Purpose

This workflow explains and probes a cross-level interaction in a linear growth model. It asks whether students’ rates of change differ across levels of a person-level moderator, then translates the interaction into conditional trajectories, simple slopes, and a continuous region-of-significance review.

The included data are entirely synthetic. An interaction is a difference in slopes, not proof that one variable causes the other. Specify the moderator, centering choice, time origin, and probing values before interpreting the model.

Interaction model

The fixed-effects portion is:

\[ Y_{it}=\beta_0+\beta_1(Time_{it})+\beta_2(Moderator_i)+\beta_3(Time_{it}\times Moderator_i)+\cdots \]

The conditional time slope is \(\beta_1+\beta_3(Moderator_i)\). Consequently, the time main effect is the rate of change when the centered moderator equals zero.

User settings

Simulate and import the example

Rows: 1,560
Columns: 6
$ student_id      <chr> "G001", "G001", "G001", "G001", "G001", "G001", "G002"~
$ time            <dbl> 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, 0, 1, 2, 3, 4, 5, ~
$ progress        <dbl> 33.5, 35.2, 37.9, 41.6, 32.9, 38.5, 59.8, 58.3, 71.0, ~
$ self_efficacy   <dbl> 30.9, 30.9, 30.9, 30.9, 30.9, 30.9, 51.5, 51.5, 51.5, ~
$ self_efficacy_c <dbl> -18.68, -18.68, -18.68, -18.68, -18.68, -18.68, 1.92, ~
$ program         <fct> Enhanced, Enhanced, Enhanced, Enhanced, Enhanced, Enha~

Data audit

students records observed_outcomes missing_outcomes students_with_changing_moderator
260 1560 1495 65 0

The moderator is grand-mean centered. Zero therefore represents the sample-average self-efficacy score, making the time main effect the expected rate of change for an average student.

Observe the trajectories

Fit the growth models

term estimate std_error df t_value p_value conf_low conf_high
(Intercept) 48.047 0.528 1233 90.996 0 47.011 49.083
time 2.164 0.061 1233 35.543 0 2.045 2.284
self_efficacy_c 0.338 0.039 257 8.751 0 0.262 0.414
programEnhanced 3.013 0.756 257 3.986 0 1.525 4.502
time:self_efficacy_c 0.070 0.006 1233 11.691 0 0.059 0.082
model call Model df AIC BIC logLik Test L.Ratio p-value
main_effect_model lme.formula(fixed = progress ~ time + self_efficacy_c + program, data = analysis_data, random = ~time | student_id, method = “ML”, na.action = na.fail, control = lmeControl(opt = “optim”)) 1 8 8652.023 8694.503 -4318.012 NA NA
interaction_model lme.formula(fixed = progress ~ time * self_efficacy_c + program, data = analysis_data, random = ~time | student_id, method = “ML”, na.action = na.fail, control = lmeControl(opt = “optim”)) 2 9 8543.618 8591.407 -4262.809 1 vs 2 110.405 0

The interaction coefficient is the expected change in the time slope for a one-point increase in self-efficacy. Its practical meaning is clearer after probing values that fall within the observed moderator distribution.

Probe simple slopes

label self_efficacy self_efficacy_c estimate std_error t_value p_value conf_low conf_high
Low (-1 SD) 39.412 -10.168 1.449 0.086 16.920 0 1.281 1.618
Mean 49.580 0.000 2.164 0.061 35.603 0 2.045 2.284
High (+1 SD) 59.748 10.168 2.879 0.087 33.237 0 2.709 3.049

Simple slopes describe change at selected moderator values. They do not test whether those selected slopes differ from each other; the interaction coefficient provides that test.

Conditional trajectories

Region-of-significance review

This is a grid of pointwise conditional-slope intervals across the observed moderator range. It is not a simultaneous confidence band or an exact Johnson-Neyman boundary. The pattern is sensitive to model form and standard errors; do not extrapolate beyond observed support or use the plot as a substitute for the interaction test.

Diagnostics

Probe only after checking the functional form of time and moderator, random-effects structure, residual covariance, influential students, overlap across moderator values, and missing-data assumptions.

Dynamic results template

A linear mixed-effects growth model was fitted to 1495 observations from 260 students. Baseline self-efficacy was grand-mean centered using one value per student. The time-by-self-efficacy interaction was 0.07, 95% CI [0.059, 0.082], p < .001. The interval supported a more positive rate of change at higher self-efficacy.

The estimated change per wave was 1.45 at one standard deviation below the mean, 95% CI [1.28, 1.62], p < .001; 2.16 at the mean, 95% CI [2.05, 2.28], p < .001; and 2.88 at one standard deviation above the mean, 95% CI [2.71, 3.05], p < .001. These conditional estimates should be interpreted with the trajectory plot, moderator distribution, diagnostics, and study design.

Reporting checklist

  • Define the time origin, time unit, moderator, and centering method.
  • Report the interaction estimate and confidence interval before simple slopes.
  • State why the probing values are meaningful and within the data range.
  • Show model-predicted trajectories and uncertainty where possible.
  • Distinguish a slope differing from zero from slopes differing from each other.
  • Document functional-form, residual, covariance, influence, and missing-data checks.

Exports

file
synthetic_growth_interaction_data.csv
interaction_model_fixed_effects.csv
simple_slopes.csv
conditional_slope_grid.csv
model_comparison.csv
conditional_trajectories.csv
run_settings.csv

References

  • Bauer, D. J., & Curran, P. J. (2005). Probing interactions in fixed and multilevel regression. Multivariate Behavioral Research, 40(3), 373–400.
  • Preacher, K. J., Curran, P. J., & Bauer, D. J. (2006). Computational tools for probing interactions in multiple linear regression, multilevel modeling, and latent curve analysis. Journal of Educational and Behavioral Statistics, 31(4), 437–448.
Dissertation Stats Helper · Reproducible statistical guidance for doctoral research · Interpret interactions with conditional estimates, observed support, and model diagnostics.