In a wider scope, trajectories are also seen in the pattern of change referring to such phenomena as the decaying quality over time of a commercial product or the collapse of a political system in a country. In the field of business management, change in consumer purchasing behavior is generally linked both with individual characteristics and with competing products. In population studies, demographers are concerned with such longitudinal processes as internal and international migration, and intervals between successive births. In cases such as these events and in others, the pattern of change over time can be influenced and determined by various factors, such as genetic predisposition, illness, violence, environment, medical and social advancements, or the like. Therefore, each trajectory can differ significantly among individuals and other observational units, or by the variables that govern the timing and rate of change in a period of time.

Data available at a single point of time does not suffice to analyze change and its pattern over time. Cross-sectional data, traditionally so popular and so widely used in a wide variety of applied sciences, only designates a snapshot of a course and thus does not possess the capacity to reflect change, growth, or development. Aware of the limitations in cross-sectional studies, many researchers have advanced the analytic perspective by examining data with repeated measurements. By measuring the same variable of interest repeatedly at a number of times, the change is displayed, its pattern over time revealed and constructive findings are derived with regard to the significance of change. Data with repeated measurements are referred to as longitudinal data. In many longitudinal data designs, subjects are assigned to the levels of a treatment or of other risk factors over a number of time points that are separated by specified intervals.

Analyzing longitudinal data poses considerable challenges to statisticians and other quantitative methodologists due to several unique features inherent in such data. First, the most troublesome feature of longitudinal analysis is the presence of missing data in repeated measurements. In a longitudinal survey, the loss of observations on the variables of interest frequently occurs. For example, in a clinical trial on the effectiveness of a new medical treatment for disease, patients may be lost to a follow-up investigation due to migration or health problems. In a longitudinal observational survey, some baseline respondents may lose interest in participating at subsequent times. These missing cases may possess unique characteristics and attributes, resulting in the fact that data collected at later time points may bear little resemblance to the sample initially gathered. Second, repeated measurements for the same observational unit are usually related because average responses usually vary randomly between individuals or other observational units, with some being fundamentally high and some being fundamentally low. Consequently, longitudinal data are clustered within observational units. In the meantime, an individual’s repeated measurements may be a response to a time-varying, systematic process, resulting in serial correlation. Third, longitudinal data are generally ordered by time either in equal space or by unequal intervals, with each scenario calling for a specific analytic approach. Sometimes, even with an equal-spacing design, some respondents may enter a follow-up investigation after a specified survey date, which, in turn, imposes unequal intervals for different individuals.

Over the years, scientists have developed a variety of statistical models and methods to analyze longitudinal data. Most of these advanced techniques are built upon biomedical and psychological settings, and therefore, these methodologically advanced techniques are relatively unfamiliar to researchers of other disciplines. To date, many researchers still use incorrect statistical methods to analyze longitudinal data without paying sufficient attention to the unique features of longitudinal data. For these researchers, the advanced models and methods developed specifically for longitudinal data analysis can be readily borrowed for use after careful verification, evaluation, and modification. In health and aging research, for example, the pattern of change in health status is generally the main focus. In analyzing such longitudinal courses, failure to use correct, appropriate methods can result in tremendous bias in parameter estimates and outcome predictions. In these areas, the application of advanced models and methods is essential.

Given the substantial limitations and constraints in the traditional approaches in repeated measures analysis, many methodologists expressed grave concerns regarding how to measure and analyze the pattern of change over time correctly (see the summary in Singer and Willett, 2003, Chapter 1). Over the past 30 years, longitudinal data analysis has grown tremendously as a consequence of the rapid developments in mixed-effects modeling, multilevel analysis, and individual growth perspectives. Accompanying these developments in statistical models and methods are the equally important advancements in computer science, particularly the powerful statistical software packages. The convenience of using computer software packages to create and utilize complex statistical models has made it possible for many scientists to analyze longitudinal data by applying complex, efficient statistical methods and techniques, once considered impossible to accomplish (Singer and Willett, 2003).

As applications of various statistical techniques on longitudinal data have grown, methodological innovation has accelerated at an unprecedented pace over the past three decades. The advent of the modern mixed-effects modeling and the various approaches for the analysis of longitudinal data triggered the advancement of a large number of statistical models and methods, characterized by the complex procedures of multivariate regression. The major contribution of mixed-effects models, with their capability of containing both fixed and the random effects, is the provision of a flexible statistical approach to model the autoregressive process involved in the trajectory of individuals, both for average change across time and change for each observational unit. Given such a powerful perspective, both the measurable covariates and the unobservable characteristics can be incorporated in the model simultaneously, thereby deriving more reliable analytic results for the description of a longitudinal process. Being robust to missing data in general circumstances, mixed-effects models also have the added advantage of permitting irregularly spaced measurements across time.

More recently, a variety of Bayes-type approximation methods have been advanced to estimate parameters in the analysis of longitudinal data characterized of nonlinear functions such as proportions and counts. Given their flexibility in modeling nonnormal outcome data, these approximation techniques have enabled researchers to estimate random effects with complex structures and to correctly perform nonlinear predictions. To date, the various approximation methods have been applied by statisticians and some other quantitative methodologists to develop more statistically refined longitudinal models, thus expanding the capacity of mixed-effects modeling in longitudinal data analysis to a new dimension. In a different track, some other methodologists have advanced growth curve modeling by introducing latent factors or/and latent classes within the framework of structural equation modeling (SEM).

In this section, longitudinal data structures are delineated. I first review the multivariate data format used traditionally in the classical repeated-measures analysis and applied presently in latent growth modeling. Second, the univariate data structure is introduced, in which an individual or an observational unit (such as a commercial product) has multiple rows of data to record repeated measurements at a limited number of time points. Lastly, balanced and unbalanced longitudinal data are defined and described.

Missing data can be grouped according to the missing data pattern, which describes which values are observed and which values are missing in the data matrix. In general terms, missing data patterns can be roughly classified into a variety of groups, such as univariate, multivariate, monotone, nonmonotone, and file matching (Little and Rubin, 2002). A univariate missing pattern indicates the situation where missing data occur only in a single variable. As an extension of the univariate case, the multivariate missing pattern refers to missing data in a set of variables, either for the entire unit or for particular items in a questionnaire. If a variable is missing for a particular subject not only at a specific time point but also at all subsequent time occasions, the missing data pattern for this individual is said to be a monotone missing pattern. In contrast, if a case is missing at a given time point and then returns at a later follow-up investigation, then the missing data pattern for this subject is referred to as the nonmonotone missing data. In longitudinal data analysis, the nonmonotone missing pattern can cause more problems than does the monotone pattern, and thus deserves close attention. Sometimes, variables are not observed together, and such missing data are referred to as a file-matching pattern. There are more missing data patterns such as the latent-factor patterns with variables that are never observed. For each missing data patterns, there are corresponding statistical techniques to handle its impact on the quality of a longitudinal data analysis.

Missing data mechanisms concern the relationship between missing data and the values of variables in the data matrix. Given this focus, missing data mechanisms can be categorized into three classes: missing completely at random (MCAR), missing at random (MAR), and missing not at random (MNAR). If missing data are unrelated to both the missing responses and the set of observed responses, the observed values are representative of the entire sample without missing values. This missing data mechanism is referred to as MCAR. If missing data depend on the set of observed responses but unrelated to the missing values, the missing data are said to be MAR. In longitudinal data analysis, the majority of missing data are categorized as MAR. Correspondingly, most longitudinal models are based on the MAR hypothesis in handling missing values.

In some special situations, missing data are related to specific missing values, referred to as MNAR. In longitudinal data analysis, ignoring the impact of this missing data mechanism can result in serious bias in analytic results and erroneous predictions. Over the years, scientists have developed a variety of empirically sound models and methods to account for MNAR in longitudinal data analysis. Compared to the methods for MAR, however, the statistical models handling MNAR are considered less mature, and research in this regard is ongoing.

With the importance of missing data analysis on longitudinal data, in this book an entire chapter (Chapter 14) is devoted to this area. In that chapter, a section is provided on the mathematical definitions and conditions associated with MCAR, MAR, and MNAR, respectively. Various statistical models handling different missing data mechanisms are described extensively in Chapter 14.

Between-subjects variability reflects individual heterogeneity in unspecified and unrecognized characteristics on the trajectory of the response variable. While much of the overall variability can be statistically addressed by specifying observable individual and contextual covariates, some between-subjects heterogeneity may derive from unobservable biological and environmental factors, such as genetic predisposition and physiological parameters. A sample of individuals may have different reactions to the same dose of a new medication on a specific disease; a good behavioral prototype fostered in childhood can help decelerate functional disablement process as age progresses; the social environment is likely to affect drug use recidivism among adolescents released from a rehabilitative center; and so on. As well documented in the literature of longitudinal data analysis, ignoring such unobserved heterogeneity can result in considerable bias in parameter estimates, especially the estimator of standard errors (Diggle et al., 2002; Fitzmaurice et al., 2004; Verbeke and Molenberghs, 2000). In empirical research, a popular approach to handle between-subjects variability is to specify the individual-specific random effects with the assumption of a known distribution. By accounting for individual differences in the analysis, intraindividual correlation in longitudinal data is implicitly manipulated thereby resulting in conditional independence in repeated measurements of the response for the same observational unit. As will be described in the succeeding chapters, many statistical models and methods applied in longitudinal data analysis are designed to handle intraindividual correlation by the specification of such random effects.

The second component of variability in longitudinal processes is within-subject variations. Given various biological, genetic, and environmental predispositions, the values of repeated measurements for the same individual tend to be more similar than those obtained from several randomly selected individuals. Therefore, intraindividual changes over time in the response variable tend to be confined within a person’s physical and environmental mechanisms, resulting in positive correlation in longitudinal data. Another distinctive characteristic in such subject-specific dependence is that intraindividual correlation has frequently been observed to decay over time. For example, the repeated measurements of blood sugar for the same person are very likely to be more similar than those from a number of randomly selected persons. A person’s blood sugar measurements also tend to be more similar between two successive time occasions than between two nonadjacent time points. In the literature of repeated-measures analyses, this type of change in correlation is referred to as serial correlation. The presence of recognizable patterns of correlation has enabled researchers to account for intraindividual correlation by specifying within-subject covariance structures on repeated measurements.

It must be emphasized that between-subjects and within-subject variations are closely interrelated. Clustering among repeated measurements of the response for the same observational unit can be understood as reflecting differences in the pattern of change over time among individuals. Likewise, individual differences reflect homogeneity of the repeated measurements within an observational unit. Therefore, in longitudinal data analysis, the researcher often needs to consider only one source of systematic variability in statistical modeling, within-subject or between-subjects, to make the longitudinal data conditionally independent. Given the close interaction between the two sources of variability, very often longitudinal data does not support modeling of all possible components simultaneously, particularly when the outcome variable of interest is qualitative rather than quantitative. Only in some special situations do both between-subjects and within-subject variability need to be specified in the same statistical model, and some cases of this sort will be described in the succeeding chapters.

If the specification of the subject-specific random effects or/and a specific covariance structure primarily account for intraindividual correlation in longitudinal data, the remaining within-subject variability is random and noninformative. Correspondingly, this residual component can be specified as random errors in the same fashion as traditionally assumed in general linear models and generalized linear models. For qualitative response variables, however, specification of within-subject components of variability cannot be easily specified, and some complex statistical procedures are needed for the estimation of random errors.

The number of time points is another primary concern in designing a longitudinal study. While there are no clear-cut rules about the optimum number of time points, the minimum number should not be two, or even three. From a two-point longitudinal dataset, the only pattern of change over time that can be generalized is linear, no matter how different the values of the response differ between the two time points. Important information about the trajectory of individuals can be missed. Longitudinal data with a design of three time points provide the capacity to describe a possible exponential pattern of change over time, but not a more complex function. Four time points can yield a cubic trajectory of individuals. Therefore, for thoroughly reflecting the possible pattern of change in the response variable, the minimum number of time points is four. Analytically, there does not seem to be a definite maximum number of time points that can be determined for longitudinal data analysis. The desired number of time points should be evaluated on a case-by-case base. A high number of time points can probably improve the precision of estimation on the pattern of change over time if the sample size is sufficiently large. In the meantime, specifying a large number of time points can significantly increase the financial burden to conduct a longitudinal analysis and complicate statistical modeling. I personally recommend that for clinical experimental studies, four to six time points, consisting of the baseline and three to five follow-up waves, should be good enough to display the effectiveness of a new medication or treatment on disease. With respect to observational studies with a large sample size, 6 to 10 time points are ideal for the description of a time trend in the response variable and its group differences.

Given N subjects, there are N such vectors in a longitudinal dataset. Repeated measurements of the response variable are generally specified as a function of the time factor and some other theoretically relevant covariates. In the analysis of cross-sectional data, various regression models are generally performed by assuming conditional independence of observations in the presence of specified model parameters. If the response variable Y has a continuous scale, it seems that a linear regression model on the response variable Y can be written as

where X_ij is a 1 × M vector of covariates for subject i at time point j, β is a vector of the regression coefficients of M covariates with values fixed for all subjects, and ɛ_ij is the random error term conventionally assumed to be normally distributed. In the context of longitudinal data, however, this classical specification does not derive efficient, robust, and consistent parameter estimates, particularly since the error is related to the random errors at other time points, even in the presence of specified observed covariates. Therefore, some additional parameters addressing intraindividual correlation need to be specified to secure the conditional independence hypothesis in random errors.

The importance of addressing intraindividual correlation has triggered the development of many advanced methods in longitudinal data analysis. As summarized by some leading statisticians in this area (Diggle, 1988; Diggle et al., 2002), modeling a longitudinal process should at least consider three different sources of variability. First, average responses usually vary randomly between subjects, with some subjects being fundamentally high and some being low. Second, a subject’s observed measurement profile may be a response to time-varying processes. Third, as the individual measurements involve subsampling within subjects, the measurement process adds a component of variation to the data. Such decomposition of stochastic variations in repeated measurements facilitates the formulation of correlation between pairs of measurements on the same subject. Based on the linear regression model specified in Equation (1.1), the multivariate linear model, including all the aforementioned three sources of variability, can be written as

where b_i indicates the variation in the average response between subjects, the term ${\tilde{W}}_i(T_{ij})$ reflects independent stationary processes given a special form of serial correlation, T_ij is the value of time for subject i at time point j, and ɛ_ij represents the subsampling uncertainty within subject. For analytic convenience, each of the three terms, b_i, ${\tilde{W}}_i(T_{ij})$, and ɛ_ij, are usually assumed to follow a normal distribution with a well-defined variance or a covariance function. With the specification of b_i, ${\tilde{W}}_i(T_{ij})$, the random component ɛ_ij is conditionally independent of random errors at other time points. Thus, the estimate of β tends to be unbiased, and the term ${X^{\prime }}_{ij}\beta$ represents the mean response for subject i at time point j.

The three sources of variability in longitudinal data are usually intertwined, as indicated earlier. Therefore, in much of the empirical research, only one source of systematic variability needs to be considered. The most popular approach in longitudinal data analysis is perhaps the mixed-effects modeling in which the between-subjects random effects are specified to address intraindividual correlation. Let n_i be the number of repeated measurements for subject i in a sample of N individuals, and Y_i be the n_i-dimensional vector of repeated measurements for the subject. If only a single term of the random effects is considered, a typical linear mixed model is given by

where X_i is a known n_i × M matrix of covariates with the first column taking constant 1, β is an M × 1 vector of unknown population parameters, and b_i is the unknown subject’s effect. In this specification, the primary predictor time is usually specified as a single continuous variable or a set of polynomials that are included in X. As the specification of a single random effect does not confound the fixed regression coefficients, b_i is also referred to as the random effect for the intercept. Given the inclusion of b_i, linear mixed models are thought to yield more efficient and robust regression coefficients than do general linear models in which residuals are potentially dependent. Sometimes, multiple terms of the random effects (e.g., the random effects for the intercept and for the time factor) need to be specified, and then the random effects for subject i are expressed in terms of b_i in mixed-effects modeling given the specification of a design matrix. These extended mixed-effects models will be described and discussed in many of the succeeding chapters.

Given the specification of b_i or b_i, the elements in e_i are assumed to be conditionally independent. Specifically, the term e_i is an n_i × 1 column vector of random errors for subject i, given by

Another popular perspective in longitudinal modeling is to specify the covariance structure of within-subject random errors in linear regression models while leaving the between-subject random effects unspecified. The use of such a design in modeling a longitudinal process becomes necessary when the application of the random-effects approach does not yield reliable analytic results or when the within-subject variability is sizable in comparison with the between-subjects variability. In this approach, the vector e_i is assumed to follow a multivariate normal distribution with mean 0 and an n_i × n_i covariance matrix. To pattern the error covariance structure in the linear regression, the time factor must be specified as a classification factor containing n discrete levels for reflecting the repeated effects. In empirical analyses, there are a variety of covariance pattern models that have been designed over the years by statisticians. Although it bears tremendous resemblance to the classical repeated-measures ANOVA-type models, this linear regression model on repeated measurements is generally viewed as a family of linear mixed models in view of its capacity to account for between-subjects variability.

The aforementioned two perspectives on the continuous response variable can be readily extended to statistical modeling on a variety of nonnormal outcome variables, such as rate and proportion, multinomial, and count data. There are many statistically complex models and methods for those longitudinal data types; the fundamental rationale in these advanced models, however, is based on the classification of variability in longitudinal data. In the succeeding chapters, detailed specifications and statistical inferences for various longitudinal models, linear or nonlinear, will gradually unfold with empirical illustrations presented for their applications.

Chapters 8–12 are devoted to statistical models and methods for modeling nonnormal longitudinal data. In particular, Chapter 8 introduces the general inference of generalized linear mixed models given the specification of the random effects. A variety of statistical methods are delineated for the estimation of fixed and random effects in generalized linear mixed models, followed with the description of various approaches to perform nonlinear predictions of nonnormal longitudinal data. Chapter 9 is focused on the delineation of generalized estimating equations modeling the marginal mean parameters by using a quasi-likelihood method. Chapter 10 concerns the general specifications of the mixed-effects logistic regression model, including the statistical inference of the regression, a brief discussion on the interpretability of the conventional odds ratio in longitudinal data analysis, and the approximation of longitudinal trajectories of the response probability and the corresponding standard errors. Chapter 11 presents model specifications and statistical inference of the mixed-effects multinomial logit model. In this chapter, a retransformation method is developed to derive unbiased nonlinear predictions of a set of response probabilities. Based on the development of the mixed-effects multinomial logit model, Chapter 12 specifies a number of multidimensional transition models, which link categorical outcome data to the time factor, the value of one or more prior states, and other theoretically relevant covariates. In each of these chapters, a step-by-step presentation with empirical illustrations is provided, and the merits and limitations in these statistical models are discussed.

In Chapter 13, techniques of latent growth modeling are described and discussed. The latent growth models covered in this chapter consist of the latent growth model, the latent growth mixture model, and the group-based model, with each method following the SEM perspective. Substantial discussions are provided on the advantages and the existing problems in these SEM-type models with latent factors or latent classes. The last chapter, Chapter 14, is devoted to missing data analysis. Based on the mathematical definitions on MCAR, MAR, and MNAR, a variety of statistical methods are described and discussed in this chapter, dealing with ignorable (MCAR and MAR) and nonignorable missing data (MNAR), respectively. Due to consideration of coherence and conciseness of the text, some supplementary procedures, datasets, and computer programs are presented as Appendices.

As indicated in Section 1.2, longitudinal data analysis was initiated and extensively applied in clinical experimental studies, and the methods have later been extended and advanced to the realm of social and behavioral sciences. While the statistical methods applied in these two scientific domains share tremendous similarities, there are some distinct differences in the data features between them. The clinical experimental research is often based on data obtained from randomized controlled clinical trials that are generally characterized by small sample sizes and with narrowly spaced intervals between successive waves of investigation. In contrast, social and behavioral studies usually rely on large-scale survey data often with widely spaced intervals. Accordingly, in this book two longitudinal datasets are used for empirical illustrations, one from a typical randomized controlled clinical trial and one from a large-scale longitudinal survey. These two datasets represent two data types. The clinical trial dataset is of small scale, with follow-up investigation conducted at every 4 weeks, whereas the second dataset comes from a nationally representative investigation of older Americans with a large sample size at baseline and an interwave interval of every 2 years. The following paragraphs provide detailed descriptions concerning the two datasets.