Applications of Individual Growth Curve Modeling
for Pediatric Psychology Research
Christian DeLucia,1PHD, and Steven C. Pitts,2 PHD
1
University of Illinois at Chicago and 2University of Maryland, Baltimore County
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?
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
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: [email protected].
Journal of Pediatric Psychology 31(10) pp. 1002–1023, 2006
doi:10.1093/jpepsy/jsj074
Advance Access publication September 21, 2005
Journal of Pediatric Psychology vol. 31 no. 10 © The Author 2005. Published by Oxford University Press on behalf of the Society of Pediatric Psychology.
All rights reserved. For permissions, please e-mail: [email protected]
Individual Growth Curve Modeling 1003
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
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,
10
9
8
Adherence
7
6
5
β10 =
1.19
4
3
β00 =2.13
b11 = 1.75
2
or average, growth curve. Parameter estimates are presented for the population (i.e., the dashed line with β00
and β10 for intercept and slope, respectively) and for one
participant (i.e., individual 1 with b01 and b11 for intercept and slope, respectively).
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.
The Unconditional Linear Growth Model
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 interindividual change model. These names highlight the
central feature of the level 2 model—it captures
between-person variability in the growth rates. Timeinvariant 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:
1 b01 =0.00
Level 1 model: Yij = b0i + b1i (time ij ) + e ij .
0
10
11
12
13
14
Age
Figure 1. Hypothetical growth curves of treatment adherence based
on individual growth curve model.
Level 2 model: b0i = b00 + n 0i .
b1i = b10 + n1i .
1004 DeLucia and Pitts
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 = (β00 +ν0i) + [(β10 +ν1i) (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
1
The level 1 residuals (i.e., eij) are assumed to be normally
distributed with a mean of zero and common variance σe and are
conditionally independent, given the random effects (Hedeker &
Gibbons, 1996b).
function of a population intercept (i.e., β00) plus his or
her deviation from the population intercept (i.e., ν0i).
Similarly, slope of person “i” (i.e., b1i) is written as a
function of a population slope (i.e., β10) plus his or her
deviation from the population slope (i.e., ν1i).
Fixed Effects
In the level 2 model, the population-level estimates (i.e.,
β00 and β01) 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., ν0i and ν1i), 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., ν0i and ν01) 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:
Level 1: Yij = b0i + b1i (time ij ) + e ij .
2
In this model, the random effects are assumed to be bivariate
normally distributed with means of zero and population variance
σν02 for the intercept, population variance σν12 for the slope, and
population covariance σν01.
Individual Growth Curve Modeling 1005
Level 2: b0i = b00 + b01 ( baseline parent
8
7
b1i = b10 + b11 ( baseline parent
6
monitoring i ) + n1i .
5
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
(β00 for intercept and β10 for slope), (b) his or her score on
baseline parental monitoring, and (c) an individual deviation (ν0i 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 β11, is actually a test of the (baseline
parental monitoringi × 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.)
A Hypothetical Applied Research Application:
Participant Response to an Intervention
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 inter8
Treatment = solid lines
Control = dashed lines
7
6
Adherence
5
4
3
2
1
0
0
1
Measurement occasion
Figure 2. Testing main effects of treatment using individual growth
curve models.
2
Adherence
monitoring i ) + n 0i .
4
3
2
1
0
0
1
2
Measurement occasion
Figure 3. Testing moderated treatment effects using individual growth
curve models.
vention to increase treatment adherence among youth
with cystic fibrosis. For illustrative purposes, suppose
some affected youth were randomized to receive a familybased intervention to cultivate and maintain adherence
behaviors, while the remaining youth were randomized to
a no-treatment assessment (i.e., control) condition.
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 postbaseline 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
1006 DeLucia and Pitts
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).
Advantages of Individual Growth Curve
Modeling
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 nonnormally 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
10
9
8
7
Adherence
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).
6
5
β10 =
1.19
4
β00 =
2.13
3
b11 = 1.19
2
b01 = 0
1
0
10
11
12
13
14
Age
Figure 4. Hypothetical growth curves of treatment adherence based
on analysis of variance (ANOVA) model.
Individual Growth Curve Modeling 1007
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).
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
Real-World Example: Modeling the
Development of Emotional Autonomy
in Youth with Spina Bifida and Able-Bodied
Comparison Youth
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
3
On a slightly more technical note, another strength of individual growth-curve modeling is that it allows the variance/covariance matrix of the repeated observations to follow a general form.
The ANOVA model assumes this matrix is compound symmetric—
that is, all variances are equal and all covariances are equal—
which is typically unrealistic, given that variances often change
over time, and covariances for more proximally spaced repeated
observations are often larger in magnitude than are covariances for
more distally spaced repeated observations (Gibbons et al., 1993;
Rogosa, 1988).
1008 DeLucia and Pitts
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.
Methods
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 ablebodied 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.
Results
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
Individual Growth Curve Modeling 1009
Table I. Descriptive Statistics for Emotional Autonomy and Age
Measurement occasion
Time 1
(N = 133)
Statistics
Time 2
(N = 131)
Time 3
(N = 129)
Emotional autonomy
M
2.24
2.38
SD
0.45
0.51
2.58
0.53
Skew
–0.01
0.14
0.06
Kurtosis
–0.52
0.06
0.47
Correlations
Time 1
1
Time 2
0.33
1
Time 3
0.24
0.46
M
8.85
10.95
12.79
SD
0.57
0.60
0.74
Skew
0.33
0.28
0.11
–1.03
–0.82
–0.97
1
Participant age
Kurtosis
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).
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
Emotional autonomy
4
3
2
1
8
9
10
11
12
13
14
Age
Figure 5. Plot of raw data of emotional autonomy for 20 youth
selected at random.
15
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.
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
1010 DeLucia and Pitts
Table II. Model Summary Information
Model name
Level 1 models
Level 2 models
χ2
Δχ2(df)
p
Fixed intercept
Yij = b0i + eij
b0I = β00
595.8
—
—
Unconditional means
Yij = b0i + eij
b0I = β00 + ν0i
566.2
29.6(1)
<.0001a
Compound symmetry
Yij = b0i + b1i (ageij – 9) + eij
b0I = β00 + ν0i
509.7
56.5(1)
<.0001
500.9
8.8(2)b
.0061a
500.7
.2(1)
.6547
b1I = β10
Unconditional linear
Yij = b0i + b1i (ageij – 9) + eij
Fixed quadratic
Yij = b0i + b1i (ageij – 9) + b2i (ageij – 9)2 + eij
b0I = β00 + ν0i
b1I = β10 + ν1i
b0i = β00 + ν0i
b1I = β10 + ν1i
b2I = β20
Full maximum likelihood estimation was used for all models.
a
Following the methods of Snijders and Bosker (1999, p. 90), halved probability values were used to determine the significance of the variance components.
b
Two dfs distinguish the unconditional linear and compound symmetry model (one for the random linear effect and one for the covariance between the random intercept
and the random linear effect). Model 4 was retained as the final model based on relative model fit.
(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 χ2 (chi-square) distribution, we hereafter
refer to these tests as χ2 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
4
When full maximum likelihood estimation is used-as was
the case in this study—these tests can be used to evaluate the elimination of either fixed or random effects. When restricted maximum likelihood estimation is used, these tests can be used to
evaluate the elimination of random effects only (see Snijders &
Bosker, 1999, p. 89).
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 χ2 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
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
5
For all of the growth models described below, 391 of a possible maximum of 405 observations contributed to the analyses
(96.54%). As such, missing data was quite minimal. The possible
maximum number of observations is computed by multiplying the
number of participants (135) by the maximum number of repeated
observations per participant (3).
Individual Growth Curve Modeling
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, χ2(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,
χ2(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, χ2(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, χ2(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).
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
6
In this case, the autocorrelation can be computed by forming
a ratio of the population estimates of the between-person variance
(i.e., level 2 variance) to the total variance (i.e., level 1 within—
person variance plus level 2 variance): σν02/(σν02 + σe2) = .07840/
(.07840 + .1885) = .29.
7
df for the tests of individual-level effects are derived through
a somewhat complicated algorithm. We used the ddfm = bw option
on the model statement in SAS PROC Mixed. Interested readers are
encouraged to see relevant SAS Institute documentation.
1011
1012 DeLucia and Pitts
Examining Individual Growth Curves of Emotional
Autonomy as a Function of Spina Bifida Group
Membership and Child Gender
Emotional autonomy
4
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.
3
2
1
8
9
10
11
12
13
14
15
Age
Figure 6. Plot of fitted data of emotional autonomy for the same 20
youth.
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
8
We also examined several models (data not presented) to
test for the possibility of a random quadratic effect, which could
not be reliably estimated in this set of analyses.
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
Level 1 : Yij = b0i + b1i ( age ij - 9) + e ij .
(1)
Level 2 : b0i = b00 + b01 ( genderi ) + b02 ( groupi )
+ b 03 ( genderi × groupi ) + n 0i .
(2a)
b1i = b10 + b11 ( genderi ) + b12 ( groupi )
+ b13 ( genderi × groupi ) + n1i .
(2b)
None of the three predictors of intercept variability
(tested by examining the t tests for the three parameter
estimates β01, β02, and β03) were statistically significant
(ps > .10). In contrast, however, the gender by group
interaction (β13) 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.)
9
Interaction terms in individual growth-curve modeling are
estimated in analogous fashion to basic regression. Similar issues
with respect to interpretability of the simple effects (effects of
either of the first-order variables) arise; the effect is correctly interpreted as the “simple” effect of the variable when the other variable is equal to numerical zero.
Individual Growth Curve Modeling 1013
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.
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.
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 ablebodied boys; slope estimates are .126 and .128 for these
Table III. Simple Fixed Effect Estimates of Slope Variability, by
Gender and Group
Effects
Estimate
SE
df
t
p
Able-boded girls
0.126
0.026
252
4.92
<.0001
Able-bodied boys
0.128
0.024
252
5.41
<.0001
–0.002
0.035
252
–0.07
.95
<.001
Effect of gender for
comparison youth
Spina bifida girls
0.092
0.026
252
3.50
Spina bifida boys
0.003
0.023
252
0.12
.91
0.089
0.035
252
2.53
.01
Effect of gender for
spina bifida youth
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) × 2 (spina
bifida group) × 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)
10
The significance of the residualized slope variance can be
tested using nested models in an analogous fashion as done above.
11
This was true for both the multivariate F statistic as well as
the F statistic based on results applying the Huynh-Feldt correction. As well, the p level for the test of gender by group by linear
change (most consistent with the individual growth-curve model)
was p = .177.
1014 DeLucia and Pitts
4
Emotional autonomy
3
2
Solid lines = comparison group
Dashed lines = spina bifida group
Thick lines = girls
Thin lines = boys
1
8
9
10
11
12
Age
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.
13
14
15
Figure 7. Plot of the simple gender by
spina bifida interactions in predicting
slope variability.
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
Individual Growth Curve Modeling 1015
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.
Discussion
The central goal of this article was to provide a brief, nontechnical introduction to individual growth curve modeling for the analysis of longitudinal data. We introduced
several possible model applications in the areas of basic
(developmental) and more applied (intervention) pediatric psychology research. Our discussion progressed from
several hypothetical examples of model applications,
which were more conceptual in nature, to an actual analytical example in which we analyzed data from an ongoing study of the possible impact of spina bifida on youth
and family development (Holmbeck et al., 2003).
Able-bodied girls
Able-bodied boys
4
Emotional autonomy
Emotional autonomy
4
3
2
1
3
2
1
8
9
10
11
12
13
14
15
8
9
10
Age
12
13
14
15
13
14
15
Age
Spina bifida girls
Spina bifida boys
4
4
Emotional autonomy
Emotional autonomy
11
3
2
1
3
2
1
8
9
10
11
12
13
14
Age
Figure 8. Plot of fitted data from final model, by group and gender.
15
8
9
10
11
12
Age
1016 DeLucia and Pitts
Growth in Emotional Autonomy as a Function
of Spina Bifida Status and Participant Gender
From our analytical work, we were able to draw several
conclusions about growth in emotional autonomy for
male and female youth with spina bifida and able-bodied
comparison youth. First, we established that on average,
growth in emotional autonomy was linear in nature.
Second, we established that growth curves of emotional
autonomy varied across individuals. Third, we established that spina bifida status—captured by the group
variable—and participant gender interacted in predicting rates of linear change in emotional autonomy. Further probing of this complex interaction revealed that
for youth with spina bifida, girls exhibited significant
positive growth in emotional autonomy, whereas boys
exhibited nonsignificant growth in emotional autonomy. In contrast, for able-bodied comparison youth,
girls and boys both exhibited significant positive growth
in emotional autonomy. Analyses of these data using an
ANOVA model failed to capture this complex interaction, resulting in different conclusions regarding growth
over time in emotional autonomy for these four subgroups of youth.
Practical Implications of Strengths of Individual
Growth Curve Modeling
Throughout our discussion, we juxtaposed individual
growth curve models with the more traditional ANOVA
model. Although we highlighted several strengths of
individual growth curve modeling relative to ANOVA, it
is worth discussing further some important statistical
and practical implications of two particular strengths of
individual growth curve modeling.
Flexible Treatment of the Time Variable
The fact that the time variable can be treated continuously in individual growth curve modeling, rather than
as a set of fixed points (Bryk & Raudenbush, 1992), can
have some very serious implications for the analysis of
longitudinal data. Losing this flexibility can introduce
error into the growth curve estimates. Consider a very
simple example in which one individual was measured
at three time points, each separated by 1 week, and that
his/her scores on the outcome were five, six, and seven,
respectively. Assume also that a second individual was
measured at three time points, each separated by 2
weeks, and his/her scores on the outcome were five,
seven, and nine, respectively. In reality, both of these
individuals have the same growth curve, indicated by an
intercept of five and a linear weekly growth rate of one.
If it were assumed that these individuals were measured
at weekly intervals (which is true for the first individual
but false for the second), the first individual’s trajectory
would be accurately estimated, but the second individual’s trajectory would be misestimated. In addition to
this misestimation, spurious variability in growth curves
estimates results. Although not true in this example, the
actual form of growth (e.g., linear, quadratic) can be
misestimated as well.
Problems might arise even when the design and
statistical analysis is sound. For example, studies may
recruit participants of differing ages at baseline, though
subsequent measurement intervals are the same for
all participants (e.g., 1 year). Common practice is to
include baseline age as a covariate in the analysis. However, to the extent that growth is nonlinear (e.g., physical stature), this analytical solution will not result in
unbiased estimates. Individual growth curve models, by
allowing time (age) to be continuous, do not suffer from
this problem. For a general discussion of the possible
impact of age heterogeneity on the variance estimates of
the growth factors, see Mehta and West (2000).
Data are Truly Modeled at the Individual Level
As noted above, the focus on the individual achieved in
individual growth curve modeling is an important
advantage and has many possible practical applications.
For example, intervention researchers can use individual growth curves of treatment response to plan the timing of follow-up or booster sessions (e.g., individuals
whose treatment response is decaying more rapidly can
be given booster sessions earlier than individuals whose
treatment response is decaying less rapidly). Similarly,
researchers interested in studying the development of
risk behaviors (e.g., externalizing, substance use) can
use the individual growth curves derived in an earlier
period of development (e.g., preadolescence) to make
decisions about which youth to follow-up in a subsequent developmental period (e.g., researchers might want
to randomly sample among subclasses of individuals—
say those with average growth, those with very steep
growth rates, and those with negative growth rates)
(Nagin, 1999). In these examples, information concerning individual-level growth, in addition to information
about aggregate-level growth, is extremely useful.
Some Suggestions for Beginning Users
We hope this article will encourage researchers to learn
more about possible applications of individual growth
curve models in their own areas of research. Although
the early stages of such an endeavor can seem quite
overwhelming, there are many excellent articles, chapters,
Individual Growth Curve Modeling 1017
and book sections dedicated to this general class of statistical procedures. Although no single work will completely unpack the “mystery” of individual growth curve
models for beginning users, we have found the following
sources particularly helpful in increasing our knowledge
(Bryk & Raudenbush, 1987, 1992; Gibbons et al., 1993;
Littell et al., 1996; Singer & Willett, 2003; Singer, 1998;
Snijders & Bosker, 1999; Willett, 1988, 1997; Willett,
Singer, & Martin, 1998).
Interested readers might begin their study by reading the first eight chapters of Singer and Willett’s (2003)
book, which includes a wonderful mix of conceptual
and technical explanations of individual growth curve
models.12 Readers can also visit the book’s companion
website (Singer & Willett, 2005), which contains links to
downloadable data sets and syntax files that can be used
to estimate individual growth curve models in several
commonly used software programs (e.g., SAS, SPSS). We
have also found D. Hedeker’s website (2005) an excellent resource for learning more about longitudinal modeling in general and individual growth curve models
more specifically. For example, the website has links to
lecture notes and sample programs for a course he
teaches on longitudinal data analysis in the Division of
Epidemiology and Biostatistics at the University of Illinois at Chicago. We also encourage visiting D. Hedeker’s
website because it contains links to many freely downloadable software programs (and user manuals), all of
which can be used to estimate individual growth curve
models (or more generally, mixed-effects regression
models) for continuous as well as other data types (e.g.,
Hedeker & Gibbons, 1996a, 1996b; Hedeker, 1999).
Although software to estimate these kinds of models is
readily available today (Snijders & Bosker, 1999, chap.
15), free software remains the exception.
generalizations exist for nonnormal data). Throughout
the article, we have discussed many possible applications of individual growth curve models for basic and
applied pediatric psychology research. In addition to
discussion of the model’s conceptual framework, we also
estimated several models examining growth in emotional autonomy during the preadolescent and adolescent years in a sample of youth with spina bifida and
able-bodied comparison youth. Our analytical work provided examples of the kinds of interesting hypotheses;
these models can be used to examine: (a) Is there
growth in the phenomenon? (b) What is the shape of
growth? (c) Are individuals growing at different rates?
and (d) Are there person-level characteristics that can
predict variability in growth parameters (both intercepts
and slopes). In addition, our discussion also touched on
several important statistical issues worthy of consideration, some of which were specific to individual growth
curve modeling (e.g., giving the time variable a meaningful metric), and others of which were more general in
nature (e.g., the probing of interaction effects). We hope
this article will encourage researchers to learn more
about individual growth curve models and consider possible applications of these models for their work.
Acknowledgments
We thank Grayson N. Holmbeck for providing the data
analyzed in this study and Donald Hedeker for statistical
consultation.
Received February 28, 2005; revisions received May 18,
2005 and July 22, 2005; accepted August 1, 2005
References
Conclusion
Individual growth curve modeling is an extremely powerful and flexible procedure for the analysis of longitudinal data. It offers researchers several advantages over
more traditional methods (e.g., data are modeled at the
individual level, flexible treatment of the time variable,
12
It is worth noting that the latter seven chapters of this text
are dedicated to describing a general class of statistical procedures
typically referred to as survival modeling, in which the modeling
of “event” timing/occurrence is of primary interest. Although discussion of survival modeling was beyond the scope of this article,
these kinds of models might also be of interest to pediatric psychologists interested in questions that center more on the timing
of events (e.g., time to school integration following a traumatic
injury).
Aiken, L. S., & West, S. G. (1991). Multiple regression:
Testing and interpreting interactions. Newbury Park,
CA: Sage.
Baron, R. M., & Kenny, D. A. (1986). The moderatormediator variable distinction in social psychological
research: Conceptual, strategic, and statistical
considerations. Journal of Personality and Social
Psychology, 51, 1173–1182.
Biesanz, J. C., Deeb-Sossa, N., Papadakis, A. A., Bollen,
K. A., & Curran, P. J. (2004). The role of coding
time in estimating and interpreting growth curve
models. Psychological Methods, 9, 30–52.
Bryk, A. S., & Raudenbush, S. W. (1987). Application
of hierarchical linear models to assessing change.
Psychological Bulletin, 101, 147–158.
1018 DeLucia and Pitts
Bryk, A. S., & Raudenbush, S. W. (1992). Hierarchical
linear models: Applications and data analysis methods.
Newbury Park, CA: Sage.
Bryk, A. S., Raudenbush, S. W., & Congdon, R. T.
(1996). HLM: Hierarchical linear and nonlinear
modeling with the HLM/2L and HLM/3L programs.
Chicago: Scientific Software International.
Choeng, J., MacKinnon, D. P., & Khoo, S. T. (2003).
Investigation of mediational processes using parallel
process latent growth curve modeling. Structural
Equation Modeling, 10, 238–262.
Coakley, R. M., Holmbeck, G. N., Friedman, D.,
Greenley, R. N., & Thill, A. W. (2002). A longitudinal study of pubertal timing, parent-child conflict,
and cohesion in families of young adolescents with
spina bifida. Journal of Pediatric Psychology, 27,
461–473.
Ferrer, E., & McArdle, J. J. (2004). An experimental
analysis of dynamic hypotheses about cognitive
abilities and achievement from childhood to early
adulthood. Developmental Psychology, 40, 935–952.
Gibbons, R. D., Hedeker, D., Elkin, I., Watermaux, C.,
Kraemer, H. C., Greenhouse, J. B., et al. (1993).
Some conceptual and statistical issues in analysis of
longitudinal psychiatric data. Archives of General
Psychiatry, 50, 739–750.
Hedeker, D. (1999). MIXNO: A computer program for
mixed-effects nominal logistic regression. Journal of
Statistical Software, 4, 1–92.
Hedeker, D. (2005). Don Hedecker’s homepage.
Retrieved August 30, 2005, from http://tigger.uic.
edu/∼hedeker/
Hedeker, D., & Gibbons, R. D. (1996a). MIXOR:
A computer program for mixed-effects ordinal
probit and logistic regression analysis. Computer
Methods and Program in Biomedicine, 49, 157–176.
Hedeker, D., & Gibbons, R. D. (1996b). MIXREG:
A computer program for mixed-effects regression
models with autocorrelated errors. Computer Methods
and Program in Biomedicine, 49, 229–252.
Hedeker, D., & Gibbons, R. D. (1997). Application of
random-effects pattern-mixture models for missing
data in longitudinal studies. Psychological Methods,
2, 64–78.
Holmbeck, G. N. (1997). Toward terminological,
conceptual, and statistical clarity in the study of
mediators and moderators: Examples from the child
clinical and pediatric psychology literatures. Journal
of Consulting and Clinical Psychology, 65, 599–610.
Holmbeck, G. N. (2002). Post-hoc probing of significant
moderational and mediational effects in studies
of pediatric populations. Journal of Pediatric
Psychology, 27, 87–96.
Holmbeck, G. N., Coakley, R. M., Hommeyer, J. S.,
Shapera, W. E., & Westhoven, V. C. (2002).
Observed and perceived dyadic and systemic
functioning in families of preadolescents with spina
bifida. Journal of Pediatric Psychology, 27, 177–189.
Holmbeck, G. N., Westhoven, V., Shapera, W. E.,
Bowers, R., Gruse, C., Nikolopoulos, T., et al.
(2003). A multimethod, multi-informant, and
multidimensional perspective on psychosocial
adjustment in preadolescents with spina bifida.
Journal of Consulting and Clinical Psychology, 71,
782–796.
Hommeyer, J. S., Holmbeck, G. N., Wills, K. E., &
Coers, S. (1999). Condition severity and psychosocial functioning in pre-adolescents with spina
bifida: Disentangling proximal functional status and
distal adjustment outcomes. Journal of Pediatric
Psychology, 24, 499–509.
Keppel, G. (1991). Design and analysis: A researcher’s handbook (3rd ed.). Englewood Cliffs, NJ: Prentice Hall.
Krull, J. L., & MacKinnon, D. P. (2001). Multilevel
modeling of individual and group level mediated
effects. Multivariate Behavioral Research, 36, 249–277.
Laird, N. M., & Ware, J. H. (1982). Random-effects models
for longitudinal data. Biometrics, 38, 963–974.
Littell, R. C., Milliken, G. A., Stroup, W. W., &
Wolfinger, R. D. (1996). SAS system for mixed
models. Cary, NC: SAS Institute.
Longford, N. T. (1993). Random coefficient models. New
York: Oxford University Press.
MacKinnon, D. P. (1994). Analysis of mediating
variables in prevention and intervention studies.
In A. Cazares & L. Beatty (Eds.), Scientific methods
for prevention intervention research (NIDA
Monograph No. 139, pp. 127–153). Rockville, MD:
National Institute on Drug Abuse.
MacKinnon, D. P., Lockwood, C. M., Hoffman, J. M.,
West, S. G., & Sheets, V. (2002). A comparison of
methods to test mediation and other intervening
variable effects. Psychological Methods, 7, 83–104.
McArdle, J. J., Hamagami, F., Jones, K., Jolesz, F.,
Kikinis, R., Spiro, A., et al. (2004). Structural
modeling of dynamic changes in memory and brain
structure using longitudinal data from the normative
aging study. Journals of Gerontology: Series B:
Psychological Sciences and Social Sciences, 59B, 294–304.
Mehta, P. D., & West, S. G. (2000). Putting the individual back into individual growth curves. Psychological
Methods, 5, 23–43.
Individual Growth Curve Modeling 1019
Nagin, D. S. (1999). Analyzing developmental
trajectories: A semiparametric, group-based
approach. Psychological Methods, 4, 139–157.
Rogosa, D. (1988). Myths about longitudinal research.
In K. W. Schaie, R. T. Campbell, W. Meredith, &
S. C. Rawlings (Eds.), Methodological issues in aging
research (pp. 171–210). New York: Springer.
Rogosa, D., Brandt, D., & Zimowski, M. (1982).
A growth curve approach to the measurement
of change. Psychological Bulletin, 92, 726–748.
Singer, J. D. (1998). Using SAS PROC MIXED to fit
multilevel models, hierarchical models, and
individual growth models. Journal of Educational
and Behavioral Statistics, 24, 323–355.
Singer, J. D., & Willett, J. B. (2003). Applied longitudinal
data analysis: Modeling change and event occurrence.
New York: Oxford University Press.
Singer, J. D., & Willett, J. B. (2005). Applied longitudinal
data analysis: Modeling change and event occurrence.
New York: Oxford University Press. Retrieved
August 30, 2005, from http://gseacademic.harvard.
edu/alda/
Snijders, T. A. B., & Bosker, R. J. (1999). Multilevel
analysis: An introduction to basic and advanced
multilevel modeling. London: Sage.
Steinberg, L., & Silverberg, S. B. (1987). The vicissitudes
of autonomy in early adolescence. Child Development,
57, 841–851.
West, S. G., & Aiken, L. S. (1997). Toward understanding individual effects in multicomponent prevention
programs: Design and analysis strategies. In K. J.
Bryant, M. Windle, & S. G. West (Eds.), The science
of prevention: Methodological advances from alcohol and
substance abuse research (pp. 167–209). Washington,
DC: American Psychological Association.
Willett, J. B. (1988). Questions and answers in the
measurement of change. In E. Rothkopf (Ed.),
Review of research in education 1988–1989
(pp. 345–422). Washington, DC: American
Educational Research Association.
Willett, J. B. (1997). Measuring change: What individual
growth modeling buys you. In E. Arnsel &
K. A. Reninger (Eds.), Change and development
(pp. 213–243). Maywah, NJ: Erlbaum.
Willett, J. B., Singer, J. D., & Martin, N. C. (1998).
The design and analysis of longitudinal studies of
development and psychopathology in context:
Statistical models and methodological
recommendations. Development and Psychopathology,
10, 395–426.
1020 DeLucia and Pitts
Appendix
SAS Syntax and Selected Output From Final model Described in Paper
/*
This syntax and accompanying data file can be downloaded from the following website: http://userpages.umbc.edu/~spitts
SAS syntax is presented in bold-face type and comments are presented in regular type
enclosed within the following comment indicators /* */
Options nocenter;
/*This option results in output printed flush left.*/
/*
Below code reads in space-delimited raw data file, assigns
variable names for existing variables, creates three additional variables
(i.e., age91, age92, age93), and creates a new working data file (i.e., eaut).
Variables are as follow:
id
=
participant id
group =
group membership (0 = comparison, 1 = spina bifida)
gender =
gender (0 = female, 1 = male)
age1
=
age at time 1 (age2 and age3 follow same form)
age91 =
age - 9 at time 1 (which centers age variable at age 9)
this variable will be used to examine linear effect of age in growth models
(age92 and age93 follow same form)
eaut1 =
emotional autonomy from mothers at time 1 (eaut2 and eaut3 follow same form)
*/
data eaut;
infile 'C:\eaut.txt';
input id group gender age1 age2 age3 eaut1 eaut2 eaut3;
age91 = age1 - 9;
age92 = age2 - 9;
age93 = age3 - 9;
run;
/*
The working data file 'eaut' is a "wide" data file in that the repeated observations
are designated as multiple variables for each individual (e.g., age1, age2, age3).
As such, each individual takes up 1 row in the data file.
To execute the mixed procedure, the data file must be converted to a "long" file,
in which the repeated observations are designated as multiple linked records for the same
individual. As such, each individual has multiple rows in the data file.
For example, if we consider a hypothetical participant id# 1000 who is 8-years-old at T1,
10 at T2, and 12 at T3, his/her id and age variables would be represented in a wide
data file as follows
1000 8 10 12
and in a long data file as follows
1000 8
1000 10
1000 12
The below array statement converts the wide file 'eaut' to the long file 'eautlong',
renames the "linear" age variables (i.e., age91, age92, age93) age9,
renames the emotional autonomy variables (i.e., eaut1, eaut2, eaut3) eaut,
and retains only the variables that will be used in the mixed procedures.
The resulting long data file has five variables in the following order:
id group gender age9 eaut.
*/
Individual Growth Curve Modeling 1021
1022 DeLucia and Pitts
Individual Growth Curve Modeling 1023