A first-difference approach to studying sequential associations with longitudinal panel data

Behavioral researchers are keenly interested in estimating lagged effects from multivariate longitudinal data to determine whether one variable leads to changes in the second variable. Historically, lagged effects were examined by collecting and analyzing time-series data. Time-series data are intensively measured at regular intervals, typically on a small number of individuals. Thus, time-series data tend to have a high number of assessments (e.g., >100) and a small sample size (e.g., < 5). Analyses of time-series data often happen at the individual level (i.e., N = 1), with lagged effects as the focal estimates.

A common model for time series data is the autoregression (AR) model, where a variable's prior instances are used to predict its current instance. An AR(1) model uses the score immediately before the current score as the explanatory variable and is written as

yti=β0+β1·yt-1i+eti,(1)

where y_*ti* is the outcome score at assessment t, y_{*t* − 1*i*} is the outcome score at the prior assessment (assessment t − 1), β₀ is the intercept, β₁ is the autoregressive effect indicating the effect of the prior score on the current score, and e_*ti* is the residual score, which is assumed to follow a normal distribution (eti ~ N(0,σe2). While the AR(1) model is common, models with additional lagged effects are possible. For example, an AR(2) model is

yti=β0+β1·yt-1i+ β2·yt-2i+eti,(2)

where β₂ is the lag-2 autoregressive effect. Models with different lags are compared to determine the optimal lag for AR models. For example, a weekly system will show a lag-7 preference, assuming data were collected daily.

The vector autoregression (VAR) model is an extension of the AR model for multiple variables. In the VAR model, each variable is predicted by its prior instance (i.e., AR) and the prior instance of the other variable in the system (i.e., cross-lag). For example, a two-variable VAR(1) model is

y1ti=β10+β11·y1t-1i+β12·y2t-1i+e1tiy2ti=β20+β21·y1t-1i+β22·y2t-1i+e2ti,(3)

where y_1*ti* and y_2*ti* are the scores for the two variables at assessment t, y_{1*t* − 1*i*} and y_2*t*−1*i* are the scores for the two variables at assessment t − 1, β₁₀ and β₂₀ are intercepts, β₁₁ and β₂₂ are the autoregressive effects, β₁₂ and β₂₁ are the cross-variable effects, and e_1*ti* and e_2*ti* are the residuals. With the VAR(1) model, researchers are primarily interested in the cross-variable effects, as these effects indicate whether the prior score in one variable predicts or explains variation in the current score of the other variable, controlling for its prior score. For example, if β₂₁ is significant, then y₁ is considered a leading indicator of y₂.

AR and VAR models rely on the stationarity assumption for valid statistical inference and predictive validity. Stationarity is when each variable has a constant mean, a constant variance, and a constant autocorrelation across time. An initial step in achieving stationarity in time-series analysis is removing any long-term trend (i.e., detrending) so that the variable's mean is constant over time. Detrending is done before fitting AR and VAR models. To illustrate the importance of detrending, we generated time-series data for two variables based on AR(1) models. That is, the two variables do not have cross-variable associations (lagged or concurrent). Time-series plots of these data are shown in Figure 1. The time-series plots show comparable means and variances across time. Estimating a VAR(1) model on these data indicates no significant lagged cross-variable effects, β^12=0.03, t₍₉₆₎ = 0.34; β^21=0.15, t₍₉₆₎ = 1.54, consistent with how the data were generated.

Trends were then added to the two variables. A small exponential trend was added to y_1*ti*, and a small linear trend was added to y_2*ti*. Time-series plots of these data are shown in Figure 2. The longitudinal trends in the data are notable and in the same direction. Estimating the VAR(1) model on these data produced significant lagged cross-variable effects, β^12=0.27, t₍₉₆₎ = 3.03; β^21=0.37, t₍₉₆₎ = 4.33. Thus, the data with trends indicated that each variable was a leading indicator of the other, whereas the simulated data had no lagged cross-variable effects in the population.

In the 1970s, AR- and VAR-type models began being applied to longitudinal panel data using the structural equation modeling (SEM) framework to estimate autoregressive and cross-lag effects for multiple variables. These models, commonly referred to as cross-lag panel models (CLPMs), have been applied in a range of social science disciplines, including education, psychology, and sociology, and are written as

y1ti=β10t+β11·y1t-1i+β12·y2t-1i+e1tiy2ti=β20t+β21·y1t-1i+β22·y2t-1i+e2ti,(4)

where y_1*ti* and y_2*ti* are the scores for the two variables at assessment t, y_{1*t* − 1*i*} and y_{2*t* − 1*i*} are the scores for the two variables at assessment t − 1, β_10*t* and β_20*t* are intercepts at each measurement occasion, β₁₁ and β₂₂ are the autoregressive effects, β₁₂ and β₂₁ are the cross-variable effects, and e_1*ti* and e_2*ti* are the residuals. One notable set of parameters in the CLPM is the time-dependent intercepts (β_10*t* and β_20*t*). These intercepts are estimated at each measurement occasion, which allows the trends in the data to be modeled; however, their estimation may also indicate that trends are present in the data. A path diagram of a CLPM specified for two variables, y₁ and y₂, assessed at four time points is shown in Figure 3 with parameter labels following Equation 4. Only the direct effects are contained in this path diagram for simplicity.

While specifying and estimating CLPMs for longitudinal panel data is straightforward, the appropriateness of the analysis has been questioned (Berry and Willoughby, 2017; Hamaker et al., 2015) because the structure of longitudinal panel data is different from that of time-series data. Longitudinal panel data typically have many more participants than repeated assessments. Often, the sample size is greater than 200, and the number of assessments is less than 10. Time-series data, as we noted, have many time points and few participants, with analyses often taking place at the individual level. When applied to individual time-series data, the cross-variable lagged effects are clearly within-person effects. For example, if the cross-variable lagged estimate is positive, then the interpretation is that when the individual's prior score on y₁ is higher relative to their own scores, then their subsequent score on y₂ is predicted to be higher relative to their own scores (controlling for the autoregressive effect). The generalizability of these within-person effects to other individuals is questionable (see Castro-Schilo and Ferrer, 2013), but the interpretation of these effects is clear.

When CLPMs are applied to longitudinal panel data, the interpretation of the autoregressive and cross-lag regression parameters is muddied because of the mix of within-person (i.e., longitudinal) data and between-person data. Thus, the cross-lag parameters are interpreted in two ways. The first follows the within-person interpretation. Assuming a positive cross-lag parameter, the interpretation is that when the individual's prior score on y₁ is higher relative to their own scores, then their subsequent score on y₂ is predicted to be higher relative to their own scores (controlling for the autoregressive effect). The second follows a between-person interpretation. Again, assuming a positive cross-lag parameter, the interpretation is that when an individual's prior score on y₁ is higher compared to other individuals, then the individual's subsequent score on y₂ is expected to be higher relative to the other individuals (controlling for the autoregressive effect). The estimates in the CLPM tend to be more influenced by the between-person data than the within-person data because the sample size is larger than the number of time points (assuming there are between-person differences).

Over the past decade, the application of CLPMs has come into question because the autoregressive and cross-lag effects represent a blend of between-person and within-person effects but are often interpreted as within-person effects (Berry and Willoughby, 2017; Hamaker et al., 2015). This has led researchers to propose adjustments to the CLPM so that the autoregressive and cross-lagged effects represent within-person effects. Here, we discuss two approaches. The first is the random intercept cross-lag panel model (RI-CLPM; Hamaker et al., 2015), and the second is the latent curve model with structured residuals (LCM-SR; Curran et al., 2014).

The RI-CLPM (Hamaker et al., 2015) was proposed to account for between-person differences to isolate the within-person nature of the cross-lagged effects. The RI-CLPM is a combination of a restricted common factor model and a CLPM. Specifically, a single factor is specified for each repeatedly measured variable with factor loadings set to 1. This is written as

y1ti=ν1t+i1i+u1tiy2ti=ν2t+i2i+u2ti,(5)

where ν_1*t* and ν_2*t* are measurement intercepts at assessment t, i_1*i* and i_2*i* are common factor or random intercept scores for individual i, and u_1*ti* and u_2*ti* are the unique factors at assessment t for individual i. The common factor scores can be thought of as the average level for each person relative to the mean at each assessment. For example, a person whose score is typically above the mean at each assessment will have a positive value for the common factor score. The unique factors indicate the distance between the observed score relative to the measurement intercept at assessment t and the individual's common factor score [i.e., u_1*ti* = y_1*ti* − (ν_1*t* + i_1*i*)]. The second part of the RI-CLPM is a CLPM for the unique factors, which is written as

u1ti=β11·u1t-1i+β12·u2t-1i+e1tiu2ti=β21·u1t-1i+β22·u2t-1i+e2ti,(6)

where β₁₁ and β₂₂ are the autoregressive effects, β₁₂ and β₂₁ are the cross-lag effects, and e_1*ti* and e_2*ti* are the residuals.

The random intercepts in the RI-CLPM serve two purposes. First, the random intercepts attempt to account for between-person differences in each series. Second, the random intercepts' specification removes any trend in the data because the measurement intercepts (i.e., ν_1*t* and ν_2*t*) are estimated at each assessment. Thus, the unique factors are specified to have means and intercepts equal to 0.

A path diagram of the RI-CLPM for two variables, y₁ and y₂, measured at four assessments is shown in Figure 4. The circles represent the latent variables in the model, which include the random intercepts (i.e., i₁ and i₂) and the unique factors (i.e., u_1*t* and u_2*t*). The factor loadings for the random intercepts, as well as the factor loadings for the unique factors, are fixed at 1. As with Figure 3, only the direct effects are shown in this path diagram. We note that the random intercepts have estimated variances, as well as a covariance, which allows for the between-person factors to be associated. Following Equation 6, the unique factors have autoregressive and cross-lag effects. The unique factors at the first time point have estimated variances and covariance. The residual variances of the unique factors for the second through fourth assessments are constrained to be equal over time, and the residual covariances of the unique factors are constrained to be equal over time.

LCM-SR

The LCM-SR is similar to the RI-CLPM; however, a latent curve model or growth model (Grimm et al., 2017; McArdle and Epstein, 1987; Meredith and Tisak, 1990) is specified to account for between-person differences and trends in the two series. The growth models are written as

y1ti=i1i+s1i·t+u1tiy2ti=i2i+s2i·t+u2ti,(7)

where i_1*i* and i_2*i* are the latent variable intercepts indicating expected values of the outcomes for individual i when t = 0, s_1*i* and s_2*i* are latent variable slopes indicating expected rates of change for a one-unit change in t for individual i, and u_1*ti* and u_2*ti* are the residuals at time t for individual i. Importantly, the latent variable intercepts and slopes have estimated means, variances, and covariances to account for between-person differences in the trajectories, as well as the trajectories' between-person associations. The time-dependent residuals then follow a cross-lag panel model, such that

u1ti=β11·u1t-1i+β12·u2t-1i+e1tiu2ti=β21·u1t-1i+β22·u2t-1i+e2ti,(8)

where β₁₁ and β₂₂ are the autoregressive effects, β₁₂ and β₂₁ are the cross-lag effects, and e_1*ti* and e_2*ti* are the residuals.

The intercepts and slopes of the LCM-SR serve the same purposes as in the RI-CLPM. That is, they account for between-person differences in the scores over time and the longitudinal trends in the data due to the inclusion of latent variable slopes. The latent variable slopes account for between-person differences in the rate of change, which is a key difference between the LCM-SR and the RI-CLPM. That is, the RI-CLPM assumes a single source of between-person differences, and the LCM-SR is open to more sources of between-person differences and more types of between-person differences (e.g., rate of change, asymptotic level, etc.) depending on the growth model.

The LCM-SR is written with linear change trajectories for y_1*ti* and y_2*ti*; however, more complex change trajectories can be specified, such as the latent basis (unstructured) growth model, where the factor loadings of the latent variable slope are estimated, except for identifying restrictions. The LCM-SR reduces to the RI-CLPM if two conditions hold: (1) the variance of the latent variable slope is 0, and (2) the LCM-SR is specified as a latent basis growth model to account for any non-linear change pattern.

A path diagram of the LCM-SR for two variables, y₁ and y₂, measured at four assessments is contained in Figure 5. The latent variable intercepts and slopes are drawn as circles with factor loadings going to the repeated measures of each variable. The factor loadings for the intercept are fixed at 1, and the factor loadings of the slope follow a linear change pattern for a linear growth model. Only the direct effects are contained in this path diagram for clarity, but we note that these latent variables have estimated means, variances, and covariances. The covariances allow the sources of between-person differences to be associated within and across variables.

As in the RI-CLPM, the unique factors are drawn as circles and have autoregressive and cross-lag effects. The unique factors at the first time point have estimated variances and covariance. The residual variances of the unique factors for the second through fourth assessments are constrained to be equal over time, and the residual covariances of the unique factors are also constrained to be equal over time.

A first-difference approach

The time-series data in Figure 2 contain trends that would be removed before analysis. Detrending these data can be done in multiple ways. A common approach is to fit a time-based regression model (e.g., linear or exponential). The residuals are then output and should have a constant mean and constant variance if the regression model captures the trend and homoscedasticity holds. An alternative approach is to difference the data. Here, the prior score is subtracted from the current score for each observation after the first assessment (i.e., t > 1). Differencing removes the level of the prior score from each score, and the differences may be stationary and suitable for estimation of VAR models. Figure 6 is a plot of the differences for the data from Figure 2. The trends of the data have been removed by differencing, and the cross-variable estimates from the VAR were 0.01 (SE = 0.08) for lagged y₂ predicting y₁ and 0.12 (SE = 0.11) for lagged y₁ predicting y₂. Thus, differencing the time-series analysis can be an effective approach to detrending, and a benefit of this approach is that it does not depend on specifying the correct (regression) model to remove the trend.

Differencing panel data to estimate cross-lag effects comes from econometrics (Wonnacott and Wonnacott, 1970; Liker et al., 1985) and attempts to remove between-person differences. This first-difference model (FDM) has been applied to two-occasion data to study within-time effects and proposed for estimating lagged effects in three-occasion data (Liker et al., 1985) to control for stable between-person differences. We extend this model to the SEM framework and then multi-wave data to examine research questions about the nature of developmental sequences.

The first part of the FDM is differencing. For all time points after the first, we calculate the difference between the current assessment and the previous assessment. That is, for t > 1,

d1ti=y1ti-y1t-1id2ti=y2ti-y2t-1i,(9)

where d_1*ti* and d_2*ti* are the first differences. Next, a cross-lag model is specified for the first differences written as

d1ti=β10t+β12·d2t-1i+f1tid2ti=β20t+β21·d1t-1i+f2ti,(10)

where β_10*t* and β_10*t* are time-dependent intercepts, β₁₂ and β₂₁ are the cross-lag effects, and f_1*ti* and f_2*ti* are the residuals. Autoregressive effects are not included in the model's specification, which follows the specification of the three-wave model presented in Liker et al. (1985).

A path diagram of the FDM is contained in Figure 7. The model only contains the model's direct effects; however, we describe the specification of variances and covariances. The observed variables are y₁₁ through y₁₄ and y₂₁ through y₂₄, and the first-difference variables are d₁₂ through d₁₄ and d₂₂ through d₂₄. Setting up the first-difference variables as latent variables follows the specification of latent change score models (McArdle and Hamagami, 2001; McArdle, 2009). This specification contains autoregressive paths for the observed variables fixed at 1 and factor loadings for the first-difference variables also fixed at 1. This defines the latent variables as differences (equal to differences calculated by subtracting observed scores). The variances of y₁₁ and y₂₁ are freely estimated; however, the residual variances of y₁₂ through y₁₄ and y₂₂ through y₂₄ are fixed at 0 because the variances of these observed variables are entirely determined by the variances of the previous observed variable and the latent difference variable. Now that the first-difference variables are defined, the cross-lag effects (β₁₂ and β₂₁) are specified as one-headed arrows from the set of difference variables to the second set of difference variables.

The variances the first set of difference variables (at time 2) are freely estimated, and these difference variables are allowed to covary with each other and the time 1 observed scores. The residual variances of difference variables after their first instance are constrained to be equal over time, and the residual covariances are also constrained to be equal over time. The difference variables within each series are allowed to covary with one another. For example, d₁₂ covaries with d₁₃ and d₁₄, and d₁₃ covaries with d₁₄.

Model comparison

The main goal of the RI-CLPM, LCM-SR, and FDM is to estimate within-person lagged effects. The RI-CLPM and LCM-SR do this by specifying a model for the between-person differences and specifying a CLPM for the unique factors. The between-person model has two goals. First, the between-person model accounts for between-person differences in each repeatedly measured variable. The RI-CLPM uses a common factor, with factor loadings set to 1, and the LCM-SR uses a growth model. Second, the between-person model accounts for the trend in each repeatedly measured variable. The RI-CLPM does this by estimating the measurement intercepts, and the LCM-SR uses a growth model. The RI-CLPM uses distinct models for these two goals, whereas the LCM-SR uses the same model for both goals. The RI-CLPM has more flexibility to account for any trend, but the LCM-SR has greater flexibility to account for the sources of between-person differences.

In both the RI-CLPM and the LCM-SR, the cross-lag model is written for the unique factors, which are typically assumed to have stationarity properties. That is, the unique factors are often assumed to have constant means (i.e., equal to 0), constant variances, and constant lagged covariances. Thus, a cross-lag model for the unique factors is in line with ideas from time-series analysis, as the between-person model yields stationary data for modeling sequential associations.

Moreover, the unique factors represent deviations from individuals' expected scores at each measurement occasion (i.e., fluctuations in scores not accounted for by the between-person model). This alters the interpretation of the autoregressive and cross-lag effects in the RI-CLPM and the LCM-SR. That is, the predictors indicate whether the observed score is higher or lower than expected for a given individual (given the between-person model). Assuming the cross-lag effect from u₁ to u₂ is positive leads to the following interpretation: if the participant's score on y₁ at the prior time point was higher than expected (i.e., positive u₁), then the subsequent score on y₂ is likely to be higher than expected (i.e., positive u₂). Because the predictors are scores relative to the individual's expected scores based on the between-person model, the between-person interpretation of the cross-lag is no longer viable.

The main limitation of the RI-CLPM is that the between-person model only consists of the random intercept. The random intercept accounts for the same amount of variance in each observed score and assumes a constant covariance for all repeatedly measured variables. That is, the covariance between the first and second variables is equal to the covariance between the first and third variables. Any residual covariance is forced into the unique factors and their autoregressive paths. The LCM-SR is a bit more flexible than the RI-CLPM as it can account for between-person differences in the level, as well as the rate of change. The main limitation of the LCM-SR is that the LCM-SR may not represent the longitudinal trajectories. That is, the data may include groups of participants with different trajectories or a complex trajectory that cannot be properly modeled given the available data. In both the RI-CLPM and the LCM-SR, the model's fit statistics may be able to indicate the model's inability to account for the between-person differences. In these cases, the cross-lag estimates should not be trusted.

The FDM uses the prior score for each individual as a centering point and removes between-person differences related to the prior assessment. This allows the between-person differences to shift over time, as in the LCM-SR but not in the RI-CLPM. A downside of the FDM is that the between-person model does not reduce the repeatedly measured variable's dimensionality. Reducing the dimensionality yields between-person factors (e.g., intercept and slope) and positive degrees of freedom. This allows researchers to examine model fit and yields latent variables that can be correlated or predicted by person-level covariates. While the FDM does not have a between-person model, the time 1 variables are correlated, and these variables can be predicted by person-level variables.

A second downside of the FDM is that differencing reduces the number of variables to estimate the cross-lag parameters. Having one less variable to estimate cross-lag parameters in a time-series (intensive longitudinal) study is not consequential; however, this loss is more significant when analyzing panel data, as there are only a few time points. Thus, FDM's statistical power is likely lower compared to that of the RI-CLPM and the LCM-SR. Additionally, decreasing the number of variables in the cross-lag model leads to a saturated model when analyzing data with three time points. Table 1 provides a summary of these benefits and limitations of the three modeling approaches.

Table 1. Benefits and limitations of the random intercept cross-lag panel model (RI-CLPM), latent curve model with structured residuals (LCM-SR), and first-difference model (FDM).

Simulation

Population models

We examine the accuracy of lagged effects of the RI-CLPM, LCM-SR, and FDM when each of these three models generated the data. The data for the RI-CLPM and LCM-SR had autoregressive parameters equal to 0.3, a cross-lag parameter equal to 0, and a cross-lag parameter equal to 0.2. We examine the Type 1 error rate for the cross-lag parameter that equals 0 in the population, and we examine raw and relative bias for the autoregressive parameters and the cross-lag parameter that equals 0.2 in the population. The random intercepts in the RI-CLPM were correlated to 0.5, and the intercepts and slopes were correlated to 0.5 in the LCM-SR. Thus, these data were simulated with between-person differences, as well as between-person associations, in the scores over time.

Similarly, in the FDM population model, one cross-lag parameter was set equal to 0.2, and the second cross-lag parameter was 0. The observed scores at the first measurement occasion were correlated (0.33), and the observed scores at the first measurement occasion correlated (−0.29) with the first latent difference scores of the same variable series. The FDM's population parameters were chosen based on the FDM's estimation of the simulated LCM-SR data.

For each model, 1,000 data sets were generated with a sample size of N = 500, with five equally spaced measurement occasions. We discuss the estimates of the autoregressive and cross-lag parameters when the fitted model was different from the population model. All data were generated and analyzed using Mplus (Muthén and Muthén, 1998–2017). All programming scripts are found at https://sites.google.com/site/longitudinalmethods/downloads.

Results

Table 2 shows the average and standard deviation for the autoregressive and cross-lag parameter estimates for the RI-CLPM, the LCM-SR, and the FDM for each of the three population models. The RI-CLPM data were generated with a linear trend, which mapped onto the LCM-SR's specification. The LCM-SR was able to capture the autoregressive and cross-lag parameters well. The cross-lag parameter that was 0 in the population had a Type 1 error rate of 6% in the LCM-SR, and the cross-lag parameter that was 0.2 in the population had 0% relative bias. The mean estimate for the slope variance was negative because the RI-CLPM had only one source of between-person differences and the LCM-SR had two (intercept and slope). When the FDM was fit, the mean estimates for the cross-lag parameters were −0.02 (13% Type 1 error rate) and 0.18 (10% relative bias). Both parameters were negatively biased, with raw bias values of −0.02.

Table 2. Mean and standard deviation (in parentheses) of the estimate for the autoregressive and cross-lag parameters for data generated by the (A) random intercept cross-lag panel model (RI-CLPM), (B) latent curve model with structured residuals (LCM-SR), and (C) first-difference model (FDM).

^aMean parameter estimate for a variance term was negative.

Next, data were generated based on the LCM-SR. Here, the RI-CLPM had positively biased estimates. The autoregressive effects were positively biased by 0.24 (20% relative bias) and 0.18 (40% relative bias). The cross-lag effects were 0.17 for the cross-lag parameter that was 0 in the population and 0.33 for the cross-lag parameter that was 0.2 in the population. The cross-lag parameter that was 0 in the population was significant in all simulated data sets (Type 1 error rate of 100%), making the dynamic association between the variables misconstrued in the RI-CLPM. While there is an obvious misspecification in the RI-CLPM for these data, the degree of misspecification did not lead to the model's fit statistics to reject the model using typical cutoff values (Hu and Bentler, 1999), as the mean comparative fit index (CFI) and Tucker–Lewis Index (TLI) were 0.99 and the mean root mean square error of approximation (RMSEA) was 0.05.

The mean cross-lag parameter estimates for the FDM were −0.00 for the cross-lag parameter that was 0 in the population (6% Type 1 error rate) and 0.19 for the cross-lag parameter that was 0.2 in the population (5% relative bias). Thus, the FDM did a fairly good job of recovering the cross-lag associations when the LCM-SR generated the data.

Finally, data were generated according to the FDM. Here, we focus on the cross-lag estimates as the FDM did not contain autoregressive effects in the population. The RI-CLPM had positively biased cross-lag estimates. The mean estimate of the cross-lag parameter that was 0 in the population was 0.21 (100% Type 1 error rate), and the mean estimate for the cross-lag parameter that was 0.2 in the population was 0.34 (70% relative bias). Thus, the RI-CLPM was unable to recover the dynamic association present in the FDM data. The mean fit statistics for the RI-CLPM were 0.98 for the CFI, 0.97 for the TLI, and 0.08 for the RMSEA. Thus, the RI-CLPM's model fit was acceptable to very good depending on the chosen fit statistic.

The LCM-SR fared better, but the cross-lag estimates remained positively biased. The mean estimate was 0.08 for the cross-lag parameter that was 0 in the population (Type 1 error rate of 36%), and the mean estimate was 0.28 (40% relative bias) for the cross-lag parameter that was 0.2 in the population. Thus, the LCM-SR struggled to recover the dynamic relations present in the population. The average model fit statistics were very strong in the LCM-SR, as the CFI and TLI were 0.99 and the RMSEA was 0.05. Thus, no serious model misfit was indicated, even though the lagged parameter estimates were positively biased.

Illustrative data analysis

Data

The data used for this illustration come from the Midlife in the United States (MIDUS) data set (Brim et al., 2004), a longitudinal survey tracking life development in a nationally representative sample of 7,108 individuals aged 25–75 years living in the United States. The first wave of data was collected using telephone interviews and mail questionnaires in 1995–1996, and follow-up waves were conducted in 2004 and 2013.

The primary variables were measures of workplace control and depression. Workplace control was a composite of six items. Five items were assessed on a 5-point Likert scale, and one item was assessed on a 0–10 scale. This item was given half the weight of the other five items. Thus, the workplace control variable scores ranged from 5 to 30. Items included “Thinking about your current job, how often do you have a say in decisions about your work?” and “Thinking about your current job, and how often do you have a say in deciding what tasks you do at work?” Depression was measured as the sum of seven items from the Center for Epidemiologic Studies Depression Scale (Radloff, 1977) that were asked in a binary fashion related to experiencing various depressive symptoms for a 2-week period over the past 12 months.

The CLPM, the RI-CLPM, the LCM-SR, and the FDM were fit to the three-wave longitudinal data composed of the measures of workplace control and depression. All analyses were conducted via the lavaan package in R (Rosseel, 2012; Rosseel et al., 2025) and Mplus (Muthén and Muthén, 1998–2017). The programming scripts are contained in the Appendix.

Results

CLPM, RI-CLPM, and LCM-SR were specified with the autoregressive and cross-lagged estimates constrained to be equal over time.

CLPM

The fit of the CLPM was good to very good based on incremental and absolute fit indices (CFI = 0.92, RMSEA = 0.05, Standardized Root Mean Square Residual (SRMR) = 0.05). The autoregressive parameter estimates were β₁₁ = 0.31 (SE = 0.01) for depression and β₂₂ = 0.50 (SE = 0.02) for workplace control, indicating a fair amount of stability in individual differences over time. The cross-lag parameter estimates were β₁₂ = 0.02 (SE = 0.01) for workplace control to depression and β₂₁ = 0.12 (SE = 0.04) for depression to workplace control. All of these estimates were significantly different from zero, suggesting a reciprocal relationship between workplace control and depression.

RI-CLPM

Next, the RI-CLPM was fit to the MIDUS data. The RI-CLPM fit the data very well based on both incremental and absolute fit indices (CFI = 0.99, RMSEA = 0.02, SRMR = 0.03). The random intercept for depression accounted for 26% of the variance in the observed depression scores, and the random intercept for workplace control accounted for 36% of the variance in the observed workplace control scores. The random intercepts were positively and significantly associated (ρ^21=.18). The estimated autoregressive parameters were β₁₁ = 0.04 (SE = 0.02) for depression and β₂₂ = 0.22 (SE = 0.05) for workplace control. Only the autoregressive estimate for workplace control was significant, indicating a level of stability in the unique factors for workplace control. The estimate for the cross-lag from workplace control to depression was β₁₂ = 0.01 (SE = 0.01), and the estimate for the cross-lag from depression to workplace control was β₂₁ = 0.07 (SE = 0.07). Neither of the cross-lag estimates was significant, indicating that depression and workplace control were not dynamically related at the within-person level.

LCM-SR

The LCM-SR was then fit, and the general set of conclusions followed those from the RI-CLPM. A linear growth model was specified for workplace control and depression; however, the estimated slope variances were negative and then fixed to zero to achieve model convergence. With slope variances fixed to zero, the model closely approximated the RI-CLPM as there was one source of between-person differences for each variable, and the slope accounted for some of the change pattern evident in the data. The model fit statistics (CFI = 0.99, RMSEA = 0.02, SRMR = 0.03), variance estimates for the random intercepts, and dynamic parameters were nearly identical to those obtained from the RI-CLPM. The estimated autoregressive parameter for depression was not significant (β₁₁ = 0.05; SE = 0.02), but the autoregressive parameter for workplace control was significant (β₂₂ = 0.22; SE = 0.05), indicating a level of stability for workplace control when accounting for the level and rate of change. The cross-lag estimates were β₁₂ = 0.01 (SE = 0.01) for workplace control to depression and β₂₁ = 0.07 (SE = 0.07) for depression to workplace control, with neither cross-lag estimate being statistically significant. Thus, based on the LCM-SR, depression and workplace control were not dynamically related when accounting for between-person differences in these constructs' levels and modeling a common rate of change in each construct.

FDM

Finally, the FDM was fit to the MIDUS data. Mirroring the model specification in Liker et al. (1985), the model was saturated with three-occasion data. Thus, model fit was not examined because the model perfectly replicated the data. The cross-lag estimate from workplace control to depression was β₁₂ = 0.01 (SE = 0.01), and the cross-lag estimate from depression to workplace control was β₂₁ = 0.05 (SE = 0.06), with neither cross-lag estimate reaching statistical significance. Thus, when accounting for stable between-person differences by examining associations among first differences, depression and workplace control were not dynamically related, mirroring the conclusions from the RI-CLPM and the LCM-SR.

Discussion

The FDM approach with latent difference scores is a novel approach to studying sequential associations that attempts to account for between-person differences. The goal of this approach mimics the goal of the RI-CLPM and the LCM-SR but uses latent differences instead of latent residuals (i.e., unique factors). The simulation indicated that the FDM was able to recover population parameters from the RI-CLPM and LCM-SR with 10% relative bias or less, highlighting the approach's potential. When applied to our illustrative data, the conclusions regarding the dynamic associations between depression and workplace control from the RI-CLPM, the LCM-SR, and FDM were highly similar and different from the conclusions from the CLPM. Moreover, the estimates between the models, along with their z-values, were very similar, indicating a similar amount of control of the between-person differences across models. We expect greater discrepancies between the approaches as the number of occasions increases because the approaches will differ in their ability to model and account for the various sources of between-person differences.

A goal of the RI-CLPM and the LCM-SR is to separate between-person differences from the estimation of within-person lagged associations. These models achieve this goal when their assumptions hold. Each model makes assumptions about the number and type of sources of between-person differences. The RI-CLPM assumes a single, constant source of between-person differences given the estimation of the measurement intercepts. This constant source of between-person differences is relative to the other individuals in the sample and not constant in terms of an unchanging level. If there is more than one source of between-person differences, then the RI-CLPM is unable to properly separate the between-person differences from the lagged effects estimation. This can bias the estimation of the autoregressive and cross-lag parameters, which was seen in the simulation when the LCM-SR generated the data.

The LCM-SR, by comparison, (typically) assumes two sources of between-person differences for each repeatedly measured variable. The LCM-SR has greater flexibility when modeling between-person differences as different growth models (e.g., linear, latent basis, quadratic, exponential) can be fit to determine an optimal number and type of between-person latent variables. However, as in the RI-CLPM, when the latent curve model is misspecified and does not properly account for the sources of between-person differences, then the lagged effects estimation can be biased. One question that remains is how much model misspecification is needed to bias the lagged effects estimation, what fit statistics should be examined, and what cutoff values should be used. The simulation highlighted instances where the model fit statistics were strong, but the lagged estimates did not map onto population values. Thus, more work on this topic is needed.

A challenge when examining the sources of between-person differences for the RI-CLPM and LCM-SR is the simultaneous estimation of the between-person model and the lagged effects. Typically, growth models, such as those specified in the LCM-SR, are estimated with the assumption that the residuals are independent and identically distributed. Thus, if autocorrelation of the residuals exists, then the growth model is misspecified; therefore, the sources of change (i.e., between-person differences) are misrepresented. Thus, the specification of an autoregressive residual structure when examining sources of between-person differences in the LCM-SR violates a common assumption of independent residuals in growth models.

The notion of specifying an autoregressive residual structure in growth models leads to a discussion of how differences arise in the lagged effects estimation when models are estimated simultaneously vs. sequentially. That is, a random intercept or latent curve model can be estimated for each repeatedly measured variable, with the residuals assumed to be independent and identically distributed. From these models, the residuals (unique factor scores) can be estimated, and a CLPM can be fit to the estimated residuals. This approach follows the typical logic in time-series analysis of detrending data before estimating a VAR model. This sequential estimation routine often leads to substantially different autoregressive and cross-lag estimates compared to when an RI-CLPM or an LCM-SR is estimated simultaneously with the between-person difference model and lagged effects. Simultaneously estimating the lagged effects affects the sources of between-person differences, as all parameters are jointly estimated. This means that the residuals in the RI-CLPM and the LCM-SR model are not left over after estimating the between-person model but are simultaneously determined with the between-person model and the specified nature of the lagged effects.

Finally, as the RI-CLPM and the LCM-SR models rely on latent residuals, the within-person model may be based on variables with low variance and potentially low reliability. The degree of variance and reliability depends on the stability of the between-person differences in the scores over time. For example, if there are high correlations over time within a repeatedly measured variable, then the latent residuals are likely to have low variance and a high degree of unreliability. For example, in our empirical data, the over-time correlations for depression were about 0.30, and the over-time correlations for workplace control were about 0.50. These values indicated that the latent residuals could have decent variance and reliability. We note that the difference variables have similar reliability issues when longitudinal correlations are high. This can be seen as the variance of the change scores is equal to σy112+σy122-2σy12y11, where σy112 is the variance of the time 1 score, σy122 is the variance of time 2 score, and 2σ_*y*12*y*11 is the covariance between the time 1 and time 2 scores. Thus, as 2σ_*y*12*y*11 increases, the amount of variance in the change scores decreases, making them less reliable.

Constraints

The simulation data and models, as well as the models applied to the MIDUS data, assumed the autoregressive and cross-lag parameters were equal over time. The idea behind these constraints comes from time series analysis, where the effects do not depend on time. Additionally, imposing equality constraints on the autoregressive and cross-lag parameters improves statistical power. Given the different structure of panel data with more time between assessments and a longer amount of time covered, it may be more reasonable for effects to differ over time. There are two approaches to examining whether the effects vary over time. The first approach is to specify the model with the constraints imposed. If the model fit is justifiable, then the model can be moved forward. If the model fit is not justifiable, then modification indices can be used to determine which parameters should be allowed to vary over time. The alternative approach is to specify the model without equality constraints and then examine whether constraining parameters significantly harms the model fit. We recommend the first approach because it provides an objective test of whether the constraints are justified given the model's fit statistics.

Final comparison and recommendation

The RI-CLPM, the LCM-SR, and the FDM use deviation scores in the cross-lag part of the model. The models differ regarding what the scores deviate from and whether they deviate from unobserved or observed variables. The RI-CLPM uses deviations from the random intercept and measurement intercept [e.g., u_1*ti* = y_1*ti* − (i_1*i* + ν_1*t*)], the LCM-SR uses deviations from the LCM [e.g., u_1*ti* = y_1*ti* − (i_1*i* + s_1*i* · t )], and the FDM uses deviations from the previous score (e.g., d_1*ti* = y_1*ti* − y_{1*t* − 1*i*}). The RI-CLPM and the LCM-SR use deviations from an unobserved quantity (i.e., i_1*i* + ν_1*t* or i_1*i* + s_1*i* · t ), whereas the FDM uses deviations from an observed quantity (i.e., y_{1*t* − 1*i*}). Thus, the question becomes which deviations should be analyzed with the cross-lag model.

Our recommendation begins with avoiding the standard CLPM. The conditions under which the CLPM provides within-person estimates are not typically seen in empirical longitudinal panel data (i.e., lack of between-person differences). We recommend beginning with the RI-CLPM but specifying it as an LCM-SR with a latent basis growth structure and setting the slope's variance to 0. This model provides initial estimates of the lagged effects. In the next model, the variance of the slope is estimated to determine if an additional source of between-person differences is viable. Changes in the estimates of the lagged effects are examined to determine the extent of their changes. From this model, additional sources of between-person differences (i.e., factors) can be specified to determine if more than two sources of between-person differences are needed. Additional factors can be added based on an exploratory approach to growth modeling (Grimm et al., 2013). Once this process is complete, we recommend fitting the FDM and comparing the cross-lag estimates. Ideally, a coherent picture of the within-person dynamics emerges.

Concluding remark

The logic behind the FDM, along with the simulation research, indicates that the FDM is a viable approach for estimating within-person lagged associations using longitudinal panel data. The model does not include a between-person component but allows for the estimation of within-person lagged effects. The simulation study, while limited, highlighted the ability of the FDM to estimate cross-lag parameters when the data conform to an RI-CLPM or an LCM-SR. More research needs to be conducted on the FDM to further evaluate its specifications and use. Following previous econometric research, the specification presented did not include autoregressive effects among the latent difference scores. Future research should consider if and when autoregressive effects should be estimated and examine non-linear change processes, more measurement occasions, and the impacts of incomplete data on the estimation of within-person lagged effects.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/forgp.2026.1864865/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Alexandre J. S. Morin, Concordia University, Canada

Reviewed by: Jose Navarro, University of Barcelona, Spain

Tim Vantilborgh, Vrije University Brussels, Belgium