The importance of integrating health trajectory modeling into an arsenal of available methods for researchers in nursing has recently been highlighted (Henly, Wyman, & Gaugler, 2011; Wyman & Henly, 2011). Scientists who are interested in modeling longitudinal patterns of change in health to better understand age-related normative dynamics or to study unfolding pathophysiological processes can choose from a number of analytical strategies. These strategies include, among others, variable-centered models (e.g., analysis of variance [ANOVA], hierarchical models, random effect models, growth curve models), person-centered models (e.g., latent class growth modeling [LCGM; Nagin 2005], and growth mixture modeling [GMM; Muthen & Shedden, 1999]).

Despite the variety of available methods, current nursing research has been dominated by use of a variable-centered approach (Henly et al., 2011). Based on the assumption that all individuals in the population follow a similar pattern of change, the variable-centered approach identifies the average longitudinal trajectory, estimates variability about the average, and explains that variability in terms of covariates of interest (Gelman & Hill, 2007; Singer & Willet, 2003). In statistical terms, variable-centered methodologies model the population distribution of trajectories of change over time based on continuous distribution functions (Singer & Willet, 2003). Correlations, regressions, and structural equation models are among the statistical techniques used in a variable-centered approach (Poncheri & Ward, 2008).

However, clinical phenomena with high inter- and intra-individual variability (e.g., depressive symptoms, obesity, disabilities), demonstrate a more multinomial pattern (Ferro, Avison, Campbell, & Speechley, 2011; Gill, Gahbauer, Han, & Allore, 2010; Liang, Bennett, Ye, & Quinones, 2010; Mustillo et al., 2003). In such circumstances, person-centered models will likely provide a better fit to the research question and to the data under investigation than variable centered models (Nagin & Odgers, 2010). Cluster analysis, latent class analysis (LCA), and latent profile analysis are the statistical techniques commonly used to analyze data from a person-centered approach (Poncheri & Ward, 2008). The purpose of this article is to demonstrate the utility of basic person-centered modeling in a real-data application using data from a large cohort of aging American women enrolled in the Women's Health Initiative Clinical Trial (WHI CT; Anderson et al., 2003).

#### Introduction to Person-Centered Models

Person-centered models belong to the class of finite mixture models, which were designed to analyze data obtained from a mixture of statistically heterogeneous groups (Nagin & Odgers, 2010). These models assume that the population is composed of a mixture of two or more groups whose longitudinal trajectories could be expressed by a distinct number of polynomial and/or non-polynomial nonlinear functions of time. Thus, the estimated group trajectories are used as a convenient statistical device to approximate and summarize an unknown but complex population distribution (Bauer & Curran, 2003). A detailed technical and mathematical overview of person-centered models can be found elsewhere (Muthen & Shedden, 1999; Nagin, 2005).

Two important results of person-centered modeling—estimates of the shape of each class's trajectory and the size of the population belonging to each trajectory—also can be used to identify those factors that predict trajectory group membership or to estimate the effect of longitudinal class membership on proximal or distal outcomes. The association between predictor variables and trajectory group membership is traditionally examined by specifying the probability of trajectory group membership to follow a multinomial logit model (Nagin, 2005). However, as we will further demonstrate, this specification could be easily extended to include ordinal models. The major benefit of using ordinal rather than multinomial models is to minimize the proliferation of regression estimates and instead apply a hierarchical or ordered structure to the estimated longitudinal trajectories.

The person-centered latent class growth model (LCGM) and the growth mixture model (GMM) differ with regard to structural assumptions of a population curve. The former does not make any assumptions about a population's distribution, whereas the latter assumes a heterogenic multimodal distribution. That is, GMM assumes that the population curve consists of a mixture of normally distributed subgroups (Nagin & Odgers, 2010). In applied analysis, among individuals composing a trajectory group, within-class variability is constrained to zero in a LCGM framework and allowed to be freely estimated in GMM analysis. Our discussion will focus on latent class growth models, given that their parsimony allows for the clearest interpretation, but all the models presented could be estimated using the GMM framework.

#### Uses of Person-Centered Analytic Approaches

In health research, and in nursing sciences in particular, LCGM can be used to address many different types of research questions. First, it can identify distinctive and prototypical growth trajectories within target populations with a variety of health conditions. These studies might range from identifying distinct trajectories of disability in the last year of life (Gill et al., 2010) to describing multiple patterns of change in physiological and psycho-emotional indicators (Payette et al., 2011; Smith, Kupper, Jonge, & Denollet, 2010; Smolderen et al., 2008). Second, LCGM provides a convenient framework for estimating the effects of interventions or individual-level characteristics (e.g., ethnicity, education, prior or current health behaviors, cognitive status) on the probability of belonging to a given trajectory. For example, Ferro et al. (2011) found that a child's cognitive function was the strongest predictor of a moderately increasing trajectory of depressive symptoms in mothers of children with newly diagnosed epilepsy. Finally, researchers who are interested in approximating the magnitude of risk carried by members of a group over time can use person-centered models to estimate the effect of membership in a growth trajectory on incidence of clinically relevant health outcomes (e.g., mortality, disability, hospitalization, falls).

#### Application of LCGM to Analysis of Energy/Fatigue in WHI CT Data

In this article we present the utility and real-data application of basic LCGM and its extensions using a large cohort of American women 65 and older enrolled in the Women's Health Initiative Clinical Trial (WHI CT). The energy/fatigue indicator was selected as a trajectory variable, given that, although its prevalence and incidence appear to increase with advancing age, the patterns of change are not homogeneous. In addition, fatigue is increasingly recognized as a specific geriatric entity (Avlund, Damsgaard, & Schroll, 2001; Moreh, Jacobs, & Stessman, 2010; Vestergaard et al., 2009).

The objectives in using this example are to demonstrate how a LCGM framework can describe longitudinal grouping of self-reported energy/fatigue levels in older women, to relate membership in these longitudinal trajectories to baseline characteristics (e.g., ethnicity, education, health status, behavior), to and estimate the effect of their membership in a particular longitudinal pattern on the first incident hospitalization during 5 years of WHI Extension Study follow-up. We also will compare alternative statistical perspectives on modeling trajectory group assignments as a dependent variable.

### Discussion

- Top of page
- Abstract
- Methods
- Results
- Discussion
- References

The purpose of this article was to provide a step-by-step example of the utility of the LCGM framework and its extensions. In this demonstration, we calculated descriptive statistics and performing normalizing data transformations; ran a LCGM-driven model selection process using a battery of indicators including, but not limited to, theoretical justifications and information criteria; linked predictor variables with trajectory group memberships using multinomial, ordered, or partial proportional logistic regression models; and used a Cox proportional hazard framework to estimate the effects of membership assignments on distal outcomes.

Four distinct trajectories of energy/fatigue scores were identified during the first 9 years of follow-up of women 65 years and older in the WHI. The number of trajectories in this sample was similar to several longitudinal patterns identified in other studies (Gill et al., 2010; Liang et al., 2010) using similar methodology. It is important to acknowledge the prospective nature of these findings, indicating, for example, that the most vigorous individuals (those in High Decliner trajectory) did not merely have elevated energy levels at baseline but sustained fairly high vitality for up to 9 years of follow-up.

Although taken together these findings suggest that there is substantial heterogeneity of indicators of functional and energy levels among aging individuals, a number of major categories of indicators could be distinguished that were in concordance with related discussions of variability in normal aging, successful aging, and pathological aging (Fried et al., 2001; Rowe & Kahn, 1997; Woods et al., 2012). In line with other studies (Avlund et al., 2001; Moreh et al., 2010; Vestergaard et al., 2009), this analysis showed that fatigue was closely linked to age, socio-demographic factors, comorbidities, health behaviors, and poor sleep quality. However, contrary to previous published work, we demonstrated that, in fully adjusted models, additional factors such as optimism and social support had significant positive and independent effects on the likelihood of membership in vigorous trajectories of energy/fatigue index scores. These findings were consistent regardless of the methodological framework used, and they hold promise for developing behavioral interventions to help the elderly maintain optimal levels of energy throughout the aging process.

Another interesting finding pertains to the negative effect of education on trajectory group assignment. One plausible explanation of this conundrum is that higher education influences an individual's sensitivity to fatigue, making these persons more likely to perceive and report lower energy scores. This assertion deserves further examination in other longitudinal studies. Worth noting is that the WHI-CT sample is in no sense a nationally representative selection of elderly women, and a quarter of the sample used here reported postgraduate education. Another important caveat is that in large cohort studies, the balances between statistical and clinical significance may be disproportional. In other words, even though the WHI cohort provided an excellent opportunity to examine the effects of intervening factors on age-sensitive indicators, the clinical significance of the results cannot be assumed.

As found in other studies (Avlund et al., 2001), we demonstrated that energy/fatigue levels predicted future hospitalization in non-disabled elderly. However, in contrast to other prospective studies in which the association between such distal outcomes (e.g., hospitalization, mortality) and cross-sectional distribution of predictors was measured, the LCGM framework enabled us to capture subtle longitudinal dynamics in energy/fatigue scores over a relatively long follow-up and relate these levels to a clinically relevant outcome. In this way, consideration of trajectory groups can be a tool for empirical partitioning of a population curve into clinically and statistically distinct longitudinal clusters. This application is especially useful, given that investigators frequently use less precise categorization criteria such as tertiles and quintiles, as well as other a priori assumptions, to assign individuals to different clinical categories. Although such a subjective categorization is methodologically reasonable, it has one major weakness, in that it provides no basis for calibrating the precision of an individual's classification into a particular category. In other words, when using traditional (i.e., variable-centered) methodology, we cannot quantify the probability of a category assignment. The LCGM-based analysis presented here supports an empirical statement that there is a 24% chance that randomly selected women age 65 years old and older who participated in the WHI CT will demonstrate an energy/fatigue-related longitudinal trend similar to that found in the Low Decliner group, a claim that would be challenging to make using a traditional variable-centered framework.

As always there are certain caveats to any analysis. Measurement error in LCGM and general models can create problems, and these issues have been thoroughly discussed in the literature (see Muthen & Shedden, 1999; Nagin, 2005). While not eliminating the problem, our use of established scales for our measures helped to minimize this issue.

Second, the pros and cons of a multi-step (used here) versus single-step approach have been recently debated in biomedical literature (Asparouhov & Muthen, ; Clark & Muthen, ). In some simulation studies it was demonstrated that multi-step analysis might result in biased standard error estimates (Clark & Muthen, ). Careful considerations using the latest development in the field are warranted in future research.

A third issue in the analysis deserves brief discussion. Women with low energy/fatigue levels presumably were less likely than less fatigued women to complete all occasions of follow-up. If that is the case, then the proportion of older women at risk of being in low-energy/fatigue-level trajectories may be misrepresented. However, the main aim of the analyses presented here was to describe the heterogeneity in energy/fatigue scores, rather than to estimate their prevalence. The LCGM framework can serve as a convenient analytic approach for outlining longitudinal patterns occurring in the population. In addition, LCGM successfully integrates full information maximum-likelihood techniques that generate consistent estimates when MAR assumptions hold (Muthén & Muthén, 1998–2011). In the analyses presented here, delayed enrollment, a flexible data collection protocol, and sparsely scheduled measurement occasions resulted in a large proportion of missing observations, supporting a MAR assumption. That is, a large proportion of missing observations were attributable to protocol-defined features of the study rather than to other confounding events.

#### Conclusion

Researchers in nursing and health can benefit from using person-centered methodologies. The estimates (i.e., trajectory functions and membership probabilities) from these basic analytic techniques could be easily used to identify factors that predict trajectory group membership and to approximate the magnitude of risk carried by membership in certain longitudinal clusters. The association between predictors and trajectory group membership could be specified using either a traditional multinomial framework or an alternative categorically ordered specification, provided that the estimated longitudinal patterns show a clear ordered structure. Sensible application of either model generates comparable estimates. However, to ensure the most parsimonious and statistically justifiable model (i.e., the model that does not violate parallel-line assumptions), a partial proportional odds model should be used. The latter provides a detailed output that is easily interpreted. Finally, either discrete- or continuous-time survival analyses can be readily fit to the data to approximate the magnitude of risk that is carried by membership in particular longitudinal clusters. In summary, person-centered methods provide unique opportunities to explore and statistically model the effects of longitudinal heterogeneity within a population and can be more fully integrated into nursing research analyses.