The k-patterns

Four ways actor and partner effects can combine. How to test which one fits your data.

The basic APIM

Let $X$ be one partner’s predictor and $Y$ be the same partner’s outcome. For a distinguishable dyad, the APIM fits two regressions:

\[ Y_A = a_A + b_A X_A + b_P X_P \\ Y_P = a_P + b_P X_A + b_A X_P \]

where $b_A$ is the actor effect (the partner of $X$ on their own $Y$) and $b_P$ is the partner effect (the partner of $X$ on the other person’s $Y$). For indistinguishable dyads, the constraint $b_A = b_P$ is imposed; the model fits a single common slope $b$.

Four patterns

Kenny & Ledermann (2010) noted that the relationship between the actor slope $b_A$ and the partner slope $b_P$ falls into one of four patterns. The ratio $k = b_P / b_A$ is what the test tracks.

Pattern	Slope configuration	Interpretation	When it arises
Actor only	$b_P = 0$	$X$ affects only its own holder	Self-contained traits (income, height)
Couple	$b_P = b_A$	$X$ has the same effect on both partners	Shared resources, joint appraisals
Contrast	$b_P = -b_A$	One partner’s $X$ increases their own $Y$ but decreases the other’s $Y$	Conflict dyads, competitive settings
Mixed	$b_P \neq 0, b_A \neq 0$, not equal or opposite	Both effects are present, with different magnitudes	Stress crossover, support transactions

The fully-constrained SEM in the SEM wide tutorial and the indistinguishability tests in the k-patterns sense generalise naturally: a non-significant departure from the actor-only pattern implies $k = 0$; from the couple pattern implies $k = 1$; from the contrast pattern implies $k = -1$.

How to test them

For each pattern, fit the constrained model and compare to a saturated model via a likelihood ratio test. A non-significant p-value supports the pattern. (See Fitzpatrick et al. (2016) for the original Mplus tutorial, replicated for lavaan in the SEM wide tutorial.)

For the actor-only pattern:

b_A * X_A + b_A * X_P
# both labels identical = constrained to be equal
# but the partner effect is allowed to be zero:
#  - "actor-only" model sets the partner effect to zero explicitly

For the couple pattern:

b_A * X_A + b_P * X_P   where b_P is constrained equal to b_A
# single label, two paths

For the contrast pattern:

b_A * X_A + (-b_A) * X_P
# the partner label is the negation of the actor label

Why the patterns matter

Two reasons.

Statistical. The partner effect is typically estimated with much less power than the actor effect, because the partner effect’s denominator is the dyad-level variance of $X$ (since the partner effect uses the partner’s $X$ as a predictor, the variation is across dyads, not within). The k-pattern tests use the actor effect as a prior and ask whether the partner effect is a specific fraction of it. This is far more powerful than asking whether the partner effect is zero.

Substantive. The pattern is a theoretical claim. “Partners buffer each other’s stress” is a couple pattern. “Conflict in one partner erodes the other’s well-being” is a contrast pattern. Saying which pattern fits your data is a meaningful scientific result, not just a statistical convenience.

Reading the partner effect

If the actor effect is $-0.30$ and the partner effect is $-0.15$, then $k = 0.5$. The partner effect is half the actor effect, and in the same direction. This is closest to a couple pattern ($k = 1$), but the partner effect is weaker than the actor effect. Whether this is “close enough” to a couple pattern to treat as one is a substantive judgment, supported by the likelihood ratio test.

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.
Fitzpatrick, J., Gareau, A., Lafontaine, M.-F., & Gaudreau, P. (2016). How to use the Actor–Partner Interdependence Model to estimate different dyadic patterns in Mplus: A step-by-step tutorial. Tutorials in Quantitative Methods for Psychology, 12(1), 74–86.

--- title: "The k-patterns" --- # The k-patterns Four ways actor and partner effects can combine. How to test which one fits your data. ## The basic APIM Let $X$ be one partner's predictor and $Y$ be the same partner's outcome. For a distinguishable dyad, the APIM fits two regressions: $$ Y_A = a_A + b_A X_A + b_P X_P \\ Y_P = a_P + b_P X_A + b_A X_P $$ where $b_A$ is the actor effect (the partner of $X$ on their own $Y$) and $b_P$ is the partner effect (the partner of $X$ on the other person's $Y$). For indistinguishable dyads, the constraint $b_A = b_P$ is imposed; the model fits a single common slope $b$. ## Four patterns Kenny & Ledermann (2010) noted that the relationship between the actor slope $b_A$ and the partner slope $b_P$ falls into one of four patterns. The ratio $k = b_P / b_A$ is what the test tracks. | Pattern | Slope configuration | Interpretation | When it arises | |---|---|---|---| | **Actor only** | $b_P = 0$ | $X$ affects only its own holder | Self-contained traits (income, height) | | **Couple** | $b_P = b_A$ | $X$ has the same effect on both partners | Shared resources, joint appraisals | | **Contrast** | $b_P = -b_A$ | One partner's $X$ *increases* their own $Y$ but *decreases* the other's $Y$ | Conflict dyads, competitive settings | | **Mixed** | $b_P \neq 0, b_A \neq 0$, not equal or opposite | Both effects are present, with different magnitudes | Stress crossover, support transactions | The fully-constrained SEM in the [SEM wide tutorial](../tutorials/distinguishable/sem-wide.html) and the indistinguishability tests in the [k-patterns sense](#) generalise naturally: a non-significant departure from the actor-only pattern implies $k = 0$; from the couple pattern implies $k = 1$; from the contrast pattern implies $k = -1$. ## How to test them For each pattern, fit the constrained model and compare to a saturated model via a likelihood ratio test. A non-significant *p*-value supports the pattern. (See Fitzpatrick et al. (2016) for the original Mplus tutorial, replicated for `lavaan` in the [SEM wide tutorial](../tutorials/distinguishable/sem-wide.html).) For the **actor-only** pattern: ``` b_A * X_A + b_A * X_P # both labels identical = constrained to be equal # but the partner effect is allowed to be zero: # - "actor-only" model sets the partner effect to zero explicitly ``` For the **couple** pattern: ``` b_A * X_A + b_P * X_P where b_P is constrained equal to b_A # single label, two paths ``` For the **contrast** pattern: ``` b_A * X_A + (-b_A) * X_P # the partner label is the negation of the actor label ``` ## Why the patterns matter Two reasons. **Statistical.** The partner effect is typically estimated with much less power than the actor effect, because the partner effect's denominator is the dyad-level variance of $X$ (since the partner effect uses the partner's $X$ as a predictor, the variation is across dyads, not within). The k-pattern tests use the actor effect as a *prior* and ask whether the partner effect is a specific fraction of it. This is far more powerful than asking whether the partner effect is zero. **Substantive.** The pattern is a theoretical claim. "Partners buffer each other's stress" is a couple pattern. "Conflict in one partner erodes the other's well-being" is a contrast pattern. Saying *which* pattern fits your data is a meaningful scientific result, not just a statistical convenience. ## Reading the partner effect If the actor effect is $-0.30$ and the partner effect is $-0.15$, then $k = 0.5$. The partner effect is half the actor effect, and in the same direction. This is closest to a couple pattern ($k = 1$), but the partner effect is *weaker* than the actor effect. Whether this is "close enough" to a couple pattern to treat as one is a substantive judgment, supported by the likelihood ratio test. ## 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. - Fitzpatrick, J., Gareau, A., Lafontaine, M.-F., & Gaudreau, P. (2016). How to use the Actor–Partner Interdependence Model to estimate different dyadic patterns in Mplus: A step-by-step tutorial. *Tutorials in Quantitative Methods for Psychology, 12*(1), 74–86.