Section 3 — Indistinguishable dyads: SEM

What you are practising: three structural equation modelling paradigms for indistinguishable dyads and the formal indistinguishability test.

Reference: Indistinguishable dyads: SEM tutorial.

Goal

Fit three SEM paradigms for each outcome (cluster-robust, two-level, wide-format equality-constrained), and then test the equality constraints with an LRT.

Tasks

Model A — cluster-robust SE (long format). For each outcome, fit a regression with all six within-dyad predictors plus the three moderators, using lavaan::sem() with cluster = "dyad_id".
Model B — two-level SEM (long format). Place the six within-dyad predictors at the within level and the three moderators at the between level.
Model C — wide-format equality-constrained. Use a single parameter label per path (so the actor and partner paths for the same predictor share a label), set equal intercepts and equal residual variances across the two dyad members. Use sem() on ddw2.
Indistinguishability test. Fit an unconstrained wide-format model for each outcome (separate labels for actor and partner paths) and use lavTestLRT() to compare against Model C. A non-significant p-value supports indistinguishability.

Reflection prompt

For each outcome, do the LRT results agree with the data-generating process (which has equal slopes across gender, with the single exception of actor_sdt × gender_male on engagement)? Which outcome’s test result is the weakest signal in favour of indistinguishability, and why?

Hints & solution outline

Tutorial reference: Indistinguishable SEM. Models A, B, and C are demonstrated in steps 1–3; the LRT is in the SEM wide tutorial, Step “Test the equality constraint”.

Substitutions: - ddl → ddl2, ddw → ddw2 - Replace the four predictors with the six predictors (3 actor + 3 partner) - Replace satisfaction_a / satisfaction_p with the wide-format outcome columns (e.g. engagement_a / engagement_p) - Add the three moderators; in Model B they go at the between level

What to expect. The DGP has only one gender interaction (actor_sdt × gender_male on engagement). For the other two outcomes, the slopes are equal across gender, so the LRT should comfortably support indistinguishability. For engagement, the LRT p-value should be the smallest (weakest signal in favour of indistinguishability) because of that one non-zero gender interaction.

What to record. For each outcome: “Unconstrained vs constrained: χ² = XX.X, df = X, p = .XXXX. Indistinguishability [is/is not] tenable for [outcome].”

--- title: "Section 3 — Indistinguishable dyads: SEM" --- # Section 3 — Indistinguishable dyads: SEM **What you are practising:** three structural equation modelling paradigms for indistinguishable dyads and the formal indistinguishability test. **Reference:** [Indistinguishable dyads: SEM tutorial](../../tutorials/indistinguishable/sem.html). ::: {.callout-note} ## Goal Fit three SEM paradigms for each outcome (cluster-robust, two-level, wide-format equality-constrained), and then test the equality constraints with an LRT. ::: ## Tasks 1. **Model A — cluster-robust SE (long format).** For each outcome, fit a regression with all six within-dyad predictors plus the three moderators, using `lavaan::sem()` with `cluster = "dyad_id"`. 2. **Model B — two-level SEM (long format).** Place the six within-dyad predictors at the within level and the three moderators at the between level. 3. **Model C — wide-format equality-constrained.** Use a single parameter label per path (so the actor and partner paths for the same predictor share a label), set equal intercepts and equal residual variances across the two dyad members. Use `sem()` on `ddw2`. 4. **Indistinguishability test.** Fit an unconstrained wide-format model for each outcome (separate labels for actor and partner paths) and use `lavTestLRT()` to compare against Model C. A non-significant *p*-value supports indistinguishability. ## Reflection prompt For each outcome, do the LRT results agree with the data-generating process (which has equal slopes across gender, with the single exception of `actor_sdt × gender_male` on engagement)? Which outcome's test result is the weakest signal in favour of indistinguishability, and why? ::: {.callout-tip collapse="true"} ## Hints & solution outline **Tutorial reference:** [Indistinguishable SEM](../../tutorials/indistinguishable/sem.html). Models A, B, and C are demonstrated in steps 1–3; the LRT is in the [SEM wide tutorial, Step "Test the equality constraint"](../../tutorials/distinguishable/sem-wide.html#the-three-nested-lrts). **Substitutions:** - `ddl` → `ddl2`, `ddw` → `ddw2` - Replace the four predictors with the six predictors (3 actor + 3 partner) - Replace `satisfaction_a` / `satisfaction_p` with the wide-format outcome columns (e.g. `engagement_a` / `engagement_p`) - Add the three moderators; in Model B they go at the between level **What to expect.** The DGP has only one gender interaction (`actor_sdt × gender_male` on `engagement`). For the other two outcomes, the slopes are equal across gender, so the LRT should comfortably support indistinguishability. For `engagement`, the LRT p-value should be the smallest (weakest signal in favour of indistinguishability) because of that one non-zero gender interaction. **What to record.** For each outcome: *"Unconstrained vs constrained: χ² = XX.X, df = X, p = .XXXX. Indistinguishability [is/is not] tenable for [outcome]."* :::