Journal of Pediatric Psychology Advance Access originally published online on September 21, 2005
Journal of Pediatric Psychology 2006 31(10):1002-1023; doi:10.1093/jpepsy/jsj074
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Applications of Individual Growth Curve Modeling for Pediatric Psychology Research
1 University of Illinois at Chicago and, 2 University of Maryland, Baltimore County
All correspondence concerning this article should be addressed to Christian DeLucia, PhD, Center for Treatment Research on Adolescent Drug Abuse, Department of Epidemiology and Public Health, University of Miami Miller School of Medicine, 1400 NW 10th Avenve Suite 1107B, Miami, Florida 33136. E-mail: cdelucia{at}med.miami.edu.
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Abstract |
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Abstract
What are Individual Growth...Objective To provide a brief, nontechnical introduction to individual growth curve modeling for the analysis of longitudinal data. Several applications of individual growth curve modeling for pediatric psychology research are discussed. Methods To illustrate these applications, we analyze data from an ongoing pediatric psychology study of the possible impact of spina bifida on child and family development (N = 135). Three repeated observations, spaced by approximately 2 years, contributed to the analyses (M age at baseline = 8.84). Results Results indicated that individual linear growth curves of emotional autonomy varied as a function of the youth gender by spina bifida group membership interaction. Conclusions Strengths of individual growth curve modeling relative to more traditional methods of analysis are highlighted (e.g., completely flexible specification of the time variable, explicit modeling of both aggregate-level and individual-level growth curves).
Key words: Growth curves; trajectories; longitudinal modeling; statistical methods.
The purpose of this article is to provide a brief, nontechnical introduction to individual growth curve modeling for the analysis of longitudinal pediatric psychology data. As will be discussed, individual growth curve models are extremely flexible and offer pediatric psychology researchers several advantages over traditional methods for analyzing longitudinal data, such as the repeated measures analysis of variance model (ANOVA). Individual growth curve modeling (e.g., Rogosa, Brandt, & Zimowski, 1982
; Willett, 1997) is one of several names used in referring to this general class of statistical techniques. Other names include random regression models (Gibbons et al., 1993; Laird & Ware, 1982), hierarchical linear models (Bryk & Raudenbush, 1987), multilevel models (Snijders & Bosker, 1999), and mixed linear models (Littell, Milliken, Stroup, & Wolfinger, 1996). Although these various models are functionally equivalent for longitudinal data analysis, nomenclature varies mainly as a function of which model characteristic is being highlighted. (We discuss these issues in more detail below.) |
What are Individual Growth Curve Models? |
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Generally speaking, individual growth curve models allow researchers to measure change over time in a phenomenon of interest (e.g., response to treatment) at both the aggregate (i.e., population) and individual (i.e., study participant) levels. Historically, researchers have been more likely to model change as an "incremental" process (e.g., through the use of change score analysis from two-wave designs, Willett, 1988
). This may have arisen due to misconceptions regarding how to adequately model change as much as due to software limitations (Gibbons et al., 1993; Rogosa, 1988; Willett, 1988). In the past 20 years, statistical procedures that model change in a completely flexible manner have been developed (Bryk & Raudenbush, 1987; Gibbons et al., 1993; Hedeker & Gibbons, 1996a; Laird & Ware, 1982; Rogosa et al., 1982; Willett, 1988). The individual growth curve model is an outgrowth of this tradition and, relative to traditional analytical techniques, represents a monumental leap in the analysis of change for several reasons: (a) change can be modeled in an extremely flexible and realistic manner; (b) in addition to aggregate-level growth curves, individual-level growth curves are explicitly modeled as well; and (c) the approach allows researchers to explore rich hypotheses, thereby helping to close the gap between conceptual and theoretical and statistical models. |
A Hypothetical Basic Research Application: Examining Treatment Adherence |
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To assist with understanding individual growth curve modeling at a conceptual level, we consider a hypothetical example in which a researcher is interested in studying the natural history of treatment adherence in a sample of youth with cystic fibrosis. By repeatedly interviewing a sample of such youth over a 5-year period, researchers could estimate developmental growth curves of adherence for each youth. Figure 1 presents hypothetical treatment adherence data for four individuals with cystic fibrosis. On the y or vertical axis is an adherence scale ranging from 0 to 10 (with higher scores indicating higher levels of adherence). On the x or horizontal axis is a "time" variable representing participant age, ranging from 10 to 14. There are five estimated growth curves displayed in the figure. The solid lines represent estimated growth curves for each individual; each curve has its own intercept (defined as score on adherence at age 10, ranging from 0 to 4) and slope (defined as rate of change in adherence per one year of age, ranging from .7 to 1.75). The dashed line is the estimated population, or average, growth curve. Parameter estimates are presented for the population (i.e., the dashed line with ß00 and ß10 for intercept and slope, respectively) and for one participant (i.e., individual 1 with b01 and b11 for intercept and slope, respectively).
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Our hypothetical data allow us to illustrate several important features of individual growth curve models. First, the population, or average, growth curve carries aggregate-level information. As such, we know that the average 10-year-old with cystic fibrosis scored a 2.13 on treatment adherence (i.e., the estimate of the population intercept) and gained 1.19 units of treatment adherence per year through age 15 (i.e., the estimate of the population slope). Second, individuals are allowed to deviate or vary from this population growth curve. In Fig. 1, all four individuals vary from the population intercept (i.e., have adherence scores other than 2.13 at age 10) and slope (i.e., have slopes other than 1.19). As we illustrate below, if individual variability in the growth curve parameters is present (as depicted in Fig. 1), this variability might be predicted from theoretically meaningful variables (e.g., parental monitoring). While considering Fig. 1, we turn to a presentation of the unconditional linear growth model.
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The Unconditional Linear Growth Model |
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A straightforward way to conceptualize growth curve models is as two levels of analysis (Bryk & Raudenbush, 1987
; Singer & Willett, 2003). The level 1 model is commonly referred to as the within-person or intra-individual change model. As we discuss in detail below, these names highlight the central feature of the level 1 model—it captures person-specific (i.e., individual) growth rates. Time-varying predictor variables (e.g., age, time elapsed since treatment, level of symptomatology) can be included in the level 1 model. The level 2 model is commonly referred to as the between-person or inter-individual change model. These names highlight the central feature of the level 2 model—it captures between-person variability in the growth rates. Time-invariant predictors (e.g., gender, ethnicity, level of symptomatology at a given point in time—say at the baseline assessment) can be included in the level 2 model. A linear growth model with no additional time varying or time invariant predictors appears as:
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Although it is useful to conceptualize the model in this "multilevel" framework, in practice a single integrated model is estimated. This integrated model can be formed by substituting the level 2 model into the level 1 model, resulting in Yij = (ß00 +
0i) + [(ß10 +1i) (timeij)] + eij. Singer and Willett (2003) provide extensive discussion of this multilevel conceptualization and the resulting integrated model in chapters three and four.
The Level 1 Model
The level 1 model resembles an ordinary least squares (OLS) regression model (OLS regression is hereafter referred to as basic regression). An outcome (i.e., Yij, where the subscripts "i" and "j" denote person and measurement occasion, respectively) is written as a function of an intercept (i.e., b0i), plus the multiplication of a slope parameter (i.e., b1i) by a predictor variable (i.e., timeij) and a residual (i.e., eij). In this model, the two regression parameters representing the intercept and slope (i.e., b0i and b1i) carry the person-level subscript "i." As such, these parameters are allowed to vary across individuals (i.e., can take on different values for different individuals). This feature of the model represents a marked departure from the basic regression model, in which parameters are assumed to be fixed for all individuals in the sample (or for all individuals belonging to a particular group, in the event that the regression lines are nonparallel across groups).1 Although we have included only a single predictor in our level 1 model (i.e., timeij), additional time-varying predictors can be included in the level 1 model (e.g., level of symptomatology), thus further reflecting the flexibility of the procedure. For a discussion of the inclusion and interpretation of time-varying predictors, see Singer and Willett (2003, chap. 5). The level 1 model can be expanded to include curvilinear growth forms as well (e.g., quadratic, cubic). For example, to examine a quadratic growth form (i.e., a curve characterized by one bend), the level 1 model could be rewritten as follows: Yij = b0i + b1i (timeij) + b2i (timeij)2 + eij. In this equation, b2i carries information about the quadratic effect.
The Level 2 Model
In the level 2 model, it is conceptually helpful to consider the individual parameter estimates from the level 1 model (i.e., b0i and b1i) being treated as outcomes. For example, intercept of person "i" (i.e., b0i) is written as a function of a population intercept (i.e., ß00) plus his or her deviation from the population intercept (i.e., 0i). Similarly, slope of person "i" (i.e., b1i) is written as a function of a population slope (i.e., ß10) plus his or her deviation from the population slope (i.e., 1i).
Fixed Effects
In the level 2 model, the population-level estimates (i.e., ß00 and ß01) are referred to as the "fixed" effects. Similar to the basic regression model, these effects are assumed fixed (i.e., constant) for all individuals in the sample.
Random Effects
The individual deviations (i.e., 0i and 1i), which can be thought of as the level 2 residuals, are referred to as the random effects.2 Indeed, it is the estimation of these deviations (of individuals from the population curve) that puts the "individual" in individual growth curve modeling. The term "random" here gives rise to another name for the procedure-random regression models. Moreover, the mixture of fixed and random effects in a single model gives rise to other names for the procedure, for example, mixed regression models, mixed linear models.
Including Predictors in the Level 2 Model
To the extent that individual growth curves vary from the population estimates, the variability associated with the random effects (i.e., 0i and 01) will be deemed statistically significant (we discuss this in more detail below). Indeed, an early task in individual growth curve modeling is establishing whether variability in the growth curves is present. Upon establishing this significant variability, the level 2 model can be expanded to include time-invariant predictors of variability in the individual growth curves. For example, in expanding our earlier hypothetical example, we might hypothesize that baseline level of parental monitoring of adherence behaviors is associated with both intercept and slope variability, such that youth exposed to higher levels of baseline parental monitoring will exhibit higher levels of adherence at age 10 (i.e., higher intercepts) and, to a lesser degree, differing growth rates of adherence (i.e., differing slopes). We could test this hypothesis empirically by expanding the level 2 model to include baseline parental monitoring as a predictor of both intercept and slope variability. The new conceptual equations appear as:
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In the above level 2 model, we are stating that each individual’s intercept (b0i) and slope (b1i) estimates are a function of three components: (a) the population estimate (ß00 for intercept and ß10 for slope), (b) his or her score on baseline parental monitoring, and (c) an individual deviation (0i for intercept and v1i for slope). An interesting observation that follows from the level 2 model for slope variability is that the predictors added in this model form interactions with the time variable from the level 1 model (this is more apparent when the level 2 model is substituted into the level 1 model). As such, in our present example, the parental monitoring effect on slope variability, captured by ß11, is actually a test of the (baseline parental monitoringi x timeij) interaction or the extent to which the linear effect of time varies as a function of baseline parental monitoring. (As shown in the Appendix, SAS syntax to request similar effects involves explicit specification of the predictor by time interaction.)
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A Hypothetical Applied Research Application: Participant Response to an Intervention |
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In Figs 2 and 3, we depict two possible applications of individual growth curve models for intervention researchers. These data are based on a hypothetical 4-week intervention to increase treatment adherence among youth with cystic fibrosis. For illustrative purposes, suppose some affected youth were randomized to receive a family-based intervention to cultivate and maintain adherence behaviors, while the remaining youth were randomized to a no-treatment assessment (i.e., control) condition.
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Testing Main Effects of Treatment
In Fig. 2, we present data that can help answer the question: "Does the intervention improve adherence behaviors?" The y axis contains a treatment adherence scale (with higher scores indicating higher levels of adherence), whereas the x axis contains the time variable (i.e., a measurement occasion scale: 0, "baseline"; 1, "four weeks post-baseline at intervention completion"; and 2, "eight weeks post-baseline at one-month follow-up"). All of the lines in the figure represent estimates of individual growth curves. The treated individuals, represented by solid lines, show greater rates of improvement than do the control participants. These data are conceptually consistent with a main effect of treatment on adherence behaviors. In the individual growth curve-modeling framework, the intervention effect could be tested by including a dichotomous treatment predictor variable (0, "control"; 1, "treatment") in the level 2 model of slope variability. Given Fig. 2, we would expect the coefficient for this predictor to be positive, suggesting that on average, treated participants are growing more rapidly in adherence than are control participants.
Testing Moderated Treatment Effects
In Fig. 3, we present data that can help answer the question: ‘For whom does the intervention work best?’ Fig. 3 takes on the same general form as Fig. 2 with one primary exception. At baseline, we depict two general "clusters" of individuals—youth who are reporting fairly low levels of adherence behaviors and youth who are reporting fairly high levels of adherence behaviors (irrespective of treatment condition). For individuals who report fairly low levels of adherence behaviors at baseline, there appears to be an intervention effect on slope variability; treated youth are growing more rapidly in adherence behaviors than are control youth. There appears to be no intervention effect, however, for youth who report higher levels of adherence at baseline (i.e., treated and control participants are growing minimally and at about the same rate in their adherence behaviors). In the context of individual growth curve modeling, we would model this intervention by baseline interaction by including three predictors in our level 2 model of slope variability: (a) treatment, (b) baseline level of adherence, and (c) the treatment by baseline-adherence interaction. Given Fig. 3, we would expect a significant treatment by baseline-adherence interaction. We would then probe the interaction following guidelines for basic (fixed effects) regression suggested by Aiken and West (1991).
Testing Mediated Treatment Effects
Individual growth curve modeling can also be used to help pediatric psychology researchers examine treatment-process questions. The question here becomes ‘How does the intervention work?’ Directly on the outcome? Indirectly through an intermediate variable? In psychology, such process questions are typically discussed in the context of mediation analysis (for examples, see Baron & Kenny, 1986; Holmbeck, 1997, 2002; MacKinnon, 1994; Mackinnon, Lockwood, Hoffman, West & Sheets, 2002; West & Aiken, 1997). For example, an intervention designed to improve treatment adherence among a sample of youth with cystic fibrosis, might attempt to improve youth adherence, at least in part, by improving parental monitoring of adherence. If youth were randomly assigned to a family-based intervention condition or to a no-intervention control condition and multiple waves of data were collected on both parental monitoring (i.e., the mediator) and youth adherence (i.e., the outcome), individual growth curve modeling could be used to explore mediational hypotheses. Although the specification of such models is more advanced and beyond the scope of this article, readers are referred to relevant illustrations by MacKinnon and colleagues (e.g., Choeng, MacKinnon, & Khoo, 2003; Krull & MacKinnon, 2001), and by McArdle and colleagues (Ferrer & McArdle, 2004; McArdle et al., 2004).
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Advantages of Individual Growth Curve Modeling |
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Relative to traditional techniques for the analysis of longitudinal data (e.g., ANOVA), individual growth curve modeling offers many advantages including analysis at the individual level, complete flexibility in the treatment of the time variable, responses to missing data, ability to handle clustered data, and generalizations to non-normally distributed outcomes.
Data are Truly Modeled at the Individual Level
First, as depicted in the above illustrations, data are truly modeled at the individual level, allowing for the examination of individual variability in intercepts and rates of change in the phenomenon under investigation (Bryk & Raudenbush, 1992; Gibbons et al., 1993; Willett, 1997). Generally speaking, the ANOVA approach focuses on growth curves at the aggregate level. Although the ANOVA model allows for individual variability in intercepts, individual variability in rates of change is not explicitly modeled (Gibbons et al., 1993). Figure 4 shows the same hypothetical data presented in Fig. 1 based on the ANOVA. The primary difference between Figs 1 and 4 is that in the latter figure all of the lines have the same slope (i.e., the rate of linear change in treatment adherence is assumed to be equal for all individuals). Given that a basic tenet of developmental theory is that individuals vary in their rates of development over time, eliminating this variability will often fail to capture the richness of the data. Intercept and slope estimates are presented again for participant 1. The estimated intercept remains the same (0.00), whereas the slope estimate has decreased from Fig. 1 by 0.56 units (a 32% reduction) to 1.19 (the population estimate).
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Flexible Treatment of the Time Variable
A second strength of individual growth curve modeling relative to the ANOVA approach is the complete flexibility with which the time variable can be treated (Bryk & Raudenbush, 1992
; Gibbons et al., 1993). For example, time can be treated continuously (e.g., representing chronological age or time since entry into the study) rather than as a factor with a relatively finite number of levels. Also, individuals can be observed at different time points (as opposed to assuming that all individuals were measured on the same occasions). (We discuss some very important implications of this added flexibility of the growth curve modeling approach below.)
Easy Handling of Missing Data
Third, in individual growth curve models, participants with missing data at one or more time points can be retained in the analysis (if the data are missing by design, completely at random, or are ignorable—i.e., can be modeled as a function of observed covariates; see Gibbons et al., 1993). (Missing data might also be able to be handled through a "pattern-mixture" approach, even if missingness is nonignorable, see Hedeker & Gibbons, 1997.) Although ANOVA models can handle cases with partial missing data, researchers often enforce listwise deletion (i.e., a participant is dropped from the analysis if she/he is missing any of the repeated observations) (Bryk & Raudenbush, 1992; Gibbons et al., 1993).
Models can Easily Incorporate Three Levels of Data Nesting or Clustering
Fourth, because the individual growth curve model is a specific form of a multilevel model—in which the repeated observations are "nested" or "clustered" within individuals—individual growth curve models can be easily generalized to include additional levels of nesting (e.g., individuals nested within broader structures, such as classrooms, clinics, hospitals, or group homes) (Bryk & Raudenbush, 1992; Singer, 1998). As mentioned above, an additional commonly used name for individual growth curve modeling is hierarchical linear modeling or HLM. This naming convention emphasizes the multilevel or hierarchical structure of the model (Bryk & Raudenbush, 1992). Bryk and colleagues have also developed software bearing the same name, which is commonly used to analyze these models (Bryk, Raudenbush, & Congdon, 1996).
Generalizations to Nonnormal Data Exist
Fifth, individual growth curve models can be easily generalized for use with nonnormal data (e.g., Hedeker & Gibbons, 1996a; Hedeker, 1999). As such, researchers could model growth in count, binary, ordered categorical, or nominal categorical data. This might be particularly useful for researchers interested in modeling growth over time in repeated observations of outcomes such as counts of pediatric admissions to hospitals over the past several years, remission status in pediatric cancer patients in the past 24 months, and monthly changes in symptom severity—coded as mild, moderate, and severe—among pediatric anxiety patients.3
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Real-World Example: Modeling the Development of Emotional Autonomy in Youth with Spina Bifida and Able-Bodied Comparison Youth |
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Next, we work through a series of relatively basic individual growth curve models to tie some of the concepts we have introduced above to actual longitudinal data in the area of pediatric psychology. These data are from an ongoing study—conducted by Holmbeck and associates—of the possible impact of spina bifida on child and family development (see Coakley, Holmbeck, Friedman, Greenley, & Thill, 2002
; Holmbeck et al., 2003; Holmbeck, Coakley, Hommeyer, Shapera, & Westhoven, 2002; Hommeyer, Holmbeck, Wills, & Coers, 1999).
Spina bifida is a relatively common birth defect, which occurs in approximately 1 in 1000 live births in the United States (Holmbeck et al., 2003). In children with spina bifida, the spinal cord fails to fully develop, resulting in exposure of a portion of the cord at birth (Holmbeck et al., 2003). In addition to common physical problems (e.g., sensory loss, bladder control problems), children with spina bifida are at increased risk (relative to able-bodied peers) for psychosocial problems as well (e.g., social immaturity, attention, and concentration problems) (Holmbeck et al., 2003). In this study, we examined growth over time in emotional autonomy from mothers (Steinberg & Silverberg, 1987) as a function of both spina bifida group membership and child gender.
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Methods |
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Sample
This sample is comprised of 67 children with spina bifida (n = 31 girls, 46.3% of spina bifida group) and 68 able-bodied comparison children (n = 31 girls, 45.6% of group). Child age at baseline (i.e., T1) was similar for children with spina bifida (M = 8.98, SD = .61) and able-bodied comparison children (M = 8.72, SD = .50). Child ethnicity was also similar for children with spina bifida, n = 12 non-Caucasian children (18%), and comparison children, n = 6 non-Caucasian children (9%).
Participant Recruitment
Children with spina bifida were recruited from four sources—a children’s hospital, a children’s hospital for youth with physical disabilities, a university-based medical center, and a statewide spina bifida association. Out of a possible participant pool of 310 children, 70 families were successfully recruited into the study. Other families were excluded for various reasons (e.g., distance from research base, failure to reach, refusal). Able-bodied comparison children were recruited from the schools attended by the children with spina bifida. Out of roughly 1,700 mailed recruitment letters, 72 families agreed to participate in the study. The intensive nature of the longitudinal study—which was explained in detail in the recruitment letters—accounts for some of this low response rate. Sample sizes were reduced to 68 youth in the spina bifida and comparison groups to facilitate matching on key demographic factors (e.g., age, SES, ethnicity). (For additional information on sample recruitment procedures, see Holmbeck et al., 2003.)
Procedure
For the purposes of this study, data collected on three measurement occasions were utilized. On average, the measurement occasions were spaced by approximately 2 years. On all occasions, interviews were conducted in participants’ homes by trained undergraduate and graduate psychology students. Interviews lasted approximately 3 h, and families were paid $50 at T1, $75 at T2, and $100 at T3 for their time and effort. Although interviews were conducted with parents and children, only child data are discussed in this aeticle.
Measures
Age
A continuous measure of child age in years was obtained on each measurement occasion using the child’s birth date and relevant interview date. This variable was used as the time variable in the growth curve models.
Gender
Child gender (0, "female"; 1, "male") was the sole demographic variable used in these analyses. This variable was used as a predictor in the growth curve models.
Group
Spina bifida diagnostic status (0, "able-bodied comparison youth"; 1, "youth with spina bifida") was ascertained at baseline. This grouping variable was used as a predictor in the growth curve models.
Emotional Autonomy from Mothers (Steinberg & Silverberg, 1987)
This measure captured the degree to which "childish" dependencies on mothers were relinquished by youth. Children were asked to rate how much they agreed with each of 14 statements (e.g., "My mother and I agree on everything," "I go to my mother for help before trying to solve a problem myself," "I try to have the same opinions as my mother"). Response options included 1 (strongly agree), 2 (agree somewhat), 3 (disagree somewhat), and 4 (strongly disagree). Relevant items were reverse coded such that higher scores indicate higher levels of emotional autonomy. The measure was administered at all three interviews. Reliability coefficients (based on Cronbach’s alpha) across the three time points were .63, .75, and .80, respectively. We estimated growth in emotional autonomy, as discussed below.
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Results |
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In these analyses, we were interested in describing growth over time in emotional autonomy from mothers during a 4-year period of preadolescent and adolescent development (i.e., on average, ages 9–13). Three waves of longitudinal data contributed to the models described below.
Descriptive Data
Descriptive data for the three repeated observations of emotional autonomy and participant age are summarized in Table I. These data are presented by measurement occasion to give the reader a general feel for the data. It is helpful to keep in mind, however, that in the growth models described below, growth in emotional autonomy is modeled as a function of participant age, not measurement occasion. These descriptive data suggest that scores on the emotional autonomy measure—at least at the sample level—increased over the three measurement occasions. Skew and kurtosis values, which help assess the degree of univariate normality of the individual measures, suggest that the measures are normally distributed. Moreover, the correlations among these measures show a classic pattern of diminishing magnitude over time (i.e., a violation of compound symmetry).
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Figure 5 shows raw data trajectories for 20 individuals selected at random from the data set. The x axis contains the age variable, which ranges from 8 to 15; the y axis contains the emotional autonomy scale, which ranges from 1 to 4. Two important pieces of information can be gleaned from these data. First, there appears to be variability in individuals’ initial scores on emotional autonomy (typically measured between the ages of 8 and 9). Second, there appears to be variability in how individuals are changing over time in emotional autonomy. Some individuals are clearly increasing in their levels of emotional autonomy over time, some individuals are decreasing, and still others are showing a more stable pattern (i.e., not changing much at all). It is this variability (in starting points and rates of change) that will be explicitly modeled using individual growth curve models. Some of these individual trajectories appear to be quadratic in nature, suggesting perhaps that a curved line with a single bend might be a better representation of the data than would a straight line. As such, we will test for both linear and quadratic forms of growth—at least at the aggregate- or population-level.
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Fitting Individual Growth Curves of Emotional Autonomy from Mothers
To assess change over time in the development of emotional autonomy from mothers, we estimated individual growth curve models using the Mixed Procedure in SAS statistical software. For a basic primer on the Mixed Procedure, see Singer (1998)
; a more comprehensive discussion of the procedure is presented by Littell et al. (1996).
When estimating change in a phenomenon, it is of interpretational benefit to identify a meaningful metric of the time variable (i.e., participant age) (Biesanz, Deeb-Sossa, Papadakis, Bollen, & Curran, 2004). For example, in this study, if age was left in its original metric, the intercept would be interpreted as the level of emotional autonomy when the participant age was zero; not a substantively meaningful interpretation. We elected to scale age (the time variable) such that the zero point corresponded to a value of age 9 (i.e., subtracted 9 from all participants’ ages at each measurement). Thus, intercept estimates are interpreted as emotional autonomy at age 9; a value that corresponds to the average age of individuals at entry into the study.
Before evaluating potential predictors of change in a phenomenon, it is important to ensure that the individual growth curve model is correctly specified. This includes identification of the correct form of growth (e.g., linear, quadratic), as well as correct specification of the variance of the individual growth estimates (e.g., intercept and slope). As we discuss in detail below, it is possible to statistically test these model estimates. If significant variability in the growth curves is present (e.g., individuals vary in their intercepts and slopes), it might be possible to predict this variability from theoretically meaningful variables (e.g., spina bifida status).
Correct model specification typically includes evaluating a series of models to determine which model results in the best relative fit to the data. Similar to other analytical techniques (e.g., logistic regression, path analysis), it is possible to directly compare two nested individual growth curve models. Two models are nested when one model (the more parsimonious) can be "created" from another model (the more complex) by not estimating one or more parameters. The relative fit of two nested models can be compared by evaluating the difference in –2 log likelihood (–2LL) between the models (similar to logistic regression). The df for the test will reflect the number of restrictions (e.g., nonestimated parameters between the models). As these change statistics follow a 
2 (chi-square) distribution, we hereafter refer to these tests as 2 tests.4
Many increasingly complex hypotheses can be directly tested with individual growth curve models, thus some nested models might be estimated. Hypotheses to be tested in this example include Do individuals significantly differ from each other in their average levels of emotional autonomy? Do individuals display growth in emotional autonomy? Do all individuals grow at the same rates? and Is a linear progression an adequate representation of the average form of growth? To test these hypotheses, we estimated five models (described below). In practice, the number of hypotheses (and thus, nested models) will vary as a function of the complexity of the researcher’s questions and study design.
The approach we present is to estimate increasingly complex models (e.g., not estimating vs. estimating the variance of the slope estimates) and test the associated change in 2 value resulting from the change in model specification. This procedure is analogous to including additional predictors in hierarchical multiple regression; the distinction is that we are focused on additional model estimates, rather than on additional predictors. The five models to be estimated are (a) fixed intercept, (b) unconditional means, (c) compound symmetry, (d) unconditional linear, and (e) fixed quadratic. The associated equations and summary information for each model are summarized in Table II.5
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Model 1: Fixed Intercept
This model serves as the baseline model. In this model, no growth parameters nor variance estimates in growth are specified. This model would fit the data only to the extent that all individuals reported similar (i.e., nonsignificantly different) levels of emotional autonomy and that these scores were stable over time.
Model 2: Unconditional Means
This model modifies Model 1 by allowing (estimating) individual level variance in the intercept scores. Model 1 is nested within, and thus can be compared with, this model. Model 2 assumes that each individual has similar scores at each of the time points, but allows these values to vary across individuals (i.e., each person’s growth "curve" is a flat line at the height of their estimated average value of emotional autonomy over time). From our observations of Fig. 5 (i.e., individuals growing over time), it is not likely that this model adequately captures true growth in this sample. However, this model does provide a significant improvement in fit relative to Model 1, 2(1) = 29.6, p < .0001. This supports the observation that individuals differ in their average level of emotional autonomy. On a more technical note, this model also allows the computation of the population autocorrelation (intraclass correlation) (Singer & Willett, 2003); a population based estimate of the average correlation of the repeated observations of emotional autonomy, which is .29 in this study.6
Model 3: Compound Symmetry
This model modifies Model 2 through the inclusion of a linear slope parameter (i.e., age is included as a level 1 predictor). Relative to Model 2, this model will test for the presence of linear growth in the sample, though it is a fixed effect as variability in the slopes is not estimated (i.e., all participants are forced to have the same rate of growth). This model is analogous to the ANOVA model with two notable differences: (a) time is allowed to be a continuous (rather than discrete) measure and (b) only the linear component of growth is estimated. (Results of the ANOVA model are presented below.) Model 3 provides a significant improvement in fit relative to Model 2, 2(1) = 56.5, p < .0001, suggesting that on average individuals are changing on emotional autonomy over time.
Model 4: Unconditional Linear
This model modifies Model 3 through estimation of two additional parameters; variance in the individual slopes and covariance between intercept and slope estimates. This model allows for the possibility that individuals grow at different rates (the slope variance estimate) and acknowledges that individuals’ initial standing may be related to their own amount of change (e.g., individuals initially high on emotional autonomy at age 9 might be expected to show less absolute growth than individuals initially low on autonomy). This model was an improvement over Model 3, 2(2) = 8.8, p = 006, suggesting that individuals vary in their rates of linear change in emotional autonomy.
Model 5: Fixed Quadratic
This model modified Model 4 through inclusion of a fixed quadratic component of growth (variance of this parameter was not estimated). The purpose of this model was to test whether linear growth was an adequate representation of the form of change in emotional autonomy in this sample. This model did not provide a significant improvement in fit relative to Model 4 however, 2(1) = 0.2, p = .66, suggesting that the form of growth is linear in nature at the average level.
Based on the decisions made above, we elected to retain our fourth model, in which growth in emotional autonomy is specified as linear, and individuals are allowed to vary in their intercepts (level at age 9), slopes (rates of change over time), and the relationship between these two values (intercept-slope covariance). Considering the population-level estimates of the unconditional linear growth model, the intercept estimate is 2.23, SE = .03, t(134) = 66.01, p < .0001, suggesting that on average, 9-year-olds score 2.23 on emotional autonomy from mothers. The fact that this quantity is significantly different from zero is not of practical value, given that the scale ranges from 1 to 4. The population slope estimate is .085, SE = .01, t(255) = 6.51, p < .0001, suggesting that on average, youth gain .085 units of emotional autonomy each year during the pre- and early-adolescent years.7
As discussed earlier, a primary strength of individual growth curve modeling is the ability to move beyond general information and focus on the individual variability in the growth curves. As stated above, significant individual variability in both intercept and slope estimates is present, suggesting that growth curves of emotional autonomy vary across individuals. Figure 6 shows the model-implied individual growth curves for the same 20 individuals whose raw data trajectories were presented in Fig. 5 (the figure also includes the population growth curve, denoted by the thicker line).
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A Note on Model Specification
An important issue to consider when fitting individual growth curve models has to do with the "complexity" of the model specification. In this set of analyses, we chose to present the results of five increasingly complex model specifications. In general, model complexity is determined by adding and/or subtracting various fixed and random effects. For example, the unconditional linear model is more complex than is the compound symmetric model, because the former allows for individual variability in the linear rates of change, whereas the latter does not. As such, the unconditional linear model attempts to capture a richness in the data that is not captured by the compound symmetry model. When the repeated observations arise from fixed measurement occasions (e.g., all participants are assessed on the same days), there are relatively straightforward "rules" that allow one to determine the upper bound of model complexity (e.g., it is possible to estimate "j–1" random effects, when j is the number of measurement occasions). When the spacing of measurement occasions varies by individual, as was the case in this study, determining the upper bound of model complexity is not as straightforward. Typically, in substantive research, the complexity of the model is determined in part by extant theory and in part by possible statistical constraints. In this article, we estimated many models to give the reader a feel for many possible model specifications, although we knew a priori that some models would not fit the data well (e.g., the fixed intercept model). Although a comprehensive discussion of model specification is beyond the scope of this article, relevant discussions can be found in Snijders and Bosker (1999
, chap. 12) and Longford (1993, p. 111).8
Examining Individual Growth Curves of Emotional Autonomy as a Function of Spina Bifida Group Membership and Child Gender
Given the presence of significant variability in the individual estimates (intercept and/or slope), many hypotheses may present themselves. Most generally, can any of the variability in these estimates be attributed to individual-level variables? In our next model, we examine whether individual variability in intercept and slope estimates can be accounted for by time-invariant predictor variables—spina bifida group membership, participant gender, and their interaction.
Model 7: Predicting Intercept and Slope Variability from Spina Bifida Group Membership, Participant Gender, and Their Interaction
In our final model, the level 2 model is expanded to include gender, group, and their cross product as predictors of both intercept and slope variability.9
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None of the three predictors of intercept variability (tested by examining the t tests for the three parameter estimates ß01, ß02, and ß03) were statistically significant (ps > .10). In contrast, however, the gender by group interaction (ß13) was a marginally statistically significant predictor of slope variability, estimate = –.09, SE = .05, t(252) = –1.84, p = .067. Given the presence of this marginally significant interaction effect, all other effects, which are subsumed by the interaction, were retained in the model. (SAS syntax used to estimate this final model and selected SAS output are presented in the Appendix. The equation numbers provided above, e.g. Equation 1, can be used to link coefficients in the equations with relevant estimates in the output.)
Probing the Marginally Significant Gender by Group Interaction in Predicting Slope Variability
Conceptually, there are two "sets" of results to consider— effects based on predictors of intercept variability and effects based on predictors of slope variability. Because all intercept effects were nonsignificant, however, they are not discussed further. As in basic regression analysis, in the presence of an interaction effect, it is helpful to consider estimating simple effects of the lower-order variables (e.g., the effect of gender on slope variability for able-bodied youth and the effect of gender on slope variability for youth with spina bifida). [See Aiken and West (1991) for a detailed discussion on testing and probing interaction effects and Holmbeck (2002) for an example of post-hoc probing of interactions in pediatric psychology research.) In Table III, we present relevant simple effects on slope variability, which will aid in the interpretation of the gender by group interaction. Importantly, significance tests for this full set of effects are not produced as part of the output of a single individual growth curve model analysis. Instead, multiple models (each with the same specification) were estimated in which the coding of gender and/or group was altered to allow determination of the full set of relevant effects, appropriate standard errors, and t tests of statistical significance.
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In Table III, we elected to present the results in a format that highlights the simple effects analysis of gender (i.e., the effect of gender on growth in emotional autonomy for both able-bodied comparison children and for children with spina bifida). Table III contains the population slope estimates for four subgroups of youth: (a) able-bodied girls, (b) able-bodied boys, (c) girls with spina bifida, and (d) boys with spina bifida. Figure 7 contains a plot of the growth curves for these four groups.
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From the examination of Fig. 7 and the estimates in Table III, it is apparent that very little difference exists in the rates of growth between able-bodied girls and able-bodied boys; slope estimates are .126 and .128 for these two groups, respectively. Both groups display statistically significant positive growth in emotional autonomy. Moreover, the simple effects analysis allows a direct test of the difference in these slopes which, not surprisingly, was not statistically significant. However, the plot presented in Fig. 7 and corresponding estimates from Table III clearly suggest that girls and boys with spina bifida are growing at different rates in emotional autonomy; slope estimates of .092 and .003, respectively, for which the difference is statistically significant. Although growth is positive and significant for girls with spina bifida, there is no evidence that boys with spina bifida are growing in emotional autonomy.
From the information summarized in Table III, it is also possible to compute the simple effects of spina bifida on slope variability for each gender. These simple effects are more pronounced for boys than for girls. For example, the simple effect of spina bifida for boys (the difference in slope estimates between able-bodied boys and boys with spina bifida) is .128 – .003 = .125. Although the test of this effect is not reported in Table III, it is statistically significant, t(252) = 3.77, p < .001. The simple effect of spina bifida for girls (the difference in slope estimates between able-bodied and spina bifida girls) is .126 – .092 = .034, which is nonsignificant, t(252) = .94, p = .35. Inclusion of child gender, spina bifida group, and their interaction resulted in a 51% reduction of slope variability from the unconditional linear growth model. Although the joint effects of these predictors account for over half of the explainable slope variability, significant variability in slope estimates remain.10
ANOVA Comparison
We also analyzed the data as a 2 (gender) x 2 (spina bifida group) x 3 (time—i.e., measurement wave) using a mixed design ANOVA (Keppel, 1991). In contrast to the results from our individual growth curve analysis, the gender by group by time interaction (similar to the effect of the gender by group interaction in predicting slope variability) was nonsignificant (p > .20).11 There were significant (or marginally significant) gender by time and group by time effects. Examination of the polynomial contrast coefficients (i.e., linear and quadratic) and their interaction with each gender and group suggested no presence of quadratic effects. Examination of the linear by gender and linear by group effects further suggested that (a) females were increasing faster than were males and (b) able-bodied youth were increasing faster than were youth with spina bifida.
When juxtaposed with results from our individual growth curve analysis, these results provide a different picture of growth over time in emotional autonomy. That is, this analysis fails to account for the finding that males with spina bifida were growing at slower rates than were both males without spina bifida and females with spina bifida. This has implications for the interpretation of both the group by linear trend and the gender by linear trend effects. With respect to the group by linear trend interaction, collapsing males and females with spina bifida into a single group (even though they were growing at different rates) and comparing them with male and female able-bodied youth result in a confounding of the spina bifida by time effect. Similarly, with respect to the gender by linear trend effects, this analysis collapses males with and without spina bifida, comparing them with females with and without spina bifida. From our simple effects analysis of the results of the individual growth curve models (Table III), we observed that males with and without spina bifida were growing at different rates; this is what is meant by confounding the gender by time effect.
As suggested earlier, there are some advantages of the individual growth curve modeling approach relative to ANOVA. One of the more salient advantages in this example regards the heterogeneity in age at each measurement occasion. Although the individual growth curve model can treat age as continuous and variable across individuals, the ANOVA model assumes all individuals are measured at the same time point for any given wave. To explore whether this treatment of time was a contributing factor to the differences in inference between the two approaches, we coded age as discrete values (9, 11, and 13) and coded gender and group as effect codes before estimating the individual growth curve model. Similar to the results from ANOVA, the gender by spina bifida interaction was not a statistically significant predictor of slope variability (p = .13). There were, however, significant effects of both gender and spina bifida groups, with the same interpretation of effects as seen with ANOVA. Thus, it appears that at least one reason for the difference in findings is the inability of the ANOVA model to treat time (age) as a truly continuous variable.
Coming Full Circle: Putting the Individual Back into the Individual Growth Curve Model
As mentioned above, even though we were able to significantly predict both intercept and slope variability, our final model still allowed for an individual component to be present in both the level 2 intercept and slope variability models. As such, individuals still varied from their group-level intercept and slope estimates. In Fig. 8, we display the model-implied individual growth curves for all study participants (by spina bifida group membership and gender) as a function of the parameters included in the final model. We present this final figure to remind the reader that, even though spina bifida group status and participant gender carry meaningful information about aggregate level growth trajectories, individuals are still allowed to deviate from their respective group trajectories, resulting in the individual growth curves displayed in the figure.
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