~/nd~anJ. PIdi,,fr, 45 : 48, 1078
CORRELATES OF AGE WITH HEIGHT AND WEIGHT OF CHILDREN A.R. Ca^UgAsJA Gwalior
Correlation coefficients have a very significant importance in proving certain facts in the fields of biology and medicine. But the infe~ei~ces ba~ed on these coefficients are inconclusive in the sense tidal these coefficients do not prove actual association between the participating variables. There may be other factors which are responsible for the observed correlation, or the situation may be in t h e opposite direction in the sense that the participating variables have association between 9 but this association is not observed because o f the l~ositive correlation in one variable and a negative correlation in the other variable, the result belng that the b bserved correlation is either found to be zero or nearly to be zero showi n g that the participating variables are unassociated, while actually it is not so. Moreover, when one is interested in crosssectional studies rather than the longitudinal studi_es, that is, in postulating the hypothesis rather than in testing or proving the hypothesis, the correlation coefficients are of little imports.rice because of the fact that they explain only association and not the variation. Thus it is evident that correlation coefficients only are not sufficient in crosssectional studies and something more is needed. Correlation coefficients alone can never describe a practical situation completely. W h a t is needed are the more refined Address:--A.R. Chaurasia, Lecturer in Statistics and Demography, G.R. Medical College, Gwalior 474001. From Received on July 31, 1977.
tools of statistic.al theory by the h e ~ which we may be able to describe n postulate the hypothesis. These 9 C~ under the category of principal compo211 analysis. The importance of principal c o m p o ~ analysis lies in the fact that it descrih~i phenomenon. It neither approves.~i disapproves the phenomenon. Thls is:~{ reason why i t is important in cross-see~-'l studies where the explanation of the p r ~ cal situation in a theoretical way is prime importance. The testing of a hyl~ thesis is of importance only when "it .~ been described completely and correcUl I f the description is incomplete and fau~l the testing will automatically give wr0~ interpretations. In principal component analysis explanation of phenomena is made poss ~ by finding out tnat relationship between t l participating variables which causes tim largest variation. Then for testing a ~ other purposes we can take this relati~ only instead of several separate factors a ~ then may proceed as usual. The importance of principal c o m p o n d analysis in the bio-medical sciences can i~l shown by the following example. Supposl a physical anthropologist makes a d o z ~ measurements on each of a number ~il individuals. He may be interested M describing and analysing how tiJl individuals differ in these kinds I physiological characteristics. Eventually lia will want to "explain" these differenceaa but first of all he will want to k n o l
p.}4AUI~.ASlA--CORRELATESOF
AOlg WITH
HEIGI4T
what measurements or comblnalions of ~easurements show considerable variation; that is, which of the measurement or the combination of measurements l~eeds further study. The principal component g~ves a new set of linearly combined m e a s , r e m e n t s having an added property that they are mutually independent, while the original variables may depend on each other. It may be that most of the variation, from individual to individual, reside in tlnee or in one, as the ease may be, of the linear combinations, then the anthropologist can direct his study to these three or One quantities. T h e other c0mbinations have such a small variation that they may b e set aside only on the basis of a formal inspection. It is clear that no reich sorting out procedure exists in correlation or in regression analysis methods. Thus keeping "he above in view a study was performed among: the school chi!dren ofGv, aliorcity, in which the age, weight, and height of the children were measured. The principal components of these quantities were calculated and on the basis of these components an a t t e m p t was m a d e to txplain the growth pattern of the child in relation to age of the child as the growth in height and weight of the child is directly related with the natural growth of the child. Statistical
Theory
Let a random variable x = ( x 1, x~..... , x~) ha~ 0 for its
mean and ~ for variance
eovariance matrix, which is real and positive ~midefinite. Let the eigenvalues o f ~ be ~lz~2z ..... iz~pzo. From the theory of matrices it is well known that there exists a p x p Qrthogonal matrix [- such that
AND WE/GIIT
49
OF CHILDREN
where
.F -----[~11 ............ 1 ~ r]
and
A --= di~g [7~1......... ~p]
Now consider the orthogonal transformation T h e elements of v are called the principal components o f t ~ e v e c t o r x . The principal components of x are unique and are uncorrelated, having" the variances 7u ......... ~p respectively. T h e generalized variance of the vector of principal component is the generalized varlance of the original vector and the sum of the variances of the principal component is the .sum o f the variances of the original vector. In actual practice v. is replaced by its m a x i m u m likelihood estimate ~ and the principal components are the solution of the si,multaneous equations A
(~-~1I)
b =0
and b' b , ~ 1.
m
where I is the identity matrix and ~1 is the first root of the determinentat equation
Ik-a
l =o.
T h e variance covar,_'ance matrix for each age group was calculated separately. T h e characteristic roots of the determinental equation I ~ -- ~ I [ = 0 were calculated. The largest root for each age group was then substituted in the matrix equation (k-- ~xI) b~0 Component.
to obtain the Principal
S t a t i s t i c a l Analysis The variance covarlance matrix for each age group was calculated separately
50
Vot.. 45, Noi
I N D I A N J O U R N A l . OF P E D I A T R I C t
a n d go were the respective principal c o m p o nents. W e have the following r e s u h s : - -
Age 8 years ^
02~176 3
Age 5 years A
?~t " .t5'4, ~1 =~ 14"505;
"A~ = 2"515
the principal c o m p o n e n t is "98 x + '14 y
i = ]- 26"41
L
"A1 = 30'95
the principal c o m p o n e n t is 979 x + "61 y '61 x + "79 y
Age 9 ),ear~
Age 6 years 9"69 ] 10"64 ~2 = 6"05
Z2 = 27"2
_-[58o ~,~ ~ 81"4,
43"11 ~2 =~ 19'7
]
the principal c o m p o n e n t is 78 x q- "63 y
the principal c o m p o n e n t is 90 x + "48 y
Age 7 years
Results and Discussgons
.o45"2] 1 ;i x = 39"09
~2 = 4'27
the principal c o m p o n e n t is "86 x + "50 y
' C a b l e 1.
Age g r o u p
T h e principal c o m p o n e n t s o f h e i g h l a n d weight o f children in different agw groups are given in T a b l e 1. Here represents the height and y the weight the children.
Principal components of height and weight in different age groups.
Principal c o m p o n e n t
5 years
0.98 x + 0.14 y
6
,)
0.90 x + 0.48 y
7
,,
0.86 x § 0.50 v
8
),
0.79 x .~- 0.61 y
8
,,
0.61 x-~- 0.79 y
9
),
0.78 x nt- 0.63 y
0HAUIIAIIIA~CORRF,LATI~,S OF AGE WITII IIF;IOIIT AND WEIOIt't" OF GIIIL1)REN
Now s~tppose we take the natural growth of the child in one year to be unity. 'rites from the above analysis we can estllllate the respective shares ,,f height and weight in the natural growth. Thus at 5 yearsofsge a larger share of the child's IlltlUra| growth is spent in increasing the height of the child while at 9 years of age a larger share o'f the natural growth is spent in Increasing the Weight of the child. Thus It can be concluded that in the early llfe growth in height is more pron~inent but as the time passes, growth in weight becomes tnore and more prominent. Thus it has been shown that the d a t a gould.have arisen from the action of formulatlom shown in Table 1. It does not follow that they did, but possibly there are other hypotheses that would account for them, Nevertheless, two important things are apparent. First, the formulation makes
51
some biological sense. For example, the relation in the 5-year-age group is equivalent to 0.9 x + 0.1 y showing that the effect of natural growth in increasing tile weight of the child i~ almost nil in comparison to the increase in height of the child. Similar inteipretations can be drawn from other formulations too. There is a peculiarity in the 8-year-age group. Here two relations turn up a g a i n and again. This does not prove that they have any biological significance but can not be dismissed as chance and need further study. Thus on the whole it can be concluded that t h e existence of these formulations, whatever may be the reasons for them, is a fact of experiment. They provide a description of the phenomenon, though not necessarily an explanation of it.
CORRELATES OF AGE WITH HEIGHT AND WEIGHT OF CHILDREN A.R. Ca^UgAsJA Gwalior
Correlation coefficients have a very significant importance in proving certain facts in the fields of biology and medicine. But the infe~ei~ces ba~ed on these coefficients are inconclusive in the sense tidal these coefficients do not prove actual association between the participating variables. There may be other factors which are responsible for the observed correlation, or the situation may be in t h e opposite direction in the sense that the participating variables have association between 9 but this association is not observed because o f the l~ositive correlation in one variable and a negative correlation in the other variable, the result belng that the b bserved correlation is either found to be zero or nearly to be zero showi n g that the participating variables are unassociated, while actually it is not so. Moreover, when one is interested in crosssectional studies rather than the longitudinal studi_es, that is, in postulating the hypothesis rather than in testing or proving the hypothesis, the correlation coefficients are of little imports.rice because of the fact that they explain only association and not the variation. Thus it is evident that correlation coefficients only are not sufficient in crosssectional studies and something more is needed. Correlation coefficients alone can never describe a practical situation completely. W h a t is needed are the more refined Address:--A.R. Chaurasia, Lecturer in Statistics and Demography, G.R. Medical College, Gwalior 474001. From Received on July 31, 1977.
tools of statistic.al theory by the h e ~ which we may be able to describe n postulate the hypothesis. These 9 C~ under the category of principal compo211 analysis. The importance of principal c o m p o ~ analysis lies in the fact that it descrih~i phenomenon. It neither approves.~i disapproves the phenomenon. Thls is:~{ reason why i t is important in cross-see~-'l studies where the explanation of the p r ~ cal situation in a theoretical way is prime importance. The testing of a hyl~ thesis is of importance only when "it .~ been described completely and correcUl I f the description is incomplete and fau~l the testing will automatically give wr0~ interpretations. In principal component analysis explanation of phenomena is made poss ~ by finding out tnat relationship between t l participating variables which causes tim largest variation. Then for testing a ~ other purposes we can take this relati~ only instead of several separate factors a ~ then may proceed as usual. The importance of principal c o m p o n d analysis in the bio-medical sciences can i~l shown by the following example. Supposl a physical anthropologist makes a d o z ~ measurements on each of a number ~il individuals. He may be interested M describing and analysing how tiJl individuals differ in these kinds I physiological characteristics. Eventually lia will want to "explain" these differenceaa but first of all he will want to k n o l
p.}4AUI~.ASlA--CORRELATESOF
AOlg WITH
HEIGI4T
what measurements or comblnalions of ~easurements show considerable variation; that is, which of the measurement or the combination of measurements l~eeds further study. The principal component g~ves a new set of linearly combined m e a s , r e m e n t s having an added property that they are mutually independent, while the original variables may depend on each other. It may be that most of the variation, from individual to individual, reside in tlnee or in one, as the ease may be, of the linear combinations, then the anthropologist can direct his study to these three or One quantities. T h e other c0mbinations have such a small variation that they may b e set aside only on the basis of a formal inspection. It is clear that no reich sorting out procedure exists in correlation or in regression analysis methods. Thus keeping "he above in view a study was performed among: the school chi!dren ofGv, aliorcity, in which the age, weight, and height of the children were measured. The principal components of these quantities were calculated and on the basis of these components an a t t e m p t was m a d e to txplain the growth pattern of the child in relation to age of the child as the growth in height and weight of the child is directly related with the natural growth of the child. Statistical
Theory
Let a random variable x = ( x 1, x~..... , x~) ha~ 0 for its
mean and ~ for variance
eovariance matrix, which is real and positive ~midefinite. Let the eigenvalues o f ~ be ~lz~2z ..... iz~pzo. From the theory of matrices it is well known that there exists a p x p Qrthogonal matrix [- such that
AND WE/GIIT
49
OF CHILDREN
where
.F -----[~11 ............ 1 ~ r]
and
A --= di~g [7~1......... ~p]
Now consider the orthogonal transformation T h e elements of v are called the principal components o f t ~ e v e c t o r x . The principal components of x are unique and are uncorrelated, having" the variances 7u ......... ~p respectively. T h e generalized variance of the vector of principal component is the generalized varlance of the original vector and the sum of the variances of the principal component is the .sum o f the variances of the original vector. In actual practice v. is replaced by its m a x i m u m likelihood estimate ~ and the principal components are the solution of the si,multaneous equations A
(~-~1I)
b =0
and b' b , ~ 1.
m
where I is the identity matrix and ~1 is the first root of the determinentat equation
Ik-a
l =o.
T h e variance covar,_'ance matrix for each age group was calculated separately. T h e characteristic roots of the determinental equation I ~ -- ~ I [ = 0 were calculated. The largest root for each age group was then substituted in the matrix equation (k-- ~xI) b~0 Component.
to obtain the Principal
S t a t i s t i c a l Analysis The variance covarlance matrix for each age group was calculated separately
50
Vot.. 45, Noi
I N D I A N J O U R N A l . OF P E D I A T R I C t
a n d go were the respective principal c o m p o nents. W e have the following r e s u h s : - -
Age 8 years ^
02~176 3
Age 5 years A
?~t " .t5'4, ~1 =~ 14"505;
"A~ = 2"515
the principal c o m p o n e n t is "98 x + '14 y
i = ]- 26"41
L
"A1 = 30'95
the principal c o m p o n e n t is 979 x + "61 y '61 x + "79 y
Age 9 ),ear~
Age 6 years 9"69 ] 10"64 ~2 = 6"05
Z2 = 27"2
_-[58o ~,~ ~ 81"4,
43"11 ~2 =~ 19'7
]
the principal c o m p o n e n t is 78 x q- "63 y
the principal c o m p o n e n t is 90 x + "48 y
Age 7 years
Results and Discussgons
.o45"2] 1 ;i x = 39"09
~2 = 4'27
the principal c o m p o n e n t is "86 x + "50 y
' C a b l e 1.
Age g r o u p
T h e principal c o m p o n e n t s o f h e i g h l a n d weight o f children in different agw groups are given in T a b l e 1. Here represents the height and y the weight the children.
Principal components of height and weight in different age groups.
Principal c o m p o n e n t
5 years
0.98 x + 0.14 y
6
,)
0.90 x + 0.48 y
7
,,
0.86 x § 0.50 v
8
),
0.79 x .~- 0.61 y
8
,,
0.61 x-~- 0.79 y
9
),
0.78 x nt- 0.63 y
0HAUIIAIIIA~CORRF,LATI~,S OF AGE WITII IIF;IOIIT AND WEIOIt't" OF GIIIL1)REN
Now s~tppose we take the natural growth of the child in one year to be unity. 'rites from the above analysis we can estllllate the respective shares ,,f height and weight in the natural growth. Thus at 5 yearsofsge a larger share of the child's IlltlUra| growth is spent in increasing the height of the child while at 9 years of age a larger share o'f the natural growth is spent in Increasing the Weight of the child. Thus It can be concluded that in the early llfe growth in height is more pron~inent but as the time passes, growth in weight becomes tnore and more prominent. Thus it has been shown that the d a t a gould.have arisen from the action of formulatlom shown in Table 1. It does not follow that they did, but possibly there are other hypotheses that would account for them, Nevertheless, two important things are apparent. First, the formulation makes
51
some biological sense. For example, the relation in the 5-year-age group is equivalent to 0.9 x + 0.1 y showing that the effect of natural growth in increasing tile weight of the child i~ almost nil in comparison to the increase in height of the child. Similar inteipretations can be drawn from other formulations too. There is a peculiarity in the 8-year-age group. Here two relations turn up a g a i n and again. This does not prove that they have any biological significance but can not be dismissed as chance and need further study. Thus on the whole it can be concluded that t h e existence of these formulations, whatever may be the reasons for them, is a fact of experiment. They provide a description of the phenomenon, though not necessarily an explanation of it.
