Beyond Pre–Post: Modeling Individual Change With Latent Growth Models

A pre–post design reduces longitudinal change to two observations: where a participant began and where that participant ended. That can be sufficient when the scientific question is genuinely about one before-versus-after contrast. But once the same outcome is measured repeatedly across several occasions, researchers can ask a richer question: What trajectory of change produced the observed outcomes, and do people differ in that trajectory?

A latent growth model (LGM)—also called a latent growth curve model—uses repeated measurements to represent systematic features of change as latent growth factors. In a basic linear model, an intercept growth factor represents a person's level at a chosen reference time, while a slope growth factor represents that person's rate of change. Their means describe the average trajectory; their variances describe differences among individual trajectories (Wang & Wang, 2012; Frey, 2022).

This makes latent growth modeling useful for longitudinal data analysis when the research problem concerns not merely whether scores changed, but how they changed over time and how that change varies across individuals.

The problem with reducing a trajectory to pre–post change

Suppose a synthetic longitudinal study follows 240 university students through an academic support program. Academic self-efficacy is measured at five equally spaced occasions:

• baseline;
• 3 months;
• 6 months;
• 9 months;
• 12 months.

Consider four illustrative students:

Students A, B, and C have the same baseline score and the same final score. A pre–post difference therefore gives all three a change score of +20.

Yet their longitudinal patterns are plainly different. Student A changes steadily. Student B improves rapidly and then levels off. Student C changes slowly at first and more rapidly later.

A single difference score cannot preserve those intermediate patterns because it uses only the endpoints. Longitudinal growth modeling instead treats the repeated observations as information about an underlying trajectory. The objective is to characterize the functional form of change and the distribution of individual trajectories rather than reducing each participant's history to one endpoint contrast (Wang & Wang, 2012; Frey, 2022).

That distinction is central to modeling change over time.

Repeated observations belong to the same person

Five observations from one student are not equivalent to five observations from five independent students.

Repeated-measures data contain a within-person structure: measurements are collected repeatedly from the same observational unit. Growth modeling explicitly represents an individual's repeated outcomes as a trajectory across time, while allowing trajectories to differ between individuals (Wang & Wang, 2012; Frey, 2022).

For participant i at occasion t, a simple linear growth representation can be written conceptually as:

Y_ti = η_0i + λ_tη_1i + ε_ti

where:

• Y_ti is the observed outcome for individual i at time t;
• η_0i is that individual's intercept growth factor;
• η_1i is that individual's slope growth factor;
• λ_t represents the time score for occasion t;
• ε_ti represents occasion-specific deviation from the modeled trajectory.

The crucial feature is the subscript i on both growth factors: the model permits individuals to have different intercepts and different slopes (Wang & Wang, 2012; Frey, 2022).

Intercept and slope growth factors

The intercept: individual starting level

In the most familiar specification, the time scores for five equally spaced occasions could be:

0, 1, 2, 3, 4

The intercept-factor loadings are fixed at 1 across the repeated outcomes, while these time scores serve as loadings for the slope factor. With the first occasion coded 0, the intercept represents the predicted outcome level at baseline (Wang & Wang, 2012; Frey, 2022).

For the synthetic student study, the intercept therefore answers:

The intercept need not always mean baseline. Changing the zero point of the time scores changes the reference occasion represented by the intercept. For example, recentering time so that the final occasion equals zero makes the intercept represent predicted status at the end of follow-up (Wang & Wang, 2012; Frey, 2022).

That is an important interpretation rule: an intercept is the predicted outcome when the model's time score equals zero.

The slope: individual rate of change

In a linear latent growth model, the slope factor represents change associated with a one-unit increase on the specified time scale. When measurements are equally spaced and time scores are 0, 1, 2, 3, 4, the slope describes a constant linear rate of change per interval (Wang & Wang, 2012; Frey, 2022).

For example, suppose the synthetic fitted model estimated a mean intercept of 44.0 and a mean slope of 3.2 points per three-month interval.

They would mean that the model-implied average trajectory begins around 44 points at baseline and increases by approximately 3.2 points per measurement interval under the assumed linear trajectory.

The slope is therefore more informative than a generic statement that "scores increased." It quantifies the modeled rate of change on a specified time scale.

Average growth is only half the question

A major reason to use a latent growth model is that it can separate the average trajectory from variation around that trajectory.

The mean of the intercept growth factor represents the average intercept across individuals, and the mean of the slope growth factor represents the average rate of change. Their variances describe the extent to which individual intercepts and slopes differ from those averages (Wang & Wang, 2012; Frey, 2022).

Suppose the synthetic model produced:

A positive mean slope would describe average improvement. A nonzero slope variance addresses a different question: Do people change at different rates?

Those findings should not be conflated. A population can show substantial average growth while individuals differ markedly in how quickly they change. Conversely, a weak average slope could conceal heterogeneous individual trajectories.

Wang and Wang (2012) describe these random components as capturing variation in trajectories both within and across individuals. Frey (2022) similarly describes growth-factor variances as summarizing the distribution of individual trajectory intercepts and slopes.

Starting level and rate of change can be related

A linear growth model can also estimate covariance between its intercept and slope factors. This describes whether differences in initial status are associated with differences in subsequent rates of change (Wang & Wang, 2012; Frey, 2022).

For example, a negative intercept–slope association in the synthetic study would indicate that students beginning at higher self-efficacy levels tend to have smaller modeled increases, whereas a positive association would indicate that higher starting levels tend to accompany greater increases.

Time coding is part of the model, not housekeeping

Time scores determine important features of a latent growth curve model.

For equally spaced occasions, a linear trajectory might use:

0, 1, 2, 3, 4.

If the actual measurements occurred at baseline, month 1, month 3, month 6, and month 12, treating them as equally spaced would misrepresent elapsed time. Growth-model time scores can instead reflect the actual spacing of measurement occasions (Wang & Wang, 2012; Frey, 2022).

Time scores also determine where the intercept is centered and the scale on which the slope is interpreted. Consequently, coding time should follow the longitudinal design and substantive meaning of elapsed time rather than software convenience (Wang & Wang, 2012).

What if change is not linear?

A linear model says something substantive: it assumes that the systematic trajectory can be represented by a constant rate of change on the specified time scale.

That may be inappropriate.

Student B in the synthetic example improved quickly and then plateaued. Student C accelerated later. Those patterns illustrate why longitudinal trajectories may require nonlinear representations.

Latent growth modeling can be extended beyond a straight-line trajectory. A quadratic model, for example, introduces another growth factor representing curvature, while other nonlinear specifications can represent different hypothesized forms of change (Wang & Wang, 2012; Frey, 2022).

This is another reason four or more measurement occasions can be valuable. Multiple observations allow the researcher to investigate the shape of change rather than observing only its two endpoints.

Predicting who starts higher and who changes faster

Once an unconditional growth model provides an adequate representation of the longitudinal outcome, predictors can be introduced to explain differences in the growth factors (Wang & Wang, 2012; Frey, 2022).

Suppose the synthetic study recorded a baseline measure of prior academic preparation. A conditional growth model could ask two distinct questions:

The distinction is substantively important. A variable associated with higher baseline status need not predict faster subsequent growth, and a variable that predicts growth need not explain baseline differences.

Time-invariant characteristics such as demographic variables or treatment-group membership can similarly be used as predictors of intercept and slope factors. Wang and Wang (2012) demonstrate conditional LGMs in which time-invariant covariates predict both growth factors, while Frey (2022) likewise describes exogenous predictors of initial status and change.

Model specification comes before model fit

Latent growth modeling is an SEM application, so the analysis should begin by specifying the trajectory the research question implies.

For a basic five-wave linear LGM, the researcher must decide at least:

1. which repeated outcome constitutes the longitudinal process;
2. what time metric the slope represents;
3. where time is centered;
4. whether linear change is scientifically plausible;
5. which growth-factor means, variances, and covariances are estimated;
6. whether predictors of the growth factors belong in the planned model;
7. how occasion-specific residual variation is represented.

Wang and Wang (2012) frame SEM as a sequence of model formulation, identification, estimation, evaluation, and—when justified—modification. Model fit therefore evaluates a substantively specified model; it should not substitute for specification.

How should model fit be evaluated?

A growth model implies a structure for the observed repeated measurements. Researchers therefore need to ask whether that proposed structure provides an adequate representation of the longitudinal data.

Within the SEM framework, model evaluation can include the model chi-square and descriptive or approximate fit measures such as CFI, TLI, RMSEA, and SRMR. Model chi-square should not be treated as the sole criterion because its behavior is affected by factors including sample size and distributional conditions; fit evidence should be interpreted collectively rather than through one mechanical cutoff (Wang & Wang, 2012; Lovric, 2011).

For a growth model, poor fit can be substantively informative. A poorly fitting linear model may indicate that a constant rate of change does not adequately represent the observed longitudinal process. Frey (2022) specifically recommends considering more complex growth specifications when the linear growth model does not adequately fit the longitudinal data.

Fit is also not the only diagnostic. Researchers should inspect observed individual trajectories and the trajectory of observed means before or alongside formal modeling. A spaghetti plot can reveal heterogeneity, possible curvature, unusual trajectories, or other features that a single set of fit indices cannot communicate (Frey, 2022).

The decision is therefore not simply:

It is:

Missing longitudinal observations

Missing follow-up observations are particularly relevant in longitudinal research because requiring complete measurements at every occasion can discard information contributed by participants who completed only part of the study.

Wang and Wang (2012) discuss full-information maximum likelihood (FIML) for incomplete longitudinal data within their latent growth modeling framework. Under a missing-at-random (MAR) assumption, missingness may depend on observed information but not on the unobserved value after conditioning on the observed information; their treatment emphasizes that FIML uses the available observed information rather than requiring conventional listwise deletion (Wang & Wang, 2012).

Latent growth models versus simpler repeated-measures approaches

A latent growth model is not automatically preferable whenever data contain repeated measurements.

Traditional repeated-measures methods can answer useful questions about mean differences across occasions. Profile analysis, for example, provides a multivariate approach to repeated measures, and conventional repeated-measures analyses may be appropriate for well-structured designs when their assumptions and target hypotheses align with the research question (Tabachnick & Fidell, 2013).

Growth modeling changes the emphasis.

Instead of treating time primarily as a set of repeated conditions whose means are compared, a growth model represents time as a trajectory and estimates characteristics of that trajectory. The intercept and slope become individual-varying model parameters, allowing researchers to examine both average change and heterogeneity in change (Wang & Wang, 2012; Frey, 2022).

Latent growth modeling also has a close relationship to multilevel or mixed-effects approaches to longitudinal data. Wang and Wang (2012) describe the basic LGM as an application of multilevel modeling within the SEM framework, and Frey (2022) notes that an SEM growth model can in many situations be equivalent to a multilevel model for the same longitudinal data. Growth-factor means correspond conceptually to fixed effects, while growth-factor variances and covariances correspond to random effects (Frey, 2022).

When is a latent growth model worth considering?

A latent growth model becomes especially worth considering when the central research question concerns a trajectory, not simply an endpoint difference.

It is a strong candidate when:

• the same outcome has been observed over multiple meaningful occasions;
• the research question concerns modeling change over time;
• individual starting levels are substantively important;
• researchers expect people to differ in their rates of change;
• the shape of change—linear versus nonlinear, for example—is scientifically relevant;
• predictors of initial status or growth rate are part of the research question;
• researchers want an SEM framework in which growth factors can participate in broader latent-variable models.

These features correspond directly to the capabilities of latent growth modeling described by Wang and Wang (2012) and Frey (2022).

If the answer is yes, reducing the data to one difference score may be answering a narrower question than the study was designed to investigate.

When is it unnecessary complexity?

A latent growth model can also be more machinery than the research problem requires.

If a study genuinely has only two relevant measurement occasions and the scientific estimand is a specific pre–post contrast, there may be no meaningful intermediate trajectory to model. Similarly, if a complete repeated-measures design is intended primarily to compare a small number of planned occasion means or profiles, a supported repeated-measures approach may answer the question directly without introducing latent growth factors (Tabachnick & Fidell, 2013).

Complexity is also difficult to justify when the study has collected several occasions but there is no substantive theory or research question concerning rates or shapes of change. Fitting intercept, slope, nonlinear, and predictor structures merely because software permits them reverses the appropriate order of model building. SEM specification should follow the research problem, with identification, estimation, and model evaluation following that specification (Wang & Wang, 2012).

A practical latent growth model decision framework

The framework follows the broader principle that longitudinal model selection should begin with the scientific question and the structure of repeated observations rather than with a preferred software procedure (Wang & Wang, 2012; Tabachnick & Fidell, 2013; Frey, 2022).

What should researchers report?

A useful latent growth analysis should make the time model and interpretation visible. At minimum, researchers should explain:

1. the number and spacing of measurement occasions;
2. the time scores used in the model;
3. the centering point of the intercept;
4. the assumed functional form of change;
5. the estimated growth-factor means and variances;
6. relevant intercept–slope covariance;
7. the evidence used to evaluate model fit.

These reporting elements are consistent with Wang and Wang (2012) and Frey (2022).

When predictors are included, reports should distinguish clearly between predictors of initial status and predictors of change. When observations are incomplete, the missing-data approach and its assumptions should also be stated (Wang & Wang, 2012).

Conclusion

The question "Did the outcome change from pre to post?" can be useful. But it is not equivalent to "How did the outcome change over time?"

With several longitudinal observations, a single endpoint difference can conceal when change occurred, whether it was approximately linear, and whether individuals followed meaningfully different trajectories.

A latent growth model addresses those questions by representing repeated outcomes through growth factors. The intercept captures status at a chosen reference time; the slope captures rate of change; their means describe the average trajectory; and their variances describe individual differences in that trajectory. Predictors can then be used, where justified, to investigate who starts at different levels and who changes at different rates (Wang & Wang, 2012; Frey, 2022).

That additional structure is valuable only when the research question requires it. The purpose of growth curve analysis is not to replace every pre–post or repeated-measures analysis with SEM. It is to preserve and model trajectory information when trajectory itself is part of the scientific question.

References

Frey, B. B. (Ed.). (2022). The SAGE encyclopedia of research design (2nd ed.). SAGE Publications.

Lovric, M. (Ed.). (2011). International encyclopedia of statistical science. Springer. https://doi.org/10.1007/978-3-642-04898-2

Tabachnick, B. G., & Fidell, L. S. (2013). Using multivariate statistics (6th ed.). Pearson.

Wang, J., & Wang, X. (2012). Structural equation modeling: Applications using Mplus. John Wiley & Sons.