RESEARCH ARTICLE. Growth Rate Models: Emphasizing Growth Rate Analysis through Growth Curve Modeling

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1 RESEARCH ARTICLE Growth Rate Models: Emphasizing Growth Rate Analysis through Growth Curve Modeling Zhiyong Zhang a, John J. McArdle b, and John R. Nesselroade c a University of Notre Dame; b University of Southern California; c University of Virginia (Received 00 Month 200x; in final form 00 Month 200x) To emphasize growth rate analysis, we develop a general method to reparametrize growth curve models to analyze rates of growth for a variety of growth trajectories, such as quadratic and exponential growth. The resulting growth rate models are shown to be related to rotations of growth curves. Estimated conveniently through growth curve modeling techniques, growth rate models have advantages above and beyond traditional growth curve models. The proposed growth rate models are used to analyze longitudinal data from the National Longitudinal Study of Youth (NLSY) on children s mathematics performance scores including covariates of gender and behavioral problems (BPI). Individual differences are found in rates of growth from age 6 to 11. Associations with BPI, gender, and their interaction to rates of growth are found to vary with age. Implications of the models and the findings are discussed. Keywords: Growth curve models, growth rate models, reparameterization and rotation, mathematics ability development 1. Introduction In the past half century, growth curve models have quickly evolved into one of the most powerful tools in facilitating the analysis of change [e.g. 24, 27]. Growth curve models are also one of the few methods that can directly address a primary goal of longitudinal research - to analyze both intra-individual change and inter-individual differences in intraindividual change [1]. Many researchers have shown interest in using growth curve models to analyze rates of growth or change [e.g. 2, 22, 38]. Except for the linear one, growth curve models in their original forms are generally unable to analyze rates of growth directly. For example, in the linear growth curve model, the linear slope is equal to the growth rate. Thus, the rate of growth can be analyzed directly. However, for the other (nonlinear) models, the rate of growth does not appear in the models explicitly and therefore the rate of growth cannot be analyzed directly. The rate of growth is an important concept in studying change. If the level of growth is viewed as the current status of a process at a specific time, the rate of growth measures how fast the process is changing at that time. Individual differences in the rate of growth are of obvious importance. For example, Freeman and Flory [14] pointed out that two persons of the same ability at a given time may differ markedly in their ability at a future time if their rates of growth are different [see also 13, 34]. Cattell [9] also emphasized that even for the same individual, the rate of growth at one level of a response variable can be Correspondence should be addressed to Zhiyong Zhang (ZhiyongZhang@nd.edu), Department of Psychology, 118-B Haggar Hall, University of Notre Dame, Notre Dame, IN

2 2 Zhiyong Zhang, John J. McArdle & John R. Nesselroade very different from that at another level of the variable. Therefore, when studying growth and development, the analysis of rates of growth should provide useful information about the process under investigation. Techniques for analyzing rates of growth are still relatively rare compared to the evolvement of growth curve models. Other than the linear growth curve model, there are only a few important methods for analyzing rates of growth. Sandland and McGilchrist [33] proposed a stochastic growth curve model based on stochastic differential equations. This method focused on the analysis of a single subject. Boker and colleagues developed a set of dynamical systems models based on differential equations and the estimation of derivatives [3]. These models can be used to investigate complex dynamical systems by modeling the relationship among derivatives of a system. McArdle and colleagues also developed latent difference score models for the analysis of change [e.g. 26]. These models can be used as a method to analyze rates of growth with discrete time intervals. Finally, Ramsay and colleagues [e.g. 30] have developed functional data analysis techniques, which can also be used to analyze rates of growth. However, none of the methods has directly investigated inter-individual differences in the rates of growth. In this study, we develop a general method to reparametrize growth curve models so that rates of growth can be analyzed for a variety of growth trajectories, such as quadratic and exponential growth. The merits of this method and the resulting growth rate models include, but not limited to, the following. First, they can be used to analyze rates of growth explicitly. Second, individual differences in rates of growth and their relationship to covariates can be investigated directly. Third, it is a general form for analyzing rates of growth for both linear and nonlinear growth trajectories. Fourth, the model estimation and selection methods of growth curve models are used when analyzing rates of growth. In the following, we review a general form of growth curve models. Then, we will discuss how to reparametrize growth curve models to analyze rates of growth connecting the reparametrization method with the rotation of growth curves. After that, an empirical example is presented to demonstrate how to use the proposed method, and simulation studies are conducted to evaluate the performance of the growth rate models. Finally, implications of the models and the application are discussed. 2. Growth Curve Models In hierarchical modeling terms, a typical growth curve model is a two-level model. The first level involves fitting a growth curve with random coefficients for each individual and the second level models the relations between the random coefficients and covariates of interest. Formally, the first level of a growth curve model can be written as, y it = f(time it, b i )+e it, t =1,...,T, i=1,...,n, (1) where y it is the observed response for individual i at trial or occasion t, time it denotes the measurement time for the tth measurement occasion on the ith individual, f(time it, b i ) is a function of time representing the growth curve with individual coefficients b i =(b i1,b i2,...,b ip ),ap 1 vector of random coefficients, and e it is the residual and the whole vector e i =(e i1,e i2,...,e it ) t is assumed to follow a multivariate normal distribution with the mean vector E(e i )=0 T 1 and the covariance matrix Cov(e i )= T T. Usually, it is assumed that Cov(e i )=I 2 i or even Cov(e i )=I 2 where I is a T T identity matrix. The function f(time it, b i ) can take either a linear or a nonlinear form of time although the current study focuses on the nonlinear functions. The function f determines the growth

3 Journal of Applied Statistics 3 trajectory representing how an individual changes over time. 1 Because b i are latent random coefficients, the f(time it, b i ) can be viewed as the true score at measurement occasion t for individual i. The second level of a growth curve model can be expressed as, b i = B X i + u i p 1 p qq 1 p 1, (2) where X i is the ith column (data for the ith individual) of the design matrix X q N with ones in the first row and data for the q 1 covariates in the other rows, B is the fixed-effects coefficient matrix, and u i represents the multivariate normal errors with E(u i )=0 p 1 and Cov(u i )=D p p. The u i is also called the random effects from a mixed-effects model perspective. The estimated parameters B provide information on the estimated growth trajectory of an average person, an individual whose coefficients are equal to the population mean values [12]. Each b i determines an individual growth trajectory. However, many other features such as rates of growth related to the growth phenomenon that are important and interesting to substantive researchers are not directly emphasized. We argue that this is mainly because the parameters (random coefficients) of a growth function may not have an intuitive meaning especially in terms of rates of growth before reparametrization. This problem has also been pointed out by Cudeck and du Toit [11] and Stimson, Carmines, and Zeller [37] when studying quadratic growth. This study attempts to address the interpretation limitations by directly defining and modeling rates of growth. As mentioned earlier, rates of growth are an important feature of a growth process that are of special interest to substantive researchers and should be analyzed more broadly [e.g., 21, 38]. It is not surprising to see studies that have adopted the linear growth curve model simply because it can be easily interpreted in terms of rates of growth although the linear growth curve model does not seem to fit the data very well [e.g. 20, 32]. For quadratic growth curve models, Cudeck and du Toit [11] proposed a reparametrization method that transforms the intercept, slope, and quadratic terms to initial performance, maximizer, and maximum parameters that have attractive interpretations. Intrigued by the reparametrization idea, we propose to analyze rates of growth through the reparametrization of growth curve models. 3. Growth Rate Models and Reparametrization of Growth Curve Models With the growth function in Equation (1), rates of growth can be calculated conveniently using the first derivative of the growth function. Let r it denote the rate of growth for the ith participant at occasion t or time time it. It is easy to see that r it = y 0 it = d d f(time it, b i )+ e it. (3) dtime it dtime it If we further assume that the residuals are time-independent as in many growth curve d analysis procedures and therefore the derivative of the residuals dtime it e it is equal to zero, the above rate of growth can be simplified to r it = y 0 it = d dtime it f(time it, b i ). (4) 1 To be precise, the trajectory used in the current study is a particular path in an individual s state space. A state space is a space in which all possible states of an individual are represented, with each possible state of the individual corresponding to one unique point in the state space. A trajectory is the path that connects the points in a state space along time.

4 4 Zhiyong Zhang, John J. McArdle & John R. Nesselroade Another underlying assumption is that the growth function is differentiable at occasion t. Many widely used growth functions such as polynomial, exponential, and logistic growth functions are differentiable. Also note that r it is the instantaneous rate of growth at occasion t, and in general, it varies with time. However, one may consider it as the average rate of growth within a small time interval around occasion t. As in growth curve analysis, we are often interested in predicting rates of growth such that r it = t X i + v it where t is 1 q vector of regression coefficients of rates of growth on covariates at occasion t and v it, E(v it )=0and Var(v it )= t 2, is the normal residual part that is not explained by the covariates. Given that the focus of the analysis is on rates of growth, the key here is to estimate t and t 2. Intuitively, one can first estimate the random coefficients b i for each individual through growth curve models and then calculate rates of growth using ˆr it = d dtime it f(time it, ˆb i ). Then, in a second step, one can estimate the parameters that are related to rates of growth. However, approaches, such as the maximum likelihood method used in SEM software like Mplus and the empirical Bayesian method used in multilevel modeling software like PROC MIXED, for estimating the random coefficients are not optimal because they are sequential as opposed to simultaneous. Furthermore, the random coefficients b i are difficult to estimate accurately especially when the number of measurement occasions is small [e.g. 16, 17, 24, 40]. Given the fact that it is not necessary to estimate b i in estimating the growth curve model parameters of B and D, we seek a method to estimate t and 2 t without estimating r it. From Equation (4), the rate of growth is a function of the time variable time it and random coefficients b i which can be expressed as r it = g(time it, b i )= d dtime it f(time it, b i ), where g denotes the first derivative function. Without loss of generalization, we can find a b ik,k 2 [1,p] that can be solved from the equation above as b ik = g 1 (time it, b k i,r it ), (5) where b k i =(b i1,...,b ik 1,b ik+1,...,b ip ) 0. Equation (5) means that we can express at least one random coefficient to be a function of time it, the rate of growth r it, and the other random coefficients. By substituting b ik in the growth curve models in Equations (1) and (2), we have 8 y >< it = f (time it,r ij, b k i )+e it b k i = B k X i + u k, t =1,...,T, i=1,...,n, (6) i >: r ij = j X i + v ij where B k = (B 1,...,B k 1,B k+1,...,b p ), u k i = (u i1,...,u i,k 1,u i,k+1,...,u ip ), and f denotes a function that is in general different from f in growth curve models. Note

5 especially, r ij = r it t=j = Journal of Applied Statistics 5 d dtime it f(time it, b i ) t=j because rates of growth generally vary across time. Here j is a time index indicating that the growth rate is evaluated at occasion t = j. Note also that through the reparametrization, we can express the growth function as a function of rates of growth. Furthermore, Equation (6) has the same form as growth curve models and thus it can be estimated using growth curve modeling techniques. To facilitate presentation, the models in Equation (6) will be called growth rate models henceforth in this article although they can also be viewed as growth curve models. 3.1 Examples For illustrations, we examine several examples of constructing growth rate models from corresponding growth curve models. In these examples, we assume that there are two covariates. For comparison, we summarize the example growth curve models and corresponding growth rate models in Table Linear growth rate model A linear growth curve model with two covariates is given in (1a) in Table 1. Taking the first derivative of the linear growth function, r it = d dtime it f(time it, b i )= d dt (b i1 + b i2 time it )=b i2, which is a constant. Thus, we can directly replace the random coefficient b i2 with the rate of growth r it r i and then interpret the linear growth curve model as a growth rate model as shown in (1b) in Table 1. Actually, the b i2 in the linear growth curve model is often interpreted in terms of rates of growth [e.g. 2, 4, 29, 32] Quadratic growth rate model A quadratic growth curve model with two covariates can be written as (2a) in Table 1. The rate of growth for quadratic growth at occasion j(j =1,...,T) is r ij = d dtime it (b i1 + b i2 time it + b i3 time 2 it) t=j =0b i1 + b i2 +2time ij b i3. (7) From Equation (7), we have b i2 = r ij 2time ij b i3. Note that we can also solve for b i3 equivalently. Plugging it into the quadratic growth curve model, we can get the quadratic growth rate model as shown in (2b) in Table 1. When time ij =0, the rate of growth r ij = b i2. This is why b i2 has been interpreted as the instantaneous rate of growth at time zero [e.g. 4, 35]. A possible limitation of this instantaneous rate of growth is that time zero is usually not part of the measurement process or may not exist and thus centering has often to be used to make time zero meaningful. Instead, the quadratic growth rate model can analyze the rate of growth at any interesting time point without changing the meaning of the initial level parameter b i1 and the acceleration parameter b i3. Individual differences in the rate of growth can be also evaluated and predicted through covariates. Note that for quadratic growth phenomena, the model discussed by [2, p. 42] based on time coding method is similar to our quadratic growth rate model.

6 6 Zhiyong Zhang, John J. McArdle & John R. Nesselroade Table 1. Comparison of growth curve models and growth rate models Growth curve models < < : : < < Growth rate models 8> 8> yit = bi1 + bi2timeit + eit yit = bi1 + ritimeit + eit Linear (1a) bi1 = xi1 + 21xi2 + ui1 (1b) bi1 = xi1 + 21xi2 + ui1 > : > : bi2 = xi1 + 22xi2 + ui2 ri = xi1 + 22xi2 + ui2 = + + bi3time 8>< 8>< yit bi1 bi2timeit it + e it yit = bi1 + rijtimeit + bi3(time 2 it 2timeijtimeit)+eit bi1 = xi1 + 21xi2 + ui1 bi1 = xi1 + 21xi2 + ui1 Quadratic (2a) (2b) bi2 = xi1 + 22xi2 + ui2 bi3 = xi1 + 23xi2 + ui3 > > bi3 = xi1 + 23xi2 + ui3 rij = j0 + j1xi1 + j2xi2 + vij 8> 8> yit = bi1 bi2 exp[ (timeit timei1) ]+eit yit = bi1 rij exp[ (timeit timeij) ]/ + eit Exponential (3a) > : bi1 = xi1 + 21xi2 + ui1 (3b) > : bi2 = xi1 + 22xi2 + ui2 bi1 = xi1 + 21xi2 + ui1 rij = j0 + j1xi1 + j2xi2 + vij

7 3.1.3 Exponential growth rate model Journal of Applied Statistics 7 One form of the exponential growth curve model [5] can be expressed as (3a) in Table 1. The rate of growth for exponential growth at time j(j =1,...,T) is r ij = d dtime it (b i1 b i2 exp[ (time it time i1 ) ]) t=j =0b i1 + b i2 exp[ (time ij time i1 ) ]. (8) From Equation (8), b i2 = r ij /{ exp[ (time ij time i1 ) ]}. By plugging it into the exponential growth curve model, we can get the exponential growth rate model with the form in (3b) in Table 1. Although in the exponential growth curve analysis the parameter has been called the rate of growth, it should be clear that this rate is the instantaneous rate of growth per capita or the rate of approach to the asymptote. In this study, the parameter has been estimated as a constant in the population as typically done in the application of the exponential growth curve model. If we estimate it as a random coefficient i as in Browne [5] and Neale and McArdle [29], the relation between this rate and our rates of growth is r it = i /f(time it, b i ) [see also 7]. While the rate of growth i monitors the overall growth trajectory for person i, the rate of growth r it reveals the subtleties of growth at each developmental phase. 3.2 Centering and reparametrization for analysis of growth rates For growth curve analysis, centering the time variable can improve the interpretability of model parameters [e.g. 31, 36]. For example, if one centers time at a certain value for a linear model, then the intercept term becomes the level of attribute at that point in time which could improve the interpretability of the intercept term. For other polynomial growth curve models, one can also study rates of growth through centering. This can be illustrated using the quadratic growth curve model. From Equation (7), the rate of growth is a linear combination of the linear slope and quadratic slope terms. If the combination coefficient (time ij ) is equal to 0, then the linear slope term is equivalent to the rate of growth at occasion j. Thus, if we center time at occasion j and then let time ij =0, we can study the rate of growth through the linear slope term. Furthermore, by centering time at different time points, we can study the rate of growth across time. However, centering only works for certain models, typically polynomial models, in order to study growth rates. For other nonlinear models, centering may not achieve the goal of growth rate analysis. For example, for the exponential growth curve model, no matter how to center the time variable (according to Equation 8), the term b i2 cannot be interpreted as the rate of growth directly. However, the reparametrization method described here can be applied to analyze rates of growth for the exponential growth curve model. Even for the quadratic growth curve model where centering can be applied to study growth rates, the reparametrization method can be viewed as the underlying mechanism of the centering method for the analysis of the rate of growth. From Equation (9), to express the linear slope term b i2 as the rate of growth, one expects that r ij = b i2. Thus, we should let time ij =0and the center for the centering method should be at occasion j. Therefore, the reparametrization method proposed in this study is more general and useful for the analysis of rates of growth than the centering method.

8 8 Zhiyong Zhang, John J. McArdle & John R. Nesselroade 3.3 Growth rate models and rotation of growth curves Growth rate models can also be derived through the rotation of growth curves. Growth curve models can be viewed as confirmatory factor models with factors already identified. For example, for a linear growth curve model, we may hypothesize and fit a level factor and a slope factor. For the quadratic growth curve model, we have y it = b i1 + b i2 time it + b i3 time 2 it + e it, where b i1, b i2, and b i3 can be viewed as factor scores for the ith individual of three factors, level, linear slope, and quadratic slope terms, and (1, time it, time 2 it ) are the factor loadings. Sometimes, when the growth curve analysis is used solely to find the curves that best fit the data, the obtained factors may be difficult or even impossible to interpret in a meaningful way. Researchers have suggested rotating factors to enhance their meaning in the framework of factor analysis [see recent reviews by 6, 19]. Tucker [39] suggested that rotation to simple structure may not be meaningful in the framework of learning or growth curves because the resulting factors could still be difficult to interpret. Our construction of growth rate models provides a way to rotate growth curves by rotating factors of growth curves to predefined meaningful target factors. More specifically, after rotation, one of the factors will be the rate of growth at a specific point in time. The rotation is conducted by b k i I k = b r ij c i = Qb i, ij where Q here represents the rotation matrix and I k is a matrix with the kth row removed from the identity matrix I p p and c ij is a vector of combination coefficients which can be determined by Equation (4). Using the quadratic model as an example, the rotation procedure for the factors is b i1 r ij b i3 1 0 A time ij b 1 0 i1 b i2 A = b 1 i1 b i2 A With this rotation matrix, the factors b i1 and b i3 remain the same but the second factor now is the rate of growth at occasion j. When factors are rotated, factor loadings need to be rotated correspondingly in order to retain the covariance structure. For the quadratic growth example, one has time i1 time 2 i1 1 time i1 time 2 1 time i2 time 2 i1 i2 1 time i2 time 2 0 = 1 time i3 time 2 i2 i3 Q 1 = 1 time i3 time time ij A B C B A time it time 2 it 1 time it time 2 it time i1 time 2 i1 2time i1 time ij 1 time i2 time 2 i2 2time i2 time ij = 1 time i3 time 2 i3 2time i3 time ij, B A 1 time it time 2 it 2time it time ij b i3 b i3

9 Journal of Applied Statistics 9 where denotes the factor loadings after rotation. Note that the elements in this matrix correspond to the factor loadings (or design matrix) in Equation (3.1). The same procedure also applies to the exponential growth rate model. The growth rate factor is obtained by bi1 = r ij 1 0 bi1 bi1 = Q. 0 exp[ (time ij time i1 ) ] b i2 b i2 Thus, the factor loading matrix for the exponential growth rate model is exp[ (time i2 time i1 ) ] = 1 exp[ (time i3 time i1 ) ] 1 0 B C 0exp[(time ij time i1 ) A 1 exp[ (time it time i1 ) ] exp[ (time i1 time ij ) ]/ 1 exp[ (time i2 time ij ) ]/ = 1 exp[ (time i3 time ij ) ]/. B A 1 exp[ (time it time ij ) ]/ This factor loading matrix corresponds to the one in model (3b) in Table Estimation of growth rate models Because they are reparametrizations and rotation of growth curve models, growth rate models can be estimated using the usual methods and software for growth curve analysis. For example, one can estimate the models as mixed-effects models using the maximum likelihood estimation method in SAS PROC MIXED and NLMIXED. Alternatively, one can estimate them as structural equation models using readily available structural equation modeling software through the specification of the rotated factors and factor loading matrix. Furthermore, the growth rate model fit can be evaluated using the fit statistics and fit indices developed for growth curve models such as the chi-square statistics, the root mean squared error of approximation (RMSEA), and the goodness of fit index (GFI). When multiple competing growth rate models can be fitted to the data, one can also use the likelihood ratio test, Akaike information criterion (AIC), and Bayesian information criterion (BIC) to select the best-fitting models. Actually, the fit statistics and fit indices are identical for the growth rate models and their corresponding growth curve models. Therefore, the evaluation of model fit and model selection can be conducted directly based on growth curve models. For the linear model, the linear slope term also the linear rate of growth is constant for each person. However, for other polynomial and nonlinear models, the growth rate varies with time. There are two approaches to analyzing varying growth rates over time. First, one can investigate the rate of growth at each time point separately. In this way, at each time point, the term for the growth rate is a random effect in the usual sense. Second, one can put some equivalence constraints on the rates of growth at different times according to the dependence among growth rates so that the rates of growth can be estimated simultaneously in one model. For example, the growth rate of the quadratic model at any time point depends on the linear and quadratic terms as shown in Equation (9). In other words, give that growth is quadratic, the growth rate at one time point can be determined by those at other points in time. Therefore, Equation (9) can be used to impose

10 10 Zhiyong Zhang, John J. McArdle & John R. Nesselroade the constraints for the estimation of growth rates. Practically, the first approach is easier to implement using the available software and arrives at the same results as the second approach. In the current study, the first approach is used for data analysis. Specifically, we estimate the rate of growth at each time point and also investigate the relationship between the growth rate and covariates separately at each time point. 4. Application We have discussed how to construct growth rate models through the reparametrization of growth curve models to analyze rates of growth. Next we use empirical data to demonstrate how to apply these models to answer substantive questions. 4.1 Background Mathematical ability is an extensively studied cognitive attribute. The development of mathematics has been examined in several longitudinal studies [e.g. 23, 41]. Associations between mathematical performance and selected covariates have also been studied. Gender differences in mathematics performance, especially, have been investigated extensively in many studies. A review by Hyde, Fennema, and Lamon [18] showed that males outperformed females significantly on mathematics tests. Associations between cognitive abilities and various behavior problems have also been investigated. In a recent longitudinal study, Bub, McCartney, and Willett [8] found that children with higher initial levels of internalizing and externalizing behaviors at 24 months had lower cognitive ability and achievement scores in the first grade. However, a couple of perspectives on the nature of mathematical ability development and change are still missing from the literature. First, a systematic analysis of rates of growth of mathematical ability is needed. Previous research has found that the development of mathematics is not linear over time [e.g. 15, 23, 41]. Thus, rates of growth of mathematics cannot be adequately analyzed using linear growth curve models. To shed additional light on this matter, the proposed growth rate models will be used to analyze rates of mathematical ability development. Second, how rates of growth of mathematics are related to covariates such as gender and behavior problems also invites more investigation. Therefore, to contribute further to the substantive literature while illustrating our models, we examine individual differences in rates of mathematical ability growth and how they are related to gender and a general measure of behavior problems. 4.2 Data Data for our empirical example are drawn from the National Longitudinal Surveys of Youth. Specifically, participants were the children of female respondents to the National Longitudinal Surveys of Youth 1979 cohort [NLSY79, 10]. The survey on the children was sponsored and directed by the U.S. Bureau of Labor Statistics and the National Institute for Child Health and Human Development. The data used in the current study are from N =1, 233 children ranging in age from 6 to 15 years with a 10-year age span. One cognitive variable the Peabody Individual Achievement Test (PIAT) mathematics assessment was used as a measure of mathematical ability and analyzed by means of growth rate models. The PIAT is a wide-ranging measure of academic achievement for children aged five and over and is widely used in research. The Mathematics (PIAT Math) subtest consists of 84 multiple-choice items arranged in increasing order of difficulty. Possible scores range from 0 to 84. For computational convenience, the scores were rescaled by dividing by 10. Two covariates, gender and Behavior Problems Index (BPI), are included

11 in the analysis as time-invariant covariates. Journal of Applied Statistics Descriptive statistics Descriptive statistics for gender, BPI, and PIAT Math at each age are given in Table 2. About 48% of the children in the sample were girls. The average BPI was The mean scores for PIAT math increased with age but the rates of increase seemed to be slowing over age. A longitudinal plot of PIAT math is shown in Figure 1. From the plot, one can see that the growth trajectory of mathematics performance is not linear with age. Furthermore, it appeared that there was not much of a mean difference between boys and girls in mathematical ability; however, children with higher BPI had lower scores on PIAT math than those with lower BPI. Table 2. Descriptive Statistics Variable Age N Mean Std Min Max Gender 1, (M) 1 (F) BPI 1, Math Note. Std: standard deviation; Min: minimum; Max: maximum; M: male; F: female; Math: mathematical performance. For this application, age, instead of measurement occasions, was used as the time scale for the following reasons. First, age is a natural scale for child development analysis. Second, using age as the time scale makes the interpretation of the results easier than the use of the measurement occasions. 1 In the data set, information on children s age in months and age in years when the test was administered was available. The age of children in years was used in the current analysis. The analysis also assumes that the time interval is approximately one year. Furthermore, in order to make the growth curve analysis more interpretable, we centered the age variable at 10. Data were missing at each age. At age 8, about 48.9% of data were missing. At age 15 about 93.5% of data were missing. The total percentage of missingness was about 64.9%. Furthermore, 29 children only participated in the study at one occasion, 96 participated at two occasions, 533 participated at three occasions, 399 participated at four occasions, 183 participated at five occasions, and no children participated in 6 or more occasions. In all analyses, missing data were assumed to be missing at random [MAR; e.g., 25] and the full information maximum likelihood (FIML) estimation method was used to obtain the growth curve and growth rate model parameter estimates. 4.4 Research questions Two main questions identified earlier are answered in the empirical data analysis. First, are there individual differences in rates of growth of mathematical ability? Second, how 1 Another possibility is to use grade as the time variable. The reason to use age as the time variable instead of grade is that grade was missing for some PIAT scores, whereas age was always recorded.

12 12 Zhiyong Zhang, John J. McArdle & John R. Nesselroade PIAT math Gender M F BPI Low High Age Figure 1. The longitudinal plot of PIAT Math for a random sample of 100 children from the sample. The thick line is the average trajectory. The subplot on the top left corner gives the average trajectories for male and female groups. The subplot on the bottom right corner gives the average trajectories for children with low and high BPI. are individual differences in growth rates of mathematical ability related to gender and BPI and the interaction between the two? 4.5 Data analysis and results In order to apply growth rate models to the data, we needed to determine what kind of growth curves mathematical ability development follows. Here, we first fitted linear, quadratic, and exponential growth curve models to the data to determine the best-fitting curve for the data based on several fit indices. Fit statistics, including Chi-square, AIC, BIC, and RMSEA, are summarized in Table 3. Based on AIC, BIC and RMSEA, the quadratic growth curve model fitted the current mathematical ability data better than the alternatives. Thus, the quadratic growth rate model was used to analyze mathematical ability growth rates. 1 1 The program scripts for the data analysis can be obtained from the first author or downloaded from psychstat.org/research/johnny_zhang/grm.

13 Journal of Applied Statistics 13 Table 3. Fit indices for the linear, quadratic, and exponential growth rate models Linear Quadratic Exponential Chi-square d.f. a AIC b BIC b RMSEA c CI for RMSEA c [0.115, 0.129] [0.051, 0.065] [0.058, 0.072] a Degrees of freedom b Akaike information criterion (AIC) and Bayesian information criterion (BIC) c Root mean squares error of approximation (RMSEA) and confidence interval (CI) To investigate individual differences in rates of mathematical ability development, a quadratic growth rate model without covariates was first fitted to the data. The output of the quadratic growth rate model naturally includes all the information for the corresponding quadratic growth curve model. Table 4 summarizes the parameter estimates that are the same for both the quadratic growth rate model and the corresponding growth curve model. The estimated mean and variance of rates of growth at each observed age are summarized in Table 5. Note that in Table 4, the slope estimate was the instantaneous rate of growth, 0.56, for the quadratic growth curve model at age 10. Furthermore, because the estimated variability of the slope was significant (0.012, S.E.=0.001), there existed individual differences in terms of growth rate at age 10. More information about rates of growth can be seen in Table 5. Clearly, the average rates of growth were not a constant but declined linearly with age. Furthermore, the variability in rates of growth first decreased and then increased. There were significant individual differences in rates of growth before age 12. From age 6 to age 11, there was significant growth in mathematical ability and there were also significant individual differences in the rate because both the mean and variance of rates of growth were significant. However, after age 12 there was only significant mean growth in mathematical ability; individual differences in rates of growth disappeared. That is, different children did not develop at different rates on mathematical ability after age 12 because there was no individual difference in rates of growth of mathematical ability. Note that the slope parameter in Table 4 gives the instantaneous rate of growth at age 10 based on the quadratic growth curve model after centering the age variable at 10. The slope parameter mean and variance estimates were exactly the same as those from the growth rate analysis. This is because for polynomial growth curve models, rates of growth can also be analyzed through centering as discussed earlier. However, if we only look at the results of the instantaneous rate of growth at one certain age from quadratic growth curve model, we cannot see the dynamic change in the rates of mathematical ability development. Especially, the significant result at one time point does not imply the significance at other time points. 1 The covariates gender and BPI variables were then included to predict rates of growth in children s mathematics performance. In addition to the main effects of gender and BPI, their interaction was also included as a covariate in the models. The results are summarized in Table 6, Table 7, and Figure 2. Again, the quadratic growth rate model provides a better picture on how the covariates are related to the rates of growth over time. From the analysis, the following conclusions can be drawn. First, the effects of the covariates on the rate of growth were linearly related to age because the best-fitted model 1 For the quadratic growth curve model and more general polynomial growth curve models, one could center age at different values to study growth rates over age. However, this strategy is rarely applied in practice. A typical growth curve analysis still focuses on one certain time.

14 14 Zhiyong Zhang, John J. McArdle & John R. Nesselroade Table 4. Results same for both the quadratic growth curve model and the quadratic growth rate model (age is centered at 10) Estimates S.E. Fixed-effects parameters Level Slope Quadratic Random-effects parameters Level Level Level Slope Level Quadratic Slope Slope Slope Quadratic E-04 Quadratic Quadratic 3.73E E Residual variances Table 5. Estimated means and variances of rates of growth for mathematics performance at each age Age Mean Variance (0.012) (0.011) (0.010) (0.007) (0.007) (0.004) (0.005) (0.002) (0.005) (0.001) (0.007) (0.002) (0.009) (0.004) (0.012) (0.008) (0.015) (0.012) (0.018) (0.018) was a quadratic growth model. Second, the standard errors of the estimated coefficients first decreased and then increased. Third, BPI was negatively related to the rates of growth from age 6 to 13. Children with higher BPI scores tended to grow more slowly on mathematics performance. However, at age 14 and 15, BPI was no longer related to rates of mathematics growth. Fourth, gender differences in rates of growth appeared only at age 8 and 9. At those ages, boys seemed to show quicker improvement in their performance on mathematics. Fifth, the interaction between gender and BPI had a significant effect on the growth rate between age 8 and 11. Closer examination revealed a positive interaction. Thus, for the same amount of increase in BPI, girls showed less decrease in rates of growth. Similarly, the dynamic associations between rates of growth and gender and BPI cannot be directly studies based on the use of a quadratic growth curve model. 4.6 Illustration of the exponential growth rate model For the purpose of demonstration, we also applied the exponential growth rate model to analyze the mathematical ability data although the quadratic growth rate model fitted the data better. Figure 3 portrays the associations between rates of growth and covariates BPI and gender and their interaction across age. From Figure 3, the coefficients for the covariates and interaction were significant across the measured age range. Furthermore, the coefficients for both BPI and gender and their interaction changed in a nonlinear way with age. Note that for the exponential growth analysis, the rate of growth cannot be

15 Journal of Applied Statistics 15 Effect of BPI on growth rate Age Effect of Gender on growth rate Age Effect of BPI and Gender interaction on growth rate Age Figure 2. The estimated coefficients with error bars for the covariates and their interaction at each age from the quadratic growth rate model. The error bars are calculated as the 95% confidence intervals.

16 16 Zhiyong Zhang, John J. McArdle & John R. Nesselroade Table 6. Results same for both the quadratic growth curve model and the quadratic growth rate model with covariates Estimates S.E. Fixed-effects parameters Level on Intercept BPI Gender BPI*Gender Slope on Intercept BPI Gender BPI*Gender Quadratic on Intercept BPI Gender BPI*Gender -5.00E Random-effects parameters Level Level Level Slope Level Quadratic Slope Slope Slope Quadratic -1.55E E-04 Quadratic Quadratic 3.72E E Residual variances Table 7. The estimated coefficients for the covariates of gender and BPI and their interaction Age BPI s.e. Gender s.e. B*G s.e analyzed through centering. 5. Simulation Studies To evaluate whether parameters in growth rate models can be recovered accurately, we conducted two simulation studies, one for the quadratic growth curve model and another for the exponential growth curve model. Because of the presence of missing data in real data analysis, we also investigated the impact of missing data on growth rate model parameter estimates. For both simulation studies, the population models were set up to mimic the quadratic and exponential growth curve models used in the empirical data analysis. First, two covariates BPI and gender were generated and their interaction term was also included in the simulated models. Second, the population parameter values were determined using the parameter estimates from the empirical study. Third, 10 occasions of

17 Journal of Applied Statistics 17 Effect of BPI on growth rate Age Effect of Gender on growth rate Age Effect of BPI and Gender interaction on growth rate Age Figure 3. The estimated coefficients with error bars for the covariates and their interaction at each age from the exponential growth rate model. The error bars are calculated as the 95% confidence intervals.

18 18 Zhiyong Zhang, John J. McArdle & John R. Nesselroade data, from age 6 to 15 were simulated. Fourth, the sample size was fixed at N =1, 233. Finally, the generated missing data had exactly the same missingness patterns as the real data. For each of the two growth rate models, we first generated 1, 000 sets of complete data and then obtained the model parameter estimates for each set of data. Then, missing data for each complete data set were created according to the missingness patterns of the real data. Subsequently, the same growth rate model was used to analyze the incomplete data and obtain the model parameter estimates. In total, there were 1, 000 sets of parameter estimates for the quadratic and exponential growth rate models, respectively, under both complete and missing data conditions. For each parameter in the growth rate models, four statistics including relative bias, mean standard error, empirical standard deviation, and coverage probability were computed. Let denote the population value of a parameter in the simulation of the growth rate models and let ˆ r denote its parameter estimate and ŝ r denote the standard error estimate from the rth set of simulated data. First, relative bias was calculated as " P1000 r=1 Bias = 100 ˆ # r For a well-estimated parameter, one would expect that the relative bias would be smaller than 5. Second, the mean standard error (S.E.) can be calculated as S.E. = X ŝ r r=1 Third, the empirical standard deviation was calculated as v u S.D. = t X 999 r=1 (ˆ r X r=1 ˆ r ) 2. Fourth, the coverage probability for a 100(1 )% confidence interval was obtained by CVG = number of times 2 [ˆ r z /2 ŝ r, ˆ r + z 1 /2 ŝ r ], 1000 where z represents the cut points of a standard normal distribution. In the simulations, the 95% coverage probability was used. The results for the quadratic growth rate model simulation are summarized in Table 8. In order to save space, only the regression coefficients between the rates of growth and the covariates are reported. For complete data analysis, the relative biases were all smaller than 5% indicating that the regression coefficients at each age were very well recovered. The standard error estimates were also very close to the empirical standard deviation and thus the standard error estimates were also accurate. Furthermore, the coverage probability of each parameter was close to the nominal level Thus, the quadratic growth rate model parameters were estimated very well. With the presence of missing data, the relative biases remained small for most parameters. Although the relative biases for the gender covariance at ages 12 and 13 were slightly larger than 5%, they were still not large given that the true parameter values were very small with an absolute value of As with the complete data case, the standard er- 1 With a small parameter value, the random sample error and rounding error tend to have a big influence on the relative bias.

19 Journal of Applied Statistics 19 Table 8. Simulation results for the quadratic growth rate model without and with missing data BPI Gender Interaction Complete data Missing data Age TRUE Bias S.E. S.D. CVG Bias S.E. S.D. CVG Note. Bias: relative bias in %. S.E.: average standard error. S.D.: standard deviation. CVG: coverage probability. rors were accurately estimated and the coverage probabilities were close to the nominal level Thus, the quadratic growth rate model parameters were still estimated very well in the presence of missing data. However, the standard error estimates were notably larger than those from complete data, which was expected because missing data caused the loss of information. Table 9 presents the results for the exponential growth rate model with and without missing data. The results clearly show that both parameter estimates and their standard errors were accurately estimated regardless of missing data. Furthermore, the coverage probabilities were close to the nominal level Therefore, the parameters of both the quadratic growth rate model and the exponential growth rate model were estimated very well under the conditions of complete and missing data. 6. Discussion We proposed a systematical method to reparametrize growth curve models for analyzing rates of growth. This method provides a precise formulation on how to construct growth rate models from their corresponding growth curve models and can especially benefit substantive researchers because of its usability. First, the method directly targets the rate of growth that has been the interest of many studies. Second, growth rate models can be estimated conveniently using existing SEM and multilevel software. Third, growth rate models can provide useful information above and beyond the information provided by traditional growth curve analysis as demonstrated by the application and simulations. For example, growth rate models such as the exponential growth rate model can be used