3.1.1 Between- and within-case components in ALT models

When analyzing longitudinal data, distinguishing within- and between-case components is common (e.g., Raudenbush & Bryk, Reference Raudenbush and Bryk2002). Some studies question the applicability of this distinction in ALT models (e.g., Andersen, Reference Andersen2022; Curran & Bauer, Reference Curran and Bauer2011; Usami, Reference Usami2021). In this subsection, we introduce how to separate between- and within-case components in the ALT model. To begin, we need to define our use of between- and within-case components, and how some definitions create problems. For instance, sometimes between-case components are described as stable features among cases (e.g., Schuurman, Reference Schuurman2023). However, the term “stable” suggests unchanging values of the components, while some differences between cases are differences in trends, and trends do not imply stability. Rather, they connote nonstationarity and change.

A common approach to separating between- and within-case components uses the case-specific over-time mean of a variable as the between-component, with the case-specific deviation from its mean as the within-component. For ${Y}_{it}$ , the decomposition would be the identity of

$$\begin{align*}{Y}_{it}={\bar{Y}_i}+\left({Y}_{it}-{\bar{Y}_i}\right),\end{align*}$$

where ${\bar{Y}_i}=\frac{1}{T}{\sum}_{t=1}^T{Y}_{it}$ is the case-specific mean (between-component), and $\left({Y}_{it}-{\bar{Y}_i}\right)$ is the deviation from that mean at time $t$ (within-component).

One issue with this expression is that it ignores the sampling error in ${\bar{Y}_i}$ . In this definition, the population mean ${\mu}_{Y_i}$ should be the between-component with ${\bar{Y}_i}$ as a sample estimator of it (Curran et al., Reference Curran, Lee, Howard, Lane, McCallumn, Harring and Hancock2012, pp. 241–244). To reflect this change, we rewrite the identity to

$$\begin{align*}{Y}_{it}={\mu}_{Yi}+\left({Y}_{it}-{\mu}_{Yi}\right),\end{align*}$$

where ${\mu}_{Yi}$ is the between-component and $\left({Y}_{it}-{\mu}_{Yi}\right)$ is the within-component. Indeed, this maps nicely onto an intercept-only growth curve model of

$$\begin{align*}{Y}_{it}={\alpha}_i+{\varepsilon}_{it},\end{align*}$$

where ${\alpha}_i$ is the random intercept of case i and ${\varepsilon}_{it}$ is the random error for case i at time t. By equating the RHS of both equations, we see that ${\alpha}_i={\mu}_{Yi}$ is a between-component and ${\varepsilon}_{it}=\left({Y}_{it}-{\mu}_{Yi}\right)$ is the within one.

Although this definition creates a clean separation of between- and within-case components, it assumes an intercept-only growth curve model. Suppose that an LGCM characterizes ${Y}_{it}$ so that

(8) $$\begin{align}{Y}_{it}={\alpha}_i+{\lambda}_t{\beta}_i+{\varepsilon}_{it.}\end{align}$$

Now ${\alpha}_i={\mu}_{Yi}$ holds under special conditions, such as balanced data with centering time on the middle time point (Curran & Bauer, Reference Curran and Bauer2011, p. 15). But, under more general conditions, this equivalency no longer holds nor equates to the between-component because we have omitted the ${\lambda}_t{\beta}_i$ term. However, we can still decompose the between- and within-components in the LGCM in Equation (8). Clearly, ${\alpha}_i$ and ${\beta}_i$ differ across cases and in this sense are variables that represent between cases’ components on ${Y}_{it}$ . The error ${\varepsilon}_{it}={Y}_{it}-\left({\alpha}_i+{\lambda}_t{\beta}_i\right)$ is the deviation of case i at time t from the growth curve and hence is a within-case component. Thus, the LGCM decomposes into between- and within-effects on ${Y}_{it}$ .

What happens when we move to the classic ALT model? The ALT model as given in Equation (1) is

$$\begin{align*}{Y}_{it}={\alpha}_i+{\lambda}_t{\beta}_i+{\rho}_{t,t-1}{Y}_{i,t-1}+{\varepsilon}_{it}.\end{align*}$$

The ${\alpha}_i$ and ${\beta}_i$ remain the same over time, but differ over cases and are the between-case components in ${Y}_{it}$ . The ${Y}_{i,t-1}$ is a time-varying covariate (TVC) that the ALT model adds to the LGCM. When TVCs are added to a model, it is common to suggest that if the TVC enters in raw form (neither centered nor detrended), its coefficient reflects a mixture of between- and within-effects, unless the between-effects are taken into account. However, a characteristic of the ALT model is that the TVC ${Y}_{i,t-1}$ is an endogenous observed variable that correlates with ${\alpha}_i$ and ${\beta}_i$ . This cross-level covariance structure ensures that the coefficient ${\rho}_{t,t-1}$ is an accurate estimate of the within-effect of the lagged variable on the current ${Y}_{it}$ .Footnote ⁴ Hence, the term $\left({\alpha}_i+{\lambda}_t{\beta}_i\right)$ accounts for or controls the between-component in ${Y}_{it}$ . This means that the autoregressive coefficient ${\rho}_{t,t-1}$ captures within-effects of the lagged variable on the current ${Y}_{it}$ , while controlling for the between-component in the same variable. We could more formally represent this as ${\rho}_{t,t-1}{Y}_{i,t-1}\mid \left({\alpha}_i+{\lambda}_t{\beta}_i\right)$ providing the within-case component where “|” signifies “given” and denotes that the between-components in ${Y}_{i,t-1}$ are controlled. The remaining within-effects component is ${\varepsilon}_{it}$ .

In brief, in the ALT model Equation (1), the between-case components are reflected in $\left({\alpha}_i+{\lambda}_t{\beta}_i\right)\mid {Y}_{i,t-1}$ , while the within-case components are captured by ${\rho}_{t,t-1}{Y}_{i,t-1}\mid \left({\alpha}_i+{\lambda}_t{\beta}_i\right)$ and ${\varepsilon}_{it}$ . The straightforward division into the between- and within-components runs counter to claims of the difficulties of this decomposition in ALT models. Like all such decompositions, this separation into between- and within-case components assumes that the model specification is correct. If there are omitted time-invariant or time-varying variables, the random slope is excluded, or if analysts use the wrong functional form, then this division of between- and within-case components is ambiguous, a point to which we return. Later, we will discuss the between- and within-effects when additional covariates are added to the ALT model.

3.2.1 Predetermined Wave 1 (PW1)

In the PW1 model, the first wave equation ( ${Y}_{i1}$ ) is

(9) $$\begin{align}{Y}_{i1}={\mu}_1+{\varepsilon}_{i1},\end{align}$$

where ${\mu}_1$ is the mean of the first repeated measure ${Y}_{i1}$ , and ${\varepsilon}_{i1}$ is the deviation from the mean of the i-th case in the first wave ${Y}_{i1}$ . The first wave is taken as given, meaning it is not explained within the model and correlates with the random intercept and random slope. This is why it is called the “Predetermined Wave 1” (PW1) version of the classic ALT model.

Figure 1 is a path diagram of the PW1 form of the ALT model for four waves of data. The random intercept ( ${\alpha}_i$ ) and random slope ( ${\beta}_i$ ) are latent variables (Bollen, Reference Bollen2002) in that we do not directly observe them, but we can trace their influence through their effects on the repeated measures. As is typical in structural equation models (SEMs), they are enclosed in ellipses or circles to signify their latent status. The repeated measures are shown in boxes, which represent observed variables. The single-headed arrows symbolize the direct effects of the variable at the base of the arrow on the variable at the head of the arrow. Thus, a series of single-headed arrows originates from the random intercept and random slope and points toward the repeated measures. The error variables ( ${\varepsilon}_{it}$ ) contain all other influences on the repeated measures that are not explicitly part of their equations. Although not drawn in Figure 1, their presence is indicated by the single-headed arrows pointing to each observed endogenous variable ${Y}_{it}$ . The first wave ${Y}_{i1}$ variable is treated as given and is correlated with the random intercept and random slope, as shown by the curved two-headed arrows connecting them.Footnote ⁵

Figure 1

Path diagram of Predetermined Wave 1 (PW1) model.

Figure 1. Long description

A structural equation model diagram. On the left side, four square boxes representing observed variables are stacked vertically: Y sub i 1 at the top, followed by Y sub i 2, Y sub i 3, and Y sub i 4 at the bottom. Vertical arrows point downward from each box to the one below it and show the autoregressive relationships between the observed variables. Small diagonal arrows point into the left side of each square box and indicate error terms. On the right side, two circular nodes represent latent variables: alpha sub i at the top and beta sub i at the bottom. Comment: A double-headed curved arrow connects alpha sub i and beta sub i, indicating that they are assumed to be correlated. Another double-headed curved arrow connects Y sub i 1 to beta sub i. Straight arrows originate from alpha sub i and point to Y sub i 2, Y sub i 3, and Y sub i 4, each labeled with a coefficient of 1. Straight arrows originate from beta sub i and point to Y sub i 2, Y sub i 3, and Y sub i 4, with coefficients labeled 1, 2, and 3 respectively.

Identification of the PW1 form of the ALT model. Model identification is necessary for the successful estimation of the classic ALT model. Here, we assume a linear growth model with ${\lambda}_t=\left(t-1\right),t=1,2,\dots, T$ . Identification refers to whether it is possible to find unique values of all parameters in the model. The parameters are constants and include the means and variances of the random intercept [ ${\mu}_{\alpha },V\left({\alpha}_i\right)$ ], random slope [ ${\mu}_{\beta },V\left({\beta}_i\right)$ ], and ${Y}_{i1}$ [ ${\mu}_1,V\left({\varepsilon}_{i1}\right)$ ], the covariances among ${\alpha}_i$ , ${\beta}_i$ , and ${Y}_{i1}$ [ $C\left({\alpha}_i,{\beta}_i\right),C\left({\alpha}_i,{Y}_{i1}\right),C\left({\beta}_i,{Y}_{i1}\right)$ ], the autoregressive coefficients ( ${\rho}_{t,t-1}$ ), and the error variances [ $V\left({\varepsilon}_{it}\right),t=1,\dots, T$ ]. These parameters are identified if it is possible to find unique solutions for each that are functions of one or more means, variances, or covariances of the observed repeated measures.Footnote ⁶

The number of time points of data in the PW1 model helps to determine whether it is identified. Bollen and Curran (Reference Bollen and Curran2004, pp. 362–363) prove that the PW1 form of the ALT model is identified with four waves of data if the autoregressive coefficient is the same over time ( ${\rho}_{t,t-1}$ = ρ and $\mid \rho \mid <1$ ) and a linear growth curve applies. With five or more waves of data, the autoregressive coefficients could differ over time, and whether they are equal is testable.

3.2.2 Endogenous Wave 1 (EW1)

The EW1 is a second form of the classic ALT model. Under this specification, the ALT equation for the first time period is given by

(10) $$\begin{align}{Y}_{i1}={\Lambda}_{1\alpha }{\alpha}_i+{\Lambda}_{1\beta }{\beta}_i+{\varepsilon}_{i1},\end{align}$$

where ${\Lambda}_{1\alpha }$ is a coefficient that gives the effect of ${\alpha}_i$ on ${Y}_{i1}$ , and ${\Lambda}_{1\beta }$ is a coefficient that indicates the effect of ${\beta}_i$ on ${Y}_{i1}$ . In general, we can interpret ${\Lambda}_{1\alpha }$ as the total effect (reduced form) coefficient of ${\alpha}_i'\mathrm{s}$ impact on ${Y}_{i1}$ and the earlier waves of ${Y}_i$ , and similarly, ${\Lambda}_{1\beta }$ as the total effect (reduced form) coefficient of ${\beta}_i'\mathrm{s}$ effects on ${Y}_{i1}$ and previous values.

The two new coefficients for ${\alpha}_i$ and ${\beta}_i$ in Wave 1 differ for the EW1 equation compared to the other waves in Equation (1). Another difference is that Equation (10) excludes the ${\rho}_{t,t-1}{Y}_{i,t-1}$ term that is present in Equation (1). A practical reason for this exclusion is that the lagged value of ${Y}_{i,t-1}$ is not available before the first wave of data.

Equation (10) for Wave 1 contrasts with Equation (9) for Wave 1 in PW1 in several ways. As their names suggest, ${Y}_{i1}$ is dependent (endogenous) on the random intercept ( ${\alpha}_i$ ) and random slope ( ${\beta}_i$ ) in the first wave, similar to Wave 2 and later waves of ${Y}_{it}$ in Equation (1). This explains the name “Endogenous Wave 1” for this form of the ALT model, in which the first wave of the repeated measure is explained within the model. In contrast, in the PW1 form of the ALT model, ${Y}_{i1}$ is treated as given and not explained by ${\alpha}_i$ and ${\beta}_i$ . Furthermore, in the PW1 form, ${Y}_{i1}$ , ${\alpha}_i$ , and ${\beta}_i$ correlate with one another, while in the EW1 model, only ${\alpha}_i$ and ${\beta}_i$ correlate as exogenous variables, and their association with ${Y}_{i1}$ results from the influence of ${\alpha}_i$ and ${\beta}_i$ on ${Y}_{i1}$ , not by correlating these three variables.

Figure 2 is the path diagram of the EW1 form of the ALT model. The symbols and notation in the path diagram were defined in Figure 1. In many ways, Figure 2 is like Figure 1 for the PW1 form. The primary difference is that the first wave of ${Y}_{i1}$ is treated as endogenous, like the other waves in the repeated measures. Labeling the paths of the random intercepts and random slopes to ${\Lambda}_{1\alpha }$ and ${\Lambda}_{1\beta }$ contrasts with the fixed constants for the paths to the other repeated measures.

Figure 2

Path diagram of Endogenous Wave 1 (EW1) model.

Figure 2. Long description

A structural equation model path diagram. On the right side are two latent variables in circles, alpha sub i at the top and beta sub i at the bottom, connected by a double-headed curved arrow indicating covariance. On the left side is a vertical column of four observed variables in squares, labeled from top to bottom as Y sub i 1, Y sub i 2, Y sub i 3, and Y sub i 4.

Directional arrows flow as follows:

* Vertical arrows point downward from Y sub i 1 to Y sub i 2, Y sub i 2 to Y sub i 3, and Y sub i 3 to Y sub i 4, indicating autoregressive relationships.

* Short diagonal arrows point into the left side of each Y square, representing error terms.

* From alpha sub i, four arrows point to the Y variables. The path to Y sub i 1 is labeled Lambda 1 1. The paths to Y sub i 2, Y sub i 3, and Y sub i 4 are each labeled with the value 1.

* From beta sub i, four arrows point to the Y variables. The path to Y sub i 1 is labeled Lambda 2 1. The path to Y sub i 2 is labeled 1, the path to Y sub i 3 is labeled 2, and the path to Y sub i 4 is labeled 3.

A special and commonly applied formulation of Equation (10) arises under the assumption that the autoregressive coefficients are time-invariant, ${\rho}_{t,t-1}=\rho$ , and in absolute value less than 1 ( $\left|\;\rho\;\right|<1$ ). Bollen and Curran (Reference Bollen and Curran2004, pp. 363–365) prove that this is equivalent to assuming that Equation (1) holds not only for the observational period of $t=2,3,\dots, T$ , but also extends to the past, such that the first wave ( ${Y}_{i1}$ ) results

(11) $$\begin{align}{Y}_{i1}={\left(1-\rho \right)}^{-1}{\alpha}_i-\rho {\left(1-\rho \right)}^{-2}{\beta}_i+{\varepsilon}_{i1},\end{align}$$

that is, ${\varLambda}_{1\alpha }={\left(1-\rho \right)}^{-1}$ and ${\varLambda}_{1\beta }=-\rho {\left(1-\rho \right)}^{-2}$ .

Another special case of the EW1 is the so-called History Independent (HI1) specification of the classic ALT process for the first wave (Ou et al., Reference Ou, Chow, Ji and Molenaar2017). It consists of setting ${\varLambda}_{1\alpha }$ to 1 and ${\varLambda}_{1\beta }$ to 0, such that

(12) $$\begin{align}{Y}_{i1}={\alpha}_i+{\varepsilon}_{i1},\end{align}$$

where the intercept, ${\alpha}_i$ , has the same conceptual meaning as the intercept term in a latent growth model. No explicit assumptions of stationarity and time invariance of the ALT process at the subsequent waves are required.

Following Ou et al. (Reference Ou, Chow, Ji and Molenaar2017), it can be demonstrated that the EW1 ALT model, along with its special cases, is nested within the predetermined ALT model. The constraints required to specify such nested relationships are detailed in the Supplementary Material.

Identification of the EW1 model. As we stated earlier, identification concerns the possibility of finding unique values for each model parameter. We addressed that question for the PW1 form in a previous section. Here, we examine the EW1 form while maintaining the assumption of a linear growth model [ ${\lambda}_t=\left(t-1\right),t=1,2,\dots, T$ ]. The unknown parameters are the means and variances of the random intercept [ ${\mu}_{\alpha },V\left({\alpha}_i\right)$ ], random slope [ ${\mu}_{\beta },V\left({\beta}_i\right)$ ], and their covariance [ $C\left({\alpha}_i,{\beta}_i\right)$ ]. Additional parameters include the factor loadings of ${Y}_{i1}$ [ ${\varLambda}_{1\alpha }$ and ${\varLambda}_{1\beta }$ ], the autoregressive coefficients ( ${\rho}_{t,t-1}$ ), and the error variances [ $V\left({\varepsilon}_{it}\right),t=1,\dots, T$ ]. In the case of four waves of data, assuming a linear growth applies, the EW1 ALT model has 14 free parameters, which exactly equals the 14 observed means, variances, and covariances. The model is just identified in that all model parameters have a complex closed-form expression based on the observed moments (see Supplementary Material). If the autoregressive coefficients are constrained to be equal over time ( ${\rho}_{t,t-1}=\rho$ ), the model becomes overidentified, allowing us to formally test whether the data support this assumption.

3.4.1 Time-varying covariates

TVCs are dynamic predictors that fluctuate over time within individuals. In our earlier empirical illustration, an example is the adolescent’s perceived exposure to smoking in movies. When analyzing its influence on adolescent smoking intensity ( ${Y}_{it}$ ), researchers typically incorporate it in Equation (1), allowing it to have a contemporaneous influence on the outcome. Denoting ${X}_{it}$ as a TVC for the i-th case, Equation (1) results

(13) $$\begin{align}{Y}_{it}={\alpha}_i+{\lambda}_t{\beta}_i+{\rho}_{t,t-1}{Y}_{i,t-1}+{\gamma}_{yt}{\mathrm{X}}_{it}+{\varepsilon}_{it},\end{align}$$

for $t=2,3,\dots, T,$ where ${\gamma}_{yt}$ is the coefficient representing the expected effect of ${X}_{it}$ on ${Y}_{it}$ while controlling for all other variables on the RHS. Note that ${Y}_{i,t-1}$ is a time-varying variable, so ${X}_{it}$ is the second time-varying variable in the equation. In the previous ALT model, the coefficient ${\rho}_{t,t-1}$ was a within-case effect because ( ${\alpha}_i+{\lambda}_t{\beta}_i$ ) controlled the between effects of ${Y}_{i,t-1}$ . The inclusion of an additional TVC in the model changes the interpretation of the model parameters. Since ${X}_{it}$ directly influences the outcome variable, the growth curve parameters (e.g., intercept and rate of change) should be interpreted as net of the effect of this TVC. Additionally, the time-specific effects of ${X}_{it}$ on ${Y}_{it}$ also need to be interpreted as effects net of the influence of the growth process.

Can time-varying explanatory variables enter the random intercept and random slope equations? Generally, the answer is no. This can be best illustrated with an example. Suppose the random intercept equation includes the TVC at Wave 3 ( ${X}_{i3}$ ) as

(14) $$\begin{align}{\alpha}_i={\mu}_{\alpha }+{\gamma}_{\alpha 3}{\mathrm{X}}_{i3}+{\varepsilon}_{\alpha i.}\end{align}$$

In Equation (1), ${\alpha}_i$ directly affects the repeated measures, ${Y}_{i2}$ , ${Y}_{i3}$ , ${Y}_{i4}$ , …. Any variable that directly affects ${\alpha}_i$ would indirectly affect these same repeated measures. There is no temporal problem for the Ys that come at Wave 3 or later, but this is an issue for ${Y}_{i2}$ and earlier values in that this implies that a future value of the covariate at Wave 3 ( ${X}_{i3}$ ) indirectly influences a Wave 2 variable ( ${Y}_{i2}$ ) from the past.

3.4.2 Between- and within-effects in time-varying covariates

Several authors have emphasized the importance of disentangling two distinct sources of variability when incorporating TVCs ${X}_{it}$ into longitudinal models (Curran & Bauer, Reference Curran and Bauer2011; Curran et al., Reference Curran, Lee, Howard, Lane, McCallumn, Harring and Hancock2012; Hoffman & Stawski, Reference Hoffman and Stawski2009; Hori & Miyazaki, Reference Hori and Miyazaki2025; Muniz-Terrera et al., Reference Muniz-Terrera, Robitaille, Kelly, Johansson, Hofer and Piccinin2017). In the earlier subsection, we explained how in the ALT model ${\rho}_{t,t-1}{Y}_{i,t-1}\mid \left({\alpha}_i+{\lambda}_t{\beta}_i\right)$ and ${\varepsilon}_{it}$ capture the within-components while $\left({\alpha}_i+{\lambda}_t{\beta}_i\right)\mid {Y}_{i,t-1}$ is the between-component.

How does the decomposition into between- and within-components differ if we introduce the TVC variable X_it, into the ALT model? Consider,

(15) $$\begin{align}{Y}_{it}={\alpha}_{Yi}+{\lambda}_{Yt}{\beta}_{Yi}+{\rho}_{t,t-1}{Y}_{i,t-1}+{\gamma}_{xt}{\mathrm{X}}_{it}+{\gamma}_{\alpha X}{\alpha}_{Xi}+{\gamma}_{\beta X}{\beta}_{Xi}+{\varepsilon}_{it},\end{align}$$

which is the previous ALT model with the addition of $\ {X}_{it}$ , ${\alpha}_{Xi}$ , ${\beta}_{Xi},$ and their corresponding coefficients of ${\gamma}_{xt}$ , ${\gamma}_{\alpha X}$ , and ${\gamma}_{\beta X}$ . We are assuming that the TVC ${X}_{it}$ follows a linear growth curve model with ${\alpha}_{Xi}$ its random intercept and ${\beta}_{Xi}$ its random slope. We could simplify this if only the random intercept were needed, or complicate it if more complicated functional forms described ${X}_{it}`\mathrm{s}$ trajectory. But the linear trajectory is sufficient to make our points. The ${\alpha}_{Xi}$ , ${\beta}_{Xi}$ , ${\alpha}_{Yi}$ , and ${\beta}_{Yi}$ are correlated exogenous variables. If analysts specify the predetermined variable ALT form, then the first wave of the repeated measures, ${X}_{i1}$ and ${Y}_{i1}$ , correlate with these growth parameters. The ${\alpha}_{Xi}$ and ${\beta}_{Xi}$ are constant over time but differ over cases. They are the between-components of ${X}_{it}$ , and their coefficients give the between-effects of ${X}_{it}$ on ${Y}_{it}$ as captured by ${\alpha}_{Xi}$ and ${\beta}_{Xi}$ while holding the other variables in the equation constant. Because we are controlling for the between-components (i.e., ${\alpha}_{Xi},{\beta}_{Xi}$ ) of ${X}_{it}$ , the coefficient of ${X}_{it}$ is the within-effects of ${\mathrm{X}}_{it}\mid \left({\alpha}_{Xi}+{\lambda}_t{\beta}_{Xi}\right)$ . Note also that Equation (15) contains the original (uncentered) ${X}_{it}$ .

Equation (15) assumes that the between-components of ${X}_{it}$ have direct effects on ${Y}_{it}$ Footnote ⁷. What happens when these direct effects are zero ( ${\gamma}_{\alpha X}={\gamma}_{\beta X}=0$ )? This implies that the ${\gamma}_{\alpha X}{\alpha}_{Xi}$ and ${\gamma}_{\beta X}{\beta}_{Xi}$ are unnecessary in Equation (15), but ${X}_{it}$ maintains a direct effect and remains in the equation. As before, ${X}_{it}$ correlates with the other RHS variables except the error. Under this scenario, the coefficient ${\gamma}_{xt}$ remains the same with or without ${\alpha}_{Xi}$ and ${\beta}_{Xi}$ in the equation. This implies that ${\gamma}_{xt}$ , the coefficient of ${X}_{it}$ , continues to give the within-effects even using the raw rather than the deviation score of ${X}_{it}$ .

3.4.3 Time-invariant covariates

When analyzing individuals, time-invariant variables include characteristics like race, ethnicity, gender, or place of birth. For each case, the time-invariant variables assume the same value over the observation period covered by the longitudinal study. At the same time, the time-invariant variables can take different values across cases.

In the ALT model, the random intercept ${\alpha}_i$ and random slope ${\beta}_i$ become the dependent variables with respect to the time-invariant covariates, leading to

(16) $$\begin{align}{\alpha}_i={\mu}_{\alpha }+{\gamma}_{\alpha Z}{Z}_{i1}+{\varepsilon}_{\alpha i},\end{align}$$

(17) $$\begin{align}{\beta}_i={\mu}_{\beta }+{\gamma}_{\beta Z}{Z}_{i1}+{\varepsilon}_{\beta i},\end{align}$$

where the μs are intercept constants, and the γs are regression coefficients. As in Equations (2) and (3), the errors ${\varepsilon}_{\alpha i}$ and ${\varepsilon}_{\beta i}$ have means of zero and can correlate with each other, but do not correlate with the time-invariant ${Z}_{i1}$ variable. The covariate ${Z}_{i1}$ can impact the random intercept and random slope, and these are between-effects in that ${Z}_{i1}$ takes the same value for each case over time but typically differs over cases.

When using the PW1 form of the ALT model, not only the random intercepts ( ${\alpha}_i$ ) and random slopes ( ${\beta}_i$ ), but also ${Y}_{i1}$ are dependent variables concerning the time-invariant covariates. In other words, together with Equations (16) and (17), we add

(18) $$\begin{align}{Y}_{i1}={\mu}_1+{\gamma}_{Y1}{Z}_{i1}+{\varepsilon}_{i1},\end{align}$$

where ${\gamma}_{Y1}$ is a regression coefficient, and the error ${\varepsilon}_{i1}$ has zero mean and is uncorrelated with the time-invariant ${Z}_{i1}$ variable. However, ${\varepsilon}_{i1}$ , ${\varepsilon}_{\alpha i}$ , and ${\varepsilon}_{\beta i}$ correlate with each other. In other words, the errors of these three equations are associated. The label “Predetermined Wave 1” (PW1) is less accurate once we introduce covariates to predict ${Y}_{i1}$ , and hence ${Y}_{i1}$ becomes a dependent variable. However, to avoid confusing matters, we continue to use the term “predetermined” and the acronym PW1 to refer to this form of the ALT model.

Bianconcini and Bollen (Reference Bianconcini and Bollen2018) highlight a difference in how time-invariant covariates enter the model within the growth curve model literature compared to fixed and random effects models in economics and other social sciences. Equations (16) and (17) are typical in LGCM, where time-invariant covariates directly influence the growth parameters. In contrast, in many econometric—specifically random effects—models (e.g., Bollen & Brand, Reference Bollen and Brand2010), time-invariant variables enter the model by directly affecting the repeated measures. For instance, in Wave 2 and beyond, the equation for the repeated measure would be

(19) $$\begin{align}{Y}_{it}={\alpha}_i+\lambda_t\;\beta_i+{\rho}_{t,t-1}{Y}_{i,t-1}+{\gamma}_{zt}{Z}_{i1}+{\varepsilon}_{it},\end{align}$$

for $t=2,3,\dots, T,$ where the time-invariant variable, ${Z}_{i1}$ , has a direct influence of ${\gamma}_{zt}$ on the repeated measures ${Y}_{it}$ , rather than having a direct effect on the growth parameters ${\alpha}_i$ and ${\beta}_i$ . Often, the effect of the time-invariant variable is constrained to be equal over time ( ${\gamma}_{zt}={\gamma}_z$ ). Furthermore, the definition of between- and within-case components requires modifications in Equation (19). Now ( ${\alpha}_i+{\lambda}_t{\beta}_{i1}+{\gamma}_{zt}{Z}_{i1}$ ) contains between-components and ( ${\rho}_{t,t-1}{Y}_{i,t-1}+{\varepsilon}_{it}$ ) contains within-case components.

The decision of whether to include the time-invariant variable in Equation (19) or in Equations (16) and (17) is not straightforward. It raises the question of whether the analyst believes all effects of the time-invariant variables are indirect through the growth parameters of ${\alpha}_i$ and ${\beta}_i$ , or if these variables directly influence the repeated measures even after accounting for the growth curve parameters. Thus, analysts need to decide where the time-invariant covariates should enter the model.

4.1.1 Predetermined Wave 1 (PW1) in the LV-ALT model

For the PW1 form of the LV-ALT model, the Wave 1 equation for ${L}_{i1}$ is

(26) $$\begin{align}{L}_{i1}={\mu}_{L1}+{\varepsilon}_{iL1},\end{align}$$

where ${\mu}_{L1}$ is the mean of the first repeated latent variable ${L}_{i1}$ and ${\varepsilon}_{L1i}$ is the deviation from the mean of the i-th case in the first wave. As is true for the PW1 form of the classic ALT model, the first wave of the PW1 form of the LV-ALT is taken as given or predetermined.

Figure 4 shows the path diagram of the PW1 form of the LV-ALT model. As before, the ellipses or circles represent latent variables such as the repeated latent variables, ${L}_{it}$ , the random intercept, ${\alpha}_i$ , and the random slope, ${\beta}_i$ . To simplify the model, there is a single indicator ( ${Y}_{it}$ ) for the latent variable ( ${L}_{it}$ ) at each of the four waves of data, with factor loadings set to 1 and intercepts set to 0. The coefficients on the paths from the random intercept ( ${\alpha}_i$ ) to the repeated latent variables ( ${L}_{i2},\dots, {L}_{i4}$ ) are also fixed at one. In contrast, the paths from the random slope ( ${\beta}_i$ ) to the repeated latent variables ( ${L}_{i2},\dots, {L}_{i4}$ ) are fixed at ( $t-1$ ), based on the assumption of linear change over time. Curved two-headed arrows connect the first wave of the latent variable ${L}_{i1}$ , the random intercept ${\alpha}_i$ , and the random slope ${\beta}_i$ , indicating the covariances among these three variables.

Figure 4

Path diagram of PW1 in LV-ALT model.

Figure 4. Long description

The diagram consists of three vertical columns of nodes connected by directional and bidirectional arrows.

1. Left Column: Four square nodes representing observed variables are stacked vertically, labeled Y sub i 1, Y sub i 2, Y sub i 3, and Y sub i 4 from top to bottom. Each square has a short incoming arrow from the left representing error terms.

2. Middle Column: Four circular nodes representing latent variables are stacked vertically, labeled L sub i 1, L sub i 2, L sub i 3, and L sub i 4. Each L node points to its corresponding Y node to the left. Additionally, a vertical sequence of arrows points from L sub i 1 down to L sub i 2, then to L sub i 3, and finally to L sub i 4, representing autoregressive paths.

3. Right Column: Two circular nodes representing growth factors. The top node is labeled alpha sub i (intercept) and the bottom node is labeled beta sub i (slope). A curved double-headed arrow connects alpha sub i and beta sub i, indicating covariance.

4. Structural Paths:

- Alpha sub i has directional arrows pointing to L sub i 2, L sub i 3, and L sub i 4, each labeled with a fixed value of 1.

- Beta sub i has directional arrows pointing to L sub i 2, L sub i 3, and L sub i 4, labeled with values 1, 2, and 3 respectively, representing time scores.

- A curved double-headed arrow also connects L sub i 1 to both alpha sub i and beta sub i, indicating covariances between the initial latent state and the growth factors.

Identification of PW1 LV-ALT model. Following Bianconcini and Bollen (Reference Bianconcini and Bollen2018), we can apply the two-step identification rule (Bollen, Reference Bollen1989) to assess model identification. They demonstrate that if we assume the single indicator ( ${Y}_{it}$ ) has its intercepts set to zero, factor loadings set to one, and equal measurement error variances [ $V({\varepsilon}_{it})=V({\varepsilon}_i)$ ], then the measurement model is identified in the first step. In the second step, they assume a linear growth curve ( ${\varLambda}_{2t}=t-1$ ), and equal variances of the errors of the latent variables [V( ${\varepsilon}_{iLt}$ ) = V( ${\varepsilon}_{iL}$ ) for $t=2,3,\dots, T$ ], and this, along with the first step identification, shows that the PW1 form of the LV-ALT model is identified with four waves with equal autoregressive coefficients and five or more waves of data when they are not equal.

The identification process varies depending on whether multiple indicators are included in the model or if different functional forms of the LGCM are used. Nonetheless, the two-step identification rule can be helpful in those cases as well.

4.1.2 Endogenous Wave 1 (EW1) LV-ALT model

The equation for the first repeated latent variable ( ${L}_{i1}$ ) in the EW1 form of the LV-ALT Model is given by

(27) $$\begin{align}{L}_{i1}={\mu}_{L1}+{\Lambda}_{1\alpha }{\alpha}_i+{\Lambda}_{1\beta }{\beta}_i+{\varepsilon}_{iL1},\end{align}$$

where ${\varLambda}_{1\alpha }$ and ${\varLambda}_{1\beta }$ are the coefficients for the random intercept and slope, respectively, for the first wave of data. Equation (27) in the EW1 form of the LV-ALT model is analogous to Equation (10) in the EW1 form of the classic ALT model. One difference is that the latent variable ${L}_{i1}$ is the dependent variable in the LV-ALT rather than the observed variable ${Y}_{i1}$ , as it was for the classic ALT model.

Figure 5 shows the path diagram of the EW1 form of the LV-ALT Model. As before, we assume a single indicator for the repeated latent variable. The diagram is like the one for the PW1 form of the LV-ALT, with a few exceptions. The most noticeable difference is that the first wave of the latent variable, ${L}_{i1}$ , depends on the random intercept ( ${\alpha}_i$ ) and random slope ( ${\beta}_i$ ), rather than these three variables covarying. The rest of the path diagrams are quite similar.

Figure 5

Path diagram of EW1 in LV-ALT model.

Figure 5. Long description

The diagram consists of three vertical columns of nodes connected by arrows.

1. Left Column: Four square nodes representing observed variables labeled Y sub i 1, Y sub i 2, Y sub i 3, and Y sub i 4 from top to bottom. Each square has a small incoming arrow from the left indicating an error term.

2. Middle Column: Four circular nodes representing latent states labeled L sub i 1, L sub i 2, L sub i 3, and L sub i 4 from top to bottom. Horizontal arrows point from each L node to its corresponding Y node in the left column. Vertical arrows point downward from L sub i 1 to L sub i 2, L sub i 2 to L sub i 3, and L sub i 3 to L sub i 4.

3. Right Column: Two circular nodes representing growth factors. The top node is alpha sub i and the bottom node is beta sub i. A curved double-headed arrow connects alpha sub i and beta sub i, indicating covariance.

4. Path Relationships:

- From alpha sub i: Arrows point to all four L nodes. The path to L sub i 1 is labeled Lambda sub 1 alpha. The paths to L sub i 2, L sub i 3, and L sub i 4 are all labeled with the value 1.

- From beta sub i: Arrows point to all four L nodes. The path to L sub i 1 is labeled Lambda sub 1 beta. The paths to L sub i 2, L sub i 3, and L sub i 4 are labeled with values 1, 2, and 3 respectively.

As discussed for the classic ALT model, a special case of the EW1 LV-ALT model arises by assuming the autoregressive coefficients are time-invariant, ${\rho}_{t,t-1}=\rho$ , and in absolute value less than one (| ρ| <1), such that

(28) $$\begin{align}{L}_{i1}={\left(1-\rho \right)}^{-1}{\alpha}_i-\rho {\left(1-\rho \right)}^{-2}{\beta}_i+{\varepsilon}_{iL1},\end{align}$$

that is, ${\varLambda}_{1\alpha }={\left(1-\rho \right)}^{-1}$ and ${\varLambda}_{1\beta }=-\rho {\left(1-\rho \right)}^{-2}$ .

Also, for the LV-ALT model, the so-called History Independent (HI1) specification for the first wave is possible, and given by

(29) $$\begin{align}{L}_{i1}={\alpha}_i+{\varepsilon}_{iL1},\end{align}$$

where ${\varLambda}_{1\alpha }$ is set to 1 and ${\varLambda}_{1\beta }$ to 0. No explicit assumptions of stationarity and time invariance of the LV-ALT process at the subsequent waves are required.

Identification of EW1 in LV-ALT model. The identification conditions we apply to the measurement model for the EW1 form of the LV-ALT are the same as those for the PW1 form. Refer to the earlier section on identification of the PW1 LV-ALT. The identification of the remaining parts of the model in Equations (25) and (27) is similar to that described for the EW1 form of the classic ALT. If we assume a linear growth curve ( ${\varLambda}_{2t}=t-1$ for $t>1$ ) and that the autoregressive coefficients are consistent over time ( ${\rho}_{t,t-1}=\rho$ ) and that the absolute value of the autoregression is less than 1 ( $\mid \rho \mid <1$ ), then the model is identified with three or more waves of data.

4.4.1 Cross-lagged panel models (CLPM)

The autoregressive model is one of the simplest longitudinal models, where the current value of the repeated measure depends on its previous value and an error term. That is,

(35) $$\begin{align}{Y}_{it}={\mu}_t+{\rho}_{t,t-1}{Y}_{i,t-1}+{\varepsilon}_{it},\end{align}$$

for $t=2,3,\dots, T,$ where ${\mu}_t$ is a constant intercept at time t, and the other terms have already been defined. At the first occasion, ${Y}_{i1}={\mu}_1+{\varepsilon}_{i1}$ .

Recall that the LV-ALT, as defined in Equation (25), is

(36) $$\begin{align}{L}_{it}={\mu}_{Lt}+{\alpha}_i+{\varLambda}_{\beta t}{\beta}_i+{\rho}_{t,t-1}{L}_{i,t-1}+{\varepsilon}_{it},\end{align}$$

for $t=2,3,\dots, T$ . To derive Equation (35) from Equation (36), requires constraints. First, Equation (35) assumes no measurement error, meaning that ${Y}_{it}$ equals ${L}_{it}$ . This leads to the classic ALT model with the addition of a constant intercept at time t ( ${\mu}_{Lt}$ ):

(37) $$\begin{align}{Y}_{it}={\mu}_{Lt}+{\alpha}_i+{\varLambda}_{\beta t}{\beta}_i+{\rho}_{t,t-1}{Y}_{i,t-1}+{\varepsilon}_{it},\end{align}$$

for $t=2,3,\dots, T$ . Second, Equation (35) assumes that there are no growth curve parameters, meaning the means and variances of ${\alpha}_i$ and ${\beta}_i$ are zero. If this holds, then these parameters can be omitted, leading to

(38) $$\begin{align}{Y}_{it}={\mu}_{Lt}+{\rho}_{t,t-1}{Y}_{i,t-1}+{\varepsilon}_{it},\end{align}$$

for $t=2,3,\dots, T$ , which is identical to Equation (35) once we realize that ${\mu}_{Lt}={\mu}_t$ . Additionally, for the first wave of data, ${Y}_{i1}$ is treated as predetermined.

Suppose we have two repeated measures and hypothesize cross-lagged effects. Two equations to represent this model are

(39) $$\begin{align}{Y}_{it}={\mu}_{yt}+{\rho}_{yt,y\left(t-1\right)}{Y}_{i,t-1}+{\rho}_{yt,x\left(t-1\right)}{\mathrm{X}}_{i,t-1}+{\varepsilon}_{yit},\end{align}$$

(40) $$\begin{align}{X}_{it}={\mu}_{xt}+{\rho}_{xt,y\left(t-1\right)}{Y}_{i,t-1}+{\rho}_{xt,x\left(t-1\right)}{\mathrm{X}}_{i,t-1}+{\varepsilon}_{xit},\end{align}$$

for $t=2,3,\dots, T$ . On the first occasion,

$$\begin{align*}{Y}_{i1}={\mu}_{Y1}+{\varepsilon}_{yi1},\\ {}{X}_{i1}={\mu}_{x1}+{\varepsilon}_{xi1},\end{align*}$$

with the errors ${\varepsilon}_{yit}$ and ${\varepsilon}_{xit}$ correlating at each wave of the data, $t=1,\dots, T$ .

The LV-ALT model in the PW1 form for the Y repeated measures is

(41) $$\begin{align}{L}_{yi t}={\mu}_{Lyt}+{\alpha}_{yi}+{\varLambda}_{y\beta t}{\beta}_{yi}+{\rho}_{yt,y\left(t-1\right)}{L}_{yi,t-1}+{\rho}_{yt,x\left(t-1\right)}{L}_{xi,t-1}+{\varepsilon}_{Lit},\end{align}$$

for $t=2,3,\dots, T,$ with ${L}_{yi1}={\mu}_{Ly1}+{\varepsilon}_{Li1}$ . To align this more closely with Equation (38), we assume each latent variable is measured with a single indicator with no measurement error ( ${Y}_{it}={L}_{yit}$ , ${X}_{it}={L}_{xit}$ for all t), resulting in the classic PW1 ALT model for ${Y}_{it}$

(42) $$\begin{align}{Y}_{it}={\mu}_{Lyt}+{\alpha}_{yi}+{\varLambda}_{y\beta t}{\beta}_{yi}+{\rho}_{yt,y\left(t-1\right)}{Y}_{i,t-1}+{\rho}_{yt,x\left(t-1\right)}{X}_{i,t-1}+{\varepsilon}_{yi t},\end{align}$$

for $t=2,3,\dots, T,$ with ${Y}_{i1}={\mu}_{Ly1}+{\varepsilon}_{yi1}$ .

The conversion assumes that the growth parameters are zero ( ${\alpha}_{yi}=0,{\beta}_{yi}=0$ ):

(43) $$\begin{align}{Y}_{it}={\mu}_{Lyt}+{\rho}_{yt,y\left(t-1\right)}{Y}_{i,t-1}+{\rho}_{yt,x\left(t-1\right)}{X}_{i,t-1}+{\varepsilon}_{yit},\end{align}$$

for $t=2,3,\dots, T,$ and setting ${\mu}_{Lyt}$ to ${\mu}_{yt}$ completes it. Making analogous constraints on the X series results in both Equations (39) and (40), showing that these equations are constrained versions of the LV-ALT.

Beyond demonstrating the restrictions needed to convert the LV-ALT model into the CLPM, these results also indicate potential modifications that could enhance the fit of the CLPM to our data. For instance, if fitting the CLPM Equations (39) and (40) produces a poor fit, we could improve the model by adding only the random intercept ${\alpha}_{yi}$ or both the random intercept and slope ( ${\alpha}_{yi}+{\varLambda}_{y2t}{\beta}_{yi}$ ) to see if this results in a better fit.

4.4.2 Longitudinal models with autoregressive or cross-lagged errors

Several longitudinal models specify autoregressive or cross-lagged relationships among the errors of repeated measures rather than directly between the observed variables. Several authors, from Hamaker (Reference Hamaker2005) to Andersen (Reference Andersen2022), have demonstrated that most of these models with direct relations between errors are statistically equivalent to restricted versions of ALT models.

By being statistically equivalent, we mean that a constrained ALT model generates the same model-implied moment matrices—and consequently the same fit statistics, indexes, and degrees of freedom—as other longitudinal models that permit complex relationships among the errors of repeated measures (Bianconcini & Bollen, Reference Bianconcini and Bollen2025; Lee & Hershberger, Reference Lee and Hershberger1990). However, statistical equivalence does not imply substantive equivalence. Researchers still need to justify why one model is more appropriate than another based on theory or context.

Because restricted ALT models can be statistically equivalent to models with more complex error structures, we can assess whether a simpler ALT model fits the data as well as a more flexible one. This allows us to indirectly evaluate certain autoregressive and cross-lagged error models. We explain how to perform these tests in the following sections.

4.4.2.1 Latent growth curve model with autoregressive disturbances.

LGCMs with autoregressive disturbances have a long history (e.g., Browne & du Toit, Reference Browne, du Toit, Collins and Horn1991; Chi & Reinsel, Reference Chi and Reinsel1989; Diggle et al., Reference Diggle, Liang and Zeger1994). These models have features such as random intercepts and slopes, like an LGCM, but they also include a separate equation for the disturbance that accounts for the autoregressive relationship. Hamaker (Reference Hamaker2005) shows that the classic linear ALT in its EW1 form—with ${\rho}_{t,t-1}=\rho$ and $\mid \rho \mid <1$ (Bollen & Curran, Reference Bollen and Curran2004), for $t=2,3,\dots, T$ ,

(44) $$\begin{align}{Y}_{it}&={\alpha}_i+t{\beta}_i+\rho {Y}_{i,t-1}+{\varepsilon}_{it}, \end{align}$$

(45) $$\begin{align}{Y}_{i1}&={\left(1-\rho \right)}^{-1}{\alpha}_i-\rho {\left(1-\rho \right)}^{-2}{\beta}_i+{\varepsilon}_{i1},\end{align}$$

is equivalent to an LGCM with autoregressive disturbances written as, for $t=2,3,\dots, T$ ,

(46) $$\begin{align}{Y}_{it}&={\eta}_{1i}+t\;{\eta}_{2i}+{z}_{it},\end{align}$$

(47) $$\begin{align}{z}_{it}&=\rho {z}_{i,t-1}+{\delta}_{it},\end{align}$$

where ${\eta}_{1i}$ is a random intercept, and ${\eta}_{2i}$ is a random slope for case i, ${z}_{it}$ is a random error with a mean of zero and an autoregressive relation to ${z}_{i,t-1}$ , whereas the errors ${\delta}_{it}$ , $t=2,\dots, T$ , have zero mean, constant variance, and are uncorrelated over time. The initial value ${Y}_{i1}$ is given by

(48) $$\begin{align}{Y}_{i1}={\eta}_{1i}+{z}_{i1}.\end{align}$$

Hamaker (Reference Hamaker2005) shows the relationships between the variables and parameters in the classic ALT model and those in the latent growth curve model with autoregressive errors (LGCM-AEs). She established that the random intercept from the latter model is

$$\begin{align*}{\eta}_{1i}={\alpha}_i{\left(1\hbox{-} \rho \right)}^{-1}\hbox{-} {\beta}_i\rho {\left(1\hbox{-} \rho \right)}^{-2},\end{align*}$$

the random slope equals

$$\begin{align*}{\eta}_{2i}={\beta}_i{\left(1- \rho \right)}^{-1},\end{align*}$$

and the errors ${z}_{it}=\sum \limits_{j=0}^{\infty }{\rho}^j\;{\varepsilon}_{i,t-j}$ .

This result implies that the LGCM-AEs is a restrictive version of the classic EW1 ALT model. We cannot empirically distinguish between a classic ALT model with equality restrictions on the autoregressive coefficients of the lagged repeated measure and the LGCM with the same equality restrictions on the autoregressive coefficients of the lagged disturbances. The two models will have the same fit, as indicated by the usual LR chi-square test and degrees of freedom from SEM software, when applied to the same data. The classic ALT model does not require these restrictions, but if they are valid, then the LGCM with equal autoregressive coefficients of the errors is statistically equivalent to it.

This statistical equivalency between the LGCM-AEs and the classic ALT—where both models have ${\rho}_{t,t-1}=\rho$ for all t and $\left|\rho \right|<1$ —provides an opportunity to compare the fit of the classic ALT model with these restrictions on ρ to one without them because they are nested and enable an LR chi-square test. Furthermore, due to the equivalence of the classic ALT and the LGCM with autoregressive errors under the same restrictions on ρ, the LR chi-square test allows us to compare the fit of the ALT model without restrictions to the two models with restrictions. Therefore, we can empirically compare the unrestricted classic ALT with the LGCM with restricted autoregressive errors.

4.4.2.2 Latent curve model with structured residuals (LCM-SR).

The LCM-SR (Curran et al., Reference Curran, Howard, Bainter, Lane and McGinley2014) is the same as the LGCM with autoregressive disturbances when analyzing a single repeated measure, so the results from the previous subsection apply to the LCM-SR model with one repeated measure. However, the LCM-SR generally applies to two or more repeated measures, and the equations are

(49) $$\begin{align} {Y}_{it}&={\eta}_{Y1i}+{\lambda}_t{\eta}_{Y2i}+{z}_{Yit}, \end{align}$$

(50) $$\begin{align} {X}_{it}&={\eta}_{X1i}+{\lambda}_t{\eta}_{X2i}+{z}_{Xit},\end{align}$$

(51) $$\begin{align}{z}_{Yi t}&={\rho}_{YY}{z}_{Yi,t-1}+{\rho}_{YX}{z}_{Xi,t-1}+{\delta}_{Yi t},\end{align}$$

(52) $$\begin{align}{z}_{Xi t}&={\rho}_{XY}{z}_{Yi,t-1}+{\rho}_{XX}{z}_{Xi,t-1}+{\delta}_{Xi t}.\end{align}$$

For the Y and X repeated measures, the random intercepts are ${\eta}_{Y1i}$ and ${\eta}_{X1i}$ , the random slopes are ${\eta}_{Y2i}$ and ${\eta}_{X2i}$ , and the equation errors are ${z}_{Yit}$ and ${z}_{Xit}$ . These errors have autoregressive and cross-lagged relationships, where the subscript of ρ indicates the variable receiving and sending effects. That is, ${\rho}_{XY}$ is the coefficient for the influence of Y on X.

As for the LCM with autoregressive disturbances, the LCM-SR has a restricted ALT counterpart—that is, the EW1 ALT model for two repeated measures with constant autoregressive and cross-lagged coefficients—which shares the same statistical fit and degrees of freedom (Andersen, Reference Andersen2022). The detailed derivation of the relationship between these two models is provided in Appendix A. It follows that an LR chi-square test can compare the unrestricted ALT with its restricted ALT counterpart and the LCM-SR as a fit diagnostic. If these restrictions and stationarity assumptions are not valid for the LCM-SR, then parameter estimates are likely to be biased.

4.4.2.3 Random intercept–cross-lagged panel model (RI-CLPM).

The RI-CLPM (Hamaker et al., Reference Hamaker, Kuiper and Grasman2015) is another longitudinal model that handles errors more complexly. This applies to one or more repeated measures, but here we will focus on the most applied case of two series. A key difference between the RI-CLPM and the LCM-SR is that the RI-CLPM includes only random intercepts, not random slopes. We define the RI-CLPM for two repeated measures as

(53) $$\begin{align} {Y}_{it}&={\mu}_{Yt}+{\eta}_{Y1i}+{z}_{Yit}, \end{align}$$

(54) $$\begin{align} {X}_{it}&={\mu}_{Xt}+{\eta}_{X1i}+{z}_{Xit},\end{align}$$

(55) $$\begin{align}{z}_{Yi t}&={\rho}_{YY}{z}_{Yi,t-1}+{\rho}_{YX}{z}_{Xi,t-1}+{\delta}_{Yi t},\end{align}$$

(56) $$\begin{align}{z}_{Xi t}&={\rho}_{XY}{z}_{Xi,t-1}+{\rho}_{XX}{z}_{Xi,t-1}+{\delta}_{Xi t},\end{align}$$

where, for the Y and X repeated measures, ${\mu}_{Yt}$ and ${\mu}_{Xt}$ are time-dependent intercepts, the random intercepts are ${\eta}_{Y1i}$ and ${\eta}_{X1i}$ , assumed to have zero means, and the equation errors are ${z}_{Yit}$ and ${z}_{Xit}$ . As before, the ρ coefficients represent the autoregressive and cross-lagged effects, with the first subscript indicating the error of the repeated measure receiving the effect, and the second subscript indicating the error of the variable sending the effect. Andersen (Reference Andersen2022) demonstrates that this RI-CLPM is a restricted version of the EW1 ALT model without the random slope. Details on the relationships are reported in Appendix B. Using an analogous logic to that described in the previous section, this would enable a comparison of the unrestricted ALT model with the restricted ALT and the corresponding RI-CLPM, providing evidence on whether the unrestricted ALT better fits the data.

Two additional issues with using the RI-CLPM are how analysts can introduce bias if they use it without diagnostics and fit assessment. One issue is that the RI-CLPM assumes stationarity of the repeated measures, meaning the autoregressive coefficients for the same series are equal and have absolute values less than 1 (see Appendix B). A second issue is that the RI-CLPM typically omits random slopes for the repeated measures. Although it already incorporates time-varying intercepts to capture occasion-specific shifts in the overall mean, those intercepts often act as a stand-in for genuine slope heterogeneity when random slopes are absent. Time-varying intercepts absorb some of the unmodeled individual differences in change that an explicit slope term would capture, thereby reducing bias in the cross-lagged estimates. However, if substantive slope variability exists, intercepts alone can only partially correct for it, leaving residual confounding. Best practice is therefore to formally test for random slopes and, where supported by the data, include latent random slope or growth curve factors alongside time-varying intercepts to ensure both between-person and within-person dynamics are modeled without bias. This diagnostic check appears rare in practice.

4.4.3 Empirical application

In this section, we use the MACC data—where Y represents the adolescents’ smoking intensity and X their perceived exposure to smoking in movies—to compare the bivariate models introduced in this section with the corresponding restricted ALT model together with more general versions of the ALT.

Table 5 illustrates how the four models provide a consistent explanation of how accounting for stable traits and systematic growth affects our estimates of autoregression, reciprocal effects (cross-lags), and individual heterogeneity. In the CLPM, which omits any trait or growth factors, both processes show high persistence: smoking intensity at one wave predicts itself with ${\rho}_{YY}$ equal to 0.514, and perceived movie smoking exposure predicts itself with ${\rho}_{XX}$ equal to 0.806. Small negative cross-lagged effects ( ${\rho}_{YX}=-0.013$ and ${\rho}_{XY}=-0.040$ ) indicate slight reciprocal suppression; however, these estimates combine within-person dynamics with unmodeled between-person differences and growth.

Table 5

Parameter estimates (standard errors in parentheses) for the bivariate models fitted to smoke intensity (Y) and exposure to smoking in movies (X)

Table 5. Long description

The table compares four models: C L P M, L C M dash S R, R I dash C L P M, and P W 1 A L T.

* Autoregressive parameters for smoke intensity (rho y sub t y sub t minus 1): C L P M is 0.514. L C M dash S R is 0.092. R I dash C L P M is 0.170. P W 1 A L T ranges from 0.115 to 0.023 across five time points.

* Autoregressive parameters for movie exposure (rho x sub t x sub t minus 1): C L P M is 0.806. L C M dash S R is 0.322. R I dash C L P M is 0.632. P W 1 A L T ranges from 0.367 to 0.125.

* Cross-lagged parameters (rho y sub t x sub t minus 1): C L P M is minus 0.013. L C M dash S R is minus 0.000. R I dash C L P M is 0.004. P W 1 A L T ranges from minus 0.002 to minus 0.005.

* Cross-lagged parameters (rho x sub t y sub t minus 1): C L P M is minus 0.040. L C M dash S R is minus 0.017. R I dash C L P M is minus 0.014. P W 1 A L T ranges from 0.060 to minus 0.042.

* Variance and Covariance of latent factors:

- V(alpha sub yi): L C M dash S R 0.623, R I dash C L P M 0.142, P W 1 A L T 0.168.

- V(beta sub yi): L C M dash S R 0.031, P W 1 A L T 0.005.

- C(alpha sub yi, beta sub yi): L C M dash S R 0.013, P W 1 A L T minus 0.013.

- V(alpha sub xi): L C M dash S R 0.201, R I dash C L P M 0.652, P W 1 A L T 0.218.

- V(beta sub xi): L C M dash S R 0.005, P W 1 A L T 0.026.

- C(alpha sub xi, beta sub xi): L C M dash S R minus 0.016, P W 1 A L T 0.033.

Introducing random intercepts and expressing the autoregressive and cross-lagged on the errors rather than on the observed variables (RI-CLPM) absorbs stable between-person differences, and the apparent persistence shrinks to ${\rho}_{XX}=0.632$ and ${\rho}_{YY}=0.170$ . Cross-lagged effects become effectively zero, indicating that the CLPM’s reciprocal effects were largely artifacts of trait variance. The remaining intercept variances, equal to 0.142 and 0.652 for ${\alpha}_{yi}$ and ${\alpha}_{xi}$ , respectively, confirm substantial stable heterogeneity in both processes.

Analyzing the estimated time-varying intercepts— ${\mu}_{Y1}=1.863$ , ${\mu}_{Y2}=1.881$ , ${\mu}_{Y3}=1.927$ , ${\mu}_{Y4}=1.939$ , ${\mu}_{Y5}=1.961$ , ${\mu}_{Y6}=1.974$ for Y, and ${\mu}_{X1}=1.317$ , ${\mu}_{X2}=1.432$ , ${\mu}_{X3}=1.568$ , ${\mu}_{X4}=1.786$ , ${\mu}_{X5}=1.851$ , ${\mu}_{X6}=1.864$ for X (all significantly different from zero)—reveal an upward trend in both processes, confirming that mean levels of Y and X increase monotonically over time.

Adding random slopes along with structured residual dynamics in the LCM-SR gives us a clearer picture of the within-person processes: residual autoregressive effects decrease to 0.322 for ${\rho}_{XX}$ and to 0.092 for ${\rho}_{YY}$ , while residual cross-lagged effects become essentially zero. The slope variances emerge, showing that adolescents differ significantly in their linear trends of smoking and movie exposure once trait effects are accounted for. A small negative covariance between the intercept and slope for movie exposure suggests that individuals who start at a higher level tend to increase less over time.

Finally, the unrestricted PW1 ALT reveals that within-person persistence in both processes decays over time: smoking intensity autoregressive coefficients decline from 0.367 in Wave 2 down to 0.125 by Wave 6, and movie exposure autoregressive pattern from 0.115 down to 0.023. Cross-lagged effects now fluctuate around zero: the only moderately sized effect is the early influence of smoking on subsequent exposure ( ${\rho}_{x2y1}=0.060$ ), but later intervals show small positive or negative effects that never resulted significantly different from zero. Growth heterogeneity remains evident, and interestingly, the covariance between the intercept and slope in smoking intensity flips negative, indicating that adolescents who start as more intense smokers tend to escalate slightly slower over time. These results differ from Usami et al. (Reference Usami, Murayama and Hamaker2019). They also fitted a bivariate ALT model to these MAAC data to test autoregressive and cross-lagged effects between smoking intensity and exposure to smoking in movies. However, they reported that “…the ALT model…resulted in an improper solution.” In their version, they constrained all autoregressive and cross-lagged coefficients, residual variances, and the residual covariances between smoking intensity and movie-related smoking at each wave to be time-invariant and homoscedastic—restrictions not supported by the data. By contrast, our PW1 ALT specification allows autoregressive paths, cross-lags, residual variances, and concomitant covariances to vary across waves, rather than assuming they are time-invariant. This flexible setup eliminates improper solutions throughout. Particularly, the chi-square is statistically significant while the other fit measures are excellent ( $\chi {}^2$ (28) = 66.147, p < .001; CFI = 0.997; TLI = 0.994; RMSEA = 0.017; BIC = −170.153).

Because most cross-lagged effects were nonsignificant across models, we also estimated constrained versions of the LCM-SR, RI-CLPM, and unrestricted PW1-ALT models in which all cross-lagged parameters were fixed to zero. The substantive conclusions based on the coefficient estimates remained unchanged.

Table 6 shows the fit indices for the remaining models. However, unlike Usami et al. (Reference Usami, Murayama and Hamaker2019), who further constrained concomitant residual covariances and repeated-measure variances to be constant over time, we follow Andersen (Reference Andersen2022) in allowing all residual covariances and variances to vary freely on each occasion, maintaining consistency with the generality of the unrestricted PW1 ALT model.

Table 6

Fit statistics of different bivariate models for smoking intensity (Y) and exposure to smoking in movies (X)

Table 6. Long description

The table consists of six columns and seven rows of data.

Columns from left to right are:

1. Fit Statistic (unlabeled header)

2. L C M dash S R

3. E W 1 A L T equivalent to the L C M dash S R

4. R I dash C L P M

5. E W 1 A L T equivalent to the R I dash C L P M

6. Unrestricted P W 1 A L T

Rows from top to bottom are:

- Chi-squared: 174.719, 174.719, 395.214, 395.214, 71.349.

- d f: 48, 48, 55, 55, 38.

- p-Value: 0.000 for all models.

- C F I: 0.991, 0.991, 0.977, 0.977, 0.998.

- T L I: 0.988, 0.988, 0.972, 0.972, 0.996.

- R M S E A: 0.024, 0.024, 0.036, 0.037, 0.014.

- B I C: minus 230.828, minus 230.828, minus 69.476, minus 69.476, minus 249.709.

The overall fit of the CLPM was poor even allowing its cross-lagged effects (e.g., $\chi {}^2$ (56) = 1,730.231, CFI = 0.887, BIC = 1,256), so we do not include its overall fit statistics. Introducing random intercepts and an autoregressive pattern among the errors in the RI-CLPM markedly improves fit, demonstrating that modeling stable between-person differences removes much of the misfit. The LCM-SR further enhances model fit by adding random slopes and structured residual autoregressive paths, while still assuming time-invariant lagged effects. For both the RI-CLPM and LCM-SR, the corresponding restricted-ALT equivalent (as discussed in the Appendix) reproduces these exact fit indices, confirming their statistical equivalence. By contrast, the unrestricted PW1 ALT, which frees all autoregressive coefficients to vary across time, achieves the best overall fit statistics, underscoring the benefit of relaxing time-invariance to capture genuine within-person dynamics. Indeed, when each restrictive model is compared to the unrestricted PW1 ALT via a LR test, the loss of fit is highly statistically significant in every case. The RI-CLPM has a $\Delta \chi {}^2$ of 323.865 on $\Delta df=17$ (p < 0.001), and the LCM-SR a $\Delta \chi {}^2$ of 103.37 on $\Delta df=10$ (p < 0.001). These results confirm that each relaxation—from omitting growth (CLPM), to adding only random intercepts (RI-CLPM), to incorporating random slopes and structured residuals (LCM-SR)—makes a statistically significant contribution, and that freeing all autoregressive effects over time in the unrestricted PW1 ALT helps capture the within-person dynamics in the MACC data.

In summary, shifting from the CLPM to the RI-CLPM/EW1-ALT and then to the LCM-SR/EW1-ALT (see Table 6) gradually separates trait and growth from state dynamics, significantly reducing apparent inertia and spurious cross-lagged effects. Allowing each lag to vary in the PW1 ALT reveals the time-varying decay of persistence and the lack of strong reciprocal effects once all heterogeneity is accounted for. This would be hidden under any model that assumes constant lagged parameters.

4.4.4 Statistical versus substantive equivalence

In this section, we have discussed how constraints on ALT models can produce statistically equivalent versions of the LCM-ARE, LCM-SR, and the RI-CLPM. This allows us to compare these restrictive ALT models to less restrictive versions and to contrast models with different structures. At the same time, we argue that statistical equivalence does not mean substantively equivalent. Here, we briefly describe how the ALT differs from these other models.

The LCM-SR and RI-CLPM are sometimes described as automatically separating the between- and within-components in longitudinal data. This description appears based on the model using either the random intercept alone (RI-CLPM) or with a random slope (LCM-SR) to represent the between-effects, while considering the error structure as the within-effects. Because the random intercepts and slopes are uncorrelated with the errors, this appears to create a clean separation of between- and within-components. However, this clear separation becomes less certain once we recognize that the uncorrelatedness of the errors and the intercepts and/or slopes is an assumption, not an inherent property of these models. There could be good reasons to question this assumption.

The error terms in LCM-ARE, LCM-SR, and RI-CLPM include all other influences on the repeated measures that are not part of the RHS of the model. Typically, when only the random intercepts/slopes are explicitly included as RHS variables, this implies that any time-varying variables affecting ${Y}_{it}$ are part of the error. The random intercepts and random slopes are between-case components and might capture time-invariant variables affecting the repeated measures. But if there are other time-invariant influences on the repeated measures not represented by the random intercepts and slopes, then these could also be part of the error.

Could these omitted variables lead to a correlation between the errors and the random intercepts/slopes, thereby violating the assumption that errors are uncorrelated? If so, we must accept that the clear separation of between- and within-components is compromised due to potentially biased estimates of the LCM parameters. Therefore, the automatic distinctions in between- and within-components claimed by the LCM-ARE, LCM-SR, and RI-CLPM depend on the assumption that errors are uncorrelated with the random intercepts/slopes, which is questionable in many applications.

Substantively, the ALT model makes different assumptions. Its equation indicates that the current repeated measure ( ${Y}_{it}$ ) depends on ${Y}_{i,t-1}$ , even after controlling for the growth curve parameters. It allows ${Y}_{i,t-1}$ to correlate with the growth curve parameters rather than assuming no association. However, the ALT model does assume that the errors in its equation are uncorrelated with the RHS random intercepts/slopes and with ${Y}_{i,t-1}$ . This assumption can also be violated by omitted time-varying or time-invariant variables in an ALT model. The difference is that analysts presenting the ALT model make no claim of automatic separation of between- and within-effects, whereas writings on LCM-ARE, LCM-SR, and RI-CLPM often do and view this as an advantage over the ALT. But as we have discussed, there is nothing automatic about the separation of between- and within-effects in any longitudinal models.

4.4.5 Summary

All three of the longitudinal models with complex errors discussed in this section share the characteristic that they equate to restrictive versions of ALT models. In fact, Andersen (Reference Andersen2022, p. 749), who presents and illustrates these statistically equivalent model findings, states: “The point of all this algebra is to make clear that the residual-level models are just the observation-level models [i.e., ALT models] with constraints on the relationships between the individual effects variable(s) and the initial observations.”

Furthermore, these three longitudinal models with complex errors make strong assumptions about stationarity, equal autoregressive coefficients, and ρs that meet specific conditions. A simpler model is applicable when these assumptions are valid, but biased coefficients result if they are not. Unfortunately, current applications of these models rarely test whether these assumptions are valid when testing is feasible.

Despite these issues, some argue that these models are easier to interpret. The next section of the article examines these and other claims about the ALT and other longitudinal models.

5.7.1 Model selection and flexibility

Another criticism of ALT models concerns the ambiguous model selection that arises from their hybrid nature. Because ALT combines growth and autoregressive processes, it can closely approximate a wide range of data-generating mechanisms. Ou et al. (Reference Ou, Chow, Ji and Molenaar2017, p. 178) warn that “the ALT model is flexible enough to produce a variety of discrepant model-implied change trajectories. … this may confound interpretations of the model’s parameters”. The empirical examples from our article illustrate how to fit and compare different longitudinal models, and we have done the same with other examples on which we have worked. Furthermore, Ye and Bollen’s (Reference Ye and Bollen2022, p. 7) simulation research suggests that the ALT is useful in choosing a longitudinal model under many conditions. They conclude that “a good empirical practice is to start with the most general model and reduce it to more restrictive forms until one reaches the most parsimonious representation with roughly or very consistent goodness-of-fit results from the most general model and applying robust criteria”. We also showed that several longitudinal models were restricted versions of the ALT model, such that LR tests and fit indexes have been used in our empirical application to choose among alternative models. As previously discussed, we recall that this raises an additional point regarding the distinction between statistical equivalence and substantive equivalence. The choice among different model specifications should consider the underlying assumptions of each model.

5.7.2 Parsimony

Hamaker (Reference Hamaker2005, p. 414) noted that treating the first observed score in the ALT model as predetermined “three extra parameters must be estimated” compared to an LCM. However, as we discussed earlier, under the constraints of time-invariant autoregressive coefficients ${\rho}_{t,t-1}=\rho$ and $\mid \rho \mid <1$ , the EW1 ALT model and the LGCM-AE have an identical number of parameters. Thus, a restricted ALT model need not be less parsimonious than an LCM with structured residuals; it only becomes more complex when one allows time-varying or higher-order autoregressive effects. Researchers should therefore test whether such constraints are tenable before adopting a more general ALT specification.

5.7.3 Nonlinearity obscuration

A common criticism in several studies is that a linear ALT model can inadvertently absorb true nonlinear growth into its autoregressive component. Voelkle (Reference Voelkle2008, p. 564) showed through a simulation study that “the autoregressive process incorrectly accounts for part of this nonlinearity, thus rendering any substantive interpretation of parameter estimates virtually impossible.” Bauldry and Bollen (Reference Bauldry and Bollen2018) found that fitting a linear ALT to data with underlying curvature caused the model to underestimate the intercept and overestimate the slope mean to “compensate for an unmodeled nonlinear pattern” (p. 285). To address this issue, these authors introduce nonlinear ALT models to “..address a concern with linear ALT models that they may be misspecified when fit to nonlinear trajectories” (p. 297). These nonlinear ALT specifications help researchers distinguish genuine curvature from carryover effects, reducing bias and enhancing interpretability. It also seems likely that nonlinearity could manifest in autoregressive and cross-lagged errors with the LGCM-AE, LCM-SR, and RI-CLPM when analysts fail to check for it. So, checks for nonlinearity are prudent regardless of model choice.