Simultaneous Fixed and Random Effects Selection in Order-restricted
Mixed Effects Models
HAIFANG SHI, JIAJIA GE
School of Science,
Civil Aviation University of China,
Tianjin,
CHINA
Abstract: - When dealing with longitudinal data, if we directly select a specific model for modeling without any
prior information about the existence of significant random effects before utilizing the mixed model, it may result
in the misuse of the model, thereby affecting the final estimation results. This paper investigates a variable
selection method that can jointly select both fixed and random effects in Bayesian mixed model under order
constraints. This method can effectively prevent model misuse. A computationally feasible Gibbs algorithm is
proposed for posterior inference. The performance of our proposal is evaluated by simulated data and two real
applications related to Blood lead levels and Ramus bone heights. Results show that the proposed approaches
perform very well in various situations.
Key-Words: - Random subject effect, Longitudinal data, Order restriction, Ramus bone heights, Model selection,
Two-way ANOVA, Blood lead levels.
Received: April 13, 2024. Revised: September 8, 2024. Accepted: September 29, 2024. Published: October 25, 2024.
1 Introduction
In many applications, researchers have prior
knowledge about the underlying parameters that
satisfy an order restriction before the data are
collected. For example, the researchers measured
Ramus bone heights of 20 boys at four time points
over 18 months, a natural assumption is that the
means of ramus bone sizes () satisfy the simple
order . [1], proposed a one-way
ANOVA model with order constraints for this data
and find evidence that there are only two growth
spurts during the 18 months. However, [2] and [3]
argued that random subject effect cannot be easily
excluded from the model, especially when time is an
explanatory variable. They developed a Bayesian
hierarchical mixed model for multiple comparisons
of fixed effects with a simple order restriction using
mixtures of an exponential distribution and a discrete
distribution. This matter raises a challenging problem
of performing joint fixed and random effects
selection in mixed models under order restriction.
Model selection in mixed models without constraints
has received substantial interest in recent years.
Based on a penalized adaptive likelihood, [4]
developed Bayesian variable selection by allowing
fixed effects or standard deviations of random effects
to be exactly zero in linear mixed models. [5],
proposed a nested EM algorithm for variable
selection of linear mixed effects. [6], proposed a
simple iterative penalized procedure that is capable
of simultaneously selecting and estimating both fixed
effects and random effects in linear mixed-effects
models. [7], integrated the penalized quasi-likelihood
estimation framework with a penalization approach
that enables simultaneous estimation of model
parameters while automatically selecting important
variables by imposing sparsity constraints on the
coefficients of both fixed effects and random effects.
[8], reviewed the methods for variable selection in
linear mixed-effects models proposed in recent
literature and compared the strengths and
weaknesses of various approaches through extensive
simulations.
However, to our knowledge, there is little
literature on the model selection of fixed effects
together with the random effects in the mixed model
with order restriction. [2], [3] proposed a Bayesian
hierarchical mixed model for repeated measures data
with missing values and a simple order restriction,
but they did not consider the selection of random
effects in the model. This paper develops a novel
Bayesian variable selection approach for two-way
ANOVA mixed model accounting for order
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restrictions. Compared with existing literature, our
greatest contribution is that our method can
determine whether the fixed effects and random
effects in the mixed model are significant, and
simultaneously estimate the significant effects. In
practical data modeling, when significant prior
information is lacking, this can prevent misuse of the
model and thereby improve the estimation accuracy
of the model. A simple and efficient Gibbs sampler is
proposed for posterior inference. The paper is
organized as follows. Within Section 2, we describe
the vectorized form of the two-way ANOVA mixed
model under a simple order. We also propose
variable selection procedures for both fixed and
random effects in the model. Section 3 develops
computational strategies for posterior inference.
Section 4 conducts simulation studies to evaluate the
performance of the proposed method. An analysis of
two real applications is presented in Section 5.
Section 6 concludes the paper.
2 Model Description
Suppose there are n subjects under investigation, and
there are k treatment for each subject. Let be the
observation of the response variable, the two-way
ANOVA mixed model is then expressed as follows:
where is the fixed treatment effect (mean) for the
i-th treatment, is a random subject effect which is
N random variable, and is measurement
error which is N(0, ) random variable. Suppose
that the random subject effects and the measurement
errors are all independent. In practical applications,
there are generally the following three types of order
constraints.
(i) The simple order ≤ · · · ≤
(ii) The simple tree order
(iii) The umbrella order ≤ · · · ≤ · ·
· .
For the simple order ≤ · · · ≤ , let
. Thus we will
have .Let be a
standard normal variable, the vectorized form of the
model can be written as:
where , and
= (1, . . . , 1)′ which is a k × 1 vector. If the
parameter means satisfy the simple tree order or the
umbrella-order, we can also obtain a model similar to
(2) through transformation. The detailed
transformation method can be referred to in reference,
[1].
By introducing indicator variables, we adopt the
method proposed by [9] to select fixed and random
effects simultaneously and fit the model. Bayesian
variable selection received large attention in recent
years, a nice review can be found in [10]. Setting
and , we can rewrite model (2)
as:
，
where are binary indicator variables (0
or 1) signifying which predictors are active in the
model. The indicator variables are assumed
independent Bernoulli prior distributions:
Following [11], we use a weak prior for , i.e.
Uniform(0,1).We further assign the following
hierarchical prior distribution for ,
where are constants, denotes
a truncated normal distribution on the interval (a,b),
IGamma(a, b) denotes an inverse-gamma
distribution with density function:
，
To implement the Bayesian model, we further set
a conjugate norm distribution for ,
where is a constant, and a noninformative joint
prior for and ,
3 Posterior
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We outline the Gibbs sampler used to obtain
posterior samples. The detailed algorithm is as
follows.
Let ,by integrating out ,the
marginal likelihood is:
where
and
3.1.1 Step 1–Sampling and :
For the convenience of implementation, we let
.
Update from its conditional distribution, an
inverse-gamma distribution:
Update from its conditional distribution, an
inverse-gamma distribution:
The variance of random subject effect can be
computed by
after knowing and .
3.1.2 Step 2–Sampling :
Update from its conditional distribution, a
truncated norm distribution:
Setting
and
.
Setting
,
and
,
we then have:
(5)
.
Then the full conditional posterior distributions
of is a truncated norm distribution,
where
and
.
3.1.3 Step 3–Sampling :
Update from its conditional distribution, a norm
distribution:
,
where
.
Setting
and
, the conditional posterior distributions of
is then a norm distribution:
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3.1.4 Step 4–Sampling :
Update from its conditional distribution, an
inverse-gamma distribution:
3.1.5 Step 5-Sampling :
Let . It can be
demonstrated that the indicator variable follows a
Bernoulli with probability parameter:
where
and
3.1.6 Step 6-Sampling :
Conditional posterior for . By the prior on and
the prior on , the full conditional distribution of
is given by:
.
4 Simulation Studies
In this section, we demonstrate the performance of
our methods (BMS) and compare it with the
Bayesian procedure for order restricted mixed model
proposed by [2], [3] using a series of simulations. We
first perform simulations with independent data and
then those with dependent data.
4.1 Dependent Data
The data are generated from the model given by:
,
where random subject effect and error term
are generated independently from . Under the
simple order and k =4 there are 8 candidate models
on the equality/inequality of fixed treatment effects:
Following [12], [1] ,there are three scenarios:
• Case 1: μ = (0, 0, 0, 0)′ , that is, there exists equal
fixed treatment effects,
• Case 2: μ = (0, 0, 0, 1)′ , that is, the last group has a
different fixed treatment effects,
• Case 3: μ = (1, 2, 3, 4)′ , that is, the fixed treatment
effects satisfy simple ordering.
4.2 Independent Data
The model is the same as that in dependent data,
except for random effects: . We
consider three sample sizes n= 10, n = 30, n = 100
and repeat 500 times in each example. In all
simulation settings, we suppose independent flat
inverse Gamma prior distribution IGamma(2.2, 20)
for , i = 1, 2, 3 and , choose hyper-parameter
for such that we obtain weakly
informative priors. We run our Gibbs sampler for
10000 iterations with 3000 for burn-in. For fixed
treatment effects, Table 1-2 list average posterior
probabilities of all the possible models and the
percentages of selecting the correct fixed parameters
from 500 data sets for dependent data and
independent data respectively. The true model is
marked as ’*’.The bold font is used to highlight the
posterior probabilities of the true model. Comparing
the conclusions from Table 1, when μ = (0, 0, 0, 0),
[2], [3] offer larger percent- ages of selecting the
correct model than BMS, but when μ = (0, 0, 0, 1)
and μ = (1, 2, 3, 4), our proposed methods BMS
generally perform better than the [2], [3] method,
showing the good performance of the method. It is
also clearly seen from Table 1 that BMS tends to
provide a larger average posterior probability of the
true model than [2], and [3] in nearly all cases across
the examples. Furthermore, for independent data, we
can observe similar results from Table 2 as
independent data. Moreover, as expected, we see that
the average posterior probabilities of the correct
model and the percentages of selecting the correct
model values for both methods increase as the
sample size increases, especially in Case 3 where the
fixed treatment effects satisfy simple ordering.
To check the performance of the proposed
methods in identifying the correct model for random
effect, Table 3 summarizes the percentages of
selecting the correct random effect parameters from
500 data sets for dependent data and independent
data. Overall, Table 3 indicates that BMS yields
promising results in both cases, even for a small
sample size when = 10, suggesting good
performance of our method.
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5 Real Data Example
5.1 Treatment of Lead-exposed Children
Trial
We first apply the proposed methodology to the
Treatment of Lead- exposed Children trial data
(TLC), [13]. In this study, Blood lead levels for 50 of
the children who did not receive the succimer
capsules were measured at week 0 (baseline), week
1, week 4, and week 6. We let ,,and
denote the mean blood lead levels corresponding to
week 6, week 4,week 1, and baseline, respectively.
Because the homes of these children were cleaned
using an established TLC regimen, it is reasonable to
assume that the mean blood lead levels satisfy the
simple order restriction, i.e.,.So
there are eight candidate models:
Table 1. Results of the average posterior probabilities
of all the possible models and the percentages of
selecting the correct model from 500 repetitions for
dependent data
Hypothesis
BMS
Shang
BMS
Shang
BMS
Shang
case 1
0.560
0.328
0.703
0.335
0.779
0.356
0.019
0.062
0.009
0.063
0.004
0.060
0.025
0.068
0.010
0.067
0.005
0.062
0.020
0.075
0.008
0.070
0.004
0.066
0.127
0.131
0.094
0.138
0.068
0.137
0.108
0.142
0.081
0.141
0.069
0.14
0.137
0.158
0.094
0.152
0.070
0.147
0.003
0.036
0.001
0.035
0.000
0.033
Percentages
0.820
0.998
0.924
0.994
0.962
0.994
case 2
0.233
0.240
0.070
0.134
0.000
0.010
0.075
0.100
0.092
0.153
0.083
0.199
0.083
0.100
0.095
0.149
0.077
0.204
0.025
0.068
0.008
0.040
0.000
0.003
0.370
0.187
0.644
0.297
0.831
0.475
0.116
0.126
0.056
0.076
0.003
0.006
0.086
0.121
0.024
0.064
0.000
0.004
0.013
0.059
0.010
0.087
0.005
0.101
Percentages
0.514
0.244
0.874
0.794
0.978
0.994
case 3
0.004
0.065
0.000
0.004
0.000
0.000
0.179
0.152
0.141
0.205
0.009
0.107
0.236
0.134
0.166
0.142
0.009
0.05
0.155
0.139
0.158
0.164
0.007
0.042
0.057
0.111
0.001
0.031
0.000
0.000
0.139
0.162
0.043
0.141
0.000
0.005
0.056
0.093
0.001
0.022
0.000
0.000
0.174
0.145
0.490
0.293
0.975
0.796
Percentages
0.086
0.036
0.582
0.398
0.998
0.962
Table 2. Results of the average posterior probabilities
of all the possible models and the percentages of
selecting the correct model from 500 repetitions for
independent data
Hypothesis
BMS
Shang
BMS
Shang
BMS
Shang
case 1
0.561
0.326
0.703
0.35
0.78
0.372
0.019
0.062
0.008
0.060
0.004
0.057
0.025
0.068
0.010
0.063
0.005
0.058
0.020
0.072
0.008
0.063
0.004
0.06
0.127
0.137
0.093
0.144
0.067
0.142
0.108
0.144
0.081
0.141
0.069
0.142
0.137
0.159
0.095
0.149
0.070
0.144
0.003
0.032
0.001
0.028
0.000
0.025
Percentages
0.832
0.906
0.924
0.940
0.962
0.96
case 2
0.232
0.190
0.071
0.071
0.000
0.001
0.074
0.118
0.093
0.175
0.083
0.199
0.083
0.115
0.095
0.171
0.077
0.202
0.025
0.066
0.008
0.028
0.000
0.001
0.370
0.221
0.643
0.377
0.831
0.509
0.116
0.121
0.057
0.055
0.003
0.002
0.087
0.106
0.024
0.038
0.000
0.000
0.013
0.063
0.010
0.084
0.005
0.086
Percentages
0.510
0.386
0.872
0.828
0.978
0.974
case 3
0.003
0.010
0.000
0.000
0.000
0.000
0.180
0.152
0.144
0.129
0.008
0.008
0.235
0.183
0.169
0.163
0.009
0.008
0.155
0.172
0.157
0.189
0.006
0.011
0.057
0.052
0.001
0.002
0.000
0.000
0.142
0.147
0.043
0.056
0.000
0.000
0.054
0.072
0.001
0.003
0.000
0.000
0.173
0.212
0.485
0.458
0.976
0.974
Percentages
0.088
0.096
0.574
0.502
0.998
0.994
Table 3. Results of the percentages of selecting the
correct random effect parameters from 500
repetitions
n =10
n =30
n =100
dependent
case 1
0.962
0.988
1.000
case 2
0.954
0.990
1.000
case 3
0.940
0.996
1.000
independent
case 1
0.852
0.932
0.994
case 2
0.874
0.952
0.986
case 3
0.896
0.952
1.000
We consider the similar prior specifications as in
Section 4 and generate 100, 000 samples with an
initial burn-in of 20, 000 iterations. The posterior
probabilities of the indicator variable for random
subject eect is 1.0, indicating that there is a high
probability of non-negligible random subject eects
in the data. We also compare our method with [2],
[3].The posterior probabilities of all the possible
models for each method are listed in Table 4.We
observe Shang method chooses ,
whereas is more supported by
BMS. Table 5 also summarizes the posterior
estimates for the parameters. It can be seen from
Table 4 that both methods yield similar posterior
point estimates, however ,BMS tends to oer more
narrow credible intervals than Shang for most
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parameters, except , showing an improvement
over Shang.
5.2 Ramus Bone Heights
In this section, we illustrate the proposed method to
the Ramus bone heights data (Ramus), which are
given by [14]. In this study, the Ramus bone heights
of 20 boys were measured at 8 years, 8 .5 years, 9
years, 9.5 year over an 18 month period. We also let
, , and denote the mean ramus bone
heights corresponding to the four time points,
respectively. [1] has applied a one-way ANOVA
model with no random subject effects to analyze this
dataset. The results of [1] show that has the
largest posterior probability.
Table 4. Results of the posterior probabilities of all
the possible models for blood lead level data
Model
TLC
Ramus
BMS
Shang
BMS
Shang
0.0000
0.0535
0.0000
0.0176
0.3946
0.1890
0.0100
0.0870
0.2624
0.1898
0.0155
0.1679
0.0021
0.0584
0.0330
0.1950
0.2012
0.1846
0.0148
0.0236
0.0049
0.0633
0.0000
0.0709
0.000
0.0478
0.0006
0.0918
0.1349
0.2136
0.9410
0.3462
Again, we use the same prior specifications as in
Section 4 and perform MCMC to obtain 80 000
samplers after the 20000 burn-in. Tables 4 and 5 lists
the posterior probabilities of all the possible models
and the posterior estimates for the parameters
respectively. We can clearly see that both methods
select the same model, indicating that there are three
growth spurts during the 18-month period, which is
dierent from [1]. Actually, the posterior probability
of the indicator variable for random subject eect
oered by BMS is 1.0, suggesting that it is
inappropriate to ignore random subject eects in the
model. [1] used a one-way ANOVA model without
random effects to analyze this real data, which could be
inappropriate.
Table 5 lists the posterior estimates for the
parameters. In summary, both BMS and Shang's
methods yield similar posterior estimates and
standard deviations for the parameters. However,
generally speaking, BMS provides shorter posterior
confidence intervals. For example, for the parameter
in Ramus, BMS gives a 95% posterior
confidence interval of 0.57, which is much smaller
than the 3.05 provided by Shang's method.
Table 5. Posterior means (mean), variances(SD) and
95% credible intervals (CrI) for blood lead level data
Methods
TLC
BMS
Mean
23.680
25.586
5.578
Var
0.610
32.285
0.433
CrI
(22.13,25.21)
(16.65,38.80)
(4.43,7.00)
Shang
Mean
23.535
25.424
6.217
Var
1.266
32.463
1.278
CrI
(21.01,25.46)
(16.47,38.64)
(4.66,8.87)
Ramus
BMS
Mean
48.477
6.861
0.734
Var
0.386
6.737
0.021
CrI
(47.23,49.68)
(3.46,13.30)
(0.50,1.07)
Shang
Mean
48.247
6.678
1.452
Var
1.215
6.771
0.767
CrI
(49.94,50.22)
(3.235,13.09)
(0.68,3.73)
TLC
BMS
Mean
0.278
0.427
1.740
Var
0.173
0.242
0.253
CrI
(0.00,1.31)
(0.00,1.49)
(0.71,2.67)
Shang
Mean
0.546
0.507
1.362
Var
0.836
0.654
1.289
CrI
(0.00,3.01)
(0.00,2.64)
(0,3.53)
Ramus
BMS
Mean
0.982
0.947
0.856
Var
0.085
0.093
0.097
CrI
(0.40,1.55)
(0.32,1.54)
(0.00,1.42)
Shang
Mean
1.419
0.875
0.632
Var
1.321
0.830
0.611
CrI
(0.00,3.70)
(0.00,2.82)
(0.00,2.43)
6 Discussion and Conclusion
In this article, we develop a simultaneous selection
method of fixed and random effects in a Bayesian
restricted two-way ANOVA mixed model, which can
accommodate some constraints such as simple order,
tree order umbrella order, etc. Simulation studies
show that the proposed Bayesian variable selection
approach works well in the selection of fixed and
random effects whether in dependent or independent
data. Specifically, in both simulations of dependent
and independent data, our method not only
successfully identifies the correct fixed effects but
also effectively determines the presence of random
effects. Furthermore, the accuracy of this
identification increases with the number of samples,
demonstrating the consistency of our method.
Real data examples indicate that the proposed
method is likely to provide more narrow 95%
credible intervals than the competing method.
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Moreover, we find strong evidence that there exists
significant random subject effects in Ramus bone
heights data. However, [1] analyzed the dataset by
using an unsuitable one-way ANOVA model without
random effects and selected a different model. This
shows that it is necessary to consider a simultaneous
selection of fixed and random effects for longitudinal
data under order constraints.
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https://citeseerx.ist.psu.edu/document?repid=re
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2024).
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.72
Haifang Shi, Jiajia Ge
E-ISSN: 2224-2880
712
Volume 23, 2024
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Jiajia Ge carried out the real data analysis.
- Haifang Shi was in charge of the theoretical design
and simulation.
The two authors collaboratively completed the
writing of the paper.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
The first author is supported by the Research Startup
Fund of Civil Aviation University of China(Fund
No.: 2013QD06S).
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on MATHEMATICS
DOI: 10.37394/23206.2024.23.72
Haifang Shi, Jiajia Ge
E-ISSN: 2224-2880
713
Volume 23, 2024
