J. Nig. Soc. Phys. Sci. 5 (2023) 1392

Journal of the
Nigerian Society

of Physical
Sciences

Model Fitness and Predictive Accuracy in Linear Mixed-Effects
Models with Latent Clusters

Waheed B. Yahyaa, Yusuf Bellob,∗, Abdulrazaq AbdulRaheemc

aDepartment of Statistics, , University of Ilorin, P.M.B. 1515, Ilorin, Kwara State, Nigeria.
bDepartment of Statistics, Federal University, Dutsin-Ma, P.M.B. 5001, Dutsin-Ma, Katsina State, Nigeria.

cDepartment of Statistics and Mathematical Sciences, Kwara State University, Malete, P.M.B. 1530, Ilorin, Kwara State, Nigeria.

Abstract

In clustered data, observations within a cluster show similarity between themselves because they share common features different from observa-
tions in the other clusters. In a given population, different clustering may surface because correlation may occur across more than one dimension.
The existing multilevel analysis techniques of the primal linear mixed-effect models are limited to natural clusters which are often not realistic to
capture in real-life situations. Therefore, this paper proposes dual linear mixed models (DLMMs) for modeling unobserved latent clusters when
such are present in data sets to yield appreciable gains in model fitness and predictive accuracy. The methodology explored the development
and analysis of the dual linear mixed models (DLMMs) based on the derived latent clusters from the natural clusters using multivariate cluster
analysis. A published data set on political analysis was used to demonstrate the efficiency of the proposed models. The proposed DLMMs have
yielded minimum values of the models’ assessment criteria (Akaike information criterion, Bayesian information criterion, and root mean squared
error), and hence, outperformed the classical PLMMs in terms of model fitness and predictive accuracy.

DOI:10.46481/jnsps.2023.1437

Keywords: Clustered data, Primal and dual clusters, Linear mixed-effects models, Model fitness, Predictive accuracy

Article History :
Received: 05 March 2023
Received in revised form: 13 May 2022
Accepted for publication: 14 May 2022
Published: 11 June 2023

c© 2023 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license

(https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Communicated by: O. Adeyeye

1. Introduction

The multilevel modeling technique follows a similar pro-
cess involved when fitting the Generalized Linear Model [1].
In particular, a Linear mixed model (LMM) is one of the ap-
proaches in modeling normally distributed clustered data [2].
In clustered data, observations within a cluster show similarity
between themselves because they share common features dif-
ferent from observations in the other clusters. In a given popu-

∗Corresponding author tel. no: +2348134983581
Email address: ybello@fudutsinma.edu.ng (Yusuf Bello)

lation, different clustering may surface because correlation may
occur across more than one dimension [3]. They further argued
that clustering is in essence a design problem, either a sampling
design or an experimental design issue. Even if data is col-
lected in an unclustered way, there is still natural clustering in
the population.

As an illustration from Nigeria
′

s crime analysis, the initial
dataset comprised 36 states grouped into six geo-political zones
and 12 Police Zonal Commands that share spatial and socio-
ethnic similarities. However, the optimal number of clusters
provided new structure classifications based on crime rates sim-

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Yahya et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 2

ilarities different from the initial spacial and socio-ethnic simi-
larities [4]. It is posited here that the observations in the newly
formed clusters based on multivariate clustering similarities are
more correlated than in the natural clusters based on sampling
and experimental design similarities. The former would better
account for the differences between the clusters and improve
model fitness and predictive accuracy.

The natural clusters and latent clusters are respectively de-
scribed as ‘primal clusters’ and ‘dual clusters’. The linear
mixed-effects models (LMEMs or simply LMMs) on the primal
clusters and dual clusters are respectively described as primal
linear mixed models (PLMMs) and dual linear mixed models
(DLMMs). This paper proposes efficient DLMMs for model-
ing data with latent clusters with appreciable gains in model
fitness and predictive accuracy.

2. The Concept of LMEMs on Latent Clusters and Model
Assessment Criteria

The general concept of LMEMs on latent clusters is to max-
imize correlation of observations within clusters, model fitness
and predictive accuracy. The latent clusters were formed from
natural clusters using the multivariate cluster analysis. Both
the natural and dual clusters contain the same observations, al-
though the cluster structures differ. The argument for compar-
ing models formed from same data set with differing data struc-
tures was demonstrated by [14]. Agglomerative algorithm is a
common approach in cluster analysis for classifying observa-
tions that share common properties into groups. The algorithm
starts by calculating the distances between all pairs of observa-
tions followed by stepwise agglomeration of close observations
into groups. Euclidean distance is the most commonly used
distance measure in numerical data, while Ward method is the
most frequently used linkage method [5].

LMM is a linear model with an extension of accounting for
dependency among clustered observations. In biological and
social sciences, a model-based cluster analysis utilizes LMM
in the grouping of individuals into one of two or more clusters
according to their longitudinal behaviour similarities. [6, 7]. In
contrast to those studies that utilize Expectation-Maximization
algorithms in cluster formations, this study conjoins LMEMs
and multivariate cluster analysis to develop efficient techniques
for modeling unobserved latent groupings in a data set.

The degree of clustering in a data set is measured by the
intraclass correlation (ICC). The ICC is the proportion of total
variance in the data that is due to the clusters [8]. The argument
in the multilevel analysis on latent clusters is that the increasing
clustering in DLMM would simultaneously reduce the indices
of the model assessment criteria. Many indices are available to
measure the performances of competing models [9], however,
the models’ assessment criteria in this work are the root mean
squared error (RMSE), Akaike information criterion (AIC) and
the Bayesian information criterion (BIC).

The RMSE indicates the absolute fit of the model to the
data. Smaller values of RMSE indicate better fit results. The
LMM improves model fitness and predictive performance, and
this is because a multilevel model produces fitted values, ŷ, that

are on the average closer to the observed y than those obtained
by fitting simply the fixed part of the model. Again, even in the
multilevel analysis, when the estimated random effects tend to
be biased towards zero; it pulls the fitted values in the direction
of those of the fixed part of the model that results in bias esti-
mates. Furthermore, simpler models such as random intercept
models produce larger bias relatively than the complex mod-
els such as random intercept and slope models [10]. Therefore,
by analogy, the more the estimated random effects tend to be
larger, the more the ŷ moves closer to y, and hence the lower
the RMSE.

The simplest information criterion widely applicable to
nonnested models is the AIC [11, 12]. This traditional AIC
is not appropriate in clustered data, and therefore marginal AIC
(mAIC) is the most widely used in model selection in LMMs
[12]. A related criterion to the mAIC in their marginal likeli-
hoods is the BIC [13]. The presence of random effects in LMMs
results in smaller AIC and BIC than in the LMs [14].

3. The Linear Mixed Model

Consider a vector y of data from J clusters, the LMM as
define by [2, 15] is

y j = X jβ + Z jυ j + � j (1)

where y j is the n j vector for cluster j, where j = 1, 2, · · · , J is
the cluster index, β is the p-vector of fixed effects, υ j is the q-
vector of random effects for cluster j, X j and Z j are respectively
the n j × p and n j × q matrices of covariates for the fixed and
random effects of full rank. It is assumed that υ j and � j follow
independent and multivariate Gaussian distributions such that
[16]: [

υ j
� j

]
∼ N

([
0
0

]
,

[
T 0
0 σ2 I

])
(2)

were T is q×q positive definite covariance matrix of the random
effects, υ j is assumed independently for each j, � j associated
with different clusters are assumed independent of each other,
and that � j is assumed independent of υ j [15].

In a marginal model

y j = N(X jβ, V j) V j = Z jT Z
′

j + σ
2 I (3)

with a marginal likelihood given as

l(y j/β̂, θ̂) = −
1
2

log|V̂ j| −
1
2

(y j − X jβ̂)
′

V̂ j
−1

(y j − X jβ̂) (4)

3.1. Cluster Effects

The cluster effect or the dependency among clustered ob-
servations is measured by the ICC, and is defined as

ICC =
σ2υ

σ2υ + σ
2
e

(5)

where σ2υ and σ
2
e are random effects variance and random error

variance respectively.

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Yahya et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 3

To show that the cluster effect is higher in the dual clusters
than in the primal clusters, that is, ICC(D) > ICC(P):

Let P and D describe PLMM and DLMM respectively,
σ2e(P) = σ

2
e, σ

2
e(D) = σ

2
e − δ1, σ

2
υ(P) = σ

2
υ, σ

2
υ(D) = σ

2
υ + δ2,

where δ2 ≥ δ1, δ2 and δ1 are small increment of random effects
variance and decrement random error variance, respectively,.

Case 1: If δ2 = δ1, such that δ2 −δ1 = 0, then

ICC(D) − ICC(P) =
σ2υ + δ2

(σ2υ + δ2) + (σ2e −δ1)
−

σ2υ
σ2υ + σ

2
e

=
σ2υ + δ2

(σ2υ + σ2e ) + (δ2 −δ1)
−

σ2υ
σ2υ + σ

2
e

=
σ2υ + δ2

(σ2υ + σ2e )
−

σ2υ
σ2υ + σ

2
e

=
δ2

(σ2υ + σ2e )

(6)

Since (6) results in a positive difference, then

ICC(D) > ICC(P)

Case 2: If δ2 > δ1, such that δ2 −δ1 = δ3 > 0, then

ICC(D) − ICC(P) =
σ2υ + δ2

(σ2υ + δ2) + (σ2e −δ1)
−

σ2υ
σ2υ + σ

2
e

=
σ2υ + δ2

(σ2υ + σ2e ) + (δ2 −δ1)
−

σ2υ
σ2υ + σ

2
e

=
σ2υ + δ2

(σ2υ + σ2e + δ3)
−

σ2υ
σ2υ + σ

2
e

=
σ2υ + δ2

(Σ + δ3)
−
σ2υ
Σ
, where Σ = σ2υ + σ

2
e

=
δ2σ

2
e + δ1σ

2
υ

Σ(Σ + δ3)

(7)

Since (7) results in a positive difference, then

ICC(D) > ICC(P)

3.2. Root Mean Squared Error

The RMSE is the difference between observed data and the
predicted values from the model, and it is defined as

RMS E =

√√ ∑J
j=1

∑n j
i=1(yi j − ŷi j)

2∑J
j=1 n j

(8)

where J is the number of clusters, n j is the number of obser-
vations in the jth cluster, yi j and ŷi j are the ith observed and
estimated y in jth cluster, respectively [17].

3.3. Marginal Akaike Information Criterion

The commonly used information criterion is the AIC [11].
This criterion which is based on Kullback-Leibler distance is
defined as

AIC = −2log[ f (y/ψ̂(y))] + 2k

where f (y/ψ̂(y)) is the maximized likelihood, and k is the num-
ber of parameters. This AIC is not appropriate in clustered data,
and hence the mAIC is widely used in the clustered data [12].

The mAIC in the LMM uses the likelihood of the implied
marginal model y ∼ N(Xβ, V ) with V = In + ZT Z

′

. The number
of estimable parameters then is p + q, with β = (β1, · · · ,βp) and
q the number of unknown parameters θ in V . Thus, the mAIC
is defined as

mAIC = −2log[ f (y/β̂, θ̂)] + 2( p + q) (9)

where f (y/β̂, θ̂) is the maximized marginal likelihood. How-
ever, the mAIC is positively biased, and favours smaller models
without random effects [18].

3.4. Bayesian Information Criterion
The is obtained by taking the mAIC (9) and replacing the

constant 2 in the penalty by log(n) to obtain

BIC = −2log[ f (y/β̂, θ̂)] + log(n)( p + q) (10)

This definition ensures that BIC bears the same relationship to
mAIC for model (1) as BIC bears to AIC in regression and so
should inherit some of its properties [13].

3.5. Cluster Effects on Model Assessment Criteria
Cluster effects in mixed models are explained by the ran-

dom effects variance of the models, and including the random
effects has an effect on the covariance matrix, V j. As an illustra-
tion, consider a random intercept model from a data where five
observations are taken on each cluster, so that n j = 5 for all j.
Therefore, Z j is a matrix of dimension 5 × 1 and R j = σ2×I5×5.
Then

V j =


1
1
1
1
1


×σ2υ ×

(
1 1 1 1 1

)

+ σ2 ×



1 0 · · · · · · 0

0 1
.
.
.

.

.

. 1
.
.
.

.

.

. 1 0
0 · · · · · · 0 1



=


σ2 + σ2υ σ

2
υ σ

2
υ σ

2
υ σ

2
υ

σ2υ σ
2 + σ2υ σ

2
υ σ

2
υ σ

2
υ

σ2υ σ
2
υ σ

2 + σ2υ σ
2
υ σ

2
υ

σ2υ σ
2
υ σ

2
υ σ

2 + σ2υ σ
2
υ

σ2υ σ
2
υ σ

2
υ σ

2
υ σ

2 + σ2υ



(11)

The elements in the diagonal, σ2υ + σ
2, are correlations be-

tween two observations from the same cluster, and the elements
at off diagonal, σ2υ, are the covariances between any two units
on the same cluster. By relating the two terms, the intraclass
correlation between two observations from the same cluster is
σ2υ/(σ

2
υ + σ

2) [14].
Denoting V j as V j(P) and V j(D) for the PLMM and DLMM

respectively, therefore, if the ICC(D) > ICC(P) as in (6) and (7),
then V j(D) > V j(P).

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Yahya et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 4

From (4), we respectively ascribe the marginal likelihood
for the PLMM and DLMM as l(y j/β̂, θ̂)(P) and l(y j/β̂, θ̂)(D), such
that

l(y j/β̂, θ̂)(P) = −
1
2

log|V̂ j(P)|−
1
2

(y j−X jβ̂)
′

V̂−1j(P)(y j−X jβ̂)(12)

and

l(y j/β̂, θ̂)(D) = −
1
2

log|V̂ j(D)|−
1
2

(y j−X jβ̂)
′

V̂−1j(D)(y j−X jβ̂)(13)

Similar to the basic concept of fraction that a negative frac-
tion increases with the increase in the denominator, the second
term in (13), −12 (y j − X jβ̂)

′

V̂−1j(D)(y j − X jβ̂), is relatively higher
than in (13) because of the increase of the inverse of the matrix
V̂−1j(D) relative to V̂

−1
j(P). Again, in the basic concept of logarithm

that a negative logarithm decreases with the increase of a num-
ber, the first term in (13), −12 log|V̂ j(D)|, is relatively lower than
in (12) because of the increase of the inverse of V̂ j(D) relative to
V̂ j(P). Although, the first term and second term in (13) decrease
and increase respectively, the increase outweights the decrease,
such that

l(y j/β̂, θ̂)(D) > l(y j/β̂, θ̂)(P) (14)

The presence of negative sign in the −2log[ f (y/β̂, θ̂)] for the
information criteria in (9) and (10) has changed the direction
of the inequality in (14), such that l(y j/β̂, θ̂)(D) < l(y j/β̂, θ̂)(P).
The p and q are the same in both the PLMM and DLMM, and
therefore

mAIC(D) < mAIC(P) and
mBIC(D) < mBIC(P)

(15)

The LMM improves model fitness and predictive perfor-
mances because it incorporates clustering effects when estimat-
ing the fixed parameter. This adjustment enhances it to produce
fitted values, ŷ, that are on the average closer to the observed y
than those obtained by fitting simply the fixed part of the model
[10]. In our proposal, DLMM has higher clustering effect than
PLMM that enhances it to produce fitted values, ŷ, that are on
the average closer to the observed y than those produced by the
PLMM.

4. Cluster Algorithm: The Agglomerative Algorithm

The agglomerative procedure depends on the definition of
the distance between two clusters. For a particular case where
metric A = (S −1X1 X1, · · · , S

−1
Xp Xp

) is used for the standardization
of the variables, the Euclidean distance di j between two cases
i and j with variable values xi = (xi1, xi2, · · · , xip), x j =
(x j1, x j2, · · · , x jp, ) is defined by

di j =

 p∑
k=1

(xik − x jk)2

S Xk Xk


1
2

where S Xk Xk is the variance of the kth component [19].

Ward algorithm computes the distance between groups and
joins the ones that do not increase a given measure of het-
erogeneity “too much”so the resulting groups are as homoge-
neous as possible. If two objects or groups say, P and Q, are
united, one computes the distance between this new group (ob-
ject) P + Q and group R using the following distance function

d(R, P + Q) =
nR + nP

nR + nP + nQ
d(R, P)

+
nR + nQ

nR + nP + nQ
d(R, Q)−

nR
nR + nP + nQ

d(P, Q)

(16)

The heterogeneity of group R is measured by the inertia in-
side the group. This inertia is defined as IR =

1
nR

∑nR
i=1 d

2(xi, x̄R)
[20].

5. Illustration and Analysis

A published data set on political analysis was used to
demonstrate the efficiency of the proposed models. The dataset
dcese provided with the ceser R package came from [21]. It
contains information on 299 (i = 1, 2, · · · , 299) observations
across 47 countries ( j = 1, 2, · · · , 47). The outcome variable
is the effective number of electoral parties (enep). The ex-
planatory variables are the number of presidential candidates
(enpc), the proximity of presidential and legislative elections
(proximity); the effective number of ethnic groups (eneg), the
logarithm of average district magnitudes (logmag), and an inter-
action term between the logarithm of the district magnitude and
the number of ethnic groups (logmag eneg = logmag × eneg).

5.1. Comparison between Primal and Dual Linear Mixed Mod-
els

We begin with a preliminary comparison of the PLMM and
DLMM using primal and dual cluster data sets with J = 47
number of groups, and subsequently test the significance of
the comparison. The comparison is in terms of the variance-
covariance components and their impact on model fitness and
predictive accuracy. The summary outputs of the models are
presented in Table 1.

It reveals from the summary in Table 1 that while σ2e is
higher under PLMM, the σ2υ and ICC are relatively higher un-
der DLMM. There is a 61 percent decrease of σ2e from PLMM
to DLMM, and respectively 64 and 38 percent increase in σ2υ
and ICC from PLMM to DLMM. The AIC, BIC and RMSE are
lower in DLMM than under PLMM by 18, 17 and 38 percent,
respectively. Hence, the proposed DLMM has increased the
homogeneity of the observations within clusters and the hetero-
geneity of the clusters, which in turn increased the model fitness
and predictive accuracy.

The PLMM and DLMM in Table 1 are described as
‘full models’ because they compose of significant and non-
significant explanatory variables. We shall now determine if
we can obtain similar gains in the model assessment criteria
when only significant variables are included in the models. The

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Yahya et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 5

Table 1. Summary outputs for the LM, PLMM and DLMM

Clustering Estimate
Effect LM PLMM DLMM Covariate LM PLMM DLMM
σ2υ 1.8460 5.0900 Intercept 1.2374 3.2081 4.5972
σ2e 1.4790 0.5823 enpc 0.8636 0.5092 0.1995
ICC 0.5552 0.8973 proximity -0.0173 -0.1921 0.0190
AIC 1168.80 1073.09 881.98 eneg -0.1208 -0.1764 -0.3091
BIC 1194.70 1102.69 911.58 logmag -0.1982 -0.0956 0.0413
RMSE 1.6689 1.1345 0.7024 logmag eneg 0.3663 0.0652 0.0274

models with only significant variables are described as ‘reduced
models’. The enpc is the only significant variable in both the
PLMM and DLMM. The summary of the reduced models is in
Table 2.

The ICC in DLMM has increased by 38 percent from
PLMM, and this increase is the same as it was in the full model.
The AIC, BIC, and RMSE have smaller values under DLMM
than under PLMM by 18, 18, and 38 percent, respectively. Sim-
ilarly, the percentage decrease is almost the same as it was in
the full model. Although the magnitudes of the AIC and BIC
have reduced when non-significant explanatory variables are
excluded in both the full PLMM and DLMM; however, the per-
centage difference between the PLMM and DLMM is almost
the same in both the full and reduced models.

The PLMM and DLMM in Table 2 are random intercept
models, we recast them to random intercept and slope models
to assess the effects of increasing complexity in DLMMs. The
summary of the random intercept and slope models is in Table
3.

The ICC in DLMM has increased by 20 percent from
PLMM, which is lower than in the random intercept model.
The AIC, BIC and RMSE have lower values in DLMM than
under PLMM by 17, 17 and 32 percents, respectively. A simi-
lar percentage differences are recorded between the PLMM and
DLMM as were in the random intercept models; however, the
difference is smaller in RMSE.

The comparison reveals a superiority of random intercept
and slope DLMM over random intercept DLMM in terms of
model fitness. The comparison reveals a superiority of random
intercept and slope DLMM over random intercept DLMM in
terms of model fitness. This coincides with the work of [14]
when random intercept and slope model has smaller value of
AIC than in the random intercept model. The model predic-
tive accuracy is higher in DLMM than in PLMM, and also it is
higher in random intercept and slope DLMM than in random
intercept DLMM. Higher predictive accuracy signifies smaller
RMSE.

The above comparison used single sample outcome, J =
47, and hence it has not satisfied statistical testing procedure.
Therefore, we obtained fifteen sample combinations of the
PLMMs and the corresponding DLMMs and compared their re-
spective outcomes. Some sample combinations were replicated
to explore possible outcome variability.

Figure 1. Random Effects Variance

Figure 2. Random Error Variance

5.2. Assessing Clustering Effects between the PLMMs and
DLMM

The ICC is a function of σ2υ and σ
2
e , and they are presented

in Table 4 and Figures 1 and 2.
It shows that σ2e decreases and σ

2
υ increases significantly

from PLMM to DLMM. The decrease in the σ2e signifies a ho-
mogeneity of observations; that is, the increase of correlations/
dependency of observations within the dual clusters. The in-
crease in the σ2υ signifies heterogeneity of clusters; that is, the
between cluster variations. The two variance components have
greatly affected the ICC, which is significantly higher in the
DLMMs. This signifies higher grouping structure in the dual
clusters, and higher clustering effects in the DLMMs.

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Yahya et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 6

Table 2. Significant Explanatory Variable in Random Intercept Models

ICC AIC BIC RMSE

PLMM DLMM PLMM DLMM PLMM DLMM PLMM DLMM
0.5603 0.8972 1066.84 876.36 1081.64 891.16 1.1360 0.7053

Table 3. Significant Explanatory Variable in Random Intercept and Slope Models

ICC AIC BIC RMSE

PLMM DLMM PLMM DLMM PLMM DLMM PLMM DLMM
0.7340 0.9149 1052.11 869.71 1074.31 891.91 0.9570 0.6472

Table 4. Random effects variance, random error variance and ICC from PLMMs and DLMMs

σ2υ σ
2
e ICC

Cluster PLMM DLMM PLMM DLMM PLMM DLMM
40 2.3330 3.7956 1.0920 0.6377 0.6811 0.8561
41 1.3060 3.6943 1.4520 0.5564 0.4734 0.8691
42 1.7770 4.6620 1.4990 0.5310 0.5425 0.8978
43 1.9850 4.4058 1.1760 0.5158 0.6280 0.8952
43b 1.9740 5.4642 1.4920 0.6113 0.5695 0.8994
44 1.9340 4.8178 1.4780 0.5924 0.5668 0.8905
44b 1.9570 4.8289 1.4510 0.5736 0.5743 0.8938
44c 1.9490 5.6101 1.4790 0.5677 0.5685 0.9081
45 1.9130 5.1335 1.6680 0.6533 0.5343 0.8871
45b 1.8770 5.3469 1.4900 0.5747 0.5574 0.9030
45c 1.8700 5.1142 1.5880 0.5965 0.5407 0.8956
46 1.7570 4.9810 1.6820 0.6280 0.5110 0.8881
46b 1.8580 5.1988 1.4920 0.5783 0.5546 0.9000
46c 1.8800 5.3696 1.5060 0.6018 0.5552 0.8992
47 1.8460 5.0900 1.4790 0.5823 0.5552 0.8973

Mean 1.8811 4.9008 1.4683 0.5867 0.5608 0.8920

Figure 3. The plot of ICC for the Model assessment

5.3. Assessing Model Fitness and Predictive Accuracy between
PLMMs and DLMMs

The model assessment criteria are presented in Table 5 and
Figures 4, 5 and 6.

The DLMMs have smaller AIC and BIC than PLMMs, this

Figure 4. The plot of AIC for the Model assessment

indicates a significant gain in model fitness in the DLMMs over
PLMMs. In addition to the relative selection of the best-fitted
model carried out using the AIC and BIC, we supplemented the
selection with the assessment of the model

′

s predictive accu-
racy. The RMSE is significantly lower in the DLMMs, and this

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Yahya et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1392 7

Table 5. Model Assessment Criteria from PLMMs and DLMMs

AIC BIC RMSE

Cluster LM PLMM DLMM LM PLMM DLMM LM PLMM DLMM
40 915.9 806.2 720.0 940.2 834.0 747.8 1.609 0.965 0.735
41 847.5 798.0 659.4 871.4 825.2 686.6 1.568 1.117 0.677
42 1057.1 971.1 772.1 1082.3 999.9 800.9 1.670 1.144 0.671
43 1074.0 963.8 799.4 1099.5 993.0 828.6 1.564 1.011 0.663
43b 1084.2 991.8 827.6 1109.5 1020.7 856.5 1.694 1.139 0.720
44 1143.8 1048.0 860.0 1169.5 1077.5 889.5 1.675 1.137 0.711
44b 1034.4 943.0 777.4 1059.4 971.5 805.9 1.696 1.118 0.693
44c 1138.1 1041.8 848.7 1163.8 1071.0 878.1 1.681 1.136 0.696
45 1052.5 979.8 816.0 1077.5 1008.4 844.5 1.743 1.198 0.738
45b 1154.6 1060.3 866.4 1180.4 1089.8 895.9 1.672 1.141 0.699
45c 1094.9 1011.3 826.9 1120.2 1040.3 855.9 1.715 1.174 0.709
46 1064.9 993.7 818.4 1090.0 1022.4 847.1 1.732 1.205 0.723
46b 1156.4 1060.9 868.5 1182.2 1090.4 898.0 1.678 1.140 0.701
46c 1150.6 1056.9 875.8 1176.3 1086.3 905.3 1.683 1.145 0.714
47 1168.8 1073.1 882.0 1194.7 1102.7 911.6 1.669 1.135 0.702

Mean 1075.8 986.6 814.6 1101.1 1015.5 843.5 1.670 1.127 0.704

Figure 5. The plot of BIC for the Model assessment

Figure 6. The plot of RMSE for the Model assessment

explains the significant gain in model predictive accuracy in the
proposed DLMM over the existing PLMMs.

6. Conclusion

The paper proposed the development and analysis of
DLMM on the dual clusters derived from the primal clusters.
The clustering similarity in the dual clusters was based on the
commonly occurring phenomenon or the experimental designs,
and the similarity in the dual clusters was based on the multi-
variate clustering algorithms. Findings revealed that observa-
tions in the dual clusters are more correlated than in the pri-
mal clusters. The proposed DLMM is relatively more efficient
than the classical PLMM based on the results of the models’ as-
sessment criteria (AIC, BIC, and RMSE) in which the DLMM
yielded minimum values of the assessment criteria. There-
fore, the proposed DLMM outperformed the classical PLMM
in terms of model fitness and predictive accuracy.

Acknowledgment

We acknowledge with thanks for the careful reading and
suggestions from the referees of this paper.

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