Between 1982 and 1988 a growth study was carried out at the Division of Pediatric Oncology of the University Hospital of Groningen. A special feature of the project was that sample sizes are small and that ages at entry may be very different. In addition the intended design was not fully complied with. This paper highlights some aspects of the statistical analysis which is based on (1) reference scores, (2) statistical procedures allowing for an irregular pattern of measurement times caused by missing data and shifted measurement times

Dzhaparidze K. O.

Gill R. D.

Prahl-Andersen B.

Rao C. R.

English

Poortema, Klaas

Tamminga, R.Y.J.

Schaafsma, W.

NARCIS 

Estimation in a growth study with irregular measurement times

University of Twente Research Information

89 Statistica Neerlandica (1993) Vol. 47, nr. 2, pp. 89-95 Estimation in a growth study with irregular measurement times K. Poortema Faculty of Applied Mathematics, University of Twente, P.O. Box 27 7, 7500 A€ Enschede, The Netherlands R. Y. J. Tarnminga Division of Pediatric Oncology, University Hospital of Groningen, ' , P.O. Box 30.001, 9700 RB Groningen, The Netherlands W. Schaafsma Department of Mathematics and Computing Science, University of Groningen, P.O. Box 800, 9700 A V  Groningen, The Netherlands Between 1982 and 1988 a growth study was carried out at the Division of Pediatric Oncology of the University Hospital of Groningen. A special feature of the project was that sample sizes are small and that ages at entry may be very different. In addition the intended design was not fully complied with. This paper highlights some aspects of the statistical analysis which is based on (1) reference scores, (2) statistical procedures allowing for an irregular pattern of measurement times caused by missing data and shifted measure- ment times. Key Words 8, Phrases: growth curves, expectations and variances with linear structure, iterative generalized least squares. 1 Introduction About 1980 N. M. Drayer (Division of Pediatric Endocrinology, University Hospital of Groningen) and J. A. de Vries (Division of Pediatric Oncology, University Hospital of Groningen) initiated the growth study to be discussed in this paper. To which extent growth of children may be disturbed by cancer and treatment was the subject of interest. Two subprojects were distinguished. Subproject A concerned growth retarda- tion during treatment and subproject B concerned catch-up growth after completion of the treatment. The idea was that systematic and careful collection of a number of measurements would reveal specific growth patterns. With respect to anthropometric measurements, W. J. Gerver and N. M. Drayer were carrying out a project aimed at the assessment of means, standard deviations, and percentiles of the population of healthy children in the region where the hospital is located, see GERVER (1988). The statisti- cians J. c. Akkerboom and W. Schaafsma participated in the discussions about the design of the growth study. After many discussions i t  was decided: (i) to restrict atten- tion to the anthropometric measurements investigated in the Gerver-Drayer project, 90 K. Poortema, R. Y. J. Tamminga and W. Schaafsma (ii) to measure the children 9 times with exactly 3 months between the different sessions, (iii) to refrain from measuring healthy children (It was argued that cross- sectional aspects could be inferred from the Gerver-Drayer data while longitudinal aspects could be derived from cohort analyses by VAN ’T HOF (1977) based on the Nijmegen growth study, cf. PRAHL-ANDERSEN et al. (1979)). Patient entry started in March (982 and lasted till April 1986. A special feature of the project is that sample sizes are small and that ages at entry may be very different. In order to account for differences in age and sex, the original .. scores are replaced by reference scores. If e.g. h ( t )  is a height measurement of a boy (girl) at age t then, instead of h(r),  the score x = (h( t )  - p( t ) ) /u( t )  is studied, where p ( t )  and o(t)  are (estimates of) respectively the mean height and the standard deviation of height, of boys (girls) at age t. The p ( t )  and a(t) are computed on  the basis of the data of GERVER (1988). Unfortunately, but not unexpectedly, the intended design was not fully complied with. The height measurements of the entire group of subproject B ( n  = 21) will be studied in Section 3. For this group 12% of the measurements are missing and many of the remaining measurements are carried out at shifted time points, other groups are similar in this respect. For this reason the statistical procedures to be constructed allow for measurements made at p ( i )  time points t f l .  ..., t , ,p ( f )  for child i ,  where the sets { f f  I ,  ..., may be fairly different (see the conditions ofsection 3). The time points t ,  are regarded as given numbers. One should be certain, however, that the sets ( I i I ,  ..., t I p ( , ) }  are not related to the vectors B,. The unit of time is taken to be one year and time is rescaled such that for each child ‘Lt = 0” lies at the centre of the intended two years’ period of observation. Consider some anthropometric variable and some (sub)group of patients. Let n denote the number of children and let xy represent the j - t h  reference score of child i ( i  = 1, ..., n ;  j = 1, . . . , p (  i)). We consider the xi, the outcomes of random variables (1) for some integer q, where the vectors B, = ( B l l ,  ..., B,4)T are independent and identically distributed, and where the E,, are independent of the  B,, independent of each other, and identically distributed with expectation zero. Xu= B , ,  + B,zt,, + ... + Bl4t:-’ + E,, 2 Estimation Let P = (PI,  ..., @ q ) r  and A = ( A J A )  denote respectively the expectation and the co- variance matrix of the vectors B,. Define u2 as the variance of the disturbances E,,. Assuming normality, the distribution of Xi  = ( X I  I ,  . . . ,X f ,p ( f ) ) r  given the observation pattern f , , ,  ..., is determined completely. It may be expressed concisely as follows: x,- N ( A , P , A , A A T  + ~ 2 [ p ( i ) )  (2) where A ,  is the p ( i )  x (I matrix with (j,k)-element t t - ’ .  The  covariance structure A,AAT + 02f,,(,) stems from RAO (1965). The case of identical distributions ( A ,  = A  for A growth study with irregular measurement times 91 all i: the original design) admits relatively simple estimation theory, see e.g. POORTEMA (1989). The differences between the design matrices A ,  were due to missing data and shifted measuring times. If there were no shifted measuring times the matrices A; could be constructed by deleting rows from the original design matrix A .  Then an approach like that of GILL (1988) and KLEINBAUM (1973), who deal however with other models, were feasible. Due to the shifted measuring times our approach resembles somewhat the approaches of VONESH and CARTER (1987) and Hut (1984). The estimator of p (the parameters of main interest) is a weighted mean of the individual least squares predictors/estimators of the B,. For the weights estimators of A and a2 are needed. We apply an iteration procedure which might be called iterative generalized least squares. GOLDSTEIN (1986) applies the same method to multilevel data. ANDERSON (1973) is the first author to give some version of the method. The method requires no numerically intensive calculations and in case of convergence a solution ofthe maximum likelihood equations is obtained. ANDERSON (1973) proposed an iteration procedure for estimation with respect to independent stochastic vectors X i -  N ( x s  Psas ,  zr y fGf ) ,  where the /I, and the y f  are unknown parameters and the (I, (Gr) are known vectors (matrices). This procedure can be generalized to independent stochastic vectors XI- N ( C s b s a ; s , C f y f G f f )  (3) where the known vectors a;, and matrices Gif may depend on i. Model (2) can be written in this form. Because of A = CA,,ejeif + x , < k A j k ( e j e l  + eke:) where ej is the j - th  unit vector of Iw4 we can write AiAA' + u2Z,,(;) = x r  ?(GI ,  with y = ( y I  ,..., y m ) T = ( A 1 1 r A 1 2 , A 2 2  ,..., L , , , O ' ) ~  and G. I f  =(I I J  . ( IT IJ if y ,  = Aj G, = uijark + u i k u c  if y r  = A j k  ( j  < k )  G;f = I p ( i )  if t = rn = q(4 + 1)/2 + 1 where ais  is the s-th column of A i .  We thus get (3) as model equivalent to (2). The generalized iteration procedure is based on the following equations z k  x i  (I;rrl(I;kbk = 1; (Ip,%; (4a) Ek x i  tr~:'GiiI';1Gjkyk/2 = . xi trrF'GorF1(xi - xsPsa i s ) (~ i  - xsPsa is ) ' / 2  (4b) with ri = If y,Gjr. These equations can be derived by equating the derivatives of the log-likelihood with zero. Note that (4b) admits a simplification as x k  T;IGikyk = IP(;), it is however convenient to base an iteration procedure on (4b). The iteration procedure is as follows. Each iteration step consists ofsolving the equations (4) forb = (PI, ..., P I ) T  92 K. Poortema, R. Y. J. Tamminga and W. Schaafsma and y = (y l ,  ..., ym)T  with f i  replaced by Ti = 2, f,G, where y’= (f,, ..., yl,)‘ is the esti- mate of the previous iteration step. We start the iteration procedure with an initial guess y(O). If p ( i )  = p  and f i  = f for all i then the original procedure Of ANDERSON (1973) is retained, this iteration procedure may be started with any positive definite guess f o  for f. The asymptotic properties of the modified iteration procedure, which have been established by POORTEMA (1989), resemble the properties given by ANDERSON (1973). For establishing these properties some or all of the following conditions are assumed: The numbersp(i) and the elements of the a ,  and the G,, are all smaller than some constant c‘ The eigenvalues of the matrices f, = 2, y,G,, and 2, yjo’G,, are all larger than some constant > 0. The matrix x l , 4 J A , ,  with A ,  = [ a f l  ... ATA,/n 2 dl  I4 for sufficiently large n for some constant dl. The matrix 1, D f D , ,  with D ,  = [ve~[G,~]  ... vec[G,,,,]], is invertible and X I  D J D ,  / n  2 LI,I,,, for sufficiently large n for some constant d2.  All fourth moments of the X, are uniformly bounded. If F, is the distribution function of 11 X, 11 then there exists a distribution function F such that ~ , , , , , t 2 d F , ( t ) ~ ~ I , l , e 1 2 d F ( t )  holds for arbitrary i and c>O. The distributions of the X, are normal. is invertible and Here we used the vec-operation: for a matrix 2 with rows z ,  ( i  = 1, ..., k) we have vec[Z] = (zl, z2, ..., zk)‘ .  Note that conditions (i) and (vii) together imply conditions (v) and (vi). The estimators p  ̂ and fa re  n1’2-consistent (n”2(/? - 0) and n1’2(f - y )  are bounded in probability) after one iteration step, if the first five conditions are satisfied. If condition (vi) is added then i t  can be established that p^ is A N @ ,  ( ~ i A i f ; l A i ) - l )  ifat  least two iteration steps are carried out. If finally condition (vii) is added then the asymptotic normality of (@*, fT)T can be derived and the estimators are asymptotically efficient, provided (again) that at least two iteration steps are carried out. Note that one iteration step suffices to obtain consistent estimators. Note moreover that using these consistent estimators asymptotically efficient estimators are constructed, this is in line with other general theory, see e.g. DZHAPARIDZE (1983). 3 Results With respect to starting the iteration procedure there were some difficulties. The produced estimates of A need not be positive definite matrices. Since the variances var (Bi3)  are very small sometimes a negative estimate ll3 is obtained. We firstly tried the initial guess ‘A = 0 and ci2 = 1’. We secondly tried the initial guess ‘i,, = I, ,i,k = 0, and 6’ = 1’. Assuming that the initial guess should resemble at least a little the co- variance structure studied we finally used estimates with respect to the spanwidth measurements of the entire group of Subproject B as initial guess for starting up the iteration procedure for whatever group and variable. With this initial guess (which is indeed highly arbitrary) no negative estimates for variances were found. A growth study with irregular measurement times 93 Beforehand it was expected that the reference scores would follow straight lines (q = 2). After inspection of the data it was clear that q = 3 (parabolic growth curves) provides a better fit. Whether q = 3 fits the data satisfactorily should be tested. The model specified by q = 4 (cubic growth curves) provides an alternative. In view of the rather small sample sizes we did not consider more complicated models. The model q = 3 has been tested with the model q = 4 as alternative. We present results with respect to the height measurements of the entire group ( n  = 21) of subproject B. The analyses of other variables and other groups can be carried out analogously. After testing q = 3, see the appendix, we postulated q = 3, the esti- mates of the (first) three iteration steps ofthe estimation procedure are given by Table 1 and 2. Table 1. Estimates p^* for q = 3 ~~ - ~~ ~~~ ~~ iteration 8, A 0 3  1 -.6425 .1544 -.0333 2 -.6380 .1535 -.0417 3 -.6383 .1535 -.0412 stand.error .2374 ,0458 .0255 The comparison of growth velocities is a natural point of interest. We therefore focus on the estimation of the expectation p2 of the coefficients B,2, the growth rates at r = 0. A rather simple estimator of p2 proves to be as good as the corresponding estimator of the iteration procedure. Note that the least squares estimator Bl2 of B12 is N ( P 2 , A Z 2  + o2cfz)-distributed where c,2 is the (2,2)-element of the matrix (A:A,)-'. If the individuals do not differ much with respect to the constants c f 2 ,  an appropriate esti- mator of pz is X I  B12/n;  as standard error may serve s/n"2, where s is the sample standard deviation of the B,2.  We get ,1507 f .0481; the standard error is not much larger than the corresponding standard error in Table 1. The outcome of 2, B 1 2 / n  is close to the asymptotically best estimate of the iteration procedure (the estimate /? is as a matter of fact a weighted mean); this is in agreement with the results with respect to the other variables and groups. As the estimation of p2 is of primary interest, the mean B,2/n is preferred by TAMMINGA (1990) who gives a thorough investigation ofall data. Table 2. Estimates ljA and 6' for q=3 iteration 1, I &> 1 2 2  113 123 1 3 ,  8' 1 1.1742 -.0094 .0421 -.0082 -.0132 .0072 .0068 2 ' 1.1814 -.0113 .0419 -.0160 -.0133 .0075 ,0069 3 1.1810 -.0113 .0421 -.0157 -.0133 .0075 ,0069 Using the estimates of the variances iJJ and the covariances i,*, correlations can be estimated as well by means of the iteration procedure. An interesting parameter is the correlation between B,z ,  the speed of growth at f = 0, and B, , ,  the level at r = 0; we found -.0509 f .2231. Another interesting correlation is the correlation between BJ2 94 K. Poorterna, R. Y. J. Tarnrninga and W. Schaafsrna and B,, - Bi2 + Bi3, the level at t = - 1 (the start of the study); here we found -.2909 f .2048. The  standard errors are computed using the delta method. No relation between Bi2 and the levels B I I  and Bil - Bi2 + B,, can be established. 4 Final remark W e  have been working with the actual times t i I ,  ..., t r ,p( l )  for each individual i. Note that beforehand no interpolations or  other data modifications have been applied to cope with the deviations from the original design. The  general iteration procedure for esti- mation turns out to work well. The parameter of main interest (p2),  however, can be estimated about equally well by using a much simpler estimator: our general approach seems to be too sophisticated in this respect. APPENDIX For  discussing the model 4 = 3 our starting point was the model 4 = 4: B f l  + B 1 2 f r )  + B f 3 t : ]  + B14t jJ  + EU W e  first tested the null hypothesis Ho : ,14J = 0 against the alternative hypothesis HI : ,lq4 # 0. The test statistic has been based on the least squares estimators/predictors BI4 of the B14; these are independent and BI4 is N u 4 ,  Ad4 + cr2cI4)-distributed where c14 is the (4,4)-element of the matrix (ATAl ) - l .  The test statistic used is T = S2/G2  with s2 = 21 c21(B~4 - 8 4 ) 2 / ( n  - l), 8 4  = 1 1  c > 1 B f 4 / 1 f  '>', dZ=2:, (xfJ-BfI - B 1 ~ t f J - B l ~ t ~ , - B f 4 t ~ ) 2 / f 1  wheref = El ( p ( r )  - 4). Under Ho (n - 1)S2 has the a2~: -1  distribution. The  statistics S2 and d2 are independent and it follows that T = S2/B2 has the F n - I J  distribution under H,, sincefd2 has the cr'x,' distribution. Note that the test statistic reduces to a ratio of variances if the constants cI4 don't differ. The hypothesis Ho need not be rejec- ted at level .05; the outcome of F equals 1.07 while n - 1 = 20 and f = 83. We hence postulate ,I4,, = 0, we may set equivalently 814 3 ... Bn4 p,, and test, in addition, the null hypothesis Ho: p4 = 0 against the alternative hypothesis H I  : p4 # 0. The iteration procedure for estimation was used again for re-estimating the remaining parameters; we found p4 = .0212 with standard error .0282. We hence need not reject the second null hypothesis and postulate the model with 4 = 3. A growth study with irregular measurement times 95 References ANDERSON, T. W. (1973), Asymptotically efficient estimation of covariance matrices with linear structure, The Annals of Statistics 1, 135-141. DZHAPARIDZE, K. 0. (1983). On iterative procedures of asymptotic inference, Statisticu Neer- landica 37, 181-189. GERVER, W. J. M. (1988), Measurement of the body proportions in children, The Oosterwolde study, Thesis, University of Groningen. GILL, R. D. (1988), Regression analysis for incomplete mixed cross-section and time-series data by a modified EM algorithm, Sankhyc? 50, 95-102. GOLDSTEIN, H. (1986), Multilevel mixed linear model analysis using iterative generalized least squares, Biomerrika 73, 43-56. HOF, M. A. VAN 'T (1977), Some statistical and methological aspects in the study.of growth and development: with a special emphasis on mixed longitudinal designs, Thesis, Nijmegen Catholic University. HUI, S .  L. (1984), Curve fitting for repeated measurements made at irregular time-points, Biometrics 40, 69 1-697. KLEINBAUM, D. G.  (1973), A generalization of the growth curve model which allows missing data, Journal of Multivariate Analysis 3, 117-124. POORTEMA, K. (l989), On the statistical analysis of growth, Thesis, University of Groningen. PRAHL-ANDERSEN, B., c. J.  KOWALSKI and P. HEYDENDAEL (1979), A mixed-longitudinal inter- disciplinaly study of growth and development, Academic Press, New York. tion to the analysis of growth curves, Biometriku 52, 447-458. RAO, C. R. (1965). The theory of least squares when the parameters are stochastic and its applica- TAMMINGA, R. Y. J. (1990), Growth of children with cancer, Thesis, University of Groningen. VONESH, E. F. and R. L. CARTER (1987), Efficient inference for random-coefficient growth curve models with unbalanced data. Biometrics 43, 617-628. Received: May 1990. Revised: January 1992. 

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Estimation in a growth study with irregular measurement times

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