Using growth curve model in anthropometric
data analysis
Anu Roos
University of Tartu, Institute of Mathematical Statistics, Liivi 2, 50409 Tartu,
Estonia anu.roos@ut.ee
Summary. In this paper we use an extension of the MANOVA model – the Growth
Curve model – for analyzing data on growth of the children (age 6-11). Growth Curve
model was first introduced by Potthoff & Roy (1964) [PR64], although some other
authors had earlier worked with a similar model. In the present paper we examine
the distribution of the location parameter of the Growth Curve model in a specific
case using real data.
The paper consists of four sections. In Section 2 the overview of the Growth
Curve model and the theory used are given. In Section 3, the data is shortly described. In Section 4 the Growth Curve model is fitted to the data. In Section 5
the estimated distribution of the location parameter of the Growth Curve model is
studied and compared with the normal distribution.
Key words: Growth Curve model, Kotz-type distribution, location parameter,
mixture distribution, parameter estimation
1 Introduction
Over the years linear models have been one of the most widely used tools in data
analysis. When considering repeated measurements an immediate extension is a
MANOVA model. In MANOVA model, the number of parameters to evaluate may
be large. When we assume a functional relationship between the mean parameters
within experimental units, we get the Growth Curve model. Unfortunately the distribution of the location parameter in Growth Curve model is not normal, even if the
underlying data follows normal law. In the paper we examine an approximation to
the distribution of the location parameter in the Growth Curve model and compare
it with the normal distribution (an assumption often made in practice).
2 Growth Curve model
An overview of the Growth Curve model is given, for example, by Woolson & Leeper
(1980) [WL80], von Rosen (1991) [vRo91] or Srivastava & von Rosen (1999) [SvR99].
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Anu Roos
Kshirsagar & Smith (1995) [KS95] have written a book on the model and for a recent
contribution see Kollo & von Rosen (2005, [KvR05] Chapter 4), where the model
and some extensions are presented. In Kollo, Roos & von Rosen (2005) [KRv05], the
distribution of the location parameter of the model is studied.
The Growth Curve model includes two design matrices, the model has a form
x = ABC + E,
(1)
where x : p × n is a data matrix (a column corresponds to each object and a row to
each observation-point), B : q × k is a matrix of unknown parameters, A : p × q, is
a known within individuals design matrix of full rank, C : k × n is a known between
individuals design matrix of full rank, and E = (e1 , e2 , . . . , en ) in an error matrix
, where ei ∼ Np (0, Σ) are i.i.d. and Σ is an unknown positive definite parameter
matrix.
The MLE of B (e.g. see [Kha68] or [KvR05] §. 4.1.2) equals
b = (A′−1 A)−1 A′−1 xC ′ (CC ′ )−1 ,
B
(2)
where
= x(I − C ′ (CC ′ )−1 C)x′ .
(3)
The other parameter - the covariance matrix Σ for the error matrix, can be estimated
using:
b = (x − ABC)(x
b
b ′
− ABC)
nΣ
(4)
b
To evaluate the goodness of model, we need to estimate the covariance matrix of B:
b =
DB
n−k−1
(CC ′ )−1 ⊗ (A′ ΣA)−1
n−k−p+q−1
(5)
b in (2) is a non-linear estimator and its
Note that unlike the MANOVA model, B
distribution is not easy to find, unfortunately it is not normal. One good approxib is derived in [KvR05], §4.3.2. The result is stated
mation for the distribution of B
in the next theorem.
b be given by (2). Then an Edgeworth type expansion of the denTheorem 1. Let B
b
sity of B equals
f (B 0 ) = fB E (B 0 ) + · · · ,
B
where
fB E (B 0 ) = fN (B 0 )
n
×
1 + 21 s Tr{A′ Σ −1 A(B 0 − B)CC ′ (B 0 − B)′ } − kq
o
(6)
,
with
s=
p−q
n−k−p+q−1
and fN (B 0 ) is a density function
Nq,k (B, (A′ Σ −1 A)−1 , (CC ′ )−1 ).
(7)
of
the
matrix
normal
distribution
Using growth curve model in anthropometric data analysis
745
Following the ideas of Fujikoshi (1987) [Fuj87], Kollo & von Rosen (2005, [KvR05]
§4.3.2) have shown that fB E (B 0 ) is a good approximation to the density f (B 0 ):
B
Theorem 2.
fB E (B 0 ) − f
(B 0 ) = O(n−2 ).
B
The distribution of B E (density given in (6)) is a mixture of matrix normal
distribution and another elliptical distribution - matrix Kotz type distribution with
the weights 1 − 12 skq and 21 skq respectively (here s is given in (7), k and q are the
dimensions of B). An overview of Kotz type distributions can be found in [FKN90]
§3.2, we only give here the definition of the Kotz distribution - the special case of
Kotz type distributions.
Definition 1 Matrix Y : q × k is Kotz distributed with parameter M , V and
W if its density function is in the following form:
fY (Y) =
p
n
kq
|V |− 2 |W |− 2 g(Tr{V −1 (Y − M )W −1 (Y − M )′ }),
(2π)kq/2
(8)
where
g(x) = x exp(−x/2).
We write Y ∼ Kq,k (M , V , W ) if Y has the density function (8).
Theorem 3. Let a random matrix x be a mixture of random matrices xN and xK
with weights 1 − λ and λ respectively, where xN ∼ Np,n (M , V , W ) and xK ∼
Kp,n (M , V , W ). Let
x=
x1
x̃1
x2 , M =
µ1
µ̃1
M2
and Σ = V ⊗ W =
Σ 11
Σ 21
Σ 12
Σ 22
,
where x1 : p1 × 1, x̃1 : (p − p1 ) × 1 and x2 : p × (n − 1), matrix M is with the same
structure as x and Σ 11 : p1 × p1 . Then the vector x1 is a mixture of random vectors
1
1
and λp
respectively, where x1N ∼ Np1 (µ1 , Σ 11 )
x1N and x1K with weights 1 − λp
np
np
and x1K ∼ Kp1 (µ1 , Σ 11 ).
The distribution of a marginal of x still is a mixture of the same type, but weight
of the Kotz distribution decreases. Therefore the distribution of marginals is always
closer to the normal than the distribution of the original matrix.
3 Data description
The data consists of measurements on children’s weight. The data was collected in
1997, retrospectively - from doctors over Estonia, using medical records. There are
121 girls and 88 boys observed, all measured six times, starting at the age of 6 until
the age of 11. The growth patterns of some individuals can be seen in Figure 1.
Only a few individuals are chosen to make the figure understandable. The growth
patterns of the children are quite similar in most cases - the ”starting weight” at age
of 6 is 20-30 kg and the weight at the age of 11 lies between 30 and 50 kg. There are
some exceptional cases as well. In some cases the weight decreases at some point,
which could be measuring errors.
The overall trend can be better seen in Figure 2, where the box plots for each
sex and age group are plotted, the mean weight of each group is also added.
746
Anu Roos
Some individual growth curves
50
40
20
30
weight of individuals
60
70
boys
girls
6
7
8
9
10
11
age
Fig. 1. The individual growth curves for some children
The box plots of the weigth by age groups
50
20
30
40
weight
60
70
boys
girls
group mean
6
6
7
7
8
8
9
9
10
age
Fig. 2. The box plot of weight, data grouped by age and sex
10
11
11
Using growth curve model in anthropometric data analysis
747
4 Results
In this section we present the Growth Curve model for described data. We use
statistical software R for all the calculations.
First we estimate the parameters of the model (1) using the formulas (2)-(4).
We assume the linear growth for the weight of both, boys and girls. From Figure 2
one could get the idea, that this could be the case. The design matrices A and C are
of the form:
01
B1
B
B1
A=B
B1
1
1
6 1
7 C
C
8 C
C,
9 C
A
10
11
0
1
1 1 ... 1 0 0 ... 0
C = |0 0 {z
. . . 0} 1
. . . 1} A .
| 1 {z
88
(9)
121
The estimates of B and Σ are:
0 14.27
b =
B
3.73
3.20
3.16
3.20
B 15.17
B
B 16.71
b
, Σ=B
B 19.27
23.05
27.26
15.17
19.61
21.21
23.92
29.09
33.58
16.71
21.21
26.30
29.27
35.13
40.38
19.27
23.92
29.27
36.60
42.08
48.88
23.05
29.09
35.13
42.08
54.88
61.53
27.26 1
33.58 C
C
40.38 C
C.
48.88 C
A
61.53
78.20
(10)
b is as follows:
The covariance matrix for B
0
0.39
B −0.06
d
b =B
DB
0
0
−0.06
0.01
0
0
0
0
0.28
−0.05
1
0
0 C
C.
−0.05 A
0.01
(11)
To provide some evaluation of the model, we use the simultaneous Bonferroni confidence intervals (on conf. level 0.95) for elements of B. The classical assumption,
b ∼ N4 (vecB, (A′ Σ −1 A)−1 ⊗ (CC ′ )−1 ) is used. The
that the distribution of vecB
confidence intervals are given in Table 1.
b
Table 1. The Bonferroni confidence intervals for the elements of the B
parameter
B 1,1
B 2,1
B 1,2
B 2,2
estimate
3.73
3.20
3.16
3.20
confidence interval
2.17
5.28
2.90
3.50
1.84
4.49
2.94
3.45
The confidence intervals for the intercepts (B 1,1 and B 1,2 ) are quite large, but they
are narrower for the linear parameters.
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Anu Roos
5 Approximate distribution of the location parameter
b
estimator B
We shall find the approximate distribution of the estimator of the location parameter B using Theorem 1. The distribution is a mixture of the matrix normal distribution N2,2 (B, (A′ Σ −1 A)−1 , (CC ′ )−1 ) and the matrix Kotz distribution
4
97
and 101
respectively, where A and
K2,2 (B, (A′ Σ −1 A)−1 , (CC ′ )−1 ), with weights 101
C are given in (9). We do not know two parameters of the mixture distribution, we
only know the third one, which does not depend on the model:
(CC ′ )−1 =
0.0114
0
0
0.0083
.
We can calculate the estimates of the other parameters, the estimate of the B is
given in (10), the estimate of the second parameter equals:
(A′ Σ −1 A)−1 =
33.58
−5.51
−5.51
1.25
.
The weight of the Kotz distribution is rather small in the mixture. To study the
b we need to calculate the cross-product of the
distributions of the elements of B,
second and third parameter of the component-distributions:
0
(A′ Σ −1 A)−1 ⊗ (CC ′ )−1
0.382
B −0.063
B
=
0
0
−0.063
0.014
0
0
0
0
0.278
−0.046
1
0
0 C
C.
−0.046 A
0.010
b
Using Theorem 3 we get that the distribution the element B i,j , i, j = 1, 2 of B
is also mixture of a one-dimensional normal distribution N1 (µi,j , σi,j ) and a Kotz
distribution K1 (µi,j , σi,j ), the estimates of the values of µi,j and σi,j are given in
1
Table 2. The weight of the Kotz distribution is 101
and for the normal distribution
100
.
101
Table 2. Parameters of the distributions of the components
i
1
2
1
2
j
1
1
2
2
µi,j
3.73
3.20
3.16
3.20
σi,j
0.382
0.014
0.278
0.010
b 1,1 , the first element of the estimator matrix B.
b
We take a closer look at B
According to Theorem 2 and Theorem 3, a good approximation to the density of
b 1,1 is a mixture of the normal distribution N (3.73, 0.382) and the Kotz distribution
B
1
K(3.73, 0.382) with weights 100
and 101
respectively. Classical way for estimating
101
the same distribution would be using of the normal distribution N (3.73, 0.382). It
is interesting to find out in which way these distributions differ.
Using growth curve model in anthropometric data analysis
749
Fig. 3. The difference of the density functions of first marginals - normal distribution
minus the mixture
Table 3. The values of density functions at the selected points, f (x) - density function of N (3.73, 0.389), g(x) - density function of N (3.73, 0.382) and K(3.73, 0.382)
1
and 101
with weights 100
101
√
µ − 10√ σ
µ−9 σ
√
. µ−8 σ
√
µ − 7√σ
µ−6 σ
√
µ−5 σ
x
−2.51
−1.89
−1.27
−0.641
−0.0175
0.606
f (x)
4.62779 × 10−23
7.46225 × 10−19
4.33983 × 10−15
9.10291 × 10−12
6.88642 × 10−09
1.87894 × 10−06
g(x)
9.23 × 10−23
1.35 × 10−18
7.09 × 10−15
1.35 × 10−11
9.31 × 10−09
2.33 × 10−06
√
µ − 4√σ
µ−3 σ
√
µ − 2√ σ
µ −√ σ
µ − σ/2
µ
x
1.23
1.85
2.48
3.1
3.41
3.73
f (x)
0.000185
0.00656
0.08400
0.38783
0.56850
0.64579
g(x)
0.000213
0.00709
0.08655
0.38791
0.56433
0.63943
In Figure 3 the difference of the density functions (the density function of the
mixture is subtracted from the density function of the normal distribution) is plotted
and in Table 3 some values of both density functions are calculated. It is apparent
that near the mean the density of the normal distribution is greater and then the
difference decreases. At the distance of two standard deviation from the mean, the
density function of the mixture is greater and this does not change. The mixture
distribution has slightly heavier tails than the normal distribution. In this case the
difference is very small.
This leads to the conclusion, that the normal approximation is acceptable in large
samples. With smaller sample size, the weight of Kotz distribution in the mixture
increases and the distribution deviates more from normal distribution which may
lead to mistakes.
Acknowledgements
The author is thankful for the financial support from Estonian Science Foundation
through the grant No. 5686.
750
Anu Roos
References
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