Interpreting CLPM Coefficients for Emotional Influence

From Model to Estimates: Practical Considerations

Once the RI-CLPM is specified, the next step is to estimate the path coefficients using the time-series emotional data you've collected via your ESM protocol. In practice, this is done using software packages like lavaan in R or the SEM modules in Stata or Mplus. The estimation process, typically using Maximum Likelihood Estimation (MLE), finds the set of path coefficients that makes the model-implied covariance matrix most similar to the covariance matrix observed in your data.

A significant practical challenge in any longitudinal study is missing data. It's almost certain that participants will occasionally miss an ESM prompt. The default method in many older statistical procedures, listwise deletion, discards any case with a single missing value, which is catastrophically inefficient and can introduce bias. The modern standard, which you should always use, is Full Information Maximum Likelihood (FIML). FIML uses all available data from every participant to compute the likelihood function, resulting in more power and less biased estimates under the assumption that data are Missing at Random (MAR).

This video provides a practical demonstration of fitting a CLPM and highlights the dramatic difference in sample size retention when using FIML (referred to as "Maximum Likelihood with Missing Values" or mlmv in Stata) compared to listwise deletion.

Interpreting Model Coefficients: Stability and Influence

After fitting the model, the software will produce a table of estimated coefficients, standard errors, and p-values for each path. Interpreting these values correctly is the key to unlocking the story in your data.

The following guide provides an excellent walkthrough of fitting a CLPM and interpreting its output. While the example is bivariate (mental and physical health), the principles of interpretation are identical for your triadic model.

Let's break down how to interpret these two types of coefficients in the context of your triadic RI-CLPM.

1. Autoregressive Coefficients ($\beta$): Within-Person Stability

The autoregressive paths (e.g., $H_t\to H_{t+1}$) quantify the stability of an individual's emotional state. Crucially, because we are using an RI-CLPM, these coefficients have a very specific meaning.

As this FAQ on the RI-CLPM explains, the autoregressive effect in an RI-CLPM is not about rank-order stability (how a person ranks relative to others over time). Instead, it captures within-person carry-over.

So, if the autoregressive path for Husband's Jealousy (${\beta }_H$) is 0.4, it means that if the husband's jealousy at time $t$ is one point above his own average, it is expected to be 0.4 points above his average at time $t+1$, controlling for the influence of the wife and bull at time $t$.

• A high $\beta$: Indicates high emotional inertia. The emotion is "sticky" and persists from one moment to the next.
• A low $\beta$: Indicates emotional volatility. The emotion dissipates quickly and is less dependent on its prior state.

2. Cross-Lagged Coefficients ($\lambda$): Within-Person Influence

The cross-lagged paths (e.g., $W_t\to H_{t+1}$) are the primary parameters for testing your causal hypotheses. They quantify the extent to which one person's emotional state predicts a subsequent change in another's.

Again, within the RI-CLPM, these are within-person effects. If the cross-lagged path from Wife's Guilt to Husband's Jealousy (${\lambda }_{WH}$) is 0.25 and statistically significant, the interpretation is: a one-unit increase in the wife's guilt above her personal average at time $t$ is associated with a 0.25-unit increase in the husband's jealousy above his personal average at time $t+1$, after accounting for the husband's own emotional state at time $t$.

This is a powerful piece of evidence for a directional, causal influence. By examining the six cross-lagged paths in your triadic model, you can map out the entire system of emotional contagion.

Comparing Path Strengths to Determine Causal Dominance

A common research question is not just if an effect exists, but whether one effect is stronger than another. For instance, is the influence of wife's guilt on husband's jealousy (${\lambda }_{WH}$) stronger than the influence of husband's jealousy on wife's guilt (${\lambda }_{HW}$)? Answering this requires comparing the coefficients.

Standardized vs. Unstandardized Coefficients

If all your emotional variables are measured on the exact same scale (e.g., all on a 1-7 Likert scale), you can directly compare their unstandardized coefficients. However, if the scales differ, or even if you just want to compare the relative importance of different predictors, you must use standardized coefficients. A standardized coefficient represents the change in the outcome variable (in standard deviation units) for a one standard deviation change in the predictor variable. This puts all paths on a common metric, allowing for direct comparison of their relative strength.

This part of the Stata video demonstrates obtaining standardized coefficients and using them to make informal comparisons about which paths are stronger.

Formal Test of Equality

While visual inspection of standardized coefficients is useful, your statistical background warrants a more rigorous approach. We can formally test the null hypothesis that two paths are equal (e.g., $H_0:{\lambda }_{WH}={\lambda }_{HW}$). This is done by fitting a second, constrained model where the two coefficients are forced to be equal, and then comparing the fit of this constrained model to the original, unconstrained model using a chi-square difference test. If the constrained model fits significantly worse, we reject the null hypothesis and conclude that the two paths are indeed different in strength.

This technique provides a definitive answer to the question of causal dominance. The following resource provides a clear, step-by-step guide on how to perform this test.

By applying this technique to all pairs of reciprocal paths in your triad (H-W, H-B, W-B), you can build a statistically-defended map of the dominant directions of emotional influence.

Conclusion

In this lesson, you have learned to bridge the gap between a specified model and actionable insights. You now know how to approach the practical step of estimation and, most importantly, how to meticulously interpret the resulting coefficients to understand the dynamics of stability and influence within the triad.

Key Takeaways:

• Estimation: FIML is the preferred method for handling the inevitable missing data in longitudinal research, as it maximizes power and minimizes bias.
• Autoregressive Coefficients ($\beta$): In an RI-CLPM, these represent within-person stability—the degree to which an emotion's deviation from its personal mean carries over to the next moment.
• Cross-Lagged Coefficients ($\lambda$): These represent within-person influence. They are the key to testing directional hypotheses about how one person's emotional state change predicts another's.
• Comparing Paths: Standardized coefficients are necessary for comparing the relative strength of different paths. Formal equality tests provide rigorous evidence for determining the dominant direction of influence between two variables.

You are now equipped to analyze the core dynamics of moment-to-moment influence. However, the CLPM assumes these influence patterns are stable over time. What if they aren't? For instance, participants might become desensitized (habituation) or more sensitive (sensitization) to certain emotional triggers over repeated experiences. In our next lesson, we will explore advanced models that allow us to specify and estimate how the causal effect of an activity on an emotion changes across repeated episodes.