Emerging Trends in The Social and Behavioral Sciences

Models of Nonlinear Growth

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Models of Nonlinear Growth

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Models of Nonlinear Growth
PATRICK COULOMBE and JAMES P. SELIG

Abstract
Models for nonlinear growth are not new, but have not been widely applied in the
social and behavioral sciences. In this essay, we describe the fundamental issues relevant to choosing and using a nonlinear growth model. We discuss how researchers
can go about choosing a model and then focus on the application of two specific
nonlinear models: the fractional polynomial model and the piecewise model. We
highlight recent work in reparameterization that allows researchers to choose models with parameters tailored specifically to research questions. We also review recent
work on the topic of growth rates in nonlinear models that will allow researchers to
obtain richer information from the application of nonlinear models. We conclude by
pointing out some of the unresolved issues in the use of nonlinear growth models.

MODELS OF NONLINEAR GROWTH
Models of nonlinear growth are used to represent change over time in scores
on a variable measured on several occasions. Both linear and nonlinear models can be used to examine change in a variable over time. A key difference
between linear and nonlinear change is that in a model for linear change,
the rate of change is constant over time (i.e., the slope of the straight line
is the same at all occasions); however, in a model for nonlinear change the
rate of change can vary across time (Cudeck & Harring, 2007). For example,
Figure 1 shows an individual’s proportion of successes in a learning task over
10 days. As can be seen, the individual improved her performance dramatically in the first 2 days (reaching a very high proportion of successes), but
improved much slower thereafter; in short, improvement over time was not
constant.
Throughout this paper, we refer to nonlinear growth models that take the
form:
yij = f (?1i , ?2i , … , ?ki , tij ) + ?ij ,
where subscript i refers to individual i, subscript j the jth occasion of measurement (timepoint), yij the individual i’s observed score at timepoint j, parameters ? 1i through ? ki the regression weights, tij the time distance to the origin,
Emerging Trends in the Social and Behavioral Sciences. Edited by Robert Scott and Stephen Kosslyn.
© 2015 John Wiley & Sons, Inc. ISBN 978-1-118-90077-2.

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1

% Success

0.8

0.6

0.4
Observed
Rational

0.2

Quadratic
Asymptote
0
1

2

3

4

5

6

7

8

9

10

Time

Figure 1 An individual’s hypothetical proportion of successes in a learning task
over 10 consecutive days, along with a rational and quadratic function fitted to the
data. The asymptote is associated with the rational function and has been
fixed to 1.

and ?ij the difference between the observed and predicted score for individual i at timepoint j (which is typically assumed to be normally distributed
with a mean of 0). The score yij on the dependent variable is described as a
function of time and the regression weights, which may be defined to vary or
to be the same across individuals. In a nonlinear growth model, the function
f follows a nonlinear trajectory when plotted against time. More formally, if
the function is differentiable, its first derivative ft ′ (? 1i , ? 2i , … , ? ki , tij )is itself
a function of tij . Finally, the ? parameters can be further specified to be a
function of covariates that vary or are fixed over time.
In this essay, we first discuss how a researcher can decide on a particular functional form when modeling nonlinear change over time. We then
describe two potentially useful nonlinear growth models, fractional polynomials and piecewise models. Next, we show how a researcher can tailor his
or her particular model to substantive questions of interest through reparameterization. Finally, we illustrate how one can effectively use rate of change
to give a more complete depiction of the nonlinear growth process to the
reader.

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CHOOSING A FUNCTIONAL FORM
When modeling linear change over time, the choice of the particular model
to use is easy: the linear growth model. When one wants to model nonlinear
change over time, however, a vast array of possible models exist. In this
section, we discuss how a researcher can choose the functional form of the
change over time.
There is no substitute for a researcher’s deep understanding of change in
a focal variable when choosing an appropriate functional form. This may be
especially true when modeling nonlinear change over time, because in many
instances, several functions might fit the data equally well. At the very least,
the researcher should choose a functional form for the model that is consistent with what he or she knows about change in the variable. For example,
an educational researcher examining the cumulative number of aggressive
behaviors over the school year among first graders would not choose any
function that decreases over time. The researcher’s understanding of the variables under study should also drive analytical decisions such as whether to
allow a specific regression coefficient to take different values across individuals (Bianconcini, 2012).
Cudeck and Harring (2007) provide three criteria in the selection of an
adequate nonlinear function that may support researchers in choosing a
functional form: (i) it must fit the data well, (ii) its parameters must have
an interpretation that answers interesting substantive questions, and (iii) its
characteristic shape corresponds to the hypothesized change (e.g., presence
of an upper asymptote if change over time in a variable with a maximum
is continually increasing). Only the first one is statistical, while the latter
two are grounded in the researcher’s understanding of the phenomenon
to be modeled. In our view, a common error when choosing functional
forms for nonlinear growth models is to overemphasize the first criterion.
The point here was that models that fit the data well, yet that do not have
interpretable parameters and/or that are not consistent with the researcher’s
understanding of change, are not useful [even though they fit the data well,
is what the point is].
When new variables are examined, or a researcher does not have a priori
ideas about patterns of change, one very effective way to probe the data is
through the use of plots. Cudeck and Harring (2007) mention three different
types of plots that can be used in the context of nonlinear growth modeling.
The first one is the spaghetti plot, in which the raw scores for a subset of the
sample are plotted against time, with straight segments joining two adjacent
scores for a given individual. The second one is the swarm plot, in which an
estimated function corresponding to the chosen function is plotted against
time. Finally, the third one is the trellis plot, which is actually an ensemble

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of plots arranged in a grid. In each plot, the scores for a single individual
are plotted against time, with the estimated curve for that specific individual
superimposed (see also Few, 2009, for the description of a similar plotting
technique). This quickly gives the researcher (or the reader) an idea of how
well the postulated model fits individual data, and of the variability of this
fit across individuals.
SPECIFIC CURVE SHAPES
Next, we present two particular variations on nonlinear growth models.
The first is the fractional polynomial model and the second is the piecewise
model. We chose these two because we believe they are relatively straightforward extensions of models commonly used for growth in the social and
behavioral sciences. Although these models are not new, there are relatively
few applications of these models.
FRACTIONAL POLYNOMIALS
Researchers interested in modeling nonlinear change over time frequently
apply a quadratic curve to the data. In the case of a quadratic curve, the
mathematical function fitted to the data is given as
yij = ?0i + ?1i tij + ?2i t2ij + ?ij .
In this model, ? 0 represents the predicted score when t = 0, ? 1 is the rate
of change when t = 0, and twice the value of ? 2 is the magnitude by which
the rate of change varies for each one-unit increase in the time variable. An
example of a quadratic curve is given in Figure 1 (dashed line), where we
attempt to model the individual’s increasing proportion of successes in a
learning task over time. The quadratic curve follows the observed scores
(triangles) quite well. In general, the characteristics of the quadratic curve
include presence of a maximum or minimum, symmetry of the curve about
the maximum or minimum, and absence of lower or upper asymptotes.
These characteristics can be at odds with the researcher’s understanding of
change. For example, the absence of asymptotes is not consistent with the
use of a measure which has a minimum and/or a maximum (in this case,
proportion of successes should not exceed 1), and rarely is the pattern of
change in behavior symmetric before and after attaining an extreme value.
One way to address these problems is to explicitly select an exponentiation of the time variable that yields a curve consistent with the theory of
interest. Polynomial models involving the exponentiation of the time variable with fractions or negative integers are known as fractional polynomials
(Long & Ryoo, 2010; see also Royston & Altman, 1994; Royston, Ambler, &

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5

Sauerbrei, 1999). The quadratic polynomial shown above is a special case of
a fractional polynomial model, in which the time variable is exponentiated
to the first and second power.
For example, Figure 1 shows a rational function (solid line) for the individual in attempting to model her increasing proportion of successes over time.
The rational function takes the general form:
yij = ?0i + ?1i t−1
+ ?ij .
ij
Therefore, the rational function is a polynomial function with one-time variable exponentiated to the first negative power. In this model, ? 0 represents a
horizontal asymptote and ? 1 is the negative value of the rate of change when
t = 1.
As can be seen in Figure 1, this function (with the constraint ? 0 = 1) is consistent with the data, slightly more so than the quadratic curve. Given that
we expected performance to improve over time, the quadratic function had
the drawback of predicting decreasing scores past a certain point (around day
8; see Figure 1). Therefore, we needed a function that would be monotonic,
that is, that would increase over time without ever decreasing. The sign of
the second parameter of the rational function (? 1 ) indicates whether scores
are increasing (? 1 < 0; Figure 1) or decreasing (? 1 > 0) over time. But why did
we fix the first parameter (? 0 ) to be 1 instead of letting it be freely estimated
with the data? This parameter in the rational function represents a horizontal
asymptote (dashed horizontal line in Figure 1). We needed a function which
would allow us to account for the fact that the proportion of successes cannot
exceed 1, and the rational function—but not the quadratic function—does
just that. We could have allowed for the specific value of the asymptote to
be estimated freely. This would allow us to check whether predicted maximal performance differs across individuals, and to potentially predict such
individual differences using covariates. For example, we could ask whether
maximal performance depends on the age of the person learning the task.
Long and Ryoo (2010) demonstrate graphically the impact of manipulating
the exponents of the time variable on the shape of the trajectory over time.
These authors separate the models into first-order and second-order polynomials, depending on the number of time variables included in the model. For
either type, one can obtain a “flipped” version of any of the curves by reversing the signs of the regression weights (?). And indeed, these curves can be
selected to include lower or upper asymptotes (as we did in Figure 1), to be
asymmetric about an extreme value, or to depend on scores on covariates
(see Lambert, Smith, Jones, & Botha, 2005; Royston & Altman, 1994); overall,
fractional polynomials are much more flexible than traditional polynomials
in representing a change process.

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

One difficulty that arises in the use of fractional polynomials over traditional polynomials lies in software use: Fractional polynomials are often
more difficult to estimate using standard linear multilevel or structural
equation modeling software. Another important issue with regard to the
use of fractional polynomial models is the interpretability of the parameters. Oftentimes, the function will contain parameters that do not have
straightforward interpretations, and even when the parameters have clear
interpretations, the interpretations might not be relevant to the study
research questions.
PIECEWISE MODEL
The piecewise model (also called spline or multiphase model, sometimes discontinuous model; Singer & Willett, 2003) allows one to model change over
time differently from one subset (range) of the time variable to the next subset. There are several ways to achieve this. Most authors have considered
piecewise models in which the trajectory for every time subset is exclusively
linear. For these models, the nonlinear nature of change is depicted using
two or more linear splines. However, it is entirely possible for the individual
splines to have nonlinear functional forms.
The simplest piecewise model is a linear–linear model, in which the full
range of the period covered by the study is separated into two subsets (phases
and epochs), within each of which change over time is linear. What distinguishes the linear trajectory of the two phases is the magnitude of the linear
slope. The measurement occasion at which the trajectory changes is called
the knot, or the transition point, and this knot can be fixed or variable across
individuals. The function describing the linear–linear model just described
is given as
yij = ?0i + ?1i t1ij + ?2i t2ij + ?ij .
In this model, ? 0 is the predicted score when both t1 and t2 = 0, ? 1 describes
the linear rate of change for the first phase and ? 2 is associated with the linear
rate of change for the second phase. The novelty in this model is the addition
of a second time variable. There are two ways to code the two time variables.
One way is to treat them as independent: the first time variable starts at a
given value and increases until the end of the first phase, at which point
it becomes constant for the remainder of measurement occasions, and the
second time variable is set to 0 until the last measurement occasion of the
first phase, at which point it starts increasing by 1 for each subsequent measurement occasion (Bollen & Curran, 2006). The second way to code the time
variables is to make them partially redundant: the first time variable starts at

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7

a given value and increases across all measurement occasions, and the second variable is coded in the same way as in the first method (Singer & Willett,
2003). For example, for a six-wave study with the first phase spanning times
1 through 3 and the second phase spanning times 4 through 6, the first time
variable could be coded (0, 1, 2, 2, 2, 2) in the first method, and (0, 1, 2, 3, 4, 5)
in the second method. The second time variable would be coded (0, 0, 0, 1, 2,
3) in both methods.
The interpretation of ? 1i concerns mainly the first phase, and represents
the magnitude of the change in predicted score for each one-unit increase
in time during the first phase. However, the same quantity for the second
phase depends on how the first time variable is coded. In the first method,
? 2i represents the change in predicted score for each one-unit increase in time
during the second phase. In the second method, this quantity is (? 1i + ? 2i )
instead.
Piecewise models can also allow for a change in elevation from one phase
to the next (Singer & Willett, 2003). An appropriate mathematical function
for two phases with a change in elevation is given as
yij = ?0i + ?1i t1ij + ?2i t2ij + ?3i even tij + ?ij .
The variable event is a binary variable coded as either 0 or 1, which tracks
whether a specific event has occurred at measurement occasion j for individual i (e.g., the individual has graduated, or has started an intervention).
Typically, the occurrence of the event will correspond to the change in phases,
and will change values only once during the course of the study, although
neither condition is a mathematical requirement. The parameter ? 3i corresponds to the magnitude of the sudden change in predicted score on the
dependent variable following the occurrence of the event. In addition, it is
possible to model a change in elevation without any change in linear trajectory between the two phases. To do so, one can simply use the second coding
method and omit the second time variable from the model (or, equivalently,
set ? 2i to 0).
Extensions to the models presented above are available. For instance, it is
possible to model growth over three (or more) distinct phases, which can be
particularly useful in a three-phase study involving a baseline, an intervention phase, and a follow-up; the reader is referred to Cudeck and Harring
(2007), Ram and Grimm (2007), and Singer and Willett (2003) for examples.
One recent interesting extension involved the modeling of phasic change
over time that differed across subsets of individuals within a single sample
(Kamata, Nese, Patarapichayatham, & Lai, 2012). Another extension is to estimate the timing of the knot with the sample data instead of determining it
a priori (e.g., Cudeck & Harring, 2007; Harring, Cudeck, & du Toit, 2006).
Finally, one possible extension is to allow for nonlinear trajectory within each

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

phase. In this case, the function for the piecewise model can be given by the
more general formula:
{ (
)
t<?
f1 ?0i , … , ?mi , t ,
.
yij =
f2 (?(m+1)i , … , ?(m+n)i , t), t ≥ ?
For this general model, scores on the dependent variable are a function of
a series of regression coefficients and the time variable, but the function is
allowed to change from the first to the second phase at the knot when t = ?.
Therefore, one could model a nonlinear trajectory during the first phase, the
second phase, or both the phases, particularly if there are theoretical reasons
to do so.
REPARAMETERIZATION
Once a researcher has chosen a specific functional form to use, the goal is
to make the model as useful as possible for addressing research questions.
This goal can be facilitated through reparameterization. Reparameterization is
the expression of an existing model using different parameters. The goal of
reparameterization is to make the interpretation of the reformulated model
more substantively interesting (Preacher & Hancock, in press). The reparameterized model should have the same number of parameters as the original
model, and should yield the same fit to the data; in short, the reparameterized
model is equivalent to the original model. The choice between two equivalent
models should depend on what specific research questions the researcher
wishes to address.
Recent literature has offered numerous examples of reparameterized models. Cudeck and du Toit (2002) reexpressed the quadratic model to make the
interpretation of its parameters more interesting from a substantive point of
view. The resulting function, using our notation, is
(
)2
tij
− 1 + ?ij .
yij = ?1i − (?1i − ?0i )
?2i
To illustrate the value of reparameterizing a model, let us compare the
interpretation of the parameters with those from the classic quadratic model.
The two models share one parameter (? 0 ), which is the predicted score
when time is 0. The rest of the parameters differ. In the classic quadratic
model, neither ? 1 nor ? 2 has a straightforward interpretation (except when
a researcher is interested in the rate of change at time 0 specifically).
Conversely, the reparameterized model bears parameters with interesting
interpretations: ? 1 is the predicted maximum (or minimum) score (e.g., the
maximum proportion of successes), and ? 2 is the time at which the predicted

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9

maximum (or minimum) is attained (e.g., the required time in days for the
maximum performance to be achieved; see Figure 1). Despite these differences, both models fit the data equally well. Therefore, it appears clear from
this example that one can substantially improve the usefulness of a model by
reparameterizing it so as to make its parameters more readily interpretable.
Although one primary goal of reparameterization is to make parameter
estimates more readily interpretable, other advantages can come from reformulating a model. For example, Preacher and Hancock (in press) mention
at least three advantages that result from explicitly including in the model a
parameter that is of interest to the researcher: (i) estimation will yield a point
estimate and a standard error for the parameter estimate, which allows one
to conduct hypothesis tests and construct confidence intervals for the parameter; (ii) it allows the parameter of interest to be predicted by other variables
included in the model; and (iii) the researcher has the flexibility to treat the
newly added parameter as a known or unknown value, and as a value that
varies or is fixed across individuals. Another possible advantage is that the
reparameterized model can sometimes be estimated more easily.
In addition to the quadratic model, other models have been reparameterized. Choi, Harring, and Hancock (2009) reparameterized the logistic growth
model to include parameters such as lower and upper asymptotes, the timepoint when rate of growth is maximal (surge point), and the rate of growth at
that timepoint (surge slope). Harring et al. (2006) reparameterized a piecewise
model so as to explicitly include the knot in the model, making it possible to
estimate the location of the knot directly from the sample data. This can be
useful if the researcher does not have any hypothesis as to the moment when
participants transition from one phase to the next. In the following section,
we also mention the recent reparameterization of growth models into growth
rate models (Zhang, McArdle, & Nesselroade, 2012).
Many reparameterized models can be challenging to implement in statistical software. Thankfully, some authors provide software syntax necessary
to estimate reparameterized models, making these models readily available
to researchers (e.g., Choi et al., 2009; Cudeck & du Toit, 2002; Harring et al.,
2006; see also Preacher & Hancock, in press). In upcoming years, we expect an
increasing number of specific models being reparameterized. For example,
Preacher and Hancock (in press) mentioned the possibility of explicitly modeling the angle between the lines of two adjacent phases in a piecewise model.
Furthermore, some models can be reformulated into similar, but not fully
equivalent models (Biancocini, 2012). In such situations, particularly if differences in model fit between the two models are sizeable, it remains unclear
whether a researcher should choose a model based on substantive interpretation of parameters rather than model fit.

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RATE OF CHANGE
Since a rate of change that varies over time is precisely what distinguishes
linear from nonlinear growth, much is gained from describing rate of change
as a function of time. It allows the researcher to gain insights into the growth
pattern that he or she is investigating and also provides the reader with a
more complete picture of the change over time.
Zhang et al. (2012) have reparameterized both the quadratic and exponential classic growth models into growth rate models. These models contain
parameters, which indicate the rate of change over time. And indeed,
they found in their simulation studies that estimation of such models in
traditional structural equation modeling software yields accurate parameter
estimates, even in the presence of missing data. Since rate of change is now
explicitly represented in the models, researchers can ask questions such
as “Does rate of change vary across individuals?” and “Is rate of change
predicted by covariates?” (e.g., “Is rate of change different for boys and
girls?”).
When parameter estimates do not carry interesting interpretations, one way
to make up for this is to focus on rate of change instead. Such is the approach
taken by Long and Ryoo (2010) in describing fractional polynomial models.
These authors plot rate of change as a function of time. We present such a
plot in Figure 2, corresponding to the rational function shown in Figure 1.
From this figure, we can easily see that the individual improves her performance the fastest in the first 2 days of training and with slow improvement
thereafter. An even more in-depth presentation of rate of change involves
plotting rate of change against time, but with confidence intervals around the
curve (e.g., Biesanz, Deeb-Sossa, Papadakis, Bollen, & Curran, 2004; Preacher
& Hancock, in press; Zhang et al., 2012). The width of confidence intervals
changes as a function of time, and this is reflected in the graphs. This allows
the reader to conduct his or her very own hypothesis tests of interest on the
rate of change for a series of timepoints, at the level of confidence used in
the plots. This also allows the reader to judge the precision of the estimate
of rate of change yielded by the reported model as a function of time, all of
this in a single visual representation of the growth process. An alternative
but easier-to-read version of this plot is to replace the confidence bands with
circles around rates of change that are significantly different from 0 at a given
significance level (see Preacher & Hancock, in press).
Zhang et al. (2012) mention that a task for future research is to reparameterize the nonlinear growth model into a growth acceleration model. Acceleration
of growth is the rate at which the rate of change varies as a function of time; in
mathematical terms, it is represented by the second derivative of the func′′
tion fitted to the data with respect to time (ft ), if the derivative function

Models of Nonlinear Growth

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0.8
0.7

Rate of change

0.6
0.5
0.4
0.3
0.2
0.1
0
1

2

3

4

5

6

7

8

9

10

Time

Figure 2 Rate of change in predicted proportion of successes as a function of
time.

exists. This acceleration of growth could be a function of time itself, and if
this were the case, one could conceivably plot growth acceleration against
time, possibly with confidence intervals around the curve. However, acceleration growth models still have to be developed before acceleration of growth
becomes part of the standard tools that researchers use to describe a nonlinear
growth process.
OTHER RECENT AND FUTURE DEVELOPMENTS
We have addressed several major recent and future developments regarding
nonlinear growth modeling in this essay, but this list is not exhaustive. In
conclusion, we mention other issues pertaining to nonlinear growth models
that we think are worthy of consideration.
One area that needs further work is software development. We have
already mentioned in passing that some models require more effort to
estimate than others, owing to the nature of their parameters. We can only
hope that upcoming versions of software packages will make the transition
from simpler to more complex models easier to researchers. Similarly,
current software packages cannot allow for the shape of the trajectory to
vary across individuals, although this constraint is not a mathematical
one (Kamata et al., 2012). One existing problem that we did not mention

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until now is the presence of discrepancies in results yielded by different
software packages. For example, Grimm and Ram (2009) fitted several
models using both Mplus and SAS, and found only small differences in
estimates of fixed effects, but nonnegligible ones in estimates of random
effects. Simulation studies are needed to help determine which software
packages yield accurate estimates of random effects. But in the meantime,
how is a researcher to choose between two software packages available to
them? And indeed, we have no straightforward answer—this decision is
currently mainly a matter of personal preference, although one which can
affect reported results. Simmons, Nelson, and Simonsohn (2011) have coined
such impactful yet personal decisions “researcher degrees of freedom.”
Some recent techniques that we have not mentioned until now have the
potential to be fruitful in future nonlinear growth modeling endeavors.
For instance, Royston and Sauerbrei (2003) used a resampling approach to
determine whether predictors should enter a regression model as linear
(first-order polynomial) or nonlinear (higher-order polynomial) predictors.
To our knowledge, this approach has yet to be applied to growth models.
Another novel approach that we delayed mentioning but that has been
adapted to nonlinear growth models is that of robust growth curve modeling. Zhang, Lai, Lu, and Tong (2012; see also Tong & Zhang, 2012) created
an online tool that allows researchers to estimate “robust” growth curve
models in the presence of nonnormal data. Their tool permits the estimation
of at least one type of nonlinear growth model, namely the latent basis
growth curve. Although this free software is still in its infancy, it constitutes
a fantastic first step toward dealing with violations of assumptions in
nonlinear growth modeling.
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Tong, X., & Zhang, Z. (2012). Diagnostics of robust growth curve modeling using
Student’s t distribution. Multivariate Behavioral Research, 47, 493–518.
Zhang, Z., Lai, K., Lu, Z., & Tong, X. (2012). Bayesian inference and application of
robust growth curve models using Student’s t distribution. Structural Equation
Modeling, 20, 47–78.
Zhang, Z., McArdle, J. J., & Nesselroade, J. R. (2012). Growth rate models: Emphasizing growth rate analysis through growth curve modeling. Journal of Applied
Statistics, 39, 1241–1262.

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

FURTHER READING
Cudeck, R., & Harring, J. R. (2007). Analysis of nonlinear patterns of change with
random coefficient models. Annual Review of Psychology, 58, 615–637.
Grimm, K. J., & Ram, N. (2009). Nonlinear growth models in Mplus and SAS. Structural Equation Modeling, 16, 676–701.
Kamata, A., Nese, J. F., Patarapichayatham, C., & Lai, C. F. (2012). Modeling nonlinear
growth with three data points: Illustration with benchmarking data. Assessment for
Effective Intervention, 38, 105–116.
Long, J., & Ryoo, J. (2010). Using fractional polynomials to model non-linear trends
in longitudinal data. British Journal of Mathematical and Statistical Psychology, 63(1),
177–203.
Preacher, K. J., & Hancock, G. R. (in press). On interpretable reparameterizations of
linear and nonlinear latent growth curve models. In Harring, J. R., & Hancock, G.
R., Advances in longitudinal methods in the social and behavioral sciences (pp. 25–58).
Charlotte, NC: Information Age Publishing.
Ram, N., & Grimm, K. (2007). Using simple and complex growth models to articulate developmental change: Matching theory to method. International Journal of
Behavioral Development, 31, 303–316.
Zhang, Z., McArdle, J. J., & Nesselroade, J. R. (2012). Growth rate models: Emphasizing growth rate analysis through growth curve modeling. Journal of Applied
Statistics, 39, 1241–1262.

PATRICK COULOMBE SHORT BIOGRAPHY
Patrick Coulombe is a graduate student in Quantitative Psychology at the
University of New Mexico, in Albuquerque, NM. He obtained his Bachelor of
Science in Psychology from the University of Québec at Montréal in 2010. He
is interested in statistical modeling, particularly as it applies to the analysis
of longitudinal data. He also enjoys software and web programming, which
led him to get involved in online research.
Personal webpage: http://www.patrickcoulombe.com.

JAMES P. SELIG SHORT BIOGRAPHY
James P. Selig, PhD, is an Associate Professor of Biostatistics at the University
of Arkansas for Medical Sciences. He received his doctorate in Quantitative Psychology from the University of Kansas in 2009. His research interests include longitudinal data analysis, multilevel modeling, and structural
equation modeling.
Personal webpage: http://biostatistics.uams.edu/faculty-and-staff/jamesp-selig-phd/

Models of Nonlinear Growth

15

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Models of Nonlinear Growth
PATRICK COULOMBE and JAMES P. SELIG

Abstract
Models for nonlinear growth are not new, but have not been widely applied in the
social and behavioral sciences. In this essay, we describe the fundamental issues relevant to choosing and using a nonlinear growth model. We discuss how researchers
can go about choosing a model and then focus on the application of two specific
nonlinear models: the fractional polynomial model and the piecewise model. We
highlight recent work in reparameterization that allows researchers to choose models with parameters tailored specifically to research questions. We also review recent
work on the topic of growth rates in nonlinear models that will allow researchers to
obtain richer information from the application of nonlinear models. We conclude by
pointing out some of the unresolved issues in the use of nonlinear growth models.

MODELS OF NONLINEAR GROWTH
Models of nonlinear growth are used to represent change over time in scores
on a variable measured on several occasions. Both linear and nonlinear models can be used to examine change in a variable over time. A key difference
between linear and nonlinear change is that in a model for linear change,
the rate of change is constant over time (i.e., the slope of the straight line
is the same at all occasions); however, in a model for nonlinear change the
rate of change can vary across time (Cudeck & Harring, 2007). For example,
Figure 1 shows an individual’s proportion of successes in a learning task over
10 days. As can be seen, the individual improved her performance dramatically in the first 2 days (reaching a very high proportion of successes), but
improved much slower thereafter; in short, improvement over time was not
constant.
Throughout this paper, we refer to nonlinear growth models that take the
form:
yij = f (𝛽1i , 𝛽2i , … , 𝛽ki , tij ) + 𝜀ij ,
where subscript i refers to individual i, subscript j the jth occasion of measurement (timepoint), yij the individual i’s observed score at timepoint j, parameters 𝛽 1i through 𝛽 ki the regression weights, tij the time distance to the origin,
Emerging Trends in the Social and Behavioral Sciences. Edited by Robert Scott and Stephen Kosslyn.
© 2015 John Wiley & Sons, Inc. ISBN 978-1-118-90077-2.

1

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

1

% Success

0.8

0.6

0.4
Observed
Rational

0.2

Quadratic
Asymptote
0
1

2

3

4

5

6

7

8

9

10

Time

Figure 1 An individual’s hypothetical proportion of successes in a learning task
over 10 consecutive days, along with a rational and quadratic function fitted to the
data. The asymptote is associated with the rational function and has been
fixed to 1.

and 𝜀ij the difference between the observed and predicted score for individual i at timepoint j (which is typically assumed to be normally distributed
with a mean of 0). The score yij on the dependent variable is described as a
function of time and the regression weights, which may be defined to vary or
to be the same across individuals. In a nonlinear growth model, the function
f follows a nonlinear trajectory when plotted against time. More formally, if
the function is differentiable, its first derivative ft ′ (𝛽 1i , 𝛽 2i , … , 𝛽 ki , tij )is itself
a function of tij . Finally, the 𝛽 parameters can be further specified to be a
function of covariates that vary or are fixed over time.
In this essay, we first discuss how a researcher can decide on a particular functional form when modeling nonlinear change over time. We then
describe two potentially useful nonlinear growth models, fractional polynomials and piecewise models. Next, we show how a researcher can tailor his
or her particular model to substantive questions of interest through reparameterization. Finally, we illustrate how one can effectively use rate of change
to give a more complete depiction of the nonlinear growth process to the
reader.

Models of Nonlinear Growth

3

CHOOSING A FUNCTIONAL FORM
When modeling linear change over time, the choice of the particular model
to use is easy: the linear growth model. When one wants to model nonlinear
change over time, however, a vast array of possible models exist. In this
section, we discuss how a researcher can choose the functional form of the
change over time.
There is no substitute for a researcher’s deep understanding of change in
a focal variable when choosing an appropriate functional form. This may be
especially true when modeling nonlinear change over time, because in many
instances, several functions might fit the data equally well. At the very least,
the researcher should choose a functional form for the model that is consistent with what he or she knows about change in the variable. For example,
an educational researcher examining the cumulative number of aggressive
behaviors over the school year among first graders would not choose any
function that decreases over time. The researcher’s understanding of the variables under study should also drive analytical decisions such as whether to
allow a specific regression coefficient to take different values across individuals (Bianconcini, 2012).
Cudeck and Harring (2007) provide three criteria in the selection of an
adequate nonlinear function that may support researchers in choosing a
functional form: (i) it must fit the data well, (ii) its parameters must have
an interpretation that answers interesting substantive questions, and (iii) its
characteristic shape corresponds to the hypothesized change (e.g., presence
of an upper asymptote if change over time in a variable with a maximum
is continually increasing). Only the first one is statistical, while the latter
two are grounded in the researcher’s understanding of the phenomenon
to be modeled. In our view, a common error when choosing functional
forms for nonlinear growth models is to overemphasize the first criterion.
The point here was that models that fit the data well, yet that do not have
interpretable parameters and/or that are not consistent with the researcher’s
understanding of change, are not useful [even though they fit the data well,
is what the point is].
When new variables are examined, or a researcher does not have a priori
ideas about patterns of change, one very effective way to probe the data is
through the use of plots. Cudeck and Harring (2007) mention three different
types of plots that can be used in the context of nonlinear growth modeling.
The first one is the spaghetti plot, in which the raw scores for a subset of the
sample are plotted against time, with straight segments joining two adjacent
scores for a given individual. The second one is the swarm plot, in which an
estimated function corresponding to the chosen function is plotted against
time. Finally, the third one is the trellis plot, which is actually an ensemble

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

of plots arranged in a grid. In each plot, the scores for a single individual
are plotted against time, with the estimated curve for that specific individual
superimposed (see also Few, 2009, for the description of a similar plotting
technique). This quickly gives the researcher (or the reader) an idea of how
well the postulated model fits individual data, and of the variability of this
fit across individuals.
SPECIFIC CURVE SHAPES
Next, we present two particular variations on nonlinear growth models.
The first is the fractional polynomial model and the second is the piecewise
model. We chose these two because we believe they are relatively straightforward extensions of models commonly used for growth in the social and
behavioral sciences. Although these models are not new, there are relatively
few applications of these models.
FRACTIONAL POLYNOMIALS
Researchers interested in modeling nonlinear change over time frequently
apply a quadratic curve to the data. In the case of a quadratic curve, the
mathematical function fitted to the data is given as
yij = 𝛽0i + 𝛽1i tij + 𝛽2i t2ij + 𝜀ij .
In this model, 𝛽 0 represents the predicted score when t = 0, 𝛽 1 is the rate
of change when t = 0, and twice the value of 𝛽 2 is the magnitude by which
the rate of change varies for each one-unit increase in the time variable. An
example of a quadratic curve is given in Figure 1 (dashed line), where we
attempt to model the individual’s increasing proportion of successes in a
learning task over time. The quadratic curve follows the observed scores
(triangles) quite well. In general, the characteristics of the quadratic curve
include presence of a maximum or minimum, symmetry of the curve about
the maximum or minimum, and absence of lower or upper asymptotes.
These characteristics can be at odds with the researcher’s understanding of
change. For example, the absence of asymptotes is not consistent with the
use of a measure which has a minimum and/or a maximum (in this case,
proportion of successes should not exceed 1), and rarely is the pattern of
change in behavior symmetric before and after attaining an extreme value.
One way to address these problems is to explicitly select an exponentiation of the time variable that yields a curve consistent with the theory of
interest. Polynomial models involving the exponentiation of the time variable with fractions or negative integers are known as fractional polynomials
(Long & Ryoo, 2010; see also Royston & Altman, 1994; Royston, Ambler, &

Models of Nonlinear Growth

5

Sauerbrei, 1999). The quadratic polynomial shown above is a special case of
a fractional polynomial model, in which the time variable is exponentiated
to the first and second power.
For example, Figure 1 shows a rational function (solid line) for the individual in attempting to model her increasing proportion of successes over time.
The rational function takes the general form:
yij = 𝛽0i + 𝛽1i t−1
+ 𝜀ij .
ij
Therefore, the rational function is a polynomial function with one-time variable exponentiated to the first negative power. In this model, 𝛽 0 represents a
horizontal asymptote and 𝛽 1 is the negative value of the rate of change when
t = 1.
As can be seen in Figure 1, this function (with the constraint 𝛽 0 = 1) is consistent with the data, slightly more so than the quadratic curve. Given that
we expected performance to improve over time, the quadratic function had
the drawback of predicting decreasing scores past a certain point (around day
8; see Figure 1). Therefore, we needed a function that would be monotonic,
that is, that would increase over time without ever decreasing. The sign of
the second parameter of the rational function (𝛽 1 ) indicates whether scores
are increasing (𝛽 1 < 0; Figure 1) or decreasing (𝛽 1 > 0) over time. But why did
we fix the first parameter (𝛽 0 ) to be 1 instead of letting it be freely estimated
with the data? This parameter in the rational function represents a horizontal
asymptote (dashed horizontal line in Figure 1). We needed a function which
would allow us to account for the fact that the proportion of successes cannot
exceed 1, and the rational function—but not the quadratic function—does
just that. We could have allowed for the specific value of the asymptote to
be estimated freely. This would allow us to check whether predicted maximal performance differs across individuals, and to potentially predict such
individual differences using covariates. For example, we could ask whether
maximal performance depends on the age of the person learning the task.
Long and Ryoo (2010) demonstrate graphically the impact of manipulating
the exponents of the time variable on the shape of the trajectory over time.
These authors separate the models into first-order and second-order polynomials, depending on the number of time variables included in the model. For
either type, one can obtain a “flipped” version of any of the curves by reversing the signs of the regression weights (𝛽). And indeed, these curves can be
selected to include lower or upper asymptotes (as we did in Figure 1), to be
asymmetric about an extreme value, or to depend on scores on covariates
(see Lambert, Smith, Jones, & Botha, 2005; Royston & Altman, 1994); overall,
fractional polynomials are much more flexible than traditional polynomials
in representing a change process.

6

EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

One difficulty that arises in the use of fractional polynomials over traditional polynomials lies in software use: Fractional polynomials are often
more difficult to estimate using standard linear multilevel or structural
equation modeling software. Another important issue with regard to the
use of fractional polynomial models is the interpretability of the parameters. Oftentimes, the function will contain parameters that do not have
straightforward interpretations, and even when the parameters have clear
interpretations, the interpretations might not be relevant to the study
research questions.
PIECEWISE MODEL
The piecewise model (also called spline or multiphase model, sometimes discontinuous model; Singer & Willett, 2003) allows one to model change over
time differently from one subset (range) of the time variable to the next subset. There are several ways to achieve this. Most authors have considered
piecewise models in which the trajectory for every time subset is exclusively
linear. For these models, the nonlinear nature of change is depicted using
two or more linear splines. However, it is entirely possible for the individual
splines to have nonlinear functional forms.
The simplest piecewise model is a linear–linear model, in which the full
range of the period covered by the study is separated into two subsets (phases
and epochs), within each of which change over time is linear. What distinguishes the linear trajectory of the two phases is the magnitude of the linear
slope. The measurement occasion at which the trajectory changes is called
the knot, or the transition point, and this knot can be fixed or variable across
individuals. The function describing the linear–linear model just described
is given as
yij = 𝛽0i + 𝛽1i t1ij + 𝛽2i t2ij + 𝜀ij .
In this model, 𝛽 0 is the predicted score when both t1 and t2 = 0, 𝛽 1 describes
the linear rate of change for the first phase and 𝛽 2 is associated with the linear
rate of change for the second phase. The novelty in this model is the addition
of a second time variable. There are two ways to code the two time variables.
One way is to treat them as independent: the first time variable starts at a
given value and increases until the end of the first phase, at which point
it becomes constant for the remainder of measurement occasions, and the
second time variable is set to 0 until the last measurement occasion of the
first phase, at which point it starts increasing by 1 for each subsequent measurement occasion (Bollen & Curran, 2006). The second way to code the time
variables is to make them partially redundant: the first time variable starts at

Models of Nonlinear Growth

7

a given value and increases across all measurement occasions, and the second variable is coded in the same way as in the first method (Singer & Willett,
2003). For example, for a six-wave study with the first phase spanning times
1 through 3 and the second phase spanning times 4 through 6, the first time
variable could be coded (0, 1, 2, 2, 2, 2) in the first method, and (0, 1, 2, 3, 4, 5)
in the second method. The second time variable would be coded (0, 0, 0, 1, 2,
3) in both methods.
The interpretation of 𝛽 1i concerns mainly the first phase, and represents
the magnitude of the change in predicted score for each one-unit increase
in time during the first phase. However, the same quantity for the second
phase depends on how the first time variable is coded. In the first method,
𝛽 2i represents the change in predicted score for each one-unit increase in time
during the second phase. In the second method, this quantity is (𝛽 1i + 𝛽 2i )
instead.
Piecewise models can also allow for a change in elevation from one phase
to the next (Singer & Willett, 2003). An appropriate mathematical function
for two phases with a change in elevation is given as
yij = 𝛽0i + 𝛽1i t1ij + 𝛽2i t2ij + 𝛽3i even tij + 𝜀ij .
The variable event is a binary variable coded as either 0 or 1, which tracks
whether a specific event has occurred at measurement occasion j for individual i (e.g., the individual has graduated, or has started an intervention).
Typically, the occurrence of the event will correspond to the change in phases,
and will change values only once during the course of the study, although
neither condition is a mathematical requirement. The parameter 𝛽 3i corresponds to the magnitude of the sudden change in predicted score on the
dependent variable following the occurrence of the event. In addition, it is
possible to model a change in elevation without any change in linear trajectory between the two phases. To do so, one can simply use the second coding
method and omit the second time variable from the model (or, equivalently,
set 𝛽 2i to 0).
Extensions to the models presented above are available. For instance, it is
possible to model growth over three (or more) distinct phases, which can be
particularly useful in a three-phase study involving a baseline, an intervention phase, and a follow-up; the reader is referred to Cudeck and Harring
(2007), Ram and Grimm (2007), and Singer and Willett (2003) for examples.
One recent interesting extension involved the modeling of phasic change
over time that differed across subsets of individuals within a single sample
(Kamata, Nese, Patarapichayatham, & Lai, 2012). Another extension is to estimate the timing of the knot with the sample data instead of determining it
a priori (e.g., Cudeck & Harring, 2007; Harring, Cudeck, & du Toit, 2006).
Finally, one possible extension is to allow for nonlinear trajectory within each

8

EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

phase. In this case, the function for the piecewise model can be given by the
more general formula:
{ (
)
t<𝜅
f1 𝛽0i , … , 𝛽mi , t ,
.
yij =
f2 (𝛽(m+1)i , … , 𝛽(m+n)i , t), t ≥ 𝜅
For this general model, scores on the dependent variable are a function of
a series of regression coefficients and the time variable, but the function is
allowed to change from the first to the second phase at the knot when t = 𝜅.
Therefore, one could model a nonlinear trajectory during the first phase, the
second phase, or both the phases, particularly if there are theoretical reasons
to do so.
REPARAMETERIZATION
Once a researcher has chosen a specific functional form to use, the goal is
to make the model as useful as possible for addressing research questions.
This goal can be facilitated through reparameterization. Reparameterization is
the expression of an existing model using different parameters. The goal of
reparameterization is to make the interpretation of the reformulated model
more substantively interesting (Preacher & Hancock, in press). The reparameterized model should have the same number of parameters as the original
model, and should yield the same fit to the data; in short, the reparameterized
model is equivalent to the original model. The choice between two equivalent
models should depend on what specific research questions the researcher
wishes to address.
Recent literature has offered numerous examples of reparameterized models. Cudeck and du Toit (2002) reexpressed the quadratic model to make the
interpretation of its parameters more interesting from a substantive point of
view. The resulting function, using our notation, is
(
)2
tij
− 1 + 𝜀ij .
yij = 𝛼1i − (𝛼1i − 𝛽0i )
𝛼2i
To illustrate the value of reparameterizing a model, let us compare the
interpretation of the parameters with those from the classic quadratic model.
The two models share one parameter (𝛽 0 ), which is the predicted score
when time is 0. The rest of the parameters differ. In the classic quadratic
model, neither 𝛽 1 nor 𝛽 2 has a straightforward interpretation (except when
a researcher is interested in the rate of change at time 0 specifically).
Conversely, the reparameterized model bears parameters with interesting
interpretations: 𝛼 1 is the predicted maximum (or minimum) score (e.g., the
maximum proportion of successes), and 𝛼 2 is the time at which the predicted

Models of Nonlinear Growth

9

maximum (or minimum) is attained (e.g., the required time in days for the
maximum performance to be achieved; see Figure 1). Despite these differences, both models fit the data equally well. Therefore, it appears clear from
this example that one can substantially improve the usefulness of a model by
reparameterizing it so as to make its parameters more readily interpretable.
Although one primary goal of reparameterization is to make parameter
estimates more readily interpretable, other advantages can come from reformulating a model. For example, Preacher and Hancock (in press) mention
at least three advantages that result from explicitly including in the model a
parameter that is of interest to the researcher: (i) estimation will yield a point
estimate and a standard error for the parameter estimate, which allows one
to conduct hypothesis tests and construct confidence intervals for the parameter; (ii) it allows the parameter of interest to be predicted by other variables
included in the model; and (iii) the researcher has the flexibility to treat the
newly added parameter as a known or unknown value, and as a value that
varies or is fixed across individuals. Another possible advantage is that the
reparameterized model can sometimes be estimated more easily.
In addition to the quadratic model, other models have been reparameterized. Choi, Harring, and Hancock (2009) reparameterized the logistic growth
model to include parameters such as lower and upper asymptotes, the timepoint when rate of growth is maximal (surge point), and the rate of growth at
that timepoint (surge slope). Harring et al. (2006) reparameterized a piecewise
model so as to explicitly include the knot in the model, making it possible to
estimate the location of the knot directly from the sample data. This can be
useful if the researcher does not have any hypothesis as to the moment when
participants transition from one phase to the next. In the following section,
we also mention the recent reparameterization of growth models into growth
rate models (Zhang, McArdle, & Nesselroade, 2012).
Many reparameterized models can be challenging to implement in statistical software. Thankfully, some authors provide software syntax necessary
to estimate reparameterized models, making these models readily available
to researchers (e.g., Choi et al., 2009; Cudeck & du Toit, 2002; Harring et al.,
2006; see also Preacher & Hancock, in press). In upcoming years, we expect an
increasing number of specific models being reparameterized. For example,
Preacher and Hancock (in press) mentioned the possibility of explicitly modeling the angle between the lines of two adjacent phases in a piecewise model.
Furthermore, some models can be reformulated into similar, but not fully
equivalent models (Biancocini, 2012). In such situations, particularly if differences in model fit between the two models are sizeable, it remains unclear
whether a researcher should choose a model based on substantive interpretation of parameters rather than model fit.

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

RATE OF CHANGE
Since a rate of change that varies over time is precisely what distinguishes
linear from nonlinear growth, much is gained from describing rate of change
as a function of time. It allows the researcher to gain insights into the growth
pattern that he or she is investigating and also provides the reader with a
more complete picture of the change over time.
Zhang et al. (2012) have reparameterized both the quadratic and exponential classic growth models into growth rate models. These models contain
parameters, which indicate the rate of change over time. And indeed,
they found in their simulation studies that estimation of such models in
traditional structural equation modeling software yields accurate parameter
estimates, even in the presence of missing data. Since rate of change is now
explicitly represented in the models, researchers can ask questions such
as “Does rate of change vary across individuals?” and “Is rate of change
predicted by covariates?” (e.g., “Is rate of change different for boys and
girls?”).
When parameter estimates do not carry interesting interpretations, one way
to make up for this is to focus on rate of change instead. Such is the approach
taken by Long and Ryoo (2010) in describing fractional polynomial models.
These authors plot rate of change as a function of time. We present such a
plot in Figure 2, corresponding to the rational function shown in Figure 1.
From this figure, we can easily see that the individual improves her performance the fastest in the first 2 days of training and with slow improvement
thereafter. An even more in-depth presentation of rate of change involves
plotting rate of change against time, but with confidence intervals around the
curve (e.g., Biesanz, Deeb-Sossa, Papadakis, Bollen, & Curran, 2004; Preacher
& Hancock, in press; Zhang et al., 2012). The width of confidence intervals
changes as a function of time, and this is reflected in the graphs. This allows
the reader to conduct his or her very own hypothesis tests of interest on the
rate of change for a series of timepoints, at the level of confidence used in
the plots. This also allows the reader to judge the precision of the estimate
of rate of change yielded by the reported model as a function of time, all of
this in a single visual representation of the growth process. An alternative
but easier-to-read version of this plot is to replace the confidence bands with
circles around rates of change that are significantly different from 0 at a given
significance level (see Preacher & Hancock, in press).
Zhang et al. (2012) mention that a task for future research is to reparameterize the nonlinear growth model into a growth acceleration model. Acceleration
of growth is the rate at which the rate of change varies as a function of time; in
mathematical terms, it is represented by the second derivative of the func′′
tion fitted to the data with respect to time (ft ), if the derivative function

Models of Nonlinear Growth

11

0.8
0.7

Rate of change

0.6
0.5
0.4
0.3
0.2
0.1
0
1

2

3

4

5

6

7

8

9

10

Time

Figure 2 Rate of change in predicted proportion of successes as a function of
time.

exists. This acceleration of growth could be a function of time itself, and if
this were the case, one could conceivably plot growth acceleration against
time, possibly with confidence intervals around the curve. However, acceleration growth models still have to be developed before acceleration of growth
becomes part of the standard tools that researchers use to describe a nonlinear
growth process.
OTHER RECENT AND FUTURE DEVELOPMENTS
We have addressed several major recent and future developments regarding
nonlinear growth modeling in this essay, but this list is not exhaustive. In
conclusion, we mention other issues pertaining to nonlinear growth models
that we think are worthy of consideration.
One area that needs further work is software development. We have
already mentioned in passing that some models require more effort to
estimate than others, owing to the nature of their parameters. We can only
hope that upcoming versions of software packages will make the transition
from simpler to more complex models easier to researchers. Similarly,
current software packages cannot allow for the shape of the trajectory to
vary across individuals, although this constraint is not a mathematical
one (Kamata et al., 2012). One existing problem that we did not mention

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EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

until now is the presence of discrepancies in results yielded by different
software packages. For example, Grimm and Ram (2009) fitted several
models using both Mplus and SAS, and found only small differences in
estimates of fixed effects, but nonnegligible ones in estimates of random
effects. Simulation studies are needed to help determine which software
packages yield accurate estimates of random effects. But in the meantime,
how is a researcher to choose between two software packages available to
them? And indeed, we have no straightforward answer—this decision is
currently mainly a matter of personal preference, although one which can
affect reported results. Simmons, Nelson, and Simonsohn (2011) have coined
such impactful yet personal decisions “researcher degrees of freedom.”
Some recent techniques that we have not mentioned until now have the
potential to be fruitful in future nonlinear growth modeling endeavors.
For instance, Royston and Sauerbrei (2003) used a resampling approach to
determine whether predictors should enter a regression model as linear
(first-order polynomial) or nonlinear (higher-order polynomial) predictors.
To our knowledge, this approach has yet to be applied to growth models.
Another novel approach that we delayed mentioning but that has been
adapted to nonlinear growth models is that of robust growth curve modeling. Zhang, Lai, Lu, and Tong (2012; see also Tong & Zhang, 2012) created
an online tool that allows researchers to estimate “robust” growth curve
models in the presence of nonnormal data. Their tool permits the estimation
of at least one type of nonlinear growth model, namely the latent basis
growth curve. Although this free software is still in its infancy, it constitutes
a fantastic first step toward dealing with violations of assumptions in
nonlinear growth modeling.
REFERENCES
Bianconcini, S. (2012). Nonlinear and quasi-simplex patterns in latent growth models. Multivariate Behavioral Research, 47, 88–114.
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.
Bollen, K. A., & Curran, P. J. (2006). Latent curve models: A structural equation perspective. Hoboken, NJ: John Wiley & Sons, Ltd.
Choi, J., Harring, J. R., & Hancock, G. R. (2009). Latent growth modeling for logistic
response functions. Multivariate Behavioral Research, 44, 620–645.
Cudeck, R., & Du Toit, S. H. (2002). A version of quadratic regression with interpretable parameters. Multivariate Behavioral Research, 37, 501–519.
Cudeck, R., & Harring, J. R. (2007). Analysis of nonlinear patterns of change with
random coefficient models. Annual Review of Psychology, 58, 615–637.

Models of Nonlinear Growth

13

Few, S. (2009). Now you see it: Simple visualization techniques for quantitative analysis.
Oakland, CA: Analytics Press.
Grimm, K. J., & Ram, N. (2009). Nonlinear growth models in Mplus and SAS. Structural Equation Modeling, 16, 676–701.
Harring, J. R., Cudeck, R., & du Toit, S. H. (2006). Fitting partially nonlinear random
coefficient models as SEMs. Multivariate Behavioral Research, 41, 579–596.
Kamata, A., Nese, J. F., Patarapichayatham, C., & Lai, C. F. (2012). Modeling nonlinear
growth with three data points: Illustration with benchmarking data. Assessment for
Effective Intervention, 38, 105–116.
Lambert, P. C., Smith, L. K., Jones, D. R., & Botha, J. L. (2005). Additive and multiplicative covariate regression models for relative survival incorporating fractional
polynomials for time-dependent effects. Statistics in Medicine, 24, 3871–3885.
Long, J., & Ryoo, J. (2010). Using fractional polynomials to model non-linear trends
in longitudinal data. British Journal of Mathematical and Statistical Psychology, 63(1),
177–203.
Preacher, K. J., & Hancock, G. R. (in press). On interpretable reparameterizations of
linear and nonlinear latent growth curve models. In Harring, J. R., & Hancock, G.
R., Advances in longitudinal methods in the social and behavioral sciences (pp. 25–58).
Charlotte, NC: Information Age Publishing.
Ram, N., & Grimm, K. (2007). Using simple and complex growth models to articulate developmental change: Matching theory to method. International Journal of
Behavioral Development, 31, 303–316.
Royston, P., & Altman, D. G. (1994). Regression using fractional polynomials of
continuous covariates: Parsimonious parametric modelling. Applied Statistics, 43,
429–467.
Royston, P., Ambler, G., & Sauerbrei, W. (1999). The use of fractional polynomials to
model continuous risk variables in epidemiology. International Journal of Epidemiology, 28, 964–974.
Royston, P., & Sauerbrei, W. (2003). Stability of multivariable fractional polynomial
models with selection of variables and transformations: A bootstrap investigation.
Statistics in Medicine, 22, 639–659.
Simmons, J. P., Nelson, L. D., & Simonsohn, U. (2011). False-positive psychology:
Undisclosed flexibility in data collection and analysis allows presenting anything
as significant. Psychological Science, 22, 1359–1366.
Singer, J. D., & Willett, J. B. (2003). Applied longitudinal data analysis: Modeling change
and event occurrence. New York, NY: Oxford University Press.
Tong, X., & Zhang, Z. (2012). Diagnostics of robust growth curve modeling using
Student’s t distribution. Multivariate Behavioral Research, 47, 493–518.
Zhang, Z., Lai, K., Lu, Z., & Tong, X. (2012). Bayesian inference and application of
robust growth curve models using Student’s t distribution. Structural Equation
Modeling, 20, 47–78.
Zhang, Z., McArdle, J. J., & Nesselroade, J. R. (2012). Growth rate models: Emphasizing growth rate analysis through growth curve modeling. Journal of Applied
Statistics, 39, 1241–1262.

14

EMERGING TRENDS IN THE SOCIAL AND BEHAVIORAL SCIENCES

FURTHER READING
Cudeck, R., & Harring, J. R. (2007). Analysis of nonlinear patterns of change with
random coefficient models. Annual Review of Psychology, 58, 615–637.
Grimm, K. J., & Ram, N. (2009). Nonlinear growth models in Mplus and SAS. Structural Equation Modeling, 16, 676–701.
Kamata, A., Nese, J. F., Patarapichayatham, C., & Lai, C. F. (2012). Modeling nonlinear
growth with three data points: Illustration with benchmarking data. Assessment for
Effective Intervention, 38, 105–116.
Long, J., & Ryoo, J. (2010). Using fractional polynomials to model non-linear trends
in longitudinal data. British Journal of Mathematical and Statistical Psychology, 63(1),
177–203.
Preacher, K. J., & Hancock, G. R. (in press). On interpretable reparameterizations of
linear and nonlinear latent growth curve models. In Harring, J. R., & Hancock, G.
R., Advances in longitudinal methods in the social and behavioral sciences (pp. 25–58).
Charlotte, NC: Information Age Publishing.
Ram, N., & Grimm, K. (2007). Using simple and complex growth models to articulate developmental change: Matching theory to method. International Journal of
Behavioral Development, 31, 303–316.
Zhang, Z., McArdle, J. J., & Nesselroade, J. R. (2012). Growth rate models: Emphasizing growth rate analysis through growth curve modeling. Journal of Applied
Statistics, 39, 1241–1262.

PATRICK COULOMBE SHORT BIOGRAPHY
Patrick Coulombe is a graduate student in Quantitative Psychology at the
University of New Mexico, in Albuquerque, NM. He obtained his Bachelor of
Science in Psychology from the University of Québec at Montréal in 2010. He
is interested in statistical modeling, particularly as it applies to the analysis
of longitudinal data. He also enjoys software and web programming, which
led him to get involved in online research.
Personal webpage: http://www.patrickcoulombe.com.

JAMES P. SELIG SHORT BIOGRAPHY
James P. Selig, PhD, is an Associate Professor of Biostatistics at the University
of Arkansas for Medical Sciences. He received his doctorate in Quantitative Psychology from the University of Kansas in 2009. His research interests include longitudinal data analysis, multilevel modeling, and structural
equation modeling.
Personal webpage: http://biostatistics.uams.edu/faculty-and-staff/jamesp-selig-phd/

Models of Nonlinear Growth

15

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