Willett and Singer (2004) offered a testimonial from peer review of their own research papers that reviewers and readers of journals have not been trained in contemporary methods for analyzing change over time and that the concepts and methods were felt to be difficult. Among the scientific and statistical reviewers of Nursing Research, quality of review of longitudinal aspects of design was lowest among 10 statistical aspects of submitted manuscripts (Henly, Bennett, & Dougherty, 2010). Thus, use of contemporary longitudinal models may be inhibited in many areas of research, including nursing, by lack of familiarity and mastery of the methods.
Key Elements of Health Trajectory Research
As in related disciplines such as psychology (Collins, 2006), advancing nursing research to incorporate trajectory science involves integration of theory about change, effective use of longitudinal design, and estimation of statistical models for individual growth and decline. Health trajectory science will develop effectively if traditions in nursing research evolve in supportive ways.
More than 30 years ago, Donaldson and Crowley (1978) described the recurrent themes in nursing inquiry as the processes of health and illness, patterning of health behavior in context during critical life situations, and processes by which positive changes in health are induced. Although change in health status is implicit in each theme, time as the essential variable in describing change was not addressed. Time is absent in nursing meta-theory identifying person, environment, nursing, and health as key meta-concepts (Flaskerud & Halloran, 1980), even as the purpose of nursing is to improve and sustain the health, or prevent a decline in health, of persons (individuals, families, groups, and communities) in interaction with the environment. The temporal context essential to understanding every process for affecting health and illness status is not explicit. Time is an essential meta-concept for nursing. As an example, time figured importantly in the critique of theories for symptom management of Brant, Beck, and Miaskowski (2010). Stretching nursing research tradition to include trajectory analysis is needed to understand and support patient experiences during critical life situations, including serious symptom experiences. The dynamic trajectory perspective aligns scientific and clinical perspectives by focusing on the individual person, family, group, or population in a way that is consistent with the human experience of health and illness over time.
Moving from static to dynamic theorizing about health and illness is challenging because temporal concepts are novel in nursing research, and ways to express relationships among temporally relevant ideas are unfamiliar. Health trajectory science is composed of concepts and relationships associated with intraindividual change and interindividual differences in intraindividual change (cf., Nesselroade & Ram, 2004). Specification of a theory about individual change is the first stage in health trajectory research. Describing and explaining interindividual differences in intraindividual change are the second stage. Taken together, theory for health trajectory research is dynamic, emphasizing patterns of change and the forces that produce change. This is in sharp contrast to static theoretical statements about relationships among variables at a single point in time.
Theorizing about intraindividual change (Singer & Willett, 2003) in health or illness over time requires (a) conceptualization of a health-illness dimension of interest and selection of a change-sensitive indicator of it, (b) conceptualization of time and selection of a time scale relevant to the indicator and to the context of the research question, and (c) specification of the form of the relationship between time and the indicator. Many indicators are associated with familiar health-illness constructs in nursing research, but new measurement approaches emphasizing within-person validity over time will be needed to model intraindividual change (Collins, 1991). Consideration of time and time scale, combined with the duration and frequency of measurement, requires insight into the process under investigation (McGrath & Tschan, 2004, pp. 160-161). The relationship between time and individual health and illness (change) may be discontinuous (from one state or stages to another) or continuous. Individual continuous change may show increasing or decreasing smooth patterns that are described easily using simple mathematical functions or may fluctuate in complex ways with multiple increases and decreases over time. When between-person variation in intraindividual change exists, naturally occurring personal factors and deliberate experimental or clinical actions may explain interindividual differences in trajectories in both time-invariant and time-variant ways. Time itself may moderate the impact of the relationship between a covariate and a changing health status indicator. Theory for health trajectories is inherently multilevel in nature because individual differences affect personal parameters governing change over time at the individual level. Thought involved in theorizing about individual change over time thus introduces many additional considerations into the theory-building process.
Figure 3 is an adaptation of the graph of Campbell and Stanley (1963) that shows baseline and possible outcome trajectories after the implementation of an intervention at some time t. It shows that observation over time, both before and after treatment, is critical to making inferences about treatment effects. Such temporal considerations are essential to the overall plan for experimental and observational health trajectory research. Key issues include the creation of a measurement protocol with potential to reveal patterns of change in the health indicator under study, a scheme for measuring and coding time, and use of instrumentation for measuring health status that produces scores responsive to individual change.
Careful selection of the scale (metric) and coding of time are essential to revealing the nature of the trajectory and interpretation of the growth parameters. Often, a variety of indicators for time exist. Because there is no natural origin for time, a key decision involves the determination of when time begins for a trajectory. Usually, theoretically interesting time points are selected. Examples of initial time points (t 0) include the time of a transition such as birth of an infant, admission to or discharge from hospital, time of diagnosis of a chronic or terminal disease, or the point of onset of symptoms.
Some understanding of when change occurs and the form that change takes is needed to establish a measurement protocol. The timing of the first observation, the duration of the overall observation period, and the frequency of observations are specified in a measurement protocol. More elaborate models for change require a larger number of observations, timed to capture the key features of the individual experience. Contemporary statistical methods are flexible with respect to measurement protocols, and person-level change can be modeled with data obtained using individualized protocols over time (Singer & Willett, 2003). Individual participants may be observed on variable schedules and for different numbers of occasions within the time frame of theoretical interest.
Health trajectory research requires the use of instrumentation for measurement of health indexes that produce valid scores responsive to change over time. Reliability in the sense of coefficient alpha, predicated on indexing the true differences between people on one occasion, is not necessarily expected, required, or desirable when the focus is modeling intraindividual change (Collins, 1991). Instead, construct stability and preservation of metric validity precision over time are essential (Meredith & Horn, 2001).
Statistical Models for Change
Translation of scientific propositions to testable statistical models is an essential research activity (Jöreskog, 1993). Within the health trajectory framework, the challenge is to map theory for change to testable statistical models to emphasize the individual experience over time. Change in health status occurs in many ways, ranging from not at all (stability) to constant positive or negative rates, increasingly faster or slower rates, or complex change that occurs in phases (Cudeck & Harring, 2007; Cudeck & Klebe, 2002).
Some mathematical functions that can be used to model change in health status are listed in Table 3. Within each functional family, variations shown on the graphs are governed by sets of parameters to create specific characteristics that individualize the growth curves. Constant functions all show no change, but the parameter value κ distinguishes them by level (the value at which the y intercept is crossed when time = 0). Linear functions used to model constant rates of change are governed by two change parameters: π 0 (the value of a health indicator at time = 0, or y intercept) and π 1 (the rate of change, or slope). Quadratic and higher order polynomial functions are useful for modeling phenomena with varying rates of change (acceleration or deceleration) over time. Likewise, exponential functions capture extreme values at time = 0 (ξ), values at which a health index eventually levels out over time (α, the asymptote), and variable times taken to change from the extreme value at time = 0 to the leveling-out value (ρ, the rate). Piecewise functions are used to model multiphase processes characterized by different change processes during different periods (Cudeck & Klebe, 2002).
Selection of a function to characterize change is derived from clinical practice, previous findings from systematic observation over time in research, and careful plotting of case trajectories as part of preliminary analysis of longitudinal data. For example, grief after the death of a spouse normally occurs and persists for some time. After consideration of several functional forms, exponential change was selected as the most reasonable way to model individual grief trajectories using repeated preloss and postloss Center for Epidemiological Studies-Depression scores among surviving spouses taking part in the Changing Lives of Older Couples study (Burke, Shrout, & Bolger, 2007). Preloss patterns of depression over time, intensity of the immediate response, shape of the grief experience over time, and eventual level of resolution varied. Marital quality and preloss coping efficacy predicted grief trajectories. Commonalities among individual trajectories were used analytically to identify groups of surviving spouses with similar experiences over time (spouses with chronic depression unchanged by loss, chronic unresolved grief not preceded by depression, resilient with no preloss depression and little postloss grief, common grief with an intense immediate response followed by resolution, and spouses with preloss depression that resolved after loss).
Random Coefficients Model for Linear Change
Random coefficients models (also called individual growth models, mixed models, and multilevel models for longitudinal data) are used to model individual trajectories and the impact of personal and situational factors (covariates) on the parameters of the individual trajectories (Raudenbush & Bryk, 2002; Singer & Willett, 2003). Linear models often are selected as a model for change because of their ease of use and interpretation. Adapting the notation and following the presentation of Singer and Willett, the individual-level model for a continuously varying health status indicator h for person i at time j is
The growth parameters π 0 i and π 1 i are subscripted with i to indicate that within the linear framework, each person is following a unique pattern of change. The εij in Equation 1 is the stochastic (random) components of each person i's score h at time j and is regarded for simplicity as normally distributed; that is,
More realistically, autocorrelation and heteroscedasticity over time may be modeled in the level 1 model for individual change.
Systematic interindividual differences in intraindividual change, arising from personal or situational factors, may be incorporated as predictors of the trajectories by considering the π 0 i and π 1 i (individual trajectories) as outcomes. Systematic differences in change may arise also from a randomly assigned intervention protocol. For this common research situation,
In this level 2 (structural) model, γ 00 is the grand mean for the set of individual intercepts π 0 i and γ 10 is the grand mean for the set of individual slopes π 1 i. When INTERVENTION is scored as 0 = control and 1 = treatment, the coefficients γ 01 and γ 11 give the population average difference at baseline (usually expected to be nonsignificant) and population average difference in rate of change (the treatment effect, expected to be significant) between the control and intervention groups. The stochastic terms ζ 0 i and ζ 1 i are deviations between individual growth parameters and the respective population average value and are assumed to have a bivariate normal distribution with means equal to 0, variances equal to σ 2 0 and to σ 2 1, and covariance equal to σ 01. Overall model fit, model comparison approaches, and significance of the fixed and random effects can be assessed using standard approaches (Singer & Willett, 2003).
Extensions and Complementary Approaches
Statistical advances over the past 30 years have created flexible extensions of the basic model for trajectory analysis (Skrondal & Rabe-Hesketh, 2004), some of which are listed in Table 4. One type of extension focuses on prediction of growth parameters. Incorporating time-varying covariates (predictors) to the model allows better understanding of covariate impact on trajectory parameters across the time of observation, and modeling trajectories in context adjusts growth parameters to allow for the moderating effects of variables at higher levels of a hierarchical theoretical system. Another type of extension to the basic model focuses on the trajectories themselves. The aim of latent class growth analysis and growth mixture models is to identify latent heterogeneity in growth by identifying groups of individuals (latent classes or components of mixture distributions, respectively) whose pattern of change is similar. The key difference between the two approaches is that within-class variability is modeled in the growth mixture approach but not in the latent class approach. Parallel process models involve simultaneous estimation of growth parameters in two or more trajectories, including estimation of the relationships among the growth parameters for each of the trajectories. Trajectories for variables measured with categorical responses are hierarchical generalized linear models with a link function. The link function transforms the trajectory variable in a way that is consistent with the sampling model (e.g., cumulative logit link for the multinomial sampling model associated with ordered categorical responses for the trajectory variable). Models that are complementary to the random coefficients approach for longitudinal data are also available. Generalized estimating equations incorporate dependence in observations over time in the estimation of a population-average (marginal) model for the means at each time point that maintains desirable properties of the estimates (consistency) without relying on the assumptions that support hierarchical generalized linear models. Finally, emerging dynamic process models are similar to trajectory models because they explore repeated measurements as a function of time but differ completely in the functional approach used. Dynamic process models use differential equations to explore relationships among varying rates of change over time in related variables. Exemplars for each of these approaches are provided in Table 4, and many of them are used in the primary reports included in this supplement.
Flexibility to individualize measurement protocols for participants is a desirable quality of growth models contributing to the feasibility of implementing health trajectory research. Data from every participant with at least one observation at some point in time can be incorporated into estimation of the model, provided that data are missing completely at random (MCAR) or missing at random (MAR); even nonignorable missingness may provide robust results. See Raudenbush and Bryk (2002, pp. 199-200) for a discussion about assessing missingness.
Power and Accuracy of Estimates
Asymptotic procedures used to estimate parameters in growth models assume large samples. Accuracy in growth parameter estimation and power to detect treatment effects in trials with trajectories as outcomes depend on sample size as well as the duration, interval, and frequency of measurement of the trajectory variable; the impact of these factors varies with functions used to model change (e.g., linear or higher order polynomial; Raudenbush & Liu, 2001). Costs associated with various combinations of sample size and temporal protocols must also be considered when planning studies.
Random coefficients models can be estimated using HLM, MPlus, LISREL, SAS PROC MIXED, and other programs. Some programs require special data set-ups, and not all can be used to estimate variations such as time-varying covariates, multiple covarying trajectories, or modeling indicators with discrete values. Examples (including data, input, and output) can be found online at http://gseacademic.harvard.edu/alda/.
Health trajectory science provides relevant knowledge for improved nursing services and optimal outcomes, at the point of care and beyond, for individual patients, individual families, and individual communities. Whether describing the natural history of a health experience or assessing the impact of an intervention on health over time, theory about change, temporal design of a study, and a statistical model to describe the impact of time on health are linked to emphasize the individual experience. The health trajectory perspective opens new horizons for nursing research that uses time to create a person-centered science.
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Keywords:© 2011 Lippincott Williams & Wilkins, Inc.
applied longitudinal data analysis; change; health trajectory; nursing theory