Evaluating the performance of clustering using different inputs and algorithms to group children based on early childhood growth patterns

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25 Νοε 2013 (πριν από 3 χρόνια και 10 μήνες)

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Nemours Biomedical Research

Evaluating the performance of clustering using
different inputs and algorithms to group children
based on early childhood growth patterns

Jobayer Hossain, Ph.D.

Tim Wysocki, Ph.D.

Samuel S. Gidding, MD


MingXing Gong, M.Sc.

H. Timothy Bunnell, Ph.D.


September 20, 2012


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Early Childhood Growth Pattern


Childhood growth pattern is


the pattern of the temporal change in height, weight, and head circumference


A determinant of body composition and body weight


Commonly used measures of body composition and weight are



Weight
-
for
-
length for ages < 2 years


Body mass index (BMI) for ages ≥ 2 years


Standardized scores of these two measures are weight
-
for
-
length z
-

score and BMI z
-
score (both termed as BMIz in this presentation)





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Enrollment Criteria


Inclusion Criteria:


Born between 2001 and 2005


Had first well child visit at a Nemours clinic in < 1 month of age


Had at least one well child visit each year for the next 5 years


Exclusion Criteria: Children with medical diagnoses associated
with poor growth and development such as cancer and cystic
fibrosis.


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Data Collection and BMIz Calculation


3365 children were enrolled in the study


Retrospectively collected (from Nemours electronic medical record
(EMR)) height, weight, age, other demographics, and comorbidities
of childhood obesity for children ages between 0
-
5 years


Calculated BMIz (using height, weight, age and gender data) on
their clinic visits.


Interpolated (LOESS) BMIz score for ages (months) 1, 6, 12, 18,
24, 30, 36, 42, 48, 60 for each subject



Demographics


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Variables

Number (%)

Gender



Male

1634 (48.6%)


Female

1731 (51.4%)

Ethnicity



Non
-
Hispanic/Non
-
Latino

2927 (87%)


Hispanic/Latino

404 (12%)


Missing/Refused

34 (1%)

Race



Caucasian

1429 (42.5%)


African American

1542 (45.8%)


Others

353 (10.
5%)


Missing/Refused

41(1.2%)



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Trend in Mean BMIz


A cubic polynomial trend over the ages (0
-
5 years)


Approximately linear in three intervals of time: 1
-
9 months, 9
-
27 months, and 27
-
60 months

Temporal Change in Mean BMIz

Natural Structure of the Temporal Change in
BMIz of Individual Children


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Mean BMIz Over Time: By Demographics


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A cubic polynomial of mean
change in BMIz over time



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Objectives


To group children (0
-
5 years of age) using a cluster analysis that
captures the natural structure of the temporal changes in BMIz


To acquire the objective we did
-


Perform many sets of cluster analyses using different clustering methods,
algorithms, software and cluster inputs


Use a mixed model to evaluate and compare the performance of many sets of
clusters


Select optimum clustering(s) based on the model fit statistics to the data




Rationale of Our Work


Cluster analysis is


mainly an exploratory data analysis


with limited scope of evaluation and comparison of the results of two or more
sets of clustering


A large number of clustering methods/algorithms have been
developed


Different statistical software accommodate different algorithms


Cluster inputs can be raw data or some form of standardized data


No method/algorithm is uniquely best





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Cluster Methods and Software Used


We used the following software/cluster methods with suggested
several distance/similarity measures


SAS Procedures: CLUSTER (Hierarchical), FASTCLUS (k
-
means),
MODECLUS (clusters based on non
-
parametric density estimation
using several algorithms) and some ancillary procedures such as
VARCLUS, TREE, ACECLUS, DISTANCE, PRINCOMP, STDSIZE


SPSS Cluster Analysis: Two
-
step (hierarchical clustering in two steps)
and k
-
means


R Cluster Analysis: Package mclust (model based clustering),
Package cluster (hierarchical), and function kmeans (K
-
means)


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Inputs Used for Cluster Analyses

We performed cluster analyses of


BMIz scores at different ages (every six
-
month interval )


Selected principal component (PC) /factor scores, after
PC/factor analyses of BMIz scores of 11 variables


Random coefficients of mixed effects model of BMIz


Coefficients of auto
-
regression (AR)/autocorrelation (AC) for
each individual


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Model Used for Cluster Evaluation


We used a mixed model that fits the BMIz of children, nested within a cluster
group, as a polynomial function of time






error term

random

the
is

s)
(continuou

time
the
is


group
th

the
of
intercept

the
is

cluster

of

group
th
within
nested
effect
subject

random

the
is

th time

at the

group
th
within
nested
subject
th

the
of

BMIz

the
is


Where,


*



*



*




3
2
ijk
ijk
i
i
j
ijk
ijk
ijk
i
ijk
i
ijk
i
i
i
j
ijk
e
t
i
group
i
group
S
k
i
j
Y
e
t
group
t
group
t
group
group
group
S
Y







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Cluster Analysis of BMIz score



? Natural structure of the
temporal change in BMI


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Principal Component Analysis (PCA)


PCA is a powerful exploratory tool to identify the patterns in data


PCs reflect the interrelation of the similarities and dissimilarities between
observed variables



Performed PCA of BMIz


The first 4 PCs (PC4) explain about 99.19% of the total variation in the data


PC
componen
t


Eigenvalues
/Extraction

Total

% of
Variance

cumulative

1

8.900

80.910

80.910

2

1.347

12.246

93.157

3

.511

4.644

97.801

4

.153

1.390

99.191

5

.055

.499

99.690

6

.019

.176

99.866

7

.007

.062

99.928

8

.005

.044

99.972

9

.002

.018

99.990

10

.001

.007

99.997

11

.000

.003

100.000


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Cluster Analysis of PC Scores


PC4 (the first four PCs (explain 99.19% variation in the data)): four
clusters
-

4 groups, 5 groups, 6 groups and the optimal number of
groups suggested by cluster algorithm


PC3 (the first three PCs (explain 97.80% variation in the data)): four
clusters
-

4 groups, 5 groups, 6 groups and the optimal number of
groups suggested by cluster algorithm


PC2 (the first two PCs (explain 93.16% variation in the data)): four
clusters
-

4 groups, 5 groups, 6 groups and the optimal number of
groups suggested


PC11( the all eleven PCs)): four clusters
-

4 groups, 5 groups, 6 groups
and the optimal number of groups suggested by cluster algorithm by
cluster algorithm (Perhaps cluster analysis using 11 PCs equivalent to
the cluster analysis of the raw data)

Cluster Analysis of PC4


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? Natural structure of the
temporal change in BMI


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Cluster Analysis of PC3






? Natural structure of the
temporal change in BMI


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Cluster Analysis of PC2



? Natural structure of the
temporal change in BMI


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Cluster Analysis of Factor Scores


Fac4 (the first four Factors (explain 99.19% variation in the data)):
four clusters
-

4 groups, 5 groups, 6 groups and the optimal number of
groups suggested by cluster algorithm


Fac3 (the first three Factors (explain 97.80% variation in the data)):
four clusters
-

4 groups, 5 groups, 6 groups and the optimal number of
groups suggested by cluster algorithm


Fac2 (the first two Factors (explain 93.16% variation in the data)): four
clusters
-

4 groups, 5 groups, 6 groups and the optimal number of
groups suggested by cluster algorithm


Fac11( the all eleven factors): four clusters
-

4 groups, 5 groups, 6
groups and the optimal number of groups suggested by cluster
algorithm (Perhaps cluster analysis using 11 Factors equivalent to the
cluster analysis of the raw data))



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Cluster Analysis of Fac4



? Natural structure of the
temporal change in BMI




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Cluster Analysis of Fac3



? Natural structure of the
temporal change in BMI


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Cluster Analysis of Fac2



? Natural structure of the
temporal change in BMI


Recall: the change in population mean of BMIz is a cubic polynomial trend
over the ages 0
-
5 years i.e.


Approximately linear in three intervals of time


We can model this dataset with an intercept and three slopes for three
intervals.


We used parameters (intercept and slopes) as both fixed and random effects
in the model


Fixed effects of slopes explain the rate of change in the population level of
BMIz in each interval of time


Random effects explain the individual to individual variation


at the beginning of life (age of one month)


of the change in BMIz trajectories in three intervals


That’s random effects account for the sources of heterogeneity in the
change in population BMIz


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Piece
-
wise Mixed Effects Model with Random
Coefficients


Exploratory analyses indicate that splits at 9 and 27 months of ages yield the
best fit of the piece
-
wise linear mixed effects model to the individual and
population levels of change in BMIz


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Piece
-
wise Mixed Effects Model with Random
Coefficients


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26
for

0
26
for

26


8
for

0
8
for

8


individual
ith

of

age

of
month

)
1
(

the
is



month

)
1
(
at

individual
ith

of

BMIz

the
is


Where,



)
/
(
27
9
27
4
9
3
2
1
27
3
9
3
2
1


































ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
ij
i
ij
i
ij
i
i
ij
ij
ij
i
ij
t
t
t
t
t
t
t
t
th
j
t
th
j
Y
t
b
t
b
t
b
b
t
t
t
b
Y
E




Piece
-
wise Mixed Effects Model with Random
Coefficients


Nemours Biomedical Research




interval

age

same

for the

individual




of

BMIz
in

change

of

rate

the
is


and

months

9)
-
(1


between

ages
for

BMIz

population
in

change

of

rate

the
is





month

1

of

age

at the

individual
ith

the
of
intercept

the
is




month

1

age
at

n)
(populatio
intercept

the
is


Where,

)
/
(
2
2
2
1
1
1
27
4
9
3
2
1
27
3
9
3
2
1
ith
b
b
t
b
t
b
t
b
b
t
t
t
b
Y
E
i
i
ij
i
ij
i
ij
i
i
ij
ij
ij
i
ij






















Piece
-
wise Mixed Effects Model with Random
Coefficients


Nemours Biomedical Research


age

same

for the

individual


of

BMIz
in

change

of

rate

the
is




months,

60)
-
(27
between

ages
for

BMIz


population
in

(month)

change

of

rate

the
is


Similarly,


age

same

for the

individual


of

BMIz
in

change



of

rate

the
is


and

months

27)
-
(9
between



ages
for

BMIz

population
in

change

of

rate

the
is


Where,

)
/
(
4
3
2
4
3
2
4
3
2
3
2
3
2
3
2
27
4
9
3
2
1
27
3
9
3
2
1
ith
b
b
b
ith
b
b
t
b
t
b
t
b
b
t
t
t
b
Y
E
i
i
i
i
i
ij
i
ij
i
ij
i
i
ij
ij
ij
i
ij





































Piece
-
wise Mixed Effects Model with Random
Coefficients


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Estimate of the regression coefficients (population)

Piece
-
wise Mixed Effects Model with Random
Coefficients

Effect
Estimate
SE
P-value
Intercept
0.2879
0.0138
<0.0001
1
0.102
0.0195
<0.0001
2
-0.1606
0.0215
<0.0001

3
0.1535
0.0082
<0.0001
Estimated variance of the random effects (individual)

Effect
Estimate
SE
p-value
V(intercept)
0.6305
0.0157
<0.0001
V(b1)
1.212
0.0311
<0.0001
V(b2)
1.4552
0.0381
<0.0001
V(b3)
0.2056
0.0055
<0.0001

The model we just discussed, is our initial model to track the
trajectories of individual and population level of BMIz


We also observed a significant difference in mean BMIz between
gender, race and ethnicity.


In addition to our initial model, we also fitted a model to track the
change in BMIz after adjustment for gender and race
-
ethnicity.





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Piece
-
wise Mixed Effects Model with Random
Coefficients


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Cluster Analysis of the Random Effects


We performed a cluster analysis of individual level four parameters:
intercept (
b
1i
) and three slopes (
b
2i
,
b
2i
+
b
3i
,
b
2i
+
b
3i
+
b
4i
)






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Cluster Analysis of the Random Coefficients of Piece
-
wise Mixed
Effects Model

? Natural structure of the
temporal change in BMI




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Time Series Analysis


Performed


Auto
-
regression (AR) on the BMIz of each individual and extract coefficients
(burg, Yule
-
walker, MLE methods were used) of the best model suggested
by AIC


Auto
-
correlation on the BMIz of each individual and extract autocorrelations


Spectral analysis on each individual’s BMIz and extract frequencies


Performed cluster analysis of AR coefficients, auto
-
correlations,
and frequencies

Cluster Analysis of the AR Coefficients (Yule
-
Walker)


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? Natural structure of the
temporal change in BMI


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Model Based Evaluation Ordered by Logliklihood
(Largest to Smallest)

AIC
BIC
(smaller is better)
(smaller is better)
Fac4_reg_grp_opt
10
-8152.6
-8140.4
-8156.6
4078.3
PC4corr_grp_opt
11
-7629.5
-7617.3
-7633.5
3816.75
Fac3_reg_grp_opt
7
-6391.3
-6379.1
-6395.3
3197.65
PC3cov_grp_opt
6
-5443.7
-5431.4
-5447.7
2723.85
Fac3_reg_grp6
6
-5171.4
-5159.1
-5175.4
2587.7
initmodel_grp6
6
-5066.2
-5054
-5070.2
2535.1
Model_gendrace_grp6
6
-4855.9
-4843.6
-4859.9
2429.95
PC3corr_grp6
6
-4301.1
-4288.8
-4305.1
2152.55
PC4corr_grp6
6
-4217.2
-4204.9
-4221.2
2110.6
Fac3_reg_grp5
5
-4153.4
-4141.2
-4157.4
2078.7
Fac4_reg_grp6
6
-3951.8
-3939.5
-3955.8
1977.9
PC3corr_grp_opt
5
-3595.1
-3582.9
-3599.1
1799.55
PC4corr_grp5
5
-3515.2
-3503
-3519.2
1759.6
Fac4_reg_grp5
5
-3248
-3235.8
-3252
1626
PC4cov_grp6
6
-3146.2
-3134
-3150.2
1575.1
PC3cov_grp5
5
-3125.1
-3112.9
-3129.1
1564.55
initmodelcoe_grp6
6
-2848.9
-2836.6
-2852.9
1426.45
Model_gender_ethnicity6
6
-2825.8
-2813.5
-2829.8
1414.9
PC11corr_grp11
11
-2741.6
-2729.4
-2745.6
1372.8
Model_gendrace_coe_grp6
6
-2278.4
-2266.2
-2282.4
1141.2
initmodel_grpopt4
4
-2139
-2126.8
-2143
1071.5
PC4cov_grp_opt
5
-2105.5
-2093.3
-2109.5
1054.75
PC3cov_grp4
4
-1467.1
-1454.8
-1471.1
735.55
PC4corr_grp4
4
-1429.7
-1417.5
-1433.7
716.85
PC2corr_grp_opt
9
-1187.9
-1175.6
-1191.9
595.95
Fac3_reg_grp4
4
-1040.5
-1028.3
-1044.5
522.25
PC3corr_grp4
4
-969.5
-957.3
-973.5
486.75
PC11corr_grp9
9
-655.9
-643.6
-659.9
329.95
PC11corr_grp10
10
-309.6
-297.4
-313.6
156.8
Fac4_reg_grp4
4
34.2
46.5
30.2
-15.1
PC4cov_grp4
4
167
179.2
163
-81.5
PC11corr_grp7
7
182
194.2
178
-89
Model_gendrace_coe_opt3
3
428.6
440.8
424.6
-212.3
initmodelcoe_opt3
3
631.5
643.7
627.5
-313.75
Clustering
Number of Groups
-2 Log-likelihood
Log-likelihood
Choosing an Optimum Clustering


The following few are the candidates of an optimum clustering
-


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Optimum Choice: Based on the likelihood score and
number of groups, prefer smaller number of groups with
maximum possible likelihood score

AIC
BIC
(smaller is better)
(smaller is better)
Fac4_reg_grp_opt
10
-8152.6
-8140.4
-8156.6
4078.3
PC4corr_grp_opt
11
-7629.5
-7617.3
-7633.5
3816.75
Fac3_reg_grp_opt
7
-6391.3
-6379.1
-6395.3
3197.65
PC3cov_grp_opt
6
-5443.7
-5431.4
-5447.7
2723.85
Fac3_reg_grp6
6
-5171.4
-5159.1
-5175.4
2587.7
initmodel_grp6
6
-5066.2
-5054
-5070.2
2535.1
Model_gendrace_grp6
6
-4855.9
-4843.6
-4859.9
2429.95
PC3corr_grp6
6
-4301.1
-4288.8
-4305.1
2152.55
PC4corr_grp6
6
-4217.2
-4204.9
-4221.2
2110.6
Fac3_reg_grp5
5
-4153.4
-4141.2
-4157.4
2078.7
Clustering
Number of Groups
-2 Log-likelihood
Log-likelihood
Best Clustering Based on the Model Fit
Statistics


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Mean Profile for the Best Clustering
Group
1
2
3
4
5
6
7
8
9
10
BMIz
-2
-1
0
1
2
3
AGE(month)
1
16
31
46
61

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Optimum Clustering Based on the Number of
Groups and Model Fit Statistics

Acknowledgement


Li Xie, Biostatistician



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Thank you very much


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The Fastclus Procedure


The FASTCLUS procedure combines an effective method for finding
initial clusters with a standard iterative algorithm for minimizing the
sum of squared distances from the cluster means. This kind of
clustering method is often called a k
-
means model.


By default, the FASTCLUS procedure uses Euclidean distances.


The FASTCLUS procedure can use an Lp (least pth powers)
clustering criterion instead of the least squares (L2) criterion used in
k
-
means clustering methods.


PROC FASTCLUS uses algorithms that place a larger influence on
variables with larger variance, so it might be necessary to
standardize the variables before performing the cluster analysis.


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The Modeclus Procedure


The MODECLUS procedure clusters observations in a SAS data set by
using any of several algorithms based on nonparametric density estimates.
PROC MODECLUS implements several clustering methods by using
nonparametric density estimation.


PROC MODECLUS can perform approximate significance tests for the
number of clusters


PROC MODECLUS produces output data sets containing density estimates
and cluster membership, various cluster statistics including approximate p
-
values, and a summary of the number of clusters generated by various
algorithms, smoothing parameters, and significance levels.


For nonparametric clustering methods, a cluster is loosely defined as a
region surrounding a local maximum of the probability density function.


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The Cluster Procedure


The CLUSTER procedure hierarchically clusters the observations
in a SAS data set by using one of 11 methods. The data can be
coordinates or distances.


The clustering methods are
: average linkage, the centroid method,
complete linkage, density linkage (including Wong’s hybrid and kth
-
nearest
-
neighbor methods), maximum likelihood for mixtures of
spherical multivariate normal distributions with equal variances but
possibly unequal mixing proportions, the flexible
-
beta method,
McQuitty’s similarity analysis, the median method, single linkage,
two
-
stage density linkage, and Ward’s minimum
-
variance method.


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The SPSS TwoStep Cluster


Handles both continuous and categorical variables by extending the model
-
based distance


Utilizes a two
-
step clustering approach similar to BIRCH (Zhang et al. 1996)


Step1(Pre
-
cluster): Uses a sequential clustering approach (Theodoridis and
Koutroumbas 1999). It scans the records one by one and decides if the
current record should merge with the previously formed clusters or start a
new cluster based on the distance criterion.


Step2 (group data in to sub
-
cluster): Use the resulting sub
-
clusters in step1
and groups them into the desired number of clusters


Provides the capability to automatically find the optimal number of clusters.


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R
-
mclust package


mclust is a contributed R package for model
-
based clustering, classification,
and density estimation based on finite normal mixture modeling.


It provides functions for parameter estimation via the EM algorithm for normal
mixture models with a variety of covariance structures, and functions for
simulation from these models.


Also included are functions that combine model
-
based hierarchical clustering,
EM for mixture estimation and the Bayesian Information Criterion (BIC) in
comprehensive strategies for clustering, density estimation and discriminant
analysis.


Provides the capability to automatically find the optimal number of clusters.



Principal Component Analysis



PCA transformed a set of interrelated variables to a new set of uncorrelated
variables called principal components (PCs).


The variance of a PC indicates the amount of total variation (information) in the
original variables conveyed by that particular PC.


The transformation is taken in such a way that the PCs are ordered and the first
PC accounts for as much of the variability in the original variables as possible,
and then each succeeding PC in turn has the highest variance possible under
the constraint that it be uncorrelated with the preceding PCs.


Thus the most informative PC is the first and the least informative is the last.


PCA is a powerful exploratory tool to identify the patterns in data because PCs
reflect the interrelation of the similarities and dissimilarities between observed
variables


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Estimate of the regression coefficients (population)

Effect
Estimate (SE)
P-value
Mean BMIz at the age of one month
0.2879 (.0138)
<0.0001
Rate of change in mean BMIz from 1-9 months
0.102 (.0195)
<0.0001
Rate of change in mean BMIz from 9-26 months
-.0586 (.0096)
<0.0001
Rate of change in mean BMIz from 26-60 months
0.0949 (.0062)
<0.0001

A significant increasing trend in population
BMIz

at ages 1 to 9 months
and at ages 27 to 60 months


A significant decreasing trend in population
BMIz

at ages 9 to 27 months


The rate of changes in
BMIz

in three pieces of time are significantly
different

Piece
-
wise Mixed Effects Model with Random
Coefficients


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Effect
Estimate (SE)
p-value
Var(BMIz at the age of one month)
0.6305(0.0157)
<.0001
Var(Rate of change in BMIz from 1-9 months)
1.212 (0.0311)
<.0001
Var(Rate of change in BMIz from 9-26 months)
0.1766 (0.0281)
<.0001
Var(Rate of change in BMIz from 26-60 months)
0.0636 (0.0045)
<.0001
Estimated variance of the random effects (individual)


A substantial individual to individual variability in


BMIz

at the age in one month


the rate of change in
BMIz

at each piece of time

Piece
-
wise Mixed Effects Model with Random
Coefficients


Nemours Biomedical Research

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