Time Dependence of Growth Parameters in Latent Growth Curve Models with Time Invariant Covariates

Transcription

1 Methods of Psychological Research Online 003, Vol.8, No., pp. -4 Department of Psychology Internet: University of Koblenz-Landau Time Dependence of Growth Parameters in Latent Growth Curve Models with Time Invariant Covariates Reinoud D. Stoel and Godfried van den Wittenboer University of Amsterdam Recently, there has been renewed interest in the lack of invariance of growth parameters in latent growth curve models. This lack of invariance is a consequence of specific values of the growth parameters, initial status and growth rate, which depend on the time scale. Different time scales lead to different estimates of the growth parameters, as well as their (residual) variances and covariances. Thus far no explicit results have been derived, however, on effects of exogenous covariates that are put into the relationship. In the paper we will show that the effect of covariates on the initial status and growth rate will depend also on the time scale involved. As a consequence, questions arise about the appropriateness of a selected time scale as will be illustrated by an empirical example. Keywords: growth curves, structural equation modeling, time invariant covariates, transformation Latent growth curve models (Meredith & Tisak, 990) represent repeated measures of dependent variables as a function of time and other measures. The relative standing of an individual at a specific time point is modeled as a function of an underlying process, the parameter values of which vary randomly across individuals. Latent growth curve methodology can be used to investigate systematic change, or growth, and interindividual variability in this change. A special topic of interest is the correlation of the growth parameters, the so-called initial status and growth rate, as well as their relation with time varying and time invariant (e.g., IQ ) covariates. Good introductions of basic, as well as more advance latent growth curve models are given by, for instance, Duncan, Correspondence concerning this article should be addressed to Reinoud D. Stoel, University of Amsterdam, Department of Education, P.O. Box 9408, NL-090 GE Amsterdam, The Netherlands. reinoud@educ.uva.nl

2 MPR-Online 003, Vol. 8, No. Duncan, Strycker, Li and Alpert (999), MacCallum, Kim, Malarkey and Kieholt-Glaser (997), Muthén and Khoo (998), Stoolmiller (995), and Willett and Sayer (994). Although latent growth curve methodology is not the only candidate for the analysis of longitudinal data from a panel design, it certainly is an elegant and parsimonious way to represent systematic change. Alternative possibilities are autoregressive models, and in particular the true individual change models of Steyer, Eid and Schwenkmezger (997) (see also McArdle, 00; Steyer, Partchev & Shanahan, 000). Judging from the amount of applications (e.g., Chan, Ramey, Ramey & Schmitt, 000; Garst, Frese & Molenaar, 000) and the amount of methodological literature, however, latent growth curve methodology provides a generally accepted framework for the analysis of longitudinal data collected from a panel design. The contention that conclusions drawn from latent growth curve analysis are sensitive to the way time is incorporated in the model, is less well documented. This lack of invariance of the growth parameters can be seen as a consequence of the time scale involved. When different time scales are incorporated in the models, different values of the initial status and the growth rate will be obtained, as well as of their (residual) variances and covariances (Garst, 000; Mehta & West, 000; Rogosa & Willett, 985; Rovine & Molenaar, 998; Rudinger & Rietz, 998; Stoolmiller, 995). In the literature on latent growth curve models, however, no attention has been paid to the fact that estimated effects of exogenous covariates (e.g., IQ) on the initial status and growth rate may also depend on the time scale involved. Merely a short comment (Stoolmiller, 995, p. 8) suggests the interference. In this paper, the mathematical relationship between the growth parameters and time invariant covariates as a function of the selected time scale will be derived. In fact, the above relationship between time and growth parameters will be generalized to time invariant covariates. Roots of the argument stem from Garst (000), who confined himself to the covariance of initial status and growth rate, and a slightly analogous problem has been discussed in the more general area of regression analysis for a variety of models (e.g., Aiken & West, 99). An interesting interactive simulation example of the correlation between intercept and slope in regression analysis can be found at the following web-address of the University of Leuven:

3 Stoel & van den Wittenboer: Time Dependence of Growth Parameters 3 Though never investigated in the area of latent growth curve analysis, consequences of the relationships are far-reaching: the time scale involved may affect the conclusions based on the model. Referring to the increasing number of applications of growth curve models, it should be noted that the opportunities of latent growth curve methodology, as well as the pitfalls, must be fully understood by researchers, when selection of the time scale arises. The subsequent paper is built up as follows. First, we provide a short description of latent growth curve models and illustrate the argument with a simple example highlighting the lack of invariance of growth parameters. Next, we formally describe the way in which relations between growth parameters and possible covariates depend on the metric of the time scale used. Subsequently, an empirical example will be presented to illustrate the theoretical findings, and the paper ends up with a discussion of the practical implications. Time Dependence in the Linear Latent Growth Curve Model Rogosa, Brandt and Zimowski (98) and Rogosa and Willett (985) were the first who pointed out, in general, that there is a relation between the initial status and the time at which it is defined. Different selections for the time of initial status in the same data led to different correlations between initial status and growth rate, ranging from zero to significantly positive or negative. In their conclusion they stated, there is no such thing as the correlation between change and initial status, although the determination of a unique correlation seems to have been the goal of much empirical research (Rogosa and Willett, p. 5). This explicit conclusion is both interesting and important, especially in the more specific area of latent growth curve modeling, where the relations between intercepts, growth rates and covariates are main topics of interest. Although mentioned briefly in several articles and handbooks (Duncan et al., 999; McArdle, 988; Rogosa, 995; Willett & Sayer, 994), a detailed analysis of the consequences and implications is still lacking. Recently, renewed interest in the lack of invariance of the growth parameters in the linear growth curve model has emerged (Garst, 000; Mehta & West, 000; Rovine & Molenaar, 998; Rudinger & Rietz, 998). Early ideas of Rogosa et al., and Rogosa and Willett have been followed up, and provided some important insights. The crux of the problem stems from the fact that the initial status is not the natural origin in many social science applications, but that it is an occasion de-

4 4 MPR-Online 003, Vol. 8, No. fined by factors, other than the origin of the process one is investigating. Often, simply the scores at first measurement occasion are taken as representing the initial status. The point of departure of this paper is the latent growth curve model with one, time invariant, covariate. Extensions to more complex growth models are straightforward, but not important to understand the main argument. The latent growth curve model with one time invariant covariate can be expressed as a confirmatory factor model: where y ij = λ 0i η 0j + λ i η j + ε ij () η 0j = ν x j + ζ 0j () η j = ν + x j + ζ j (3) In Equation, y ij refers to the measure of individual j on time point i that is predicted by two latent factors η 0j and η j, with λ 0i and λ i as their respective factor loadings and residual ε ij. The stochastic variable x j represents the perfectly measured time invariant covariate with expectation E(x) = 0. The complications induced by recognizing that x j may be fallible are not important for understanding the correlates of change (Rogosa & Willett, 985, p. 05). The factors η 0j and η j, with E(η 0 )=ν 0 ; E(η )=ν, are predicted by x j and two residuals ζ 0j and ζ j. Besides standard assumptions of structural equation modeling (Bollen, 989), the following additional assumptions are made:. The growth of each individual is linearly related to the passage of time. This assumption implies linear growth from the origin of the process (which may be outside the measured time interval) until the last measurement occasion, and the same functional form for the growth curves of all individuals.. The assumption of common causation: the sources of between-occasion variation and of individual differences are identical (Mandys, Dolan & Molenaar, 994). This assumption implies that the variation between individuals (within-group variance) must be caused by the same factors as the variation in the means (between occasion variance). Equations to 3 present the more general latent growth model (Meredith & Tisak, 990) that allows also for non-linear growth curves. Linear growth, however, requires certain parameters in the basis function (Meredith & Tisak, p. 08) to be constrained to

5 Stoel & van den Wittenboer: Time Dependence of Growth Parameters 5 specific values. To obtain the linear growth curve model, constraints have to be placed on the factor loadings λ 0i and λ i, also known as the basis functions. First, by constraining λ 0i =, η 0j becomes the initial status factor, and by constraining λ i = i, η j becomes the linear growth rate factor. If the growth rate factor represents a unit of change, then each basis function coefficient represents the number of units of change that occur between that occasion and the origin of the process (Rovine & Molenaar, 998). The implied covariance matrix Σ of Equation is: Σ = Λ y (ΓΦΓ + Ψ)Λ y + Θ ε (4) where Λ y is the (i ) matrix of basis function coefficients, Γ is the vector of regression coefficients relating the time invariant covariate to the growth parameters, Φ is a scalar representing the variance of the time invariant covariate, Ψ is the ( ) covariance matrix of the growth parameters, and Θ ε is the covariance matrix of the residual terms ε ij. As an example, Figure depicts a latent growth curve model with 4 consecutive measurements of a dependent variable y. The measurements of y are expressed as a function of a residual ε and the two latent factors η 0 and η. Fixing the basis functions at specific values, gives η 0 and η the interpretation of initial status and growth rate, respectively. Thus, η 0 represents the expected status at the start of the process, and η represents the average growth rate of the true status over time. The covariance between initial status and growth rate is modeled, as well as a perfectly measured time invariant covariate x, covarying with both η 0 and η. In Figure the λ 0 s are fixed at, and the λ s are fixed at 0,,, and 3, respectively. With this specification the first measurement occasion t corresponds to the origin of the process. Because there is seldomly a natural origin of time in social science research, the interpretation of the initial status is controlled by establishing the origin at an arbitrary point in time (Willett and Sayer, 994, p. 367). In Figure, the origin is defined at the first measurement occasion by fixing the loading of y on η to zero. The origin of the process is therefore identified at the first measurement occasion, which is the standard practice. If the loading of the first measurement occasion had been fixed at a value o- ther than zero, the origin of the process would have been identified at another point in time.

6 6 MPR-Online 003, Vol. 8, No. x 0 ζ ζ η 0 η 0 3 y y y3 y4 ε ε ε 3 ε 4 Figure. Simplified linear growth model with a fixed basis function (only the covariance structure is shown is this model). For instance, suppose that our model describes the growth process, of assertiveness in adolescents on five consecutive yearly intervals. The age of all subjects at the first measurement occasion is years 3. Fixing the loading of the first measurement to zero in this case formally means that the process of assertiveness starts at age. Another growth model for assertiveness could be a model in which the process starts right at the birth of an individual and, therefore, years prior to our first measurement. Although linearly extrapolating the growth curve in this way involves certain risks, the argument can technically be incorporated in the growth model. In order to do so, one has to fix λ to, 3, 4 and 5, respectively. Then, y represents the status of the subjects twelve years after the start of the process. In other words, the true initial status is shifted on the time axis. The origin of the process is straightforward in this example. It is less straightforward, however, in situations where there is no natural origin and researchers have to define the origin of the process themselves. The consequences of this can be serious, especially in situations where exogenous covariates are involved to explain the interindividual differences in the growth parameters. 3 Mehta and West (000) provide solutions when the age of the subjects is not equal at the first measurement occasion.

7 Stoel & van den Wittenboer: Time Dependence of Growth Parameters 7 Transformations of the Time Scale To show the effect of a different time scale on the growth parameters, we assume the time scale t i, defined by [t = 0, t =,..., t k = k-], to be transformed into a new time scale t * i by the linear function in Equation (5). Extending the ideas of Garst (000), who confined himself to the initial status and growth rate, we will investigate consequences of varying time scales of the form t * i = + t i (5) on the effect of time invariant covariates. In Equation 5, represents a shift on the time axis and will be the scaling factor that represents a change in the unit of time. The transformation of t i corresponds to a transformation of the basis function in Λ y to Λ * y :.... t t t = t + t.. =.. + k t k Λ y * (6) If P = 0 then P - = 0 and Λ y = Λ y * P - (7) Next, the implied covariance matrix Σ of Equation 4 can be written as: Σ Θ ε = Λ y * P - (Γ Φ Γ + Ψ) P - Λ y * (8) which is equal to Λ y * (Γ * Φ Γ * + Ψ * ) Λ y * (9) the implied matrix Σ Θ ε.

8 8 MPR-Online 003, Vol. 8, No. Equations 8 and Equation 9 show how the transformation P of the basis function is absorbed by the transformation P - of the matrix ΓΦΓ +Ψ. Since the matrix Γ * ΦΓ * +Ψ * provides the parameters of interest, we elaborate further on the identical matrix P - (ΓΦΓ +Ψ) P -. P - (ΓΦΓ +Ψ) P - = = [ ] +, ϕ = , ϕ (0) = Γ * ΦΓ * +Ψ * Equation 0 shows how the parameters in the growth model change as a function of a transformation of the metric of the time scale. The transformed parameters can now be expressed as a function of the original estimates: 0 0 * = () * = () * + = (3) * = (4) * = (5)

9 Stoel & van den Wittenboer: Time Dependence of Growth Parameters 9 of which Equations and refer to time invariant covariates and Equations 3 to 5 specify the relationship between initial status and growth rate already shown in Garst (000). From Equations to 5 we conclude that:. Multiplying the basis function with a constant leads to a change of all the parameters. Changing the time unit involved, leads to a change in the values of all relevant parameters.. Adding a constant to the basis function, or subtracting, changes the residual variance of the initial status, the covariance between initial status and growth rate, and the effect of the covariate on initial status. Stated more explicitly in terms of the topic of this paper, the relationship between the covariate and the initial status depends linearly on shifts of the time scale according to Equation if the scaling factor has been fixed. 3. The dependence of the growth parameters on the time scale holds only if there is variance in growth rates between subjects. If all individuals grow at the same rate, the growth parameters do not change, because the variance of the growth rate then equals zero (c.f. Rogosa, 995). If the measurement unit () of the time factor changes, in the case of nonzero variance of the growth rate, all the growth parameters change proportionally. A shift of the origin of the process (), on the other hand, results in changes of some of the growth parameters only. Both changes are equally straightforward, but the effect of changes in is less predictable on intuitive grounds. It may potentially be more dangerous in applied settings than the change of. Therefore, the remainder of this paper will mainly focus on change, and will be kept at a value of. 4. There exists a point in time t 0 (Rogosa & Willett, 985) where the correlation between initial status and growth rate equals zero. This point can be found, since rewriting Equation 4 with = 0 and =, * gives: 0 = 0 (6a) or 0 = (6b)

10 30 MPR-Online 003, Vol. 8, No. Equation 6b shows that the number of time units we have to shift the time axis 0, is equal to the ratio of the covariance between initial status and growth rate, and the (residual) variance of the growth rate. To compute t 0, we simply subtract 0 from the first element of the basis function: t 0 = t 0 (7) In any collection of linear growth curves, the variance of true scores will be a quadratic function of time, with a minimum at t 0 (Mehta & West, 000, p. 6), the time point where the residual covariance between initial status and growth rate is equal to zero. This is however not true if the growth rate is the same for all individuals (i.e. if the variance of the growth rate factor equals zero). Growth Curve Analysis of Example Data To illustrate the consequences of change in growth parameters, we analyze data taken from the National Longitudinal Survey of Youth (NLSY) of Labor Market Experience in Youth; a study initiated in 979 by the U.S. Department of Labor to examine the transition of young people into the labor force. The data were collected using faceto-face interviews of both child and mother taken in two-year intervals between 986 and 99, making the measurement unit of time equal to two years. A detailed description of the data and data collecting procedures can be found in Baker, Keck, Mott and Quinlan (993), and Curran (997). The measurements in the present example are from a battery of assessments of Curran 4, who presents the complete data of 6 children on four consecutive measures of children s antisocial behavior, and one (time invariant) measure of the degree of cognitive stimulation provided to the child at home. Antisocial behavior was measured using the Behavior Problems Index (BPI) antisocial behavior subtest (Baker et al.). It consists of the mother s report on six items concerning the child s antisocial behavior during the last three months. The degree of cognitive stimulation provided to the child was assessed measured using the cognitive stimulation subscale of the modified version of the Home Inventory (Baker et al.). It consists of 4 dichotomously scored items as reported by the mother. Although we would preferably carry out the analysis using the observed item scores, we were not able to retrieve the 4 The data can be found at

11 Stoel & van den Wittenboer: Time Dependence of Growth Parameters 3 raw data, leaving us with the sum scores, obtained by Curran. The general substantive goal of the study from which the data were derived was to get a better understanding of the relationship between parental emotional and cognitive support, antisocial behavior of the child, and child reading recognition. In this way they attempted to draw conclusions that might be interesting for developmental theory and may guide treatment intervention research in this area. Bearing in mind that we have used a sub-sample consisting of subjects with complete records on the sum scores of a selected set of variables only, substantive conclusions based on the results presented here should be treated with care. On the other hand, our main objective is not a substantive one, but to illustrate the relationship that has been derived in the previous paragraph. Table shows the sample mean vector and the covariance matrix for the four measurement occasions of antisocial behavior and the measure of cognitive stimulation. Table Sample Means and Covariance Matrix of Antisocial Behavior (t - t 4 ) and Cognitive Stimulation. Antisocial behavior t t t 3 t 4 Cognitive Stimulation Mean vector Covariance matrix t.707 t t t Cognitive stimulation Note. N = 6; Source: Curran (997). The growth curve models with different time axes are fit to the covariance matrix and mean vector of Table using Mplus.04 (Muthén & Muthén, 998). If the values of the basis function are changed systematically, Table shows that the growth parameters change accordingly. It displays the relevant parameter estimates of the six fitted

12 3 MPR-Online 003, Vol. 8, No. models with different specifications of the time axis in the basis function. The fitted growth models are equivalent, since they all yield the same expected covariance matrix. As a consequence, they all have the same fit measures [χ (7, N = 6) = 6.95, p =.44; RMSEA =.000]. The difference between the models is found in the specification of the intercept of the basis function for the growth rate (λ i ). Before interpreting the results, we first test the assumption of common causation as suggested by Mandys et al. (994). Latent growth curve analysis implicitly assumes that both the mean structure and the covariance structure are to be produced by the same factors. This implies that the variation between individuals (within-group variance) must be produced by the same factors as the variation between the means (betweenoccasion variance). Common causation may not always be present, however. As advocated by Mandys et al., the assumption should always be tested whenever means are included in the model, as is the case in latent growth curve analysis. The test of common causation is performed by comparing the goodness of fit of a model with, and without a mean structure. Since models with mean structures are nested in models without mean structures, the χ difference goodness-of-fit test can be used. Using goodness-of-fit indices of a model with and without a mean structure specified, the hierarchical likelihood ratio may test the assumption of common causation. The model without mean structure has the following fit measure χ (5) = 4.59, p =.47. When computing the difference with the mean structure (from Table ), a chisquare difference of χ () =.34, p =.3 results for the test of common causation. Therefore we may conclude that there is no serious deterioration of the model fit when means are included into the model, and that the assumption of common causation becomes plausible. Given common causation, we may focus on transformations of the basis function by adding a constant. In other words, the effect of changing the point in time of the initial status of the growth process gets into focus. The transformation of the basis function with a constant may have several rationales. If the start of the growth process corresponds to the time of the first measurement occasion, then Model has to be used. Model, 3 and 4 specify models in which the origin of the growth process is located prior to the first measurement occasion. Technically it is possible also to center the time scale (model 5), or to use the last measurement occasion as the initial status (model 6).

13 Stoel & van den Wittenboer: Time Dependence of Growth Parameters 33 As can be estimated (or computed by hand from Equations to 5), the growth parameters change if the basis functions take other values. Table shows clearly that a corresponding change can be observed in the residual variance of the initial status ( ), the residual covariance between initial status and growth rate ( ), and the effect of the covariate on the initial status ( 0 ). Also included in the model are the expectations of the initial status and growth rate (ν 0 and ν respectively). Change of ν 0 is shown as well. Table. Maximum Likelihood Estimates of the Parameters of the Fitted Growth Models with Different Basis Functions Parameter Model Model Model 3 Model 4 Model 5 Model 6 λ i [0,,, 3] [.57,.57,.57, 3.57] [,, 3, 4] [0,,, 3] [-.5, -.5,.5,.5] [-3, -, -, 0].4 (5.3).8 (3.74).05 (3.0).85 (.99).745 (9.34).857 (7.99).3 (.6).3 (.6).3 (.6).3 (.6).3 (.6).3 (.6).075 (.88).000 (.00) (-.45) -.38 (-.9).7 (4.9).469 (4.0) (-.4) (-.43) (.97).76 (.55) -.33 (3.7) -.87 (-3.86) (-.) (-.) (-.) (-.) (-.) (-.) ν 0.65 (7.09).59 (4.56).465 (.69).6 (.6).839(9.9).064 (6.53) ν.50 (3.68).50 (3.68).50 (3.68).50 (3.68).50 (3.68).50 (3.68) Note. Residual variances are invariant across the 6 models [ε =.50 (6.57); ε =.64 (9.37); ε 3 =.36 (8.5); ε 4 =.43(5.7)]. The models are identified with their basis function [t, t, t 3, t 4 ]. The first element of the basis function equals of equation 5. The column of Table, belonging to Model, shows the parameter estimates for the model in which the first measurement occasion gets the value of zero. It represents the growth model with the origin of the process (the true initial status) defined at the first measurement occasion. We take this model as the base model to compare with. Since the values of the basis function are specified as [0,,, 3], the parameters have the straightforward interpretation that the first measurement occasion corresponds with the true initial status. In addition, no assumption has to be made about the growth curve

14 34 MPR-Online 003, Vol. 8, No. outside the observed time interval. From a statistical perspective this specification delivers the least complicated model, and with respect to substantive goals it has the easiest interpretation. The true growth curve has expectations.65 for the initial status (ν 0 ) and 0.50 (ν ) for growth rate. Table shows significant variation from these expectations ( =.4 and =.3), and a significant negative effect of the covariate on both the initial status and growth rate ( 0 = and = -.036). Using the parameter estimates of model, it is possible to compute t 0, the time point where equals zero (see Equations 6b and 7): =.075/.3 =.57; t 0 = = As can be seen t 0 is located 0.57 time units prior to the first measurement occasion. Column Model of Table shows the parameter estimates for the growth model with t 0 = -.57 defined as the true origin of the process. The estimated parameter values could also be computed by hand from Equations to 5. As can be seen, equals zero, and the residual variance of the initial status factor reaches the minimum value of.8, as predicted. The effect of the covariate on the initial status is not significantly different from zero ( 0 = -.060), and the estimate of the initial status is smaller (ν 0 =.59) than in the base model. Thus, in contrast to model, model shows no effect of the covariate on the initial status. Model 3 and Model 4 of Table define the initial status prior to the first measurement occasion and prior to t 0. The basis function of Model 4 represents a time scale with an initial status of 0 time units prior to the first measurement occasion. This would be a setting where the measures are taken at ages 0,,, and 4. As shown by Equations to 5, the residual variance of the initial status increases, and the residual covariance between growth rate and initial status as well as the estimate of the initial status change from negative to positive. A similar pattern emerges for Models 3, though at intermediate values in which the estimate of the initial status is not negative yet. On the other hand, if the initial status is defined later than t 0 (Model 5 and Model 6), the residual variance of the initial status increases again (showing that t 0 has minimum variance indeed). The residual covariance and the estimate of the initial status turn out to be increasingly positive by now, and the effect of the covariate on the initial status turns out to be negative. A more detailed picture of the relationship is found in Figure to 4, where plots based on the data analysis are shown of estimated parameter values (y-axis) against values of (x-axis). The relations of 0 and with in Figure and 3 are perfectly linear. They show that the relationship, already postulated in Equations and 4 re-

15 Stoel & van den Wittenboer: Time Dependence of Growth Parameters 35 spectively, will be found exactly when is shifted on the time axis. Figure 4 depicts the relation between and, which is exactly the relationship according to Equation 3. reaches a minimum value of.8 at =.57. The residual covariance between the growth rate and the initial status, as well as the residual variance of the initial status and the relation of a covariate with the intercept react exactly on shifts of on the time axis as predicted by the derived model equations Figure. Plot of the parameter estimates of against Figure 3. Plot of the parameter estimates of against.

16 36 MPR-Online 003, Vol. 8, No Figure 4. Plot of the parameter estimates of 0 against. Discussion Rogosa and Willett (985) present a detailed discussion of the correlation between initial status and growth rate (see also Garst, 000; Mehta & West, 000; Rovine & Molenaar, 998). The argument is elaborated further in Rovine and Molenaar, showing that the covariance also depends on the time scale if basis function values are used to estimate the shape, or basis function, of the latent growth curve (Meredith & Tisak, 990). The analysis in this paper shows explicitly, that the argument must be extended to the effects of covariates: effects of covariates on the growth parameters in linear latent growth curve models depend on the time scale involved. Although latent growth curve methodology has been presented as a flexible framework to analyze longitudinal data, with the possibility to analyze systematic differences in growth by including covariates, one should apparently bear in mind that the estimates of the population parameters are sensitive to the incorporated time scale. Since maximum likelihood is a scale-free estimation method (Long, 984, p. 58), transformations of the factor loadings will completely be absorbed by corresponding changes in the factor (co)variances. They, therefore, yield the same expected covariance matrix, and thus the same model fit. This type of factor indeterminacy problems have been voiced in the past for the general latent variable model (McArdle & Cattell, 994; McDonald, 985). However, we believe we make a unique contribution by showing how, and why, this problem persists in linear latent growth curve modeling.

17 Stoel & van den Wittenboer: Time Dependence of Growth Parameters 37 While the time scale may be selected arbitrarily, for example by taking the first measurement occasion as representing the initial status, the interpretation of the corresponding parameter estimates is certainly not arbitrary. It is inextricably bound to a specific time scale and a linear transformation of that scale leads to a change of all parameters involved. It creates the same sort of confusion that has already been noted in the area of regression analysis (Aiken & West, 99; Mehta & West, 000). Besides effects on the residual variance and the grand mean of the initial status factor, as well as the residual covariance between the initial status and the growth rate, the results of this paper show clearly that effects of exogenous covariates on the initial status have a conditional interpretation; i.e. conditional on the selected time of initial status. Statistically these parameters can be estimated easily, but their theoretical interpretation is ambiguous. Why is the residual covariance between growth rate and the level computed at this specific point in time? Why is the first measurement occasion taken as representing the initial status? A strong substantive argument is needed to answer these questions. An example might come from intervention studies, where change as a result of the intervention is expected only after the intervention. If such an argument cannot be presented, the parameters represent nothing more than the estimates at an arbitrary selected level of the process. A change in the time of the first measurement might have serious consequences for the substantive conclusions. Since part of the confusion might be due to the term initial status, it may be useful to employ the alternative term level (McArdle & Hamagami, 99; Rovine & Molenaar, 998). Although one can use this, in principle, as an advantage by studying how the relationship of the estimated level of an outcome with a covariate changes over time. The gain in knowledge is limited because the changing relationship of the level and the covariate is completely determined by its prior estimated value and the change of the time scale (Equation ). In addition, once the growth model has been specified, and estimates of the level and growth rate factor have been obtained. The variance of the level, at any point of the process, can be computed by hand (Equation 3). If one is really interested in investigating the relation of the level of a variable and covariates, autoregressive models might be more appropriate. In addition, if the interest is in the true change over consecutive occasions and interfering covariates, true individual change models (McArdle, 00; Steyer et al., 997; Steyer et al., 000) might even be a better alternative. In true individual change models, the growth curve parameters are replaced by latent difference variables (e.g., the latent difference between adjacent occasions).

18 38 MPR-Online 003, Vol. 8, No. Subsequently, these latent difference variables can be explained by covariates. True individual change models are beyond the scope of this paper, however, because time is incorporated in a different way than in latent growth curve models. In linear latent growth curve models, the question is, essentially, whether there exists a real initial status, or worded differently, if time is measured on a ratio scale. If the assumption of time being measured on a ratio scale has been violated, the initial status can only be identified arbitrarily. In this case, growth already took place before the first measurement occasion. In other words, subjects will have reached their level at the time of the initial status, partly because of their past growth. The situation is much different, however, if time is measured on a ratio scale. Then, the process under investigation has a natural origin corresponding to the first occasion of measurement (e.g., intervention research). If the subjects do not have the same level at the outset, this will show up in the residual variance of the initial status. In that case, the effect of covariates, as well as the residual variance of initial status and the residual covariance with growth rate and the effect of covariates, have meaningful interpretations. As other researchers already pointed out in the case of non-equal linear growth rates (Garst, 000; Rogosa & Willett, 985) there is no such thing as the covariance between initial status and change. Any estimate of the residual covariance between initial status and growth rate depends also on the selected basis function. Whether one assumes that the data are generated by an underlying process that has a particular covariance structure for the latent variables, one can only expect the correct correlation when the basis vector has been scaled in an opportunistic fashion (Rovine & Molenaar, 998, p. 0). Over and above that, this paper shows that the effects of covariates must be included in this argument. The effects of covariates on the growth parameters in the linear latent growth curve model depend also on the time scale of the growth model. Furthermore, it has been made clear that there exists a point in time t 0 in any collection of growth curves, where the residual covariance between initial status and growth rate must be equal to zero. At this point t 0 the residual variance of the initial status reaches its minimum value. So we provide an extra argument for Mehta and West (000) who propose t 0 to be thought of as an additional parameter of any collection of linear growth curves. Using the results of Rovine and Molenaar (998), and the equations of this paper for the linear latent growth curve model, it can be easily shown that the argument holds true also for latent growth curve models with an estimated basis function. For latent

19 Stoel & van den Wittenboer: Time Dependence of Growth Parameters 39 growth curve models including higher-order polynomials (e.g., a quadratic term) to control for the nonlinearity, the situation is more complicated, however. The equations of this paper will not be sufficient to describe the effect of changing the initial status on the model parameters. More specifically, in a quadratic model the variance of the level, the covariance between the level and linear component, and the effect of the covariate on the level will change more complicated than described by Equations, 3 and 4. Also, in a quadratic model, the variance of the linear component, and the effect of a covariate on the linear component will change as a function of the time of initial status. To conclude, it is apparent that problems of arbitrary time axes in the linear latent growth curve model only arise if researchers overreach their goals in interpreting the parameters related to the initial status. The initial status, however, is meaningful only for growth processes with a natural origin. If the first measurement occasion (or any other occasion) is arbitrarily defined as the initial status, it will be advisable to abandon substantive interpretation of the corresponding parameters, and to focus exclusively on the growth rate. After all, it is growth, what we are after in latent growth curve modeling. References Aiken, L. S., & West, S. G. (99). Multiple regression: testing and interpreting interactions. Newbury Park, CA: Sage. Baker, P. C., Keck, C. K., Mott, F. L., & Quinlan, S. V. (993). NLSY child handbook: a guide to the National Longitudinal Survey of Youth child data. Columbus, OH: Center for Human Resource Research. Bollen, K. A. (989). Structural equations with latent variables. New York: Wiley. Chan, D., Ramey, S., Ramey, C., & Schmitt, N. (000). Modeling intraindividual changes in children's social skills at home and at school: a multivariate latent growth approach to understanding between-settings differences in children's social skill development. Multivariate Behavioral Research, 35, Curran, P. J. (997). The bridging of quantitative methodology and applied developmental research. In P. J. Curran (Chair), Comparing three modern approaches to longitudinal data analysis: An examination of a single developmental sample. Symposium conducted at the 997 meeting of the Society for Research on Child Development, Washington, DC.

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