{\varSigma}^{-1/2}\bm{\epsilon}_{k}$. Thus using
the expression of the information matrix from Kunert,  (1983), we get that
under the proportional structure, the information matrix for the direct
effects can be expressed as

$\displaystyle\bm{C}_{d(s1)}$
$\displaystyle=\left(\bm{I}_{g}\otimes\bm{T}_{d}^{{}^{\prime}}\right)\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\begin{bmatrix}\bm{Z}_{1}&\bm{I}_{g}\otimes\bm{F}_{d}\end{bmatrix}\right)\bm{\varSigma}^{-1/2}\left(\bm{I}_{g}\otimes\bm{T}_{d}\right).$
(S.4)

From (S.4) and using the result
$pr^{\perp}\left(\begin{bmatrix}\bm{\zeta}_{1}&\bm{\zeta}_{2}\end{bmatrix}\right)=pr^{\perp}\left(\bm{\zeta}_{1}\right)-pr^{\perp}\left(\bm{\zeta}_{1}\right)\bm{\zeta}_{2}\left(\bm{\zeta}^{{}^{\prime}}_{2}pr^{\perp}\left(\bm{\zeta}_{1}\right)\bm{\zeta}_{2}\right)^{-}\times$
$\bm{\zeta}^{{}^{\prime}}_{2}pr^{\perp}\left(\bm{\zeta}_{1}\right)$ (Kunert,
1983), we get

$\displaystyle\bm{C}_{d(s1)}$
$\displaystyle=\bm{C}_{d(s1)(11)}-\bm{C}_{{d(s1)}(12)}\bm{C}^{-}_{{d(s1)}(22)}\bm{C}_{{d(s1)}(21)},$
(S.5)

where
$\bm{C}_{d(s1)(11)}=\left(\bm{I}_{g}\otimes\bm{T}^{{}^{\prime}}_{d}\right)\bm{A}^{*}\left(\bm{I}_{g}\otimes\bm{T}_{d}\right)$,
$\bm{C}_{d(s1)(12)}=\bm{C}^{{}^{\prime}}_{d(s1)(21)}=\left(\bm{I}_{g}\otimes\bm{T}^{{}^{\prime}}_{d}\right)\bm{A}^{*}\left(\bm{I}_{g}\otimes\bm{F}_{d}\right)$
and
$\bm{C}_{d(s1)(22)}=\left(\bm{I}_{g}\otimes\bm{F}^{{}^{\prime}}_{d}\right)\bm{A}^{*}\left(\bm{I}_{g}\otimes\bm{F}_{d}\right)$.
Here,
$\bm{A}^{*}=\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\bm{Z}_{1}\right)\bm{\varSigma}^{-1/2}$.
∎

######  Lemma S.2.

Under the proportional structure, for $g>1$, the matrix $\bm{A}^{*}$ as given
in Lemma S.1 can be expressed as

$\displaystyle\bm{A}^{*}$
$\displaystyle=\bm{\varGamma}^{-1}\otimes\left(\bm{H}_{n}\otimes\bm{V}^{*}\right),$
(S.6)

where
$\bm{V}^{*}=\bm{V}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{V}^{-1}\bm{J}_{p\times
p}\bm{V}^{-1}$, and $\bm{\varGamma}$ is a $g\times g$ positive definite and
symmetric matrix with non-zero off-diagonal matrix. Here,
$\bm{H}_{n}=\bm{I}_{n}-\frac{1}{n}\bm{{1}}_{n}\bm{{1}}_{n}^{{}^{\prime}}$.

###### Proof.

Under the proportional structure, for $g>1$, the dispersion matrix of the
vector of error terms is given as

$\displaystyle\bm{\varSigma}$
$\displaystyle=\bm{\varGamma}\otimes\left(\bm{I}_{n}\otimes\bm{V}\right),$
(S.7)

where $\bm{V}$ is a known positive definite and symmetric matrix.
$\bm{\varGamma}$ is a $g\times g$ positive definite and symmetric matrix with
non-zero off-diagonal elements.

Using (2.3) and properties of kronecker product, we get

$\displaystyle\begin{split}\bm{\varSigma}^{-1}&=\bm{\varGamma}^{-1}\otimes\left(\bm{I}_{n}\otimes\bm{V}^{-1}\right),\\\
\bm{\varSigma}^{-1}\bm{Z}_{1}&=\bm{\varGamma}^{-1}\otimes\left(\bm{I}_{n}\otimes\bm{V}^{-1}\right)\bm{X}_{1},\text{
and}\\\
\bm{Z}^{{}^{\prime}}_{{1}}\bm{\varSigma}^{-1}\bm{Z}_{1}&=\bm{\varGamma}^{-1}\otimes\bm{X}^{{}^{\prime}}_{{1}}\left(\bm{I}_{n}\otimes\bm{V}^{-1}\right)\bm{X}_{1}.\end{split}$
(S.8)

From Bose and Dey,  (2009), we get that
$\left[\left(\bm{I}_{n}\otimes\bm{V}^{-1/2}\right)pr^{\perp}\left(\left(\bm{I}_{n}\otimes\bm{V}^{-1/2}\right)\bm{X}_{1}\right)\left(\bm{I}_{n}\otimes\bm{V}^{-1/2}\right)\right]=\bm{H}_{n}\otimes\bm{V}^{*}$,
where
$\bm{V}^{*}=\bm{V}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{V}^{-1}\bm{J}_{p\times
p}\bm{V}^{-1}$. Here, $\bm{J}_{p\times
p}=\bm{{1}}_{p}\bm{{1}}_{p}^{{}^{\prime}}$ and
$\bm{H}_{n}=\bm{I}_{n}-\frac{1}{n}\bm{{1}}_{n}\bm{{1}}_{n}^{{}^{\prime}}$.
Thus from (S.8) and using the expression
$\bm{A}^{*}=\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\bm{Z}_{1}\right)\bm{\varSigma}^{-1/2}$
as given in Lemma S.1, we get that for $g>1$

$\displaystyle\bm{A}^{*}$
$\displaystyle=\bm{\varGamma}^{-1}\otimes\left[\left(\bm{I}_{n}\otimes\bm{V}^{-1/2}\right)pr^{\perp}\left(\left(\bm{I}_{n}\otimes\bm{V}^{-1/2}\right)\bm{X}_{1}\right)\left(\bm{I}_{n}\otimes\bm{V}^{-1/2}\right)\right]$
$\displaystyle=\bm{\varGamma}^{-1}\otimes\left(\bm{H}_{n}\otimes\bm{V}^{*}\right).$

∎

### S.3 Results for the generalized markov-type covariance structure

######  Lemma S.3.

Under the generalized markov-type structure, the information matrix for the
direct effects can be expressed as

$\displaystyle\bm{C}_{d(s2)}$
$\displaystyle=\bm{C}_{d(s2)(11)}-\bm{C}_{{d(s2)}(12)}\bm{C}^{-}_{{d(s2)}(22)}\bm{C}_{{d(s2)}(21)}$
(S.9)
$\displaystyle=\left(\bm{I}_{2}\otimes\bm{T}_{d}^{{}^{\prime}}\right)\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\begin{bmatrix}\bm{Z}_{1}&\bm{I}_{2}\otimes\bm{F}_{d}\bm{H}_{t}\end{bmatrix}\right)\bm{\varSigma}^{-1/2}\left(\bm{I}_{2}\otimes\bm{T}_{d}\right)$
(S.10)
$\displaystyle=\bm{C}_{{d(s2)}(11)(1)}-\bm{C}_{{d(s2)}(12)(1)}\bm{C}^{-}_{{d(s2)}(22)(1)}\bm{C}_{{d(s2)}(21)(1)},$
(S.11)

where
$\bm{C}_{d(s2)(11)}=\left(\bm{I}_{2}\otimes\bm{T}^{{}^{\prime}}_{d}\right)\bm{A}^{*}\left(\bm{I}_{2}\otimes\bm{T}_{d}\right)$,
$\bm{C}_{d(s2)(12)}=\bm{C}^{{}^{\prime}}_{d(s2)(21)}=\left(\bm{I}_{2}\otimes\bm{T}^{{}^{\prime}}_{d}\right)\bm{A}^{*}\left(\bm{I}_{2}\otimes\bm{F}_{d}\right)$,
$\bm{C}_{d(s2)(22)}=\left(\bm{I}_{2}\otimes\bm{F}^{{}^{\prime}}_{d}\right)\bm{A}^{*}\left(\bm{I}_{2}\otimes\bm{F}_{d}\right)$,
$\bm{C}_{{d(s2)}(11)(1)}=\left(\bm{I}_{2}\otimes\bm{T}_{d}^{{}^{\prime}}\right)\bm{A}^{*}\left(\bm{I}_{2}\otimes\bm{T}_{d}\right)$,
$\bm{C}_{{d(s2)}(12)(1)}=$
$\bm{C}^{{}^{\prime}}_{{d(s2)}(21)(1)}=\left(\bm{I}_{2}\otimes\bm{T}_{d}^{{}^{\prime}}\right)\bm{A}^{*}\left(\bm{I}_{2}\otimes\bm{F}_{d}\bm{H}_{t}\right)$
and
$\bm{C}_{{d(s2)}(22)(1)}=\left(\bm{I}_{2}\otimes\bm{F}^{{}^{\prime}}_{d}\bm{H}_{t}\right)\bm{A}^{*}\left(\bm{I}_{2}\otimes\bm{F}_{d}\bm{H}_{t}\right)$.
Here,
$\bm{A}^{*}=\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\bm{Z}_{1}\right)\bm{\varSigma}^{-1/2}$
and $\bm{H}_{t}=\bm{I}_{t}-\frac{1}{t}\bm{{1}}_{t}\bm{{1}}_{t}^{{}^{\prime}}$.

###### Proof.

Under the generalized markov-type structure, following the proof of Lemma S.1
till (S.4) by taking $g=2$, we get

$\displaystyle\bm{C}_{d(s2)}$
$\displaystyle=\left(\bm{I}_{2}\otimes\bm{T}_{d}^{{}^{\prime}}\right)\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\begin{bmatrix}\bm{Z}_{1}&\bm{I}_{2}\otimes\bm{F}_{d}\end{bmatrix}\right)\bm{\varSigma}^{-1/2}\left(\bm{I}_{2}\otimes\bm{T}_{d}\right).$
(S.12)

From (S.12) and using the result
$pr^{\perp}\left(\begin{bmatrix}\bm{\zeta}_{1}&\bm{\zeta}_{2}\end{bmatrix}\right)=pr^{\perp}\left(\bm{\zeta}_{1}\right)-pr^{\perp}\left(\bm{\zeta}_{1}\right)\bm{\zeta}_{2}\left(\bm{\zeta}^{{}^{\prime}}_{2}pr^{\perp}\left(\bm{\zeta}_{1}\right)\bm{\zeta}_{2}\right)^{-}\times$
$\bm{\zeta}^{{}^{\prime}}_{2}pr^{\perp}\left(\bm{\zeta}_{1}\right)$ (Kunert,
1983), we get

$\displaystyle\bm{C}_{d(s2)}$
$\displaystyle=\bm{C}_{d(s2)(11)}-\bm{C}_{{d(s2)}(12)}\bm{C}^{-}_{{d(s2)}(22)}\bm{C}_{{d(s2)}(21)},$
(S.13)

where
$\bm{C}_{d(s2)(11)}=\left(\bm{I}_{2}\otimes\bm{T}^{{}^{\prime}}_{d}\right)\bm{A}^{*}\left(\bm{I}_{2}\otimes\bm{T}_{d}\right)$,
$\bm{C}_{d(s2)(12)}=\bm{C}^{{}^{\prime}}_{d(s2)(21)}=\left(\bm{I}_{2}\otimes\bm{T}^{{}^{\prime}}_{d}\right)\bm{A}^{*}\left(\bm{I}_{2}\otimes\bm{F}_{d}\right)$
and
$\bm{C}_{d(s2)(22)}=\left(\bm{I}_{2}\otimes\bm{F}^{{}^{\prime}}_{d}\right)\bm{A}^{*}\left(\bm{I}_{2}\otimes\bm{F}_{d}\right)$.
Here,
$\bm{A}^{*}=\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\bm{Z}_{1}\right)\bm{\varSigma}^{-1/2}$.

Since
$\bm{F}_{d}\bm{{1}}_{t}=\bm{{1}}_{n}\otimes\begin{bmatrix}0&\bm{{1}}_{p-1}^{{}^{\prime}}\end{bmatrix}^{{}^{\prime}}=\bm{P}\begin{bmatrix}0&\bm{{1}}_{p-1}^{{}^{\prime}}\end{bmatrix}^{{}^{\prime}}$,
we get
$\left(\bm{I}_{2}\otimes\frac{1}{t}\bm{F}_{d}\bm{{1}}_{t}\bm{{1}}_{t}^{{}^{\prime}}\right)=\left(\bm{I}_{2}\otimes\bm{P}\right)\times$
$\left(\bm{I}_{2}\otimes\frac{1}{t}\begin{bmatrix}0&\bm{{1}}_{p-1}^{{}^{\prime}}\end{bmatrix}^{{}^{\prime}}\bm{{1}}_{t}^{{}^{\prime}}\right)$.
Hence,
$\bm{I}_{2}\otimes\frac{1}{t}\bm{F}_{d}\bm{{1}}_{t}\bm{{1}}_{t}^{{}^{\prime}}$
belongs to the column space of $\bm{I}_{2}\otimes\bm{P}$. So,
$\bm{\varSigma}^{-1/2}\left(\bm{I}_{2}\otimes\frac{1}{t}\bm{F}_{d}\bm{{1}}_{t}\bm{{1}}_{t}^{{}^{\prime}}\right)$
belongs to the column space of $\bm{\varSigma}^{-1/2}\bm{Z}_{1}$. Thus from
(S.12), we can prove (S.10).

Further using the result
$pr^{\perp}\left(\begin{bmatrix}\bm{\zeta}_{1}&\bm{\zeta}_{2}\end{bmatrix}\right)=pr^{\perp}\left(\bm{\zeta}_{1}\right)-pr^{\perp}\left(\bm{\zeta}_{1}\right)\bm{\zeta}_{2}\left(\bm{\zeta}^{{}^{\prime}}_{2}pr^{\perp}\left(\bm{\zeta}_{1}\right)\bm{\zeta}_{2}\right)^{-}\bm{\zeta}^{{}^{\prime}}_{2}pr^{\perp}\left(\bm{\zeta}_{1}\right)$
(Kunert,  1983), we get

$\displaystyle\bm{C}_{d(s2)}$
$\displaystyle=\bm{C}_{{d(s2)}(11)(1)}-\bm{C}_{{d(s2)}(12)(1)}\bm{C}^{-}_{{d(s2)}(22)(1)}\bm{C}_{{d(s2)}(21)(1)},$
(S.14)

where
$\bm{C}_{{d(s2)}(11)(1)}=\left(\bm{I}_{2}\otimes\bm{T}_{d}^{{}^{\prime}}\right)\bm{A}^{*}\left(\bm{I}_{2}\otimes\bm{T}_{d}\right)$,
$\bm{C}_{{d(s2)}(12)(1)}=\bm{C}^{{}^{\prime}}_{{d(s2)}(21)(1)}=$
$\left(\bm{I}_{2}\otimes\bm{T}_{d}^{{}^{\prime}}\right)\bm{A}^{*}\left(\bm{I}_{2}\otimes\bm{F}_{d}\bm{H}_{t}\right)$
and
$\bm{C}_{{d(s2)}(22)(1)}=\left(\bm{I}_{2}\otimes\bm{F}^{{}^{\prime}}_{d}\bm{H}_{t}\right)\bm{A}^{*}\left(\bm{I}_{2}\otimes\bm{F}_{d}\bm{H}_{t}\right)$.
Here,
$\bm{H}_{t}=\bm{I}_{t}-\frac{1}{t}\bm{{1}}_{t}\bm{{1}}_{t}^{{}^{\prime}}$. ∎

######  Lemma S.4.

Under the generalized markov-type structure, the matrix $\bm{A}^{*}$ as given
in Lemma S.3 can be expressed as

$\displaystyle\bm{A}^{*}$
$\displaystyle=\begin{bmatrix}\bm{H}_{n}\otimes\bm{\varOmega}_{1}&-\bm{H}_{n}\otimes\bm{\varOmega}_{2}\\\
-\bm{H}_{n}\otimes\bm{\varOmega}_{2}&\bm{H}_{n}\otimes\bm{\varOmega}_{4}\end{bmatrix},$
(S.15)

where

$\displaystyle\begin{split}\bm{\varOmega}_{1}&=\bm{V}_{1}^{*}+\frac{\bar{\rho}^{2}}{\sigma_{12}}\bm{V}_{R}^{*},\\\
\bm{\varOmega}_{2}&=\frac{\bar{\rho}}{\sigma_{12}}\bm{V}_{R}^{*},\text{
and}\\\ \bm{\varOmega}_{4}&=\frac{1}{\sigma_{12}}\bm{V}_{R}^{*}.\end{split}$
(S.16)

Here,
$\bm{H}_{n}=\bm{I}_{n}-\frac{1}{n}\bm{{1}}_{n}\bm{{1}}_{n}^{{}^{\prime}}$,
$\bm{V}_{1}^{*}=\bm{V}_{1}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}_{1}^{-1}\bm{{1}}_{p}\right)^{-1}\bm{V}_{1}^{-1}\bm{J}_{p\times
p}\bm{V}_{1}^{-1}$,
$\bm{V}_{R}^{*}=\bm{V}_{R}^{-1}-\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{V}_{R}^{-1}\bm{{1}}_{p}\right)^{-1}\times$
$\bm{V}_{R}^{-1}\bm{J}_{p\times p}\bm{V}_{R}^{-1}$, $\bm{J}_{p\times
p}=\bm{{1}}_{p}\bm{{1}}_{p}^{{}^{\prime}}$,
$\bar{\rho}=\rho\sqrt{\frac{\sigma_{22}}{\sigma_{11}}}$ and
$\sigma_{12}=\sigma_{22}\left(1-\rho^{2}\right)$.

###### Proof.

From (2.4), under the generalized markov-type structure,

$\displaystyle\bm{\varSigma}=\begin{bmatrix}\bm{\varSigma}_{11}&\bm{\varSigma}_{12}\\\
\bm{\varSigma}_{21}&\bm{\varSigma}_{22}\end{bmatrix},$ (S.17)

where $\bm{\varSigma}_{11}=\bm{I}_{n}\otimes\bm{V}_{1}$,
$\bm{\varSigma}_{12}=\bm{\varSigma}_{21}=\rho\sqrt{\frac{\sigma_{22}}{\sigma_{11}}}\bm{\varSigma}_{11}$,
$\bm{\varSigma}_{22}=\rho^{2}\frac{\sigma_{22}}{\sigma_{11}}\bm{\varSigma}_{11}+\sigma_{22}(1-\rho^{2})\bm{\varSigma}_{R}$
and $\bm{\varSigma}_{R}=\bm{I}_{n}\otimes\bm{V}_{R}$. Here, $\bm{V}_{1}$ is a
known positive definite and symmetric matrix with each diagonal element as
$\sigma_{11}>0$, and $\bm{V}_{R}$ is a known correlation matrix. Also,
$\sigma_{11},\sigma_{22}>0$ and $0<|\rho|<1$. So,

$\displaystyle\bm{\varSigma}^{-1}$
$\displaystyle=\begin{bmatrix}\bm{\varSigma}^{11}&\bm{\varSigma}^{12}\\\
\bm{\varSigma}^{21}&\bm{\varSigma}^{22}\end{bmatrix},$ (S.18)

where
$\bm{\varSigma}^{11}=\bm{\varSigma}_{11}^{-1}+\bm{\varSigma}_{11}^{-1}\bm{\varSigma}_{12}\bm{\varSigma}^{22}\bm{\varSigma}_{21}\bm{\varSigma}_{11}^{-1}$,
$\bm{\varSigma}^{12}=\left(\bm{\varSigma}^{21}\right)^{{}^{\prime}}=-\bm{\varSigma}_{11}^{-1}\bm{\varSigma}_{12}\bm{\varSigma}^{22}$
and $\bm{\varSigma}^{22}=$
$\left(\bm{\varSigma}_{11}-\bm{\varSigma}_{12}\bm{\varSigma}^{-1}_{22}\bm{\varSigma}_{21}\right)^{-1}$.
Let $\bar{\rho}=\rho\sqrt{\frac{\sigma_{22}}{\sigma_{11}}}$ and
$\sigma_{12}=\sigma_{22}\left(1-\rho^{2}\right)$. Note that since
$\sigma_{11},\sigma_{22}>0$ and $0<|\rho|<1$, $|\bar{\rho}|>0$ and
$\sigma_{12}>0$. By calculation we get

$\displaystyle\begin{split}\bm{\varSigma}^{11}&=\bm{I}_{n}\otimes\bm{K}_{1}^{-1},\text{
and}\\\
\bm{\varSigma}^{12}&=\bm{\varSigma}^{21}=-\bar{\rho}\bm{\varSigma}^{22}=-\bm{I}_{n}\otimes\bar{\rho}\bm{K}_{2}^{-1}.\end{split}$
(S.19)

where $\bm{K}_{1}^{-1}=\bm{V}_{1}^{-1}+\bar{\rho}^{2}\bm{K}_{2}^{-1}$ and
$\bm{K}_{2}=\sigma_{12}\bm{V}_{R}$. Note that $\bm{K}_{1}$ and $\bm{K}_{2}$
are symmetric matrices. Since, $\bm{V}_{R}$ is positive definite matrix,
$\sigma_{12}>0$ and $|\bar{\rho}|>0$, we get that $\bm{K}_{2}$ is a positive
definite matrix. Since $\bm{\varSigma}$ is positive definite,
$\bm{\varSigma}^{11}$ is also a positive definite matrix. Thus $\bm{K}_{1}$ is
a positive definite matrix.

Now, we will find
$\left(\bm{Z}_{1}^{{}^{\prime}}\bm{\varSigma}^{-1}\bm{Z}_{1}\right)^{-}$.
Since from (S.19), $\bm{\varSigma}^{12}=\bm{\varSigma}^{21}$, using (2.3) and
(S.18), we get

$\displaystyle\bm{Z}^{{}^{\prime}}_{1}\bm{\varSigma}^{-1}\bm{Z}_{1}$
$\displaystyle=\begin{bmatrix}\bm{X}^{{}^{\prime}}_{1}\bm{\varSigma}^{11}\bm{X}_{1}&\bm{X}^{{}^{\prime}}_{1}\bm{\varSigma}^{12}\bm{X}_{1}\\\
\bm{X}^{{}^{\prime}}_{1}\bm{\varSigma}^{12}\bm{X}_{1}&\bm{X}^{{}^{\prime}}_{1}\bm{\varSigma}^{22}\bm{X}_{1}\end{bmatrix}.$

Let $\bm{A}^{(1)}=\bm{X}^{{}^{\prime}}_{1}\bm{\varSigma}^{11}\bm{X}_{1}$,
$\bm{B}^{(1)}=\bm{X}^{{}^{\prime}}_{1}\bm{\varSigma}^{12}\bm{X}_{1}$ and
$\bm{D}^{(1)}=\bm{X}^{{}^{\prime}}_{1}\bm{\varSigma}^{22}\bm{X}_{1}$. So,

$\displaystyle\left(\bm{Z}^{{}^{\prime}}_{1}\bm{\varSigma}^{-1}\bm{Z}_{1}\right)^{-}$
$\displaystyle=\begin{bmatrix}\bm{L}_{1}&\bm{L}_{2}\\\
\bm{L}_{3}&\bm{L}_{4}\end{bmatrix},$ (S.20)

where
$\bm{L}_{1}=\left(\bm{A}^{(1)}\right)^{-}+\left(\bm{A}^{(1)}\right)^{-}\bm{B}^{(1)}\left(\bm{Z}^{(1)}\right)^{-}\bm{B}^{(1)}\left(\bm{A}^{(1)}\right)^{-}$,
$\bm{L}_{2}=\bm{L}_{3}^{{}^{\prime}}=-\left(\bm{A}^{(1)}\right)^{-}\bm{B}^{(1)}\left(\bm{Z}^{(1)}\right)^{-}$
and $\bm{L}_{4}=\left(\bm{Z}^{(1)}\right)^{-}$. Here,
$\bm{Z}^{(1)}=\bm{D}^{(1)}-\bm{B}^{(1)}\left(\bm{A}^{(1)}\right)^{-}\bm{B}^{(1)}$.
From (2.3) and (S.19), by calculation we get

$\displaystyle\bm{A}^{(1)}$
$\displaystyle=\begin{bmatrix}\bm{A}^{(2)}&\bm{B}^{(2)}\\\
\bm{C}^{(2)}&\bm{D}^{(2)}\end{bmatrix},$ (S.21)

where $\bm{A}^{(2)}=n\bm{K}_{1}^{-1}$,
$\bm{B}^{(2)}=\left(\bm{C}^{(2)}\right)^{{}^{\prime}}=\bm{{1}}_{n}^{{}^{\prime}}\otimes\bm{K}_{1}^{-1}\bm{{1}}_{p}$
and $\bm{D}^{(2)}=\delta_{1}^{-1}\bm{I}_{n}$, where
$\delta_{1}=1/\bm{{1}}_{p}^{{}^{\prime}}\bm{K}_{1}^{-1}\bm{{1}}_{p}$. Note
that since $\bm{K}_{1}$ is a positive definite matrix, $\bm{K}_{1}^{-1}$ is a
positive definite matrix, and thus $\delta_{1}>0$.

Hence,

$\displaystyle\left(\bm{A}^{(1)}\right)^{-}$
$\displaystyle=\begin{bmatrix}\left(\bm{A}^{(2)}\right)^{-}+\left(\bm{A}^{(2)}\right)^{-}\bm{B}^{(2)}\left(\bm{Z}^{(2)}\right)^{-}\bm{C}^{(2)}\left(\bm{A}^{(2)}\right)^{-}&-\left(\bm{A}^{(2)}\right)^{-}\bm{B}^{(2)}\left(\bm{Z}^{(2)}\right)^{-}\\\
-\left(\bm{Z}^{(2)}\right)^{-}\bm{C}^{(2)}\left(\bm{A}^{(2)}\right)^{-}&\left(\bm{Z}^{(2)}\right)^{-}\end{bmatrix},$
(S.22)

where
$\bm{Z}^{(2)}=\bm{D}^{(2)}-\bm{C}^{(2)}\left(\bm{A}^{(2)}\right)^{-}\bm{B}^{(2)}$.
By calculation we get

$\displaystyle\begin{split}\bm{Z}^{(2)}&=\delta_{1}^{-1}\bm{H}_{n},\\\
\left(\bm{A}^{(2)}\right)^{-}\bm{B}^{(2)}\left(\bm{Z}^{(2)}\right)^{-}&=\frac{1}{n}\bm{{1}}_{n}^{{}^{\prime}}\otimes\delta_{1}\bm{{1}}_{p},\\\
\left(\bm{Z}^{(2)}\right)^{-}\bm{C}^{(2)}\left(\bm{A}^{(2)}\right)^{-}&=\frac{1}{n}\bm{{1}}_{n}\otimes\delta_{1}\bm{{1}}^{{}^{\prime}}_{p},\text{
and}\\\
\left(\bm{A}^{(2)}\right)^{-}+\left(\bm{A}^{(2)}\right)^{-}\bm{B}^{(2)}\left(\bm{Z}^{(2)}\right)^{-}\bm{C}^{(2)}\left(\bm{A}^{(2)}\right)^{-}&=\frac{1}{n}\bm{K}_{1}+\frac{1}{n}\delta_{1}\bm{J}_{p\times
p}.\end{split}$ (S.23)

Here $\bm{J}_{p\times p}=\bm{{1}}_{p}\bm{{1}}_{p}^{{}^{\prime}}$.

So, using (S.23), (S.22) becomes

$\displaystyle\left(\bm{A}^{(1)}\right)^{-}$
$\displaystyle=\begin{bmatrix}\frac{1}{n}\bm{K}_{1}+\frac{1}{n}\delta_{1}\bm{J}_{p\times
p}&-\frac{1}{n}\bm{{1}}_{n}^{{}^{\prime}}\otimes\delta_{1}\bm{{1}}_{p}\\\
-\frac{1}{n}\bm{{1}}_{n}\otimes\delta_{1}\bm{{1}}^{{}^{\prime}}_{p}&\delta_{1}\bm{I}_{n}\end{bmatrix}.$
(S.24)

From (2.3) and (S.19), by calculation we get

$\displaystyle\bm{B}^{(1)}$
$\displaystyle=-\bm{D}^{(1)}\bar{\rho}=\begin{bmatrix}-n\bar{\rho}\bm{K}_{2}^{-1}&-\bm{{1}}_{n}^{{}^{\prime}}\otimes\bar{\rho}\bm{K}_{2}^{-1}\bm{{1}}_{p}\\\
-\bm{{1}}_{n}\otimes\bar{\rho}\bm{{1}}^{{}^{\prime}}_{p}\bm{K}_{2}^{-1}&-\bar{\rho}\delta_{2}^{-1}\bm{I}_{n}\end{bmatrix},$
(S.25)

where $\delta_{2}=1/\bm{{1}}^{{}^{\prime}}_{p}\bm{K}_{2}^{-1}\bm{{1}}_{p}$.
Note that since $\bm{K}_{2}$ is a positive definite matrix, $\bm{K}_{2}^{-1}$
is a positive definite matrix, and thus $\delta_{2}>0$.

Let $\bm{K}_{3}=\bar{\rho}^{2}\bm{V}_{1}+\bm{K}_{2}$. Note that $\bm{K}_{3}$
is a symmetric matrix. Since $\bm{V}_{1}$, $\bm{K}_{1}$ and $\bm{K}_{2}$ are
positive definite matrices, using the result that
$\bm{I}_{p}+\bm{\zeta}_{3}\bm{\zeta}_{4}$ is invertible if and only if
$\bm{I}_{p}+\bm{\zeta}_{4}\bm{\zeta}_{3}$ is invertible, where
$\bm{\zeta}_{3}$ and $\bm{\zeta}_{4}$ are matrices such that
$\bm{\zeta}_{3}\bm{\zeta}_{4}$ is a $p\times p$ matrix, we get that
$\bm{K}_{3}$ is a invertible matrix. Using the result
$\left(\bm{I}_{p}+\bm{\zeta}_{3}\bm{\zeta}_{4}\right)^{-1}=\bm{I}_{p}-\bm{\zeta}_{3}\left(\bm{I}_{p}+\bm{\zeta}_{4}\bm{\zeta}_{3}\right)^{-1}\bm{\zeta}_{4}$,
where $\bm{\zeta}_{3}$ and $\bm{\zeta}_{4}$ are matrices such that
$\bm{\zeta}_{3}\bm{\zeta}_{4}$ is a $p\times p$ matrix, we get that
$\bm{K}_{2}^{-1}-\bar{\rho}^{2}\bm{K}_{2}^{-1}\bm{K}_{1}\bm{K}_{2}^{-1}=\bm{K}_{3}^{-1}$.
Thus from (S.24) and (S.25), by calculation we get

$\displaystyle\bm{Z}^{(1)}$
$\displaystyle=\bm{D}^{(1)}-\bm{B}^{(1)}\left(\bm{A}^{(1)}\right)^{-}\bm{B}^{(1)}$
$\displaystyle=\begin{bmatrix}n\bm{K}_{3}^{-1}&\bm{{1}}_{n}^{{}^{\prime}}\otimes\bm{K}_{3}^{-1}\bm{{1}}_{p}\\\
\bm{{1}}_{n}\otimes\bm{{1}}_{p}^{{}^{\prime}}\bm{K}_{3}^{-1}&\delta_{4}\bm{H}_{n}+\frac{1}{n}\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{K}_{3}^{-1}\bm{{1}}_{p}\right)\bm{J}_{n\times
n}\end{bmatrix},$ (S.26)

where $\delta_{4}=\delta_{2}^{-1}-\bar{\rho}^{2}\delta_{1}\delta_{2}^{-2}$.
Here, $\bm{J}_{n\times n}=\bm{{1}}_{n}\bm{{1}}_{n}^{{}^{\prime}}$ and
$\bm{H}_{n}=\bm{I}_{n}-\frac{1}{n}\bm{{1}}_{n}\bm{{1}}_{n}^{{}^{\prime}}$.
Using the expression of $\bm{K}_{1}^{-1}$, $\delta_{1}$ and $\delta_{2}$, we
get

$\displaystyle\delta_{1}\bm{{1}}_{p}^{{}^{\prime}}\bm{V}_{1}^{-1}\bm{{1}}_{p}+\bar{\rho}^{2}\delta_{1}\delta_{2}^{-1}=1.$
(S.27)

From (S.27) and using the expression of $\delta_{4}$, we get

$\displaystyle\delta_{4}=\delta_{1}\delta_{2}^{-1}\bm{{1}}_{p}^{{}^{\prime}}\bm{V}_{1}^{-1}\bm{{1}}_{p}.$
(S.28)

Since $\delta_{1},\delta_{2}>0$ and $\bm{V}_{1}$ is a positive definite
matrix, from (S.28), we get that $\delta_{4}>0$.

Let $\bm{A}^{(3)}=n\bm{K}_{3}^{-1}$,
$\bm{B}^{(3)}=\left(\bm{C}^{(3)}\right)^{{}^{\prime}}=\bm{{1}}_{n}^{{}^{\prime}}\otimes\bm{K}_{3}^{-1}\bm{{1}}_{p}$
and
$\bm{D}^{(3)}=\delta_{4}\bm{H}_{n}+\frac{1}{n}\left(\bm{{1}}_{p}^{{}^{\prime}}\bm{K}_{3}^{-1}\bm{{1}}_{p}\right)\bm{J}_{n\times
n}$. So,

$\displaystyle\left(\bm{Z}^{(1)}\right)^{-}$
$\displaystyle=\begin{bmatrix}\left(\bm{A}^{(3)}\right)^{-}+\left(\bm{A}^{(3)}\right)^{-}\bm{B}^{(3)}\left(\bm{Z}^{(3)}\right)^{-}\bm{C}^{(3)}\left(\bm{A}^{(3)}\right)^{-}&-\left(\bm{A}^{(3)}\right)^{-}\bm{B}^{(3)}\left(\bm{Z}^{(3)}\right)^{-}\\\
-\left(\bm{Z}^{(3)}\right)^{-}\bm{C}^{(3)}\left(\bm{A}^{(3)}\right)^{-}&\left(\bm{Z}^{(3)}\right)^{-}\end{bmatrix},$
(S.29)

where
$\bm{Z}^{(3)}=\bm{D}^{(3)}-\bm{C}^{(3)}\left(\bm{A}^{(3)}\right)^{-}\bm{B}^{(3)}$.
Using the expression of the matrices $\bm{A}^{(3)}$, $\bm{B}^{(3)}$,
$\bm{C}^{(3)}$ and $\bm{D}^{(3)}$, by calculation we get

$\displaystyle\begin{split}\bm{Z}^{(3)}&=\delta_{4}\bm{H}_{n},\\\
\left(\bm{A}^{(3)}\right)^{-}\bm{B}^{(3)}\left(\bm{Z}^{(3)}\right)^{-}&=\frac{1}{n}\bm{{1}}_{n}^{{}^{\prime}}\otimes\delta_{4}^{-1}\bm{{1}}_{p},\\\
\left(\bm{Z}^{(3)}\right)^{-}\bm{C}^{(3)}\left(\bm{A}^{(3)}\right)^{-}&=\frac{1}{n}\bm{{1}}_{n}\otimes\delta_{4}^{-1}\bm{{1}}_{p}^{{}^{\prime}},\text{
and}\\\
\left(\bm{A}^{(3)}\right)^{-}+\left(\bm{A}^{(3)}\right)^{-}\bm{B}^{(3)}\left(\bm{Z}^{(3)}\right)^{-}\bm{C}^{(3)}\left(\bm{A}^{(3)}\right)^{-}&=\frac{1}{n}\bm{K}_{3}+\frac{1}{n}\delta_{4}^{-1}\bm{J}_{p\times
p}.\end{split}$ (S.30)

From (S.29) and (S.30), we get

$\displaystyle\left(\bm{Z}^{(1)}\right)^{-}$
$\displaystyle=\begin{bmatrix}\frac{1}{n}\bm{K}_{3}+\frac{1}{n}\delta_{4}^{-1}\bm{J}_{p\times
p}&-\frac{1}{n}\bm{{1}}_{n}^{{}^{\prime}}\otimes\delta_{4}^{-1}\bm{{1}}_{p}\\\
-\frac{1}{n}\bm{{1}}_{n}\otimes\delta_{4}^{-1}\bm{{1}}_{p}^{{}^{\prime}}&\delta_{4}^{-1}\bm{I}_{n}\end{bmatrix}.$
(S.31)

Using the expression of matrices $\bm{K}_{1}$, $\bm{K}_{2}$ and $\bm{K}_{3}$,
we get

$\displaystyle\bm{K}_{1}\bm{K}_{2}^{-1}=\bm{V}_{1}\bm{K}_{3}^{-1}.$ (S.32)

So, from (S.24), (S.25) and (S.31), by calculation we get

$\displaystyle\left(\bm{A}^{(1)}\right)^{-}\bm{B}^{(1)}\left(\bm{Z}^{(1)}\right)^{-}=\left(\bm{Z}^{(1)}\right)^{-}\bm{B}^{(1)}\left(\bm{A}^{(1)}\right)^{-}=\begin{bmatrix}-\frac{1}{n}\bar{\rho}\bm{V}_{1}&\bm{{0}}_{p\times
n}\\\ \bm{{0}}_{n\times
p}&-\bar{\rho}\delta_{4}^{-1}\delta_{1}\delta_{2}^{-1}\bm{H}_{n}\end{bmatrix}$
(S.33)

and

$\left(\bm{A}^{(1)}\right)^{-}+\left(\bm{A}^{(1)}\right)^{-}\bm{B}^{(1)}\left(\bm{Z}^{(1)}\right)^{-}\bm{C}^{(1)}\left(\bm{A}^{(1)}\right)^{-}\\\
=\begin{bmatrix}\frac{1}{n}\bm{K}_{1}+\frac{1}{n}\delta_{1}\bm{J}_{p\times
p}+\frac{1}{n}\bar{\rho}^{2}\bm{V}_{1}\bm{K}_{2}^{-1}\bm{K}_{1}&-\frac{1}{n}\bm{{1}}_{n}^{{}^{\prime}}\otimes\delta_{1}\bm{{1}}_{p}\\\
-\frac{1}{n}\bm{{1}}_{n}\otimes\delta_{1}\bm{{1}}^{{}^{\prime}}_{p}&\delta_{1}\bm{I}_{n}+\bar{\rho}^{2}\delta_{4}^{-1}\delta_{1}^{2}\delta_{2}^{-2}\bm{H}_{n}\end{bmatrix}.$
(S.34)

From (S.31), (S.33) and (S.34), (S.20) becomes

$\displaystyle\left(\bm{Z}^{{}^{\prime}}_{1}\bm{\varSigma}^{-1}\bm{Z}_{1}\right)^{-}$
$\displaystyle=\begin{bmatrix}\bm{L}_{1}&\bm{L}_{2}\\\
\bm{L}_{3}&\bm{L}_{4}\end{bmatrix},$ (S.35)

where

$\displaystyle\begin{split}\bm{L}_{1}&=\begin{bmatrix}\frac{1}{n}\bm{K}_{1}+\frac{1}{n}\delta_{1}\bm{J}_{p\times
p}+\frac{1}{n}\bar{\rho}^{2}\bm{V}_{1}\bm{K}_{2}^{-1}\bm{K}_{1}&-\frac{1}{n}\bm{{1}}_{n}^{{}^{\prime}}\otimes\delta_{1}\bm{{1}}_{p}\\\
-\frac{1}{n}\bm{{1}}_{n}\otimes\delta_{1}\bm{{1}}^{{}^{\prime}}_{p}&\delta_{1}\bm{I}_{n}+\bar{\rho}^{2}\delta_{4}^{-1}\delta_{1}^{2}\delta_{2}^{-2}\bm{H}_{n}\end{bmatrix},\\\
\bm{L}_{2}&=\bm{L}_{3}=\begin{bmatrix}\frac{1}{n}\bar{\rho}\bm{V}_{1}&\bm{{0}}_{p\times
n}\\\ \bm{{0}}_{n\times
p}&\bar{\rho}\delta_{4}^{-1}\delta_{1}\delta_{2}^{-1}\bm{H}_{n}\end{bmatrix},\text{
and}\\\
\bm{L}_{4}&=\begin{bmatrix}\frac{1}{n}\bm{K}_{3}+\frac{1}{n}\delta_{4}^{-1}\bm{J}_{p\times
p}&-\frac{1}{n}\bm{{1}}_{n}^{{}^{\prime}}\otimes\delta_{4}^{-1}\bm{{1}}_{p}\\\
-\frac{1}{n}\bm{{1}}_{n}\otimes\delta_{4}^{-1}\bm{{1}}_{p}^{{}^{\prime}}&\delta_{4}^{-1}\bm{I}_{n}\end{bmatrix}.\end{split}$
(S.36)

Now, we find $\bm{\varSigma}^{-1}\bm{Z}_{1}$. From (2.3), (S.18) and (S.19),
by calculation we get

$\displaystyle\bm{\varSigma}^{-1}\bm{Z}_{1}$
$\displaystyle=\begin{bmatrix}\bm{\varSigma}^{11}\bm{X}_{1}&\bm{\varSigma}^{12}\bm{X}_{1}\\\
\bm{\varSigma}^{12}\bm{X}_{1}&\bm{\varSigma}^{22}\bm{X}_{1}\end{bmatrix},$
(S.37)

where

$\displaystyle\begin{split}\bm{\varSigma}^{11}\bm{X}_{1}&=\begin{bmatrix}\bm{{1}}_{n}\otimes\bm{K}_{1}^{-1}&\bm{I}_{n}\otimes\bm{K}_{1}^{-1}\bm{{1}}_{p}\end{bmatrix},\text{
and}\\\
\bm{\varSigma}^{12}\bm{X}_{1}&=-\bar{\rho}\bm{\varSigma}^{22}\bm{X}_{1}=\begin{bmatrix}-\bm{{1}}_{n}\otimes\bar{\rho}\bm{K}_{2}^{-1}&-\bm{I}_{n}\otimes\bar{\rho}\bm{K}_{2}^{-1}\bm{{1}}_{p}\end{bmatrix}.\end{split}$
(S.38)

Now, we find
$\bm{\varSigma}^{-1}\bm{Z}_{1}\left(\bm{Z}^{{}^{\prime}}_{1}\bm{\varSigma}^{-1}\bm{Z}_{1}\right)^{-}\bm{Z}^{{}^{\prime}}_{1}\bm{\varSigma}^{-1}$.
Since $\bm{V}_{1}$, $\bm{K}_{1}$, $\bm{K}_{2}$ and $\bm{K}_{3}$ are symmetric
matrices, from (S.32), we get that
$\bm{K}_{2}^{-1}\bm{K}_{3}=\bm{K}_{1}^{-1}\bm{V}_{1}$ and
$\bm{K}_{1}^{-1}\bm{V}_{1}\bm{K}_{2}^{-1}\bm{K}_{1}=\bm{K}_{2}^{-1}\bm{V}_{1}$.
So, using the expression of matrix $\bm{K}_{3}$, we get that
$\bm{K}_{2}^{-1}\bm{K}_{3}=\bm{I}_{p}+\bar{\rho}^{2}\bm{K}_{2}^{-1}\bm{V}_{1}$
and
$\bm{K}_{2}^{-1}\bm{K}_{1}+\bar{\rho}^{2}\bm{K}_{2}^{-1}\bm{V}_{1}\bm{K}_{2}^{-1}\bm{K}_{1}=\bm{K}_{2}^{-1}\bm{V}_{1}$.
Using the expression of $\delta_{4}$, by solving we get
$\delta_{1}+\bar{\rho}^{2}\delta_{4}^{-1}\delta_{1}^{2}\delta_{2}^{-2}=\delta_{4}^{-1}\delta_{1}\delta_{2}^{-1}$.
Thus from (S.35), (S.36), (S.37) and (S.38), by calculation we get

$\displaystyle\bm{\varSigma}^{-1}\bm{Z}_{1}\left(\bm{Z}^{{}^{\prime}}_{1}\bm{\varSigma}^{-1}\bm{Z}_{1}\right)^{-}\bm{Z}^{{}^{\prime}}_{1}\bm{\varSigma}^{-1}$
$\displaystyle=\begin{bmatrix}\bm{A}^{(4)}&\bm{B}^{(4)}\\\
\bm{B}^{(4)}&\bm{D}^{(4)}\end{bmatrix},$ (S.39)

where

$\bm{A}^{(4)}=\frac{1}{n}\bm{J}_{n\times
n}\otimes\bm{K}_{1}^{-1}+\bm{H}_{n}\otimes\delta_{4}^{-1}\delta_{1}\delta_{2}^{-1}\bm{K}_{1}^{-1}\bm{J}_{p\times
p}\bm{K}_{1}^{-1}-\bm{H}_{n}\otimes\bar{\rho}^{2}\delta_{4}^{-1}\delta_{1}\delta_{2}^{-1}\bm{K}_{2}^{-1}\bm{J}_{p\times
p}\bm{K}_{1}^{-1}\\\
+\bm{H}_{n}\otimes\bar{\rho}^{2}\delta_{4}^{-1}\bm{K}_{2}^{-1}\bm{J}_{p\times
p}\bm{K}_{2}^{-1}-\bm{H}_{n}\otimes\bar{\rho}^{2}\delta_{4}^{-1}\delta_{1}\delta_{2}^{-1}\bm{K}_{1}^{-1}\bm{J}_{p\times
p}\bm{K}_{2}^{-1},\text{ and}$ $\displaystyle\bm{B}^{(4)}$
$\displaystyle=-\bar{\rho}\bm{D}^{(4)}=-\frac{1}{n}\bm{J}_{n\times
n}\otimes\bar{\rho}\bm{K}_{2}^{-1}+\bm{H}_{n}\otimes\bar{\rho}^{3}\delta_{4}^{-1}\delta_{1}\delta_{2}^{-1}\bm{K}_{2}^{-1}\bm{J}_{p\times
p}\bm{K}_{2}^{-1}-\bm{H}_{n}\otimes\bar{\rho}\delta_{4}^{-1}\bm{K}_{2}^{-1}\bm{J}_{p\times
p}\bm{K}_{2}^{-1}.$

Thus from (S.27), (S.28) and using the expression of $\bm{A}^{*}$ from Lemma
S.3, by calculation we can prove (S.15). ∎

######  Remark S.1.

The matrices $\bm{\varOmega}_{1}$, $\bm{\varOmega}_{2}$, and
$\bm{\varOmega}_{4}$ have row sums and column sums as zero.

###### Proof.

Using the expression of the matrices $\bm{\varOmega}_{1}$,
$\bm{\varOmega}_{2}$, and $\bm{\varOmega}_{4}$, as given in Theorem S.4, we
get that
$\bm{{1}}_{p}^{{}^{\prime}}\bm{\varOmega}_{1}=\bm{{1}}_{p}^{{}^{\prime}}\bm{\varOmega}_{2}=\bm{{1}}_{p}^{{}^{\prime}}\bm{\varOmega}_{4}=\bm{{0}}_{1\times
p}$ and
$\bm{\varOmega}_{1}\bm{{1}}_{p}=\bm{\varOmega}_{2}\bm{{1}}_{p}=\bm{\varOmega}_{4}\bm{{1}}_{p}=\bm{{0}}_{p\times
1}$. ∎

######  Lemma S.5.

Under the generalized markov-type structure, the information matrix for the
direct effects in the absence of the period effects can be expressed as

$\displaystyle\begin{split}\bm{\tilde{C}}_{d(s2)}&=\left(\bm{I}_{2}\otimes\bm{T}_{d}^{{}^{\prime}}\right)\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\begin{bmatrix}\bm{I}_{2}\otimes\bm{U}&\bm{I}_{2}\otimes\bm{F}_{d}\bm{H}_{t}\end{bmatrix}\right)\bm{\varSigma}^{-1/2}\left(\bm{I}_{2}\otimes\bm{T}_{d}\right)\\\
&=\bm{\tilde{C}}_{d(s2)(11)}-\bm{\tilde{C}}_{d(s2)(12)}\bm{\tilde{C}}^{-}_{d(s2)(22)}\bm{\tilde{C}}_{d(s2)(21)},\end{split}$
(S.40)

where
$\bm{\tilde{C}}_{d(s2)(11)}=\left(\bm{I}_{2}\otimes\bm{T}_{d}^{{}^{\prime}}\right)\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\begin{bmatrix}\bm{I}_{2}\otimes\bm{U}\end{bmatrix}\right)\bm{\varSigma}^{-1/2}\left(\bm{I}_{2}\otimes\bm{T}_{d}\right)$,
$\bm{\tilde{C}}_{d(s2)(12)}=\bm{\tilde{C}}^{{}^{\prime}}_{d(s2)(21)}=\left(\bm{I}_{2}\otimes\bm{T}_{d}^{{}^{\prime}}\right)\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\begin{bmatrix}\bm{I}_{2}\otimes\bm{U}\end{bmatrix}\right)\bm{\varSigma}^{-1/2}\left(\bm{I}_{2}\otimes\bm{F}_{d}\bm{H}_{t}\right)$,
and
$\bm{\tilde{C}}_{d(s2)(22)}=\left(\bm{I}_{2}\otimes\bm{H}_{t}\bm{F}_{d}^{{}^{\prime}}\right)\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\begin{bmatrix}\bm{I}_{2}\otimes\bm{U}\end{bmatrix}\right)\bm{\varSigma}^{-1/2}\left(\bm{I}_{2}\otimes\bm{F}_{d}\bm{H}_{t}\right)$.
Here,
$\bm{H}_{t}=\bm{I}_{t}-\frac{1}{t}\bm{{1}}_{t}\bm{{1}}_{t}^{{}^{\prime}}$.

###### Proof.

Following the same approach as to obtain the information matrix for the direct
effects in the presence of the period effects as disucssed in Lemma S.3, we
can prove (S.40). ∎

######  Lemma S.6.

The matrix
$\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\begin{bmatrix}\bm{I}_{2}\otimes\bm{U}\end{bmatrix}\right)\bm{\varSigma}^{-1/2}$
can be expressed as

$\displaystyle\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\begin{bmatrix}\bm{I}_{2}\otimes\bm{U}\end{bmatrix}\right)\bm{\varSigma}^{-1/2}$
$\displaystyle=\begin{bmatrix}\bm{I}_{n}\otimes\bm{\varOmega}_{1}&-\bm{I}_{n}\otimes\bm{\varOmega}_{2}\\\
-\bm{I}_{n}\otimes\bm{\varOmega}_{2}&\bm{I}_{n}\otimes\bm{\varOmega}_{4}\end{bmatrix},$
(S.41)

where $\bm{\varOmega}_{1}$, $\bm{\varOmega}_{2}$, and $\bm{\varOmega}_{4}$ are
as given in Lemma S.4.

###### Proof.

From (2.3), (S.18) and (S.19), by calculation we get

$\displaystyle\left(\bm{I}_{2}\otimes\bm{U}^{{}^{\prime}}\right)\bm{\varSigma}^{-1}\left(\bm{I}_{2}\otimes\bm{U}\right)$
$\displaystyle=\begin{bmatrix}\bm{A}^{(5)}&\bm{B}^{(5)}\\\
\bm{B}^{(5)}&\bm{D}^{(5)}\end{bmatrix},$ (S.42)

where $\bm{A}^{(5)}=\delta_{1}^{-1}\bm{I}_{n}$,
$\bm{B}^{(5)}=-\bar{\rho}\delta_{2}^{-1}\bm{I}_{n}$, and
$\bm{D}^{(5)}=\delta_{2}^{-1}\bm{I}_{n}$. Here,
$\delta_{1}=1/\bm{{1}}_{p}^{{}^{\prime}}\bm{K}_{1}^{-1}\bm{{1}}_{p}$,
$\delta_{2}=1/\bm{{1}}_{p}^{{}^{\prime}}\bm{K}_{2}^{-1}\bm{{1}}_{p}$,
$\bm{K}_{1}^{-1}=\bm{V}_{1}^{-1}+\bar{\rho}^{2}\bm{K}_{2}^{-1}$ and
$\bm{K}_{2}=\sigma_{12}\bm{V}_{R}$,
$\bar{\rho}=\rho\sqrt{\frac{\sigma_{22}}{\sigma_{11}}}$ and
$\sigma_{12}=\sigma_{22}\left(1-\rho^{2}\right)$. So,

$\left[\left(\bm{I}_{2}\otimes\bm{U}^{{}^{\prime}}\right)\bm{\varSigma}^{-1}\left(\bm{I}_{2}\otimes\bm{U}\right)\right]^{-}=\\\
\begin{bmatrix}\left(\bm{A}^{(5)}\right)^{-}+\left(\bm{A}^{(5)}\right)^{-}\bm{B}^{(5)}\left(\bm{Z}^{(5)}\right)^{-}\bm{B}^{(5)}\left(\bm{A}^{(5)}\right)^{-}&-\left(\bm{A}^{(5)}\right)^{-}\bm{B}^{(5)}\left(\bm{Z}^{(5)}\right)^{-}\\\
-\left(\bm{Z}^{(5)}\right)^{-}\bm{B}^{(5)}\left(\bm{A}^{(5)}\right)^{-}&\left(\bm{Z}^{(5)}\right)^{-}\end{bmatrix},$
(S.43)

where
$\bm{Z}^{(5)}=\bm{D}^{(5)}-\bm{B}^{(5)}\left(\bm{A}^{(5)}\right)^{-}\bm{B}^{(5)}$.
Using the expressions of $\bm{A}^{(5)}$, $\bm{B}^{(5)}$ and $\bm{D}^{(5)}$, by
solving we get

$\displaystyle\begin{split}\left(\bm{Z}^{(5)}\right)^{-}&=\delta_{4}^{-1}\bm{I}_{n},\\\
\left(\bm{A}^{(5)}\right)^{-}\bm{B}^{(5)}\left(\bm{Z}^{(5)}\right)^{-}&=\left(\bm{Z}^{(5)}\right)^{-}\bm{B}^{(5)}\left(\bm{A}^{(5)}\right)^{-}=-\bar{\rho}\delta_{4}^{-1}\delta_{1}\delta_{2}^{-1}\bm{I}_{n},\text{
and}\\\
\left(\bm{A}^{(5)}\right)^{-}+\left(\bm{A}^{(5)}\right)^{-}\bm{B}^{(5)}\left(\bm{Z}^{(5)}\right)^{-}\bm{B}^{(5)}\left(\bm{A}^{(5)}\right)^{-}&=\left(\delta_{1}+\bar{\rho}^{2}\delta_{4}^{-1}\delta_{1}^{2}\delta_{2}^{-2}\right)\bm{I}_{n},\end{split}$
(S.44)

where $\delta_{4}=\delta_{2}^{-1}-\bar{\rho}^{2}\delta_{1}\delta_{2}^{-2}$.
Using the expression $\delta_{4}$, we get that
$\delta_{1}+\bar{\rho}^{2}\delta_{4}^{-1}\delta_{1}^{2}\delta_{2}^{-2}=\delta_{4}^{-1}\delta_{1}\delta_{2}^{-1}$.
So, from (S.43) and (S.44), we get

$\displaystyle\left[\left(\bm{I}_{2}\otimes\bm{U}^{{}^{\prime}}\right)\bm{\varSigma}^{-1}\left(\bm{I}_{2}\otimes\bm{U}\right)\right]^{-}$
$\displaystyle=\delta_{4}^{-1}\begin{bmatrix}\delta_{1}\delta_{2}^{-1}\bm{I}_{n}&\bar{\rho}\delta_{1}\delta_{2}^{-1}\bm{I}_{n}\\\
\bar{\rho}\delta_{1}\delta_{2}^{-1}\bm{I}_{n}&\bm{I}_{n}\end{bmatrix}.$ (S.45)

From (2.3), (S.18) and (S.19), by calculation we get

$\displaystyle\left(\bm{I}_{2}\otimes\bm{U}^{{}^{\prime}}\right)\bm{\varSigma}^{-1}$
$\displaystyle=\begin{bmatrix}\bm{I}_{n}\otimes\bm{{1}}_{p}^{{}^{\prime}}\bm{K}_{1}^{-1}&-\bm{I}_{n}\otimes\bar{\rho}\bm{{1}}_{p}^{{}^{\prime}}\bm{K}_{2}^{-1}\\\
-\bm{I}_{n}\otimes\bar{\rho}\bm{{1}}_{p}^{{}^{\prime}}\bm{K}_{2}^{-1}&\bm{I}_{n}\otimes\bm{{1}}_{p}^{{}^{\prime}}\bm{K}_{2}^{-1}\end{bmatrix}.$
(S.46)

From (S.45) and (S.46), by calculation we get

$\displaystyle\bm{\varSigma}^{-1}\left(\bm{I}_{2}\otimes\bm{U}\right)\left[\left(\bm{I}_{2}\otimes\bm{U}^{{}^{\prime}}\right)\bm{\varSigma}^{-1}\left(\bm{I}_{2}\otimes\bm{U}\right)\right]^{-}\left(\bm{I}_{2}\otimes\bm{U}^{{}^{\prime}}\right)\bm{\varSigma}^{-1}$
$\displaystyle=\begin{bmatrix}\bm{I}_{n}\otimes\bm{A}^{(6)}&-\bm{I}_{n}\otimes\bm{B}^{(6)}\\\
-\bm{I}_{n}\otimes\bm{B}^{(6)}&\bm{I}_{n}\otimes\bm{D}^{(6)}\end{bmatrix},$
(S.47)

where

$\bm{A}^{(6)}=\delta_{4}^{-1}\delta_{1}\delta_{2}^{-1}\bm{K}_{1}^{-1}\bm{J}_{p\times
p}\bm{K}_{1}^{-1}-\bar{\rho}^{2}\delta_{4}^{-1}\delta_{1}\delta_{2}^{-1}\bm{K}_{1}^{-1}\bm{J}_{p\times
p}\bm{K}_{2}^{-1}+\bar{\rho}^{2}\delta_{4}^{-1}\bm{K}_{2}^{-1}\bm{J}_{p\times
p}\bm{K}_{2}^{-1}\\\
-\bar{\rho}^{2}\delta_{4}^{-1}\delta_{1}\delta_{2}^{-1}\bm{K}_{2}^{-1}\bm{J}_{p\times
p}\bm{K}_{1}^{-1},\text{ and}$
$\displaystyle\bm{B}^{(6)}=\bar{\rho}\bm{D}^{(6)}=-\bar{\rho}^{3}\delta_{4}^{-1}\delta_{1}\delta_{2}^{-1}\bm{K}_{2}^{-1}\bm{J}_{p\times
p}\bm{K}_{2}^{-1}+\bar{\rho}\delta_{4}^{-1}\bm{K}_{2}^{-1}\bm{J}_{p\times
p}\bm{K}_{2}^{-1}.$

Here $\bm{J}_{p\times p}=\bm{{1}}_{p}\bm{{1}}_{p}^{{}^{\prime}}$. Thus from
(S.18), (S.19), (S.27), (S.28) and (S.47), by calculation we can prove (S.41).
∎

######  Lemma S.7.

For any design $d\in\varOmega_{t,n,p}$,
$\bm{{C}}_{d(s2)}\leq\bm{\tilde{C}}_{d(s2)}$. If $d$ is a design uniform on
periods, then $\bm{{C}}_{d(s2)}=\bm{\tilde{C}}_{d(s2)}$.

###### Proof.

Since
$pr^{\perp}\left(\bm{\varSigma}^{-1/2}\begin{bmatrix}\bm{Z}_{1}&\bm{I}_{2}\otimes\bm{F}_{d}\bm{H}_{t}\end{bmatrix}\right)=pr^{\perp}\left(\bm{\varSigma}^{-1/2}\begin{bmatrix}\bm{I}_{2}\otimes\bm{P}&\bm{I}_{2}\otimes\bm{U}&\bm{I}_{2}\otimes\bm{F}_{d}\bm{H}_{t}\end{bmatrix}\right)\leq
pr^{\perp}\left(\bm{\varSigma}^{-1/2}\begin{bmatrix}\bm{I}_{2}\otimes\bm{U}&\bm{I}_{2}\otimes\bm{F}_{d}\bm{H}_{t}\end{bmatrix}\right)$,
where
$\bm{H}_{t}=\bm{I}_{t}-\frac{1}{t}\bm{{1}}_{t}\bm{{1}}_{t}^{{}^{\prime}}$,
using the expression of the matrices $\bm{C}_{d(s2)}$ and
$\bm{\tilde{C}}_{d(s2)}$ as given in Lemma S.3 and Lemma S.5, respectively, we
get that $\bm{C}_{d(s2)}\leq\bm{\tilde{C}}_{d(s2)}$.
Since $\bm{F}_{d}=\left(\bm{I}_{n}\otimes\bm{\psi}\right)\bm{T}_{d}$, where
$\bm{\psi}=\begin{bmatrix}\bm{{0}}^{{}^{\prime}}_{p-1\times 1}&0\\\
\bm{I}_{p-1}&\bm{{0}}_{p-1\times 1}\end{bmatrix}$, using the expression of the
matrices $\bm{\tilde{C}}_{d(s2)(11)}$, $\bm{\tilde{C}}_{d(s2)(12)}$,
$\bm{\tilde{C}}_{d(s2)(21)}$ and $\bm{\tilde{C}}_{d(s2)(22)}$ as given in
Lemma S.5, and the expression of
$\bm{\varSigma}^{-1/2}pr^{\perp}\left(\bm{\varSigma}^{-1/2}\begin{bmatrix}\bm{I}_{2}\otimes\bm{U}\end{bmatrix}\right)\bm{\varSigma}^{-1/2}$
as given in Lemma S.6, we get

$\displaystyle\begin{split}\bm{\tilde{C}}_{d(s2)(11)}&=\begin{bmatrix}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{1}\right)\bm{T}_{d}&-\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{2}\right)\bm{T}_{d}\\\
-\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{2}\right)\bm{T}_{d}&\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{4}\right)\bm{T}_{d}\end{bmatrix},\\\
\bm{\tilde{C}}_{d(s2)(12)}&=\bm{\tilde{C}}^{{}^{\prime}}_{d(s2)(21)}=\begin{bmatrix}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{1}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}&-\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\\\
-\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}&\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\varOmega}_{4}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\end{bmatrix},\text{
and}\\\
\bm{\tilde{C}}_{d(s2)(22)}&=\begin{bmatrix}\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{1}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}&-\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\\\
-\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}&\bm{H}_{t}\bm{T}_{d}^{{}^{\prime}}\left(\bm{I}_{n}\otimes\bm{\psi}^{{}^{\prime}}\bm{\varOmega}_{4}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\end{bmatrix},\end{split}$
(S.48)

where $\bm{\varOmega}_{1}$, $\bm{\varOmega}_{2}$, and $\bm{\varOmega}_{4}$ are
the matrices as given in Lemma S.4.

Since $\bm{F}_{d}=\left(\bm{I}_{n}\otimes\bm{\psi}\right)\bm{T}_{d}$, from the
expression of the matrices $\bm{C}_{{d(s2)}(11)(1)}$,
$\bm{C}_{{d(s2)}(12)(1)}$, $\bm{C}_{{d(s2)}(21)(1)}$ and
$\bm{C}_{{d(s2)}(22)(1)}$ as given in Lemma S.3, and using the expression of
$\bm{A}^{*}$ as given in Lemma S.4, we get

$\displaystyle\begin{split}\bm{C}_{{d(s2)}(11)(1)}&=\begin{bmatrix}\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{1}\right)\bm{T}_{d}&-\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{2}\right)\bm{T}_{d}\\\
-\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{2}\right)\bm{T}_{d}&\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{4}\right)\bm{T}_{d}\end{bmatrix},\\\
\bm{C}_{{d(s2)}(12)(1)}&=\bm{C}^{{}^{\prime}}_{{d(s2)}(21)(1)}=\begin{bmatrix}\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{1}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}&-\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\\\
-\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{2}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}&\bm{T}_{d}^{{}^{\prime}}\left(\bm{H}_{n}\otimes\bm{\varOmega}_{4}\bm{\psi}\right)\bm{T}_{d}\bm{H}_{t}\end{bmatrix},\text{
and}\\\