Distinguishability

The first design decision in any APIM: can the two members of each dyad be told apart?

The decision

For every dyadic dataset you can name, ask one question: Is there a variable that distinguishes the two members of each dyad, in a way that the two roles are not arbitrary?

If the answer is yes	If the answer is no
Heterosexual couples (gender)	Same-sex couples (no gender)
Parent–child (role)	Siblings (no role)
Doctor–patient (role)	Strangers paired in a lab (no role)
Older–younger twin (age)	Identical twins of the same age
Therapist–client in session 1 (role)	Friends in a friendship pair (no role)

A useful asymmetry test

For a continuous distinguishing variable (e.g. age, education), a paired t-test on the variable itself, between the two roles, is a quick way to see whether the roles are systematically different. If the test is significant, the dyads are distinguishable; if not, you have indistinguishable dyads.

The dyad members may have different means (males score higher on income than females, parents score higher on work experience than children) even when the slopes are equal. Indistinguishability of slopes is the working definition of an indistinguishable dyad.

Why it matters

The model you fit changes.

Indistinguishable dyads. The actor and partner slopes for the same predictor are constrained to be equal. The reason is that there is no role to anchor them to: if dyad members are interchangeable, the slope from A’s predictor to B’s outcome and the slope from B’s predictor to A’s outcome are two estimates of the same number, and pooling them gives you more power.
Distinguishable dyads. The actor and partner slopes are free to differ. You can then ask whether the gender difference in slopes is itself significant — this is the Kenny & Ledermann (2010) k-pattern test.

An empirical question, not a data property

Here is the most important point in the foundations section. Distinguishability is not a property of the data you happened to collect. It is a property of the dyad type you are studying. You can study same-sex couples with a dataset whose members all happen to be female, and the dyads are still indistinguishable (because the two members cannot be told apart by gender). You can study heterosexual couples with a dataset whose males and females have identical means, and the dyads are still distinguishable (because the two roles exist).

The choice between the indistinguishable tutorials and the distinguishable tutorials is made before you open the data.

What if you are not sure?

Two practical tests:

Equality-constrained SEM. Fit the wide-format model with actor and partner slopes constrained to be equal. Fit the unconstrained model. Run a likelihood ratio test. A non-significant p-value supports indistinguishability. (This is the indistinguishability test in the SEM wide tutorial.)
MLM with gender interaction. Fit a multilevel model with gender interacting every actor and partner predictor. A likelihood ratio test against the gender-free model tells you whether the slopes differ by gender. (This is the MLM moderator tutorial.)

The two approaches usually agree, but if they disagree, trust the test that has more power for your specific design. The wide-format SEM test is the gold standard.

References

Kenny, D. A., & Ledermann, T. (2010). Detecting, measuring, and testing dyadic patterns in the Actor–Partner Interdependence Model. Journal of Family Psychology, 24(3), 359–366.
Olsen, J. A., & Kenny, D. A. (2006). Structural equation modeling with interchangeable dyads. Psychological Methods, 11(2), 127–141.

--- title: "Distinguishability" --- # Distinguishability The first design decision in any APIM: can the two members of each dyad be told apart? ## The decision For every dyadic dataset you can name, ask one question: *Is there a variable that distinguishes the two members of each dyad, in a way that the two roles are not arbitrary?* | If the answer is **yes** | If the answer is **no** | |---|---| | Heterosexual couples (gender) | Same-sex couples (no gender) | | Parent–child (role) | Siblings (no role) | | Doctor–patient (role) | Strangers paired in a lab (no role) | | Older–younger twin (age) | Identical twins of the same age | | Therapist–client in session 1 (role) | Friends in a friendship pair (no role) | ::: {.callout-tip} ## A useful asymmetry test For a continuous distinguishing variable (e.g. age, education), a paired *t*-test on the variable itself, between the two roles, is a quick way to see whether the roles are systematically different. If the test is significant, the dyads are distinguishable; if not, you have indistinguishable dyads. ::: The dyad members may have *different means* (males score higher on income than females, parents score higher on work experience than children) even when the *slopes* are equal. Indistinguishability of slopes is the working definition of an indistinguishable dyad. ## Why it matters The model you fit changes. - **Indistinguishable dyads.** The actor and partner slopes for the *same* predictor are constrained to be equal. The reason is that there is no role to anchor them to: if dyad members are interchangeable, the slope from A's predictor to B's outcome and the slope from B's predictor to A's outcome are two estimates of the same number, and pooling them gives you more power. - **Distinguishable dyads.** The actor and partner slopes are free to differ. You can then ask whether the gender difference in slopes is itself significant — this is the Kenny & Ledermann (2010) k-pattern test. ## An empirical question, not a data property Here is the most important point in the foundations section. Distinguishability is **not** a property of the data you happened to collect. It is a property of the *dyad type* you are studying. You can study same-sex couples with a dataset whose members all happen to be female, and the dyads are still indistinguishable (because the two members cannot be told apart by gender). You can study heterosexual couples with a dataset whose males and females have identical means, and the dyads are still distinguishable (because the two roles exist). The choice between the [indistinguishable tutorials](../tutorials/indistinguishable/index.html) and the [distinguishable tutorials](../tutorials/distinguishable/index.html) is made before you open the data. ## What if you are not sure? Two practical tests: 1. **Equality-constrained SEM.** Fit the wide-format model with actor and partner slopes constrained to be equal. Fit the unconstrained model. Run a likelihood ratio test. A non-significant *p*-value supports indistinguishability. (This is the indistinguishability test in the [SEM wide tutorial](../tutorials/distinguishable/sem-wide.html).) 2. **MLM with gender interaction.** Fit a multilevel model with `gender` interacting every actor and partner predictor. A likelihood ratio test against the gender-free model tells you whether the slopes differ by gender. (This is the [MLM moderator tutorial](../tutorials/distinguishable/mlm-moderator.html).) The two approaches usually agree, but if they disagree, trust the test that has more power for your specific design. The [wide-format SEM test](../tutorials/distinguishable/sem-wide.html) is the gold standard. ## References - Kenny, D. A., & Ledermann, T. (2010). Detecting, measuring, and testing dyadic patterns in the Actor–Partner Interdependence Model. *Journal of Family Psychology, 24*(3), 359–366. - Olsen, J. A., & Kenny, D. A. (2006). Structural equation modeling with interchangeable dyads. *Psychological Methods, 11*(2), 127–141.