Least Squares Estimation of Regression

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Unmeasured or omitted covariates in such models may be correlated with the ... Omitted covariates in regression analysis of longitudinal data can be related to ...
Least Squares Estimation of Regression Parameters in Mixed E ects Models with Unmeasured Covariates J. Shao , M. Palta, and R.P. Qu University of Wisconsin-Madison 1

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2

SUMMARY We consider mixed e ects models for longitudinal, repeated measures or clustered data. Unmeasured or omitted covariates in such models may be correlated with the included covariates, and create model violations when not taken into account. Previous research and experience with longitudinal data sets suggest a general form of model which should be considered when omitted covariates are likely, such as in observational studies. We derive the marginal model between the response variable and included covariates, and consider model tting using the ordinary and weighted least squares methods, which require simple noniterative computation and no assumptions on the distribution of random covariates or error terms. Asymptotic properties of the least squares estimators are also discussed. The results shed light on the structure of least squares estimators in mixed e ects models, and provide large sample procedures for statistical inference and prediction based on the marginal model. We present an example of the relationship between uid intake and output in very low birth weight infants, where the model is found to have the assumed structure.

KEY WORDS: Omitted covariates, repeated measures, random e ects, longitudinal data, misspeci cation, marginal models.

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Work supported by grant R01 CA-53786 from the National Cancer Institute. Work supported by grant CA-09565 from the National Cancer Institute.

1. INTRODUCTION Omitted covariates in regression analysis of longitudinal data can be related to either follow-up time or to the characteristics of individuals who enter the study. For example, a recent report (Neuscha er, Bush and Hale, 1992) showed that in a study of cholesterol levels in the elderly, there was both evidence that persons born in earlier calendar years had lower cholesterol and that cholesterol levels had risen with the calendar time at which the measurement was obtained. These trends may mark the presence of unknown individual level factors, such as childhood diet, that may cause generational di erences. Time related trends might be speculated to have arisen from population wide unmeasured covariates, such as availability of pre-processed, fat-rich foods. From an epidemiologic perspective, such factors confound regression estimators of aging e ect with commonly used models, and are well known under the terms cohort and period e ects. In other situations, such as in the example of the relationship between uid intake and output in very premature newborns given later in this paper, included covariates may simultaneously be markers for several unmeasured covariates. The weight of a neonate is a marker for body surface area, which in turn is positively related to a component of uid loss not measured. Although this component can be expected to be a major determinant of why neonates di er in measured uid output, short term day to day variation in weight may be a marker for temporary uid retention or diuresis. Thus, again omitted covariates are present, which have di erent relationships with in uential unmeasured covariates within and between individuals. From a statistical point of view, omitted covariates create violations of assumptions with commonly used methods in regression analysis. There is a rich literature on model misspeci cation due to omitted covariates. Unless one has some additional information (such as observed instrumental variables; see Bowden and Turkington, 1984), some parameters in the regression model cannot be estimated because of the presence of omitted covariates. In this paper we focus on tting the \marginal" model between the response variable and included covariates, using a model between the response variable and all covariates and a model between the omitted and included covariates. The tted marginal model can be used for statistical inference (such as to test whether there are omitted covariates) and prediction of future response values. 2

Although omitted covariates can be treated as random e ects, there is an important di erence between our approach and ordinary mixed e ects modeling: it is usually implicitly assumed in mixed e ects regression models that the distribution of the random e ects does not depend on the xed covariates. Palta and Yao (1991) and Palta and Qu (1995) demonstrated that omitted covariates are likely related to included covariates. Essentially this occurs when omitted and included covariates are correlated, but di erentially so longitudinally and cross-sectionally. Palta, Yao and Velu (1994) give several examples of such situations, derive models which may arise when conditioning only on observed covariates, and develop procedures for testing for omitted covariates. In related work, Wishart (1938) and Mundlak (1978) both addressed the implications of the independence assumption in the mixed e ect model with a random intercept, as did Dwyer et al. (1992), pages 27-33. Palta and Yao (1991) derived the correct form of the marginal model under a normality assumption on the response variable, the omitted and included covariates with a certain covariance structure. In the present investigation we generalize this form and develop procedures for tting such marginal models that make no distributional assumptions. Our regression model is particularly relevant when omitted covariates may be present, but may also arise from other circumstances. In Section 2 we give speci c forms of marginal models by studying models between the included and omitted covariates. We then develop tting procedures based on least squares and weighted least squares methods which, in contrast to likelihood based procedures, do not require distributional assumptions on the responses (Section 3). The resulting estimator generalizes a results of Hsiao (1986) and provides insight into the construction of estimators from mixed e ect models. We also derive asymptotic results which are useful for large sample statistical inference and prediction based on the marginal model. The weighted least squares method is shown to be asymptotically ecient when the covariance matrix of the response is correctly speci ed. In Section 4, we apply the method to the data set of uid output and intake in very low birth weight neonates.

2. MODELS Consider independent random vectors y ; :::; yn, where for each i, 1

y

i

= 1 + X i + Z i + ei 3

(2.1)

is a ki  1 response vector, 1 is a vector of one's (of an appropriate order), X i and Z i are ki  p and ki  q matrices of values of random covariates, respectively, , and are parameters or parameter vectors of appropriate orders, and ei is a ki  1 random error vector satisfying E (eijX i; Z i) = 0 and V (eijX i; Z i) = i (which may depend on X i and/or Z i). Note that model (2.1) is a mixed e ects model; for example, ei = X i( i ? ) + e~i , where the i are unobserved random e ects having mean , the e~i are random errors having mean 0 and V (~ei) = e I , I denotes the identity matrix (of an appropriate order), and the i and e~i are independent. In such a case i = e I + X iV ( i)X 0i. 2

2

Suppose that the Z i are values of q unmeasured covariates. Taking the average over Z i, we obtain the following marginal model: E (yi jX i ) = E [E (y i jX i ; Z i )jX i ] = 1 + X i + E (Z i jX i)

(2.2)

V (yi jX i ) = E [V (yi jX i ; Z i )jX i ] + V [E (y i jX i ; Z i )jX i ] = E (ijX i) + V (Z i jX i):

(2.3)

and

We cannot treat the omitted covariates Z i as unobserved random e ects and apply the methods for mixed e ects models, because the random e ects in mixed e ects models are usually assumed independent of the covariates X i , whereas the omitted covariates Z i are related to X i so that E (Z i jX i ) is a function of X i (see Examples 1-2).

We need a model that relates X i and Z i. The simplest model is the following linear model: E (Z i jX i ) = Ai (X i ); (2.4)

where Ai (X i) is a ki  s matrix whose elements are functions of X i, and  is an s  1 vector of unknown parameters. Two examples are given as follows.

Example 1. Consider the case of p = q = 1 and k  k. If (X ; Z ) is multivariate normal i

as described in Palta and Yao (1991), i.e., xij zij

!

=

!

x z

i

i

4

+

i

!

ex ez

ij

ij

;

i

where xij is the j th component of X i, zij is the j th component of Z i, !

x z

i

i

D

1

=

2 1

1

 N (0; D ) ;

then (2.4) holds with

ij

1

p

r 1 c1 p r1 c1 c1

!

!

ex ez

D

and

 N (0; D ) ; 2

ij

=

2

2 2

p

1

r 2 c2 p r2 c2 c2

!

;

A (X ) = (X ; x 1) i

i

i

and

i

p

!

 = (r1prc21 ?c2r2pc2) ; where xi is the average of the components of X i and  = 12=(12 + 22=k). In this case, s = 2 and

p

V (Z i jX i ) = (1 ? r22 )22 c2 I + [(1 ? r12 )c1 12 + (c2 r22 ? 2r1 r2 c1 c2 )22 =k]110 ;

which does not depend on X i. As described in Palta and Yao (1991), r is a measure of \between-individual confounding" and r is a measure of \within-individual confounding". 1

2

Example 2. Suppose that

Z = A (X ) + u ; i

i

i

i

(2.5)

where E (uijX i) = 0. That is, there is a linear model that describes the relationship between X i and Z i . Note that Example 1 is a special case of Example 2. No assumption on the joint distribution of X i and Z i is made.

The use of two equations (2.1) and (2.5) is notationwise very similar to the method of instrumental variables commonly used in econometrics (Bowden and Turkington, 1984). To apply the method of instrumental variables, one needs some additional observed covariates (called instrumental variables) uncorrelated with Z i + ei in (2.1) so that the \original" parameters and in (2.1) are estimable. In our approach, however, there may not exist instrumental variables (in the general case, Ai(X i) in (2.4) or (2.5) cannot be used as instrumental variables since it may be related to Z i + ei), and the parameters and in (2.1) may not be estimable. Model (2.4) or (2.5) is used to derive a marginal model between yi and X i which can be used for statistical inference and prediction. 5

From (2.4), the marginal model (2.2) becomes E (y i jX i ) = 1 + X i + Ai (X i ):

(2.6)

The matrix Ai (X i) may contain 1 and/or some columns of X i (see Example 1). Let A~ i(X i) be the submatrix of Ai(X i) containing columns that are not in (1; X i). Then (2.6) can be written as ~ i(X i) ; E (y i jX i ) = ~ 1 + X i ~ + A (2.7) where ~ is an unknown parameter and ~ and ~ are unknown parameter vectors. Note that the original parameter or is estimable if and only if Ai(X i) does not contain X i or 1.

The relationship between X i and Z i may be nonlinear. Nonlinear cases will be included in a future study.

3. MODEL FITTING AND TESTING If the distribution of yi jX i is known except for some unknown parameters, then one may apply the maximum likelihood method to t the marginal model (2.2)-(2.3). For example, if both eij(X i; Z i) and Z ijX i are normal, then yijX i is normal. When the type of the distribution of yi jX i is unknown, the least squares method can be used to t and test model (2.2)-(2.3) when E (Z i jX i) and V (Z i jX i) are of certain forms. There is a rich literature on the least squares method, either weighted or unweighted; see, for example, Searle (1971), Rao (1973), Judge et al. (1982), and Dielman (1995). In this section we apply this method to the marginal models derived in Section 2 and discuss some properties and results that are useful for statistical inference and prediction. We start with the ordinary least squares (OLS) method, which does not require any assumption on the covariance matrix V (yijX i) in (2.3).

3.1. The OLS Method De ne





W = 1; X ; A~ (X ) : Assume that the matrix W = (W 0 ; W 0 ; :::; W 0 )0 is of full rank. (~ ; ~ 0 ;  0 )0 in the marginal model (2.7) is ^ = (W 0W )? W 0y; i

1

i

i

i

n

2

1

OLS

6

Then the OLSE of  =

where y = (y0 ; :::; y0n)0. The OLSE ^ OLS is a weighted average of the individual OLSE's of the following models: 1

E (y i jX i ) = W i ;

(3.1)

i = 1; :::; n. Note that the W i may not be of full rank, although W is of full rank. Assume that the rank of W i is r. Let Ri be the matrix containing r independent columns of W i.

Then

W

= Ri Qi

(3.2)

= (1; X i; xi 1; :::; xip1);

(3.3)

for some matrix Qi. For example, if

W

i

i

1

where xij is the average of the j th column of X i (e.g., Example 1), then

R

i

= (1; X i)k  p i

( +1)

and

Q

i

I

=

xi1

(p+1)

0

(p+1)

p 1

   xip    0p

1

!

:

Let ^i be the OLSE under model (3.1), i.e., ^i = (R0iRi)? R0iyi. Then ^i is unbiased for i = Qi and 1

^

OLS

= (W 0W )?

1

= (W 0W )?

1

= (W 0W )?

1

n X i=1 n

X i=1 n

X i=1

W 0y i

i

Q0 R0 y i

i

i

Q0 (R0 R )^ : i

i

i

i

(3.4)

It is clear that ^ OLS is unbiased for . Assuming that C1. supi ki < 1, C2. inf i E (W 0iW i) and inf i E [W 0i V (yijX i)W i ] are positive de nite, C3. supi E kW 0iW ik  < 1 and supi E kW 0i V (yijX i)W i k  < 1 for some  > 0, we can establish the following result by directly applying the central limit theorem and the law of large numbers: V ?OLS= (^ OLS ? ) ! N (0; I ) in law as n ! 1, where V OLS = (W 0W )? W 0W (W 0W )? 1+

1+

1 2

1

7

1

and  is the covariance matrix of y, a block diagonal matrix whose ith block is V (yijX i). A consistent estimator of V OLS is

V^

OLS

= (W 0 W )? W 0^ W (W 0 W )? ; 1

(3.5)

1

where ^ is a block diagonal matrix whose ith diagonal block is rir0i with

r = y ? W ^ i

Hence

V^ ? (^ 1=2

OLS

OLS

i

i

OLS

(3.6)

:

? ) ! N (0; I ) in law:

(3.7)

Result (3.7) is useful for statistical inference and prediction. An approximate level a test for hypothesis H : C = 0 versus H : C 6= 0 (3.8) 0

rejects H if 0

1

^0 C 0[C V^ OLS

OLS

C 0]? C ^ 1

OLS

> 2a (d);

(3.9)

where C is a matrix of constants, d is the row-dimension of C , and a (d) is the 1 ? a percentile of the chi-square distribution with d degrees of freedom. For example, if W i is given by (3.3) and  = 0 (which means that the unmeasured covariates have no signi cant e ects) is to be tested, then   C = 0p p ; I pp : 2

( +1)

3.2. The WLS Method Under linear model (2.7), the eciency of the OLSE is lower than that of the best linear unbiased estimator (BLUE) of . If the V (yi jX i ) are known, then the BLUE is of the form

^ = (W 0? W )? W 0? y: Since the diagonal blocks V (y jX ) in  are unknown, replacing them by estimates V^ (y jX ) BLU

1

1

1

i

i

i

leads to the weighted least squares estimator (WLSE)

^

W LS

= (W 0GW )? W 0 Gy; 1

where G is a block diagonal matrix whose ith diagonal block is Gi = [V^ (yijX i)]? . 1

8

i

We cannot use G = ^ ? , where ^ is given by (3.5) and (3.6), since the resulting WLSE is very inecient (Chen and Shao, 1993). The main problem is that rir0i is not a consistent estimator of V (yijX i) for every xed i, although V^ OLS is consistent for V OLS . 1

Construction of consistent estimators of V (yijX i) depends on the structure of the covariance matrix V (yijX i), which can be modeled using several approaches; for example, the mixed e ect regression model (Laird and Ware, 1982; Palta and Yao, 1991) and the time series and cross-sectional model (Judge et al., 1982; Dielman, 1995). These approaches may provide very similar estimators of V (yijX i). Under normality assumptions, one can also obtain REML estimators (Searle, 1971). We adopt the mixed e ects regression model and assume that V (yi jX i ) =  I + U i DU 0i ; (3.10) where  > 0 is an unknown parameter, D is a nonnegative de nite matrix of unknown parameters, and U i is a matrix containing some columns of Ri de ned in (3.2). This variance structure can be shown to hold under quite general conditions when the variance of model (2.1) is of similar form. In Example 1, for instance, if ei in (2.1) is X i ( i ? ) + e~i with iid ( i; e~i), then V (eijX i; Z i) = e I + X iV ( )X 0i and 2

2

2

1

V (yi jX i ) = e2 I + X i V ( 1 )X 0i + (1 ? r22 )22 c2 I p + [(1 ? r12 )c112 + (c2r22 ? 2r1r2 c1c2)22 =k]110 ;

which can be written as (3.10) with U i = (1; X i),  = e + (1 ? r ) c , and 2

2

2 2

2 2 2

p (1 ? r )c  + (c r ? 2r r c c ) =k 0 : D= 0 V ( ) 2 1

1

2 1

2 2 2

1 2

2 2

1 2

!

1

Under (3.10), we may estimate V (yi jX i ) by estimating  and D. An unbiased estimator of  is 2

2

^

2

= =

since, conditional on X i's, E (^ ) = 2

n X i=1

tr

X n

n X i=1 n

(yi ? Ri^i)0(yi ? Ri^i)

X 0 [ i=1

y I ? R (R0 R )? R0 ]y i

i

i

1

i

i

X i

n

o [I ? R (R0 R )?1 R0 ]E (y y0 ) i

i

i

i

9

i

(ki ? r)

i=1 n

i

i=1

(ki ? r);

X n i=1

(ki ? r)

(3.11)

= =

n X i=1 n

X i=1

n

tr [I ? Ri(R0iRi)? R0i][ I + U iDU 0i] tr

1

2

o [I ? R (R0 R )?1 R0 ]2

n

i

i

i

X n

i

i=1

= :

n oX i=1

(ki ? r)

(ki ? r)

2

This estimator is also consistent if Pi(ki ? r) ! 1 as n ! 1.

An asymptotically unbiased and consistent estimator of D is n X 1 ^ D = (U 0iU i)? U 0irir0iU i(U 0iU i)? ? ^ 1

n i=1

1

2

n

n X i=1

(U 0iU i)? ;

(3.12)

1

where ri is the ith residual (from the OLS tting) given by (3.6). In fact

D^ ? D ! 0

in probability;

where

D = n1 X(U 0 U )? U 0 (y ? W )(y ? W )0U (U 0 U )? ? ^n X(U 0 U )? ; n

i=1

i

1

i

i

i

i

i

i

i

i

i

n

2

1

i=1

1

i

i

and, under (3.1) and (3.10), ) = E (D

n n X 1X (U 0iU i)? U 0i V (yijX i )U i(U 0iU i)? ? E (^ ) (U 0iU i)?

n i=1

= 1

1

n

1

i=1

(U 0iU i)? U 0i [ I + U iDU 0i]U i(U 0iU i)? ? 

n X

n i=1 = D:

2

1

1

2

1

2

n

n X i=1

(U 0iU i)?

1

When U i = Ri , the estimator in (3.12) becomes

D^ = n1 X(^ ? Q ^ n

i

i

i=1

OLS

)(^i ? Qi ^ OLS )0 ? ^

2

n

n X i=1

(R0iRi)? ; 1

In this case, a di erent estimator of D given by Swamy (1970) is

D~ = 1

n  X

n ? 1 i=1

 0 ^i ? ^ ^i ? ^ ? ^

2

n

n X i=1

(R0iRi)? ; 1

~ is neither asymptotically unbiased nor consistent where ^ is the average of the ^i . However, D in general. It is asymptotically unbiased and consistent in some special cases, e.g., when 10

 = 0 (which is often a null hypothesis we would like to test). This is because ~) = E (D =

n 1X 1 V (^i ) +

n i=1

D + n ?1 1

n ? 1 i=1

n X i=1

(i ? )(i ? )0 ? 

n X

2

n

(i ? )(i ? )0;

n X i=1

(R0iRi )?

1

where  is the average of the i, and under  = 0, i  . ^ in (3.12) is not necessarily nonnegative de nite, the resulting estimator of Although D V (yi jX i ), n ^ 0 = 1 X U (U 0 U )? U 0 r r0 U (U 0 U )? U 0 ^ I + U DU (3.13) 2

i

i

1

i i i i n i=1 i i i n 2 X + ^n [I ? U i(U 0i U i)?1U 0i]; i=1

i

i

1

i

is always nonnegative de nite and, hence, it can be used in the WLS tting. W W Let ^i be the WLSE of i under model (3.1), i.e., ^i = (R0iGiRi)? R0iGi yi. Then ^W LS is a weighted average of the ^Wi : 1

^

0 ? W LS = (W GW )

1

n X i=1

Q0 (R0 G R )^ i

i

i

i

W i

:

W If (3.10) holds, then ^i = ^i for all i (Lange and Laird, 1986) and consequently,

^

0 ? W LS = (W GW )

1

n X i=1

Q0 (R0 G R )^ : i

i

i

i

i

(3.14)

Comparing (3.4) with (3.14), we nd that both the OLSE and WLSE are weighted averages of the ^i , but they use di erent weights. This generalizes a result of Hsiao (1986) to the case when covariates are included which vary only between individuals. The WLSE ^ W LS may not be exactly unbiased, but is asymptotically unbiased. If conditions C1-C3 hold and max kG V (yijX i) ? I k ! 0 in probability; in i

1

(3.15)

where kAk =tr(A0 A), then 2

V ? (^ 1=2 W LS

W LS

? ) ! N (0; I ) in law 11

(3.16)

and

V^ ? (^ 1=2 W LS

where

W LS

V

and is a consistent estimator of V W LS .

W LS

V^

? ) ! N (0; I ) in law;

(3.17)

= (W 0 ? W )?

W LS

1

1

= (W 0GW )?

1

^ 0i given by (3.13), (3.15) holds under conWhen V (yijX i) is estimated by ^ I + U iDU dition (3.10) and the following mild moment conditions: (i) supi E (Ri R0i) < 1; (ii) supi E k(R0iRi)? k  < 1 and supi E ki0ik  < 1 for some  > 0, where i = yi ? E (yijX i). Result (3.16) means that ^ W LS is asymptotically as ecient as the BLUE, i.e., there is no asymptotic eciency loss due to estimating the unknown V (yijX i). Result (3.17) is useful for statistical inference and prediction. For example, hypotheses of the form (3.8) can be tested by using (3.9) with ^ OLS and V^ OLS replaced by ^ W LS and V^ W LS , respectively. 2

1

1+

1+

The WLSE is more ecient than the OLSE. But it is sensitive to correct speci cation of the matrix U i in (3.10). Our experience in applying the WLSE to actual data has shown that the WLSE may be very unstable when U i contains close to the the maximum number of columns (i.e., when D is over parameterized). For this reason, the OLS is a valuable tool.

4. AN APPLICATION The following example is taken from information up to four days of life in a data set of 756 very low birth weight (< 1501g) neonates hospitalized in seven neonatal intensive care units in Wisconsin and Iowa (Palta et al., 1991). ki = 4 repeated measurements on uid output in ml=kg=day, uid intake in ml=kg=day and infant's daily weight in grams were obtained on 714 infants but ki = 3 repeated measurements were obtained on 42 infants. We regress

uid output on uid intake and the infant's daily weight. It is known that smaller infants lose more uid through the body surface as the body surface area is larger relative to weight than for larger infants. The relationship of surface area to volume led us to use weight raised 12

to the 2/3 power in the regression. Initial regression of output on intake and body weight = yielded signi cant coecients for both by ordinary and weighted least squares including a random e ect for the intercept (p < :00001). The regression coecients with their standard errors are given below for the ordinary and weighted least squares procedures. OLS WLS Intercept -54.372 -57.466 Fluid intake (ml=kgl=day) 0.509 (0.020) 0.480 (0.014) (Weight in grams) = 0.862 (0.054) 0.939 (0.051) The within individual variation was estimated by (3.11) as 1363 and the between infant variation in intercept at 4.5. Various residual plots (not shown) indicated no violation of linearity assumptions in this model. The covariance assumption for the WLS method is given by (3.10) with U i = 1, i.e., there are random intercepts but no random slopes. It was not possible to add random slopes to the WLS model without creating oversaturation. 2 3

2 3

Comparison of the results in the above table indicates that WLS and OLS gave very similar results and that there was a small gain in standard error of the uid intake coecient when using WLS. There was little change in the standard error of the coecient of weight = . 2 3

To examine model t, and to allow for the possibility of omitted covariates, we tted the above model with the additional terms mean intake and mean (weight = ) with these means taken across all observations on a neonate, i.e., using formulas (3.4) and (3.14) with 2 3

RQ =W i

i

i

= (1; X i; xi 1; :::; xip1); 1

where xij is the average of the j th column of X i . The covariance assumption for the WLS method is the same as before. The resulting models are given below: OLS WLS Intercept -52.79 -52.24 Fluid intake (ml=kg=day) 0.515 (0.026) 0.506 (0.021) (Weight in grams) = 2.23 (0.226) 2.19 (0.202) Mean uid intake 0.033 (0.0386) 0.041 (0.0296) = Mean weight -1.445 (0.230) -1.406 (0.210) Again, WLS demonstrates a small gain in eciency. It can be shown that in this type of model where only the intercept is associated with a random e ect, errors are assumed independent and where the means of all time dependent covariates are included, WLS is more ecient than OLS only when numbers of observations ki are unequal (Lange and 2 3

2 3

13

Laird, 1986; Weerakkody and Johnson, 1992), which is the case in our example (ki = 3 or 4). The variance components estimated by our method for this model were 893 for the within infant variation and 375 for the between infant variation. It is of interest that the coecient of mean weight = is statistically signi cant and that the coecient of weight = has changed considerably when the mean weight = term is added. It is easy to hypothesize variables which may confound the weight-output relationship. Crosssectionally several treatments such as heating lamps and UV treatment for jaundice increase

uid loss through the skin. However, as these treatments would be most often administered to the smallest neonates they cannot explain the negative sign of the coecient of mean weight = . Instead, it may be suspected that within infant factors related to weight are involved. It is possible, for example, that an upward deviation in daily weight from mean weight may be a marker for uid retention at time of weighing. This may in turn lead to change of diuretic regimen or natural diuresis. The latter is expected as the child's lung function improves in the rst few days of life. Hence there may be an in uence of daily weight additional to that generated by the body surface relationship. 2 3

2 3

2 3

2 3

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