edicine and Rehabilitation

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Archives of Physical Medicine and Rehabilitation 2013;94:589-96

SPECIAL COMMUNICATION

An Introduction to Applying Individual Growth Curve Models toEvaluate Change in Rehabilitation: A National Institute onDisability and Rehabilitation Research Traumatic Brain InjuryModel Systems Report

Allan J. Kozlowski, PhD,a,b Christopher R. Pretz, PhD,c,d Kristen Dams-O’Connor, PhD,e

Scott Kreider, MS,c,d Gale Whiteneck, PhDc,d

From the aCenter for Rehabilitation Outcomes Research, Rehabilitation Institute of Chicago, and bCenter for Healthcare Studies, FeinbergMedical School, Northwestern University, Chicago, IL; cCraig Hospital and the dTraumatic Brain Injury National Statistical and Data Center,Englewood, CO; and eMount Sinai School of Medicine, New York, NY.

Abstract

The abundance of time-dependent information contained in the Spinal Cord Injury and the Traumatic Brain Injury Model Systems National

Databases, and the increased prevalence of repeated-measures designs in clinical trials highlight the need for more powerful longitudinal analytic

methodologies in rehabilitation research. This article describes the particularly versatile analytic technique of individual growth curve (IGC)

analysis. A defining characteristic of IGC analysis is that change in outcome such as functional recovery can be described at both the patient and

group levels, such that it is possible to contrast 1 patient with other patients, subgroups of patients, or a group as a whole. Other appealing

characteristics of IGC analysis include its flexibility in describing how outcomes progress over time (whether in linear, curvilinear, cyclical, or

other fashion), its ability to accommodate covariates at multiple levels of analyses to better describe change, and its ability to accommodate cases

with partially missing outcome data. These features make IGC analysis an ideal tool for investigating longitudinal outcome data and to better

equip researchers and clinicians to explore a multitude of hypotheses. The goal of this special communication is to familiarize the rehabilitation

community with IGC analysis and encourage the use of this sophisticated research tool to better understand temporal change in outcomes.

Archives of Physical Medicine and Rehabilitation 2013;94:589-96

ª 2013 by the American Congress of Rehabilitation Medicine

Evaluating change in patient functioning is a primary concern inrehabilitation clinical practice and research. Before the develop-ment of more sophisticated approaches, analysis of change reliedon linear regression methods that do not account for relationshipsthat exist across time or hierarchical levels. Investigation oflongitudinal outcomes has been conducted with linked cross-sectional analyses or pre-post treatment models in which only thebaseline status and a single endpoint are considered. However,

Supported by the National Institute on Disability and Rehabilitation Research through the

Rehabilitation Research and Training Center on Improving Measurement of Medical Rehabilitation

Outcomes (grant no. H133B090024), and the Traumatic Brain Injury Model Systems National Data

and Statistical Center (grant no. H133A110006).

No commercial party having a direct financial interest in the results of the research supporting

this article has or will confer a benefit on the authors or on any organization with which the authors

are associated.

0003-9993/13/$36 - see front matter ª 2013 by the American Congress of Re

http://dx.doi.org/10.1016/j.apmr.2012.08.199

cross-sectional analyses fail to model time explicitly; that is, time isnot directly related to outcome, but instead, data collected forindividuals are defined by “snapshots” of group means at mile-stones, such as admission, discharge, or 1-month follow-up.Consequently, cross-sectional analyses cannot model outcomes orcovariates as they relate to time. Additionally, in studies whereindividuals are assessed at several time points, cross-sectionalanalyses ignore correlations of the individuals’ repeated measures.1

If correlated measures are treated as if they were independent, thevariance of the parameter estimatesdand consequently the infer-ences based on these estimatesdwill be inaccurate and potentiallymisleading.2 Likewise, in pre-post treatment designs, informationon the nature of change is lost when interim measures are excludedfrom the analysis and change from baseline to endpoint is collapsedinto a single indicator such as a difference score.1,3 Repeated-

habilitation Medicine

590 A.J. Kozlowski et al

measures analysis of variance (ANOVA) provides a markedimprovement in comparing groups over time by accounting for thecorrelations between measures taken on the same individual.However, the focus of such a design is limited to comparison ofgroup means, and it cannot accommodate analysis at the individuallevel.4-6 Accurate documentation and modeling of change isimportant to researchers because it directly affects the veracity ofthe findings and is of critical importance to clinicians who hope toextract knowledge from the literature.

Fortunately, rehabilitation researchers have access to a longi-tudinal analytic methodology that can incorporate time, addresscorrelations between data points resulting from repeated measurestaken from the same individual, retain information about indi-vidual as well as group change, and accommodate missing datawithout excluding individuals. Although a number of modernlongitudinal analytic techniques exist, and were applied in reha-bilitation more than a decade ago,7 researchers in rehabilitationhave only recently begun to adopt these techniques.8-14

One of these powerful yet underutilized methodologies isindividual growth curve (IGC) analysis. IGC analysis is known bymany names; some of the more common ones are latent growthcurve analysis, hierarchical linear modeling, mixed-effectmodeling, random effects modeling, and multilevel modeling.These labels represent variations of multilevel modeling and meanessentially the same thing. Even though IGC analysis has beenavailable for more than 30 years, implementation in rehabilitationresearch is limited. IGC analysis has not been routinely applied fora variety of reasons including the absence of longitudinal data,necessary statistical software, computational power, and theknowledge and skills necessary to implement the analysis itself.However, with the maturation of longitudinal datasets in rehabil-itation such as the Spinal Cord Injury (SCI) and the TraumaticBrain Injury (TBI) Model Systems National Databases, accessi-bility to fast and affordable computers, enhancements in softwarepackages such as SAS,15 SPSS,16 Mplus,17 Stata,18 HLM-7,19 andthe open-source ‘R’ Project,20 and the growing prevalence ofadvanced training in longitudinal data analyses, rehabilitationresearchers are now well positioned to include IGC analysis asa key component of their methodologic repertoire.

IGC analysis: the basic idea

IGC analysis is an extension of traditional regression models; it isunique, however, in that it allows the researcher to simultaneouslymodel outcomes at the individual and group levels. In terms ofmodeling at the individual level, measurements of a particularoutcome can be directly related to time by use of a variety ofmathematical functions that range from the simple to the complex.This feature is crucial because even though outcome measuresoften change over time in a consistent fashion, they may exhibitcurvature (quadratic change), rising and falling patterns (cubicchange), floor or ceiling effects (nonlinear change), or a number ofother patterns. Although IGC analyses are suited to observational

List of abbreviations:

AIC Akaike information criterion

ANOVA analysis of variance

COWA Controlled Oral Word Association

IGC individual growth curve

SCI spinal cord injury

TBI traumatic brain injury

databases like the SCI and TBI Model Systems National Data-bases, they are also useful for longitudinal randomized controlledtrials. In the following paragraphs we expound on the subtletiesand describe the advantages of using IGC analysis as an analytictool, and provide a detailed example of IGC analysis to reinforcethe ideas discussed.

In the simplest case, suppose each individual’s response is bestdefined by a straight line. Consider the response patterns providedin figure 1, which are based on a dataset created for didacticpurposes. Using a straight line to describe the response pattern ofevery individual may not be prudent because some individualsmay be fit better by a different function, such as a curve. However,since a straight line describes the response patterns of mostindividuals fairly well, a line is used to describe all responsepatterns. As a result, each individual is represented by a lineartrajectory, or the line of best fit that passes through the data pointsof the individual. The use of linear trajectories to describe changeover time is shown in figure 2, where each individual’s trajectory,based on their unique response patterns across each time pointpresented in figure 1, is represented by a dashed straight linedefined by 2 parameters: an intercept (starting level) and a slope(rate of change).

Since time is an integral part of an IGC model, the intercept(initial status) and rate of change (slope) parameters should bedefined to facilitate interpretation of the models. For example, theintercept might be set to the date of injury or of admission torehabilitation. All subsequent time points would be defined indays, weeks, months, years, or other meaningful time unit fromthe intercept. Change is reported as an increase or decrease in theoutcome for a 1-unit increase in the selected time unit. Thus,change per day or week might not be optimal for a study thatcovers years of data.

In figure 2, each trajectory is distinct and consequently retainsthe intercept and slope growth parameters for each individualrepresented in the data. In IGC terminology, the intercept is oftendefined as a person’s initial status, while the slope is his/her rate ofchange. The trajectory that is defined by an intercept and slopedescribes how the outcome for the individual progresses overtime. The utility of these individual growth parameters arises fromthe ability to make direct comparisons between persons,subgroups, and the group average. This utility accounts for the

Fig 1 Response patterns for a sample of individuals.

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Outcome

0

5

10

15

20

25

30

35

40

Year

0 1 2 3 4 5

Group 101 302 311 348 351353 360 501 603 610

Fig 2 IGC trajectories for a sample of individuals.

Introduction to individual growth modeling 591

strength of IGC analysis in modeling rehabilitation outcomesfor research and clinical applications. In figure 2, individualscan be compared visually in terms of their initial status and rateof change estimates, but can also be compared statistically asdescribed below.

The outcome trajectories for the group are described by aver-aging the values for the individual growth parameters to create thegrowth parameters for the group, also called fixed effects. That is,the group initial status is estimated by averaging the initial statusacross individuals, while the group rate of change is determinedby calculating the mean individual rate of change, as seen in thebold red line in figure 2. It appears that the group starts with anaverage value of approximately 23.5, from which point the groupaverage value steadily decreases over time at a rate of e2.4outcome units per year. Not only does the group trajectory relatechange in outcome to time, it also serves as a basis for subsequentcomparisons. That is, an individual’s initial status and rate ofchange can be directly compared with those of the group orsubgroups. Likewise, subgroups can be compared with othersubgroups, such as women with men or strata of condition severity(mild, moderate, severe disability).

Another important component of IGC analysis is investigationof the variance of the individual growth parameters (called randomeffects) about the group means (fixed effects). Figure 2 demon-strates that individuals vary both in initial status and ratedthat is,in terms of where they start and how they change over time on theoutcome measure. The presence of significant variance in theindividual growth parameters implies that related factors repre-sented by 1 or more covariates might explain this variability. Forexample, as a covariate such as age at injury increases, rates ofchange may decrease. Thus, if the outcome measure is levelof depression, where a higher score indicates a greater degree ofdepression, such a relationship would indicate that older individ-uals tend to display less rapid decreases in depression comparedwith those who are younger. Similar statements could also bemade about the relationships between a covariate and theinitial status.

Many of the aforementioned characteristics of IGC analysismake it highly suited for rehabilitation research applications. This

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is particularly so where a patient’s impairments, activity limi-tations, participation restrictions, quality-of-life ratings, or otheroutcome measures are tracked over time in response to treatmentor as resulting from natural recovery. Perhaps greater value liesin how clinical practice can be informed by the detail inherent inIGC models. IGC models provide group change estimates basedon individual change, and retain the individual differences thatcontribute to that change. Thus, evidence from research in formsof comparative norms and prognostic models could be appliedclinically to plan interventions and evaluate patient outcomes atboth the group and individual levels. Consequently, IGC anal-ysis is a useful technique for evaluating how rehabilitationpatients progress while identifying the patient, condition, andtreatment factors that are associated with change, and permittingcomparisons among individuals, subgroups, and the groupsas a whole.

To augment the discussion above, we include an example tofacilitate a deeper understanding of the basic features of IGCanalysis. We encourage rehabilitation clinicians and researchers tobecome familiar with the equations that define the example modelbecause they are generally reported in IGC research publicationsand other IGC-related literature. Additionally, understanding howthe equations relate to the concepts of simultaneously modeling theindividual and group levels will facilitate understanding the morecomplex manifestations of IGC analysis. However, every attemptwas made to construct an example that conveys the principles ofIGC analysis without equations so that those with a limitedbackground in mathematics will be able to attain a conceptualunderstanding. Note that this example is for didactical purposesonly and is not intended to be interpreted as an actual study.

Example

TBI can result in various cognitive deficits. Reduced verbal fluency,for instance, can be measured with the Controlled Oral WordAssociation (COWA) Test,21,22 as this test is a sensitive indicator ofbrain dysfunction.23 Patients name in 1 minute as many words aspossible that begin with a given letter of the alphabet. The totalscore of all acceptable words (ie, no proper nouns or repetitions)produced across 3 trials is adjusted using nationally representativenormative data for age, sex, and education to create the adjustedCOWA score, which is our outcome variable (labeled as COWAadjin the formulas). In this example, the word association test wasadministered during rehabilitation and each consecutive postinjuryyear for 3 years in a substantial number of patients enrolled in theTBI Model Systems National Database.

Participants whose data are represented in the TBI ModelSystems National Database consented to the collection of theirdata for research purposes at the facility in which they receivedrehabilitation. The current analyses were approved by the insti-tutional review board of the TBI Model Systems National Dataand Statistical Center. All analyses were conducted at the TBIModel Systems National Data and Statistical Center usinga deidentified version of the database and SAS version 9.3a

statistical software.

Investigating response patterns and trajectories

It is common in IGC analysis to examine data graphically as aninitial step. Figure 3 depicts individual response patterns (dashedlines) for a selection of participants. Each individual response

Fig 3 Response patterns of adjusted COWA test scores over time for

a sample of individuals from the Traumatic Brain Injury Model Systems

database.

Fig 4 IGC trajectories of adjusted COWA test scores over time for

a sample of individuals from the Traumatic Brain Injury Model Systems

database.

592 A.J. Kozlowski et al

pattern is scrutinized to identify the mathematical function (ie,linear, quadratic, cubic, etc) that most appropriately describes therelationship between the outcome and time, and the type ofchange that conforms to most response patterns is chosen.Although response patterns may vary, common practice in IGCanalysis is to select the mathematical expression that best fits mostof the data. We placed the intercept (year 0) at the time of the firstCOWA assessment, which is at approximately 1 year postinjury,so the response patterns represent postrehabilitation time frames.Although some patterns curve upward and others downward,a straight line adequately describes the change for most individ-uals. Since most response patterns appear to follow a linear trend,the following equation (function) was selected to describe theadjusted COWA score:

COWAadjtiZp0i þp1iðYeartiÞ þ εti

This equation has components that are similar to those used inordinary least-squares regression, except that this equation focusesexclusively on the individual, where equations in ordinary least-squares regression define the association between predictor andoutcome variables for a group. For each individual (denoted bya subscript i), the equation linearly relates outcome ðCOWAadjtiÞto time ðYeartiÞ, where the time point in which the data for theoutcome were collected is denoted by t. Consequently, the aboveformula is often called an individual or level-1 equation. IGCparameters are interpreted similar to ordinary least-squaresregression coefficients, except p0i represents the true initialstatus of an individual (the intercept), and p1i represents theindividual’s true rate of change (the slope). In practice, theseparameters are estimated. The final term in the above equation, εti,is the error term, and for a given time point, is the portion of theindividual’s outcome that remains unaccounted for. Correctspecification of this error term can be a complicated process, anddiscussion of this process is beyond the introductory nature of thisarticle. However, many longitudinal analysis texts cover this topicextensively.24

The purpose of the equation is to provide a general represen-tation of each response pattern and, in doing so, create a commonbasis of comparison for individuals and set the stage for groupanalysis. In other words, the individuals’ response patterns are

modeled mathematically as trajectories that are defined by eachindividual’s initial status and rate of change. Thus, the initialstatus and rate of 1 individual can be compared with another, witha subgroup, or with the group as a whole. Trajectories thatcorrespond to the response patterns in figure 3 are provided infigure 4. The trajectories displayed in figure 4 indicate bothvariability in initial status and rate of change. An essentialcomponent of IGC analysis is to understand the extent to whichinitial statuses and rates differ from the group averages.

Random effects

Determining the extent to which initial statuses and rates vary isimportant because significant variability implies the existence ofa meaningful spread in initial statuses, rates of change, or both.Note that the variations in initial statuses, rates, or both could beexplained by including patient, injury, or intervention character-istics in the model as covariates. Conversely, if initial statuses andrates do not significantly vary, trajectories appear almost identicalacross individuals, and the group average provides a reasonabledescription of everyone. The variability in initial statuses and ratesare defined by the parameters u00 and u11, respectively, where thedegree to which initial statuses and rates are related, and theircovariance is denoted by the parameter u01. The initial status andrate estimates are often related because both measures are derivedfrom the same individual. Estimates of u00, u11, and u01 areprovided in table 1. In IGC terminology, u00, u11, and u01 capturewhat are called random effects, and define the individual variationfrom the group average.

Based on their respective P values (P<.05), both the variabilityin initial statuses and rates (62.1 and 3.7, respectively) arestatistically significant (see table 1). Therefore, we conclude thatindividuals differ in word fluency at initial status and in the ratesof word fluency change over time. This variability in initialstatuses and rates may be explained in part by 1 or more covar-iates. The covariance between initial statuses and rates is notsignificant in this case. If the covariance was positive and found tobe significant, this would imply that those with higher initialadjusted COWA scores would, in general, exhibit faster rates ofimprovement in word fluency.

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Table 1 Estimates of variability in initial status and rate, and

the covariance of initial status and rate

Label Parameter Estimate SE z Score P

Variability in initial

status

u00 62.1 6.4 9.7 <.0001

Variability in rate u11 3.7 1.7 2.2 .02

Covariance of initial

status and rate

u01 3.1 2.5 1.3 .21

Fig 5 Comparison of individual and group average fitted

trajectories.

Introduction to individual growth modeling 593

Fixed effects

Also included in figure 4 is the average or group trajectory indi-cated by the red line. It is common in IGC analysis to use b00 torepresent the average initial status and b10 to represent the averagerate. The individual’s initial status is equal to the average initialstatus plus a random error term ða0iÞ such that p0iZb00 þ a0i.Consequently, each individual’s initial status can be defined asvarying randomly about the average initial status of the group.Likewise, the individual’s rate of change is equal to the average rateplus a random error term ða1iÞ such that p1iZb10 þ a1i, meaningthat each individual’s rate varies randomly around the groupaverage. Together, equations p0iZb00 þ a0i and p1iZb10 þ a1i areknown as the “group” or level-2 equations because they containinformation at the group level. Additionally, both the average initialstatus and rate are known as fixed effects because they do not vary,unlike the initial statuses and rates of individuals. Estimates of thefixed effects for our example are given in table 2. Based on theirrespective P values (see table 2), the average initial status and theaverage rate both differ significantly from zero (P<.05). Specifi-cally, the average adjusted COWA initial status is 25.4, while theaverage rate suggests a gain of 4.8 adjusted COWA points per year.

Upon obtaining values for the fixed effects these values can becompared, if desired, with those of an individual or subgroup.Figure 5 depicts a comparison between trajectories of 2 individ-uals indicated by the dashed lines and the group indicated by thebold red line. The values of each individual’s initial status and rateare given in table 3 along with the corresponding P values thatcompare the individual’s initial status and rate with those of thegroup. The first individual’s initial status is similar to the groupaverage with an adjusted COWA value of 25.9. However, the rateto which the person improved (9.8 adjusted COWA points peryear) is significantly different from that of the group. The secondindividual differs from the group on initial status (13.1), thoughthis person’s rate does not differ from the rate of the group(PZ .77). Although not demonstrated here, the principles dis-cussed above can easily be used to compare 1 individual withanother or 1 subgroup with another.

The unconditional model describes the best-fit average trajec-tory for the available data, and the addition of 1 or more covariatesto explain variance of the growth parameters will produce modelsthat are conditional on the specific associations between thecovariate(s) and the growth parameters that are included.

Table 2 Estimates of group intercept and rate

Parameter Estimate SE t P

Group initial status 25.4 0.4 57.2 <.0001

Group rate 4.8 0.2 22.6 <.0001

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Covariate introduction

If the random effects are statistically significant, variability ininitial statuses, rates, or both may be explained by introducing 1 ormore covariates into the group level equations. Covariates can becontinuous, dichotomous, or categorical. It is common practice tocenter continuous covariates by calculating the difference betweenthe mean of the covariate and each individual’s value of thecovariate. This process transforms the average of the covariate tobe zero and allows for accurate interpretation of growth parame-ters, which we illustrate below. The parameter estimates forcontinuous covariates are interpreted as the amount of change inthe outcome variable for a 1-unit change in the covariate. Singer25

provides an in-depth discussion on the process and importance ofcentering covariates. Dichotomous covariates can be used byassigning a reference category as is commonly done in regressionanalysis. Similarly, categorical covariates can be included in theanalysis where the selected reference category serves as the basisof comparison between itself and the other levels of the covariate.Note that the choice of the reference category for a categoricalcovariate is arbitrary, though it should represent the level of thecovariate against which one wishes to draw contrast.

For this example, we use estimated years of education at injuryas the covariate. To maintain interpretability of the fixed effects,we centered the covariate about the mean, creating a new variable,mean-centered education (labeled in formulas as EducationMC).With the covariate, the group level equations becomethe following:

p0iZb00 þ b01ðEducationMCÞ þ a0i

and

p1iZb10 þ b11ðEducationMCÞ þ a1i

Notice that when EducationMCZ0, which is now the averagefor EducationMC, the above equations reduce to p0iZb00 þ a0i

Table 3 Individual initial statuses and rates

Individual Initial Status P Rate P

1 25.9 .89 9.8 <.0001

2 13.1 <.0001 4.6 .77

594 A.J. Kozlowski et al

and p1iZb10 þ a1i, respectively. Thus, the average trajectory withthe centered covariate set at a value of zero reflects the trajectoryfor the unconditional model. Introducing a covariate requires theaddition of 2 fixed effects, b01 and b11. The first of these fixedeffects describes the linear relationship between centered educa-tion and the individual’s initial status, and is labeled b01. Thesecond describes the effect of the covariate on the individual’srate, and is labeled b11. Thus, the initial status of an individual isa function of the average initial status plus the effect of the co-variate on the initial status, and an error term. Likewise, the rate ofchange of an individual is a function of the average rate plus theeffect of the covariate on the rate, and an error term. To assess thevariability explained by the covariate, new estimates of variabilityin initial status and rate estimates (table 4) are compared withthose from the unconditional model.

The addition of the covariate reduces the variance in bothinitial statuses and rates. To compute the variability accountedfor by grand mean-centered education, we subtract the estimateof the variance based on the inclusion of the covariate from theestimate provided by the initial model and divide this quantityby the latter.

3:7� 3:3

3:7Z0:11

Thus, years of education explained roughly 11% of the vari-ability in the rates of change, and (62.1 e 60.9)/62.1Z .02 or2.0% of the variability in initial statuses. In both instances, theamount of variability explained is relatively small. The smallcontribution of this covariate in explaining overall variance is alsoevident in the finding that variability in both initial statuses andrates remains statistically significant (P<.05). Based on theseresults, additional covariates may explain additional variance. Todemonstrate that the addition of the covariate improves overallmodel fit, model fit statistics such as the Akaike informationcriterion (AIC) can be compared between the unconditional modeland models containing a covariate(s). In this instance, the AIC forthe unconditional model is 10,485, while the AIC for the modelcontaining education is 10,387. Since a drop in the AIC greaterthan 10 indicates model separation,26 we conclude that the addi-tion of education as a covariate improves model fit.

The estimates of the fixed effects are displayed in table 5.Because of the inclusion of the covariate, the model now containsa total of 4 fixed effects: the group intercept, the group rate, therelationship between the covariate and initial status, and therelationship between the covariate and rate.

In comparison with the results in table 2, values of the esti-mates of the group’s initial status and rate have changed onlyslightly after the inclusion of the covariate, which was expected.Years of education is linearly related to initial status (PZ .01),such that every additional year of education (before TBI) above

Table 4 Estimates of variability in initial status and rate, and

the covariance of initial status and rate

Label Parameter Estimate SE z Score P

Variability in initial

status

u00 60.9 6.4 9.6 <.0001

Variability in rate u11 3.3 1.7 1.9 .0311

Covariance of initial

status and rate

u01 2.4 2.5 1.0 .3229

the average for the group is associated with a 0.4-unit increase oninitial adjusted COWA score. The significant effect of the covar-iate on rates of change indicates that each 1-year increase ineducation before TBI is associated with an increase in rate by anaverage of 0.2 adjusted COWA units per year. Thus, those withmore years of education before TBI start with higher word asso-ciation scores and demonstrate faster improvement, in comparisonto those with fewer years of education.

Had this example described an actual study, examining thesignificant associations between the COWAadj covariate and theinitial status and rate would provide an interesting side note.While scores for the adjusted COWA measure are initiallyadjusted for age, sex, and education based on norm tables,centered education was significantly associated with both theinitial status and the rate of change estimates. This could indicatethat the initial adjustment in COWA scores based on normativenon-TBI data was insufficient, or it could indicate that the IGCmodeling of associations between centered education and theinitial status and rate parameters captured variance for which theoriginal adjustment method was unable to account. This findingdemonstrates one of the clinically relevant questions that can beexplored with IGC methods.

Broader applications

This example provides only a glimpse of the possibilities offeredby the application of IGC analyses in rehabilitation. In ourexample, we considered 1 covariate for illustrative purposes,although more would likely be included in hypothesis-drivenresearch. We modeled assessments taken over time within indi-viduals; however, patients are nested within rehabilitationprograms or facilities, which may have differential effects onoutcome. The IGC model can be expanded to account for theseorganizational structures by introducing a series of level-3 equa-tions that define the effects of an additional “program” level.4

Although there is no theoretic limit to the number of levels onecan model, higher levels should have some form of clustering ofindividuals at lower levels, and as the number of levels increase,so does the computational power needed to perform the analysis.

Often individual response patterns do not conform to simplestraight lines but instead display curves, cyclical patterns, or othertypes of change. To accommodate different types of change, IGCanalysis is flexible in modeling a wide variety of responsepatterns. Covariates can also be used to explain variability in thegrowth parameters that describe more complex types of change,along with defining the trajectory of a curvilinear trend for groupcomparison. For instance, those who are 20 years of age at injurymay have a completely different curvilinear trend than thosewhose injury occurred at the age of 60. Multiple covariates canalso be modeled and assessed simultaneously.

Table 5 Estimates of group intercept and rate, effect of the

covariate on initial status and rate

Parameter Estimate SE t P

Group intercept 25.5 0.5 57.4 <.0001

Group rate 4.8 0.2 22.7 <.0001

Effect of covariate on initial status 0.4 0.2 2.5 .0137

Effect of covariate on rate 0.2 0.09 2.70 .0073

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Introduction to individual growth modeling 595

IGC analysis models time explicitly, which can be treatedeither as continuous (ie, using the actual amount of time from theinitial status in days, weeks, years, or other time unit at whicheach individual’s scores were measured) or as categorical (ie, byassigning common time points to comparable scores for individ-uals, such as week 1, week 2, week 6, etc, where the actual datacollection took place in a window of 2 or 3d). Intervals betweentime points need not be equally spaced, and as long as the data areat least missing at random, IGC analysis does not require indi-viduals to have scores for all time points.5 In addition, if data aremissing systematically, it is possible to assess the validity of theresults by use of pattern mixture modeling, meaning that in someinstances, even cases with systematically absent data may beused.6 Therefore, an individual need not be removed from theanalysis simply because 1 or more outcome measures are missing.

As discussed earlier, IGC analysis allows an individual’strajectory to be compared with that of the group; the trajectory of 1individual (or subgroup of individuals) can also be directlycompared with that of another. This capability provides advantagesto clinicians in the application of research findings to practice andin the evaluation of clinical outcomes to inform practice. Inapplying research findings, clinicians could compare individuals ora subgroup of specific patients with each other or the groupaverage. In this respect, individual growth models are more clini-cally relevant than fixed-effects-only models (eg, repeated-measures ANOVA) because of their ability to describe recoveryat the individual, subgroup, or group level. However, IGC analysisonly works well when the response patterns of most individuals fitthe selected model structure. When individual response patterns donot adhere to the model, then alternative analytic methods such asfixed-effects models should be used.

Individual-level prognoses could be generated for new patientsbased on previous patients’ data. Once an IGC model has beendeveloped from existing patient data, a trajectory can be predictedfor a new patient by substituting the values for the new patient’sdemographic and injury characteristics, and for any other covar-iates included in the IGC model, into the equations for the growthparameters. This predicted trajectory could then be used as anindividual benchmark to evaluate the new patient’s actual recovery.Decisions like estimating discharge date and disposition could bebased on the predicted magnitude and timing of a plateau ina curvilinear trajectory. In addition, where different components offunctioning demonstrate different trajectories of recovery (eg,mobility vs self-care), interventions might be sequenced rather thanprovided concurrently. Competing interventions could be evaluatedfor clinically significant differences in rate of recovery in additionto just the magnitude of posttreatment outcome differences.

Discussion

In this special communication, we have described and demonstratedthe application of IGC methods and highlighted the benefits andappropriateness of this approach in modeling rehabilitationoutcomes. In addition to accounting for interindividual differences inchange over time, IGCmodels can account for associations betweencovariates and any or all of thegrowth parameters considered, aswellas for associations between the growth parameters themselves.These features offer analytic options that are particularly well suitedfor rehabilitation research. Rehabilitation patient populationschange during and after intervention, and a variety of factors caninfluence individual outcomes. IGC models offer the flexibility and

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complexity to better understand the factors that account for indi-vidual variability in outcomes over time. The integration of IGC intorehabilitation research and practice can provide for revolutionaryadvances in our understanding of how individuals recover and livewith disabilities, and offers a powerfulmethod to evaluate the effectsof rehabilitation interventions on outcomes. Thus, application ofIGC analysis may provide the opportunity to understand intricaciesof recovery as trajectories of change not apparentwhen evaluating anoutcome at a single time point or as a set of cross-sectional timepoints, and could inform all aspects of rehabilitation science, prac-tice, administration, and policy.

Additionally, continued development and applications of IGCmethods for the evaluation of patient outcomes has the potential torevolutionize decision-making for all stakeholders in rehabilita-tion. Interpreting outcome as an ever-changing process mapped bylinear, curvilinear, or nonlinear trajectories rather than as a resultat a single point in time (or a series of points in time) couldimprove treatment and discharge planning. However, IGC andother mixed methods are not appropriate for all applications, andresearchers should ensure that their statistical methods arecompatible with their study design, particularly in the case ofsmall samples or few time points.27 Many resources are availableto readers who are interested in learning more about IGC modelsspecifically and mixed models generally.4,5,24,27-29

In summary, IGC analysis is highly versatile, offering numerousoptions for longitudinal data analysis that are unavailable in otherapproaches. The benefits of IGC analysis are as follows:

� Simultaneous evaluation of change over time at both the indi-vidual and group level

� The ability to model time as related to outcome in a flexiblemanner

� The capacity to treat time as either a continuous or a discretevariable

� The ability to model continuous or categorical covariates, or both

� The capacity to retain data for individuals with missing data at 1or more time points

� The capacity to examine the variability in the parameter esti-mates (ie, intercepts and slopes) used in describing how anoutcome changes over time

� The capability to evaluate the extent to which 1 or morecovariates explain variability in the parameter estimates ofchange of the outcome over time

� The option to include additional levels in the modeling process,by modeling factors that introduce homogeneity into otherwiseheterogeneous groups

� The ability to investigate an expansive set of hypotheses

While the benefits of IGC analysis for rehabilitation researchare considerable, these methods have limitations. Limitationsinclude the following; the assumptions needed for proper infer-ence are indicated by an asterisk:

� Data should ideally be available for at least 3 time points.� Dependent variables should be continuous, though the use ofpseudocontinuous outcomes is not uncommon in the socialsciences.

� Missing data are ideally missing at random.� The trajectories of the individual response profiles (level-1model)are correctly specified (ie, linear, quadratic, cubic change, etc).*

� Similarly, level-2 models are correctly specified.*

596 A.J. Kozlowski et al

� The structure of the residuals for the level-1 model is correctlyspecified.*

� The structure of the residuals for the level-2 models assumesa bivariate normal distribution.*

Conclusions

IGC modeling has widespread application in rehabilitationresearch and clinical practice, and can serve as a link between the2 arenas by simultaneously modeling individual and group-levelchange in outcome over time. IGC modeling is versatile, moreappropriate for longitudinal data analyses than are cross-sectionalanalyses or pre-post treatment designs, and in many cases moreappropriate than repeated-measures ANOVA, as these methodsfail to model at the individual level. IGC models should beconsidered where both the individual and group representimportant components of analysis. These situations include thosewhere data are nested within the individual (ie, multiple assess-ments over time), and where individuals are nested within higher-order structures such as treatment programs or facilities. Whileapplicable to randomized designs, the IGC model is particularlyuseful for analyses of observational longitudinal data such as thosefound in the TBI and SCI Model Systems databases. Thecombination of faster computers, advancements in statisticalsoftware packages, growing exposure to IGC analysis by reha-bilitation researchers, and the expansion of longitudinal databasesall point toward incorporating IGC analysis as an essentialcomponent of the rehabilitation researcher’s statistical arsenal.

Supplier

a. SAS Institute Inc, 100 SAS Campus Dr, Cary, NC 27513-2414.

Keywords

Longitudinal studies; Regression analysis; Rehabilitation;Treatment outcome

Corresponding author

Allan J. Kozlowski, PhD, Rehabilitation Institute of Chicago, 345E Ontario St, Chicago, IL 60611. E-mail address: akozlowski@ric.org.

Acknowledgments

We thank John D. Corrigan, PhD, Mark Sherer, PhD, JenniferBogner, PhD, Flora M. Hammond, MD, Jeffrey P. Cuthbert, PhD,and Allen W. Heinemann, PhD, for their contributions in thewriting and editing of this manuscript.

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