Dyadic Data Analysis with R

Part I: Conceptual Foundations

Pascal Küng

Turu Stadler

Introduction

Turu & Pascal
Tell us about you and your dyadic data!
- Name, affiliation & career stage
- Your dyadic data and RQ

Today’s Aims and Program

Time	Focus
11:00–12:45	Introduction and types of dyadic models
12:45–14:00	Lunch break
14:00–15:30	Estimating cross-sectional models in R
15:30–15:45	Short break
15:45–16:45	Intensive longitudinal extensions
16:45–17:00	Short break
17:00–18:00	Exercises in R

Introduction

Relevance and todays’ example dataset

Why dyadic data?

“Much of what we think, do, and feel is determined not by who we are but by whom we are with.” (Kenny & Cook, 1999, p. 446)

Partners matter: An outcome may depend on both my characteristics and my partner’s characteristics.
Some constructs and processes are dyadic: Shared contexts and interdependence processes can make partners’ outcomes correlated.
The configuration matters: Partners’ similarity, or complementarity (differences) may predict outcomes beyond either individual score.

Kenny & Cook (1999); Iida et al. (2018)

Introducing today’s dyadic dataset!

38 physically inactive opposite-sex romantic couples
Both partners intended to increase their physical activity together
Smartphone-based planning interventions and JITAIs
Daily diaries over 55 days

Küng et al. (2026); Höhener et al. (2026)

Research Question 1

RQ1: In romantic couples, how is provided health-related social support linked to physical activity?

Measurement - Predictor

Provided emotional support

Rough English translation of the original German item
“How much did you emotionally support your partner today so that they would be physically active with greater enjoyment?”

Adapted from Bolger et al. (2000)

0 = not at all today
5 = very much today

Measurement - Outcome

Moderate to vigorous physical activity (MVPA)

Rough English translation of the original German item
“How many minutes did you spend engaging in moderate-to-vigorous physical activity today?” Adapted from Amireault & Godin (2015)

Alone: _____ minutes
Together with your partner: _____ minutes

Total MVPA = solo MVPA + joint MVPA.

Aggregating for cross-sectional examples

To show simple dyadic models we aggregate the daily diary data to obtain a “cross-sectional” dataset.

library(dplyr)

pilot_aggregated <- pilot |>
  group_by(couple_id, person_id, role) |>
  summarise(
    across(where(is.numeric), ~ mean(.x, na.rm = TRUE)),
    .groups = "drop"
  )

The variables now represent person-level daily averages across the study:

provided_emotional_support: average daily support provided
mvpa_total: average daily total MVPA (minutes)

Plotting provided support by partner role

support_distribution_plot <- ggplot(
  pilot_aggregated,
  aes(provided_emotional_support, y = after_stat(density))
) +
  geom_histogram(
    bins = 15, boundary = 0,
    fill = "steelblue", color = "white", alpha = 0.75
  ) +
  geom_density(
    bounds = c(0, Inf), color = "navy", linewidth = 1.1
  ) +
  facet_wrap(~role, nrow = 1) +
  labs(x = "Average provided support", y = "Density") +
  see::theme_modern(base_size = 16)

Provided support by partner role

Histograms and boundary-corrected density curves of average provided emotional support, shown separately for female and male partners.

MVPA by partner role

Histograms and boundary-corrected density curves of average daily MVPA, shown separately for female and male partners.

Different types of dyadic models

One model dominates dyadic research

A horizontal bar showing that more than 85% of 998 coded dyadic studies used an APIM or APIM variant, leaving fewer than 15% for all other model specifications combined.

Preliminary results from an ongoing systematic review (APIM use coded for \(n = 998\) eligible articles)

(Sakaluk et al., 2025, p. 5).

Many models to choose from

Various dyadic models can be used, each answering different questions!

Actor-partner interdependence model (APIM): actor and partner associations between interdependent individuals
Dyad-individual model (DIM) and dyadic score model (DSM): associations involving dyad levels and within-dyad differences
Common fate model (CFM): latent dyad-level constructs reflected in both partners’ reports

Other approaches include the mutual-influence model, for reciprocal influence between partners (Iida et al., 2018, p. 61), and the Truth and Bias model, for separating mean-level bias from correlational accuracy in judgments of a partner (Kenny & Acitelli, 2001; West & Kenny, 2011).

Many models to choose from

Various dyadic models can be used, each answering different questions!

Actor-partner interdependence model (APIM): actor and partner associations between interdependent individuals
Dyad-individual model (DIM) and dyadic score model (DSM): associations involving dyad levels and within-dyad differences
Common fate model (CFM): latent dyad-level constructs reflected in both partners’ reports

Actor-partner interdependence models

Distinguishable & Exchangeable APIM

Research Question 1a

RQ1a: Is a person’s provided emotional support associated with (a) their own MVPA and (b) their partner’s MVPA?

The distinguishable APIM

Conceptual distinguishable APIM with actor and partner paths.

Actor and partner effects together explain some of the covariance between partners’ outcomes.

The distinguishable APIM

Conceptual distinguishable APIM with role-specific actor and partner paths.

Actor and partner effects together explain some of the covariance between partners’ outcomes. \(\rightarrow\) But some covariance may remain!

How to estimate actor and partner effects?

Conceptually, we are esimating these two equations in a single model:

\[ \begin{gathered} \\[0.5em] Y_{\mathrm{male},i} = b_{0,\mathrm{M}} + b_{1,\mathrm{M}} \times X_{\mathrm{male},i} + b_{2,\mathrm{M}} \times X_{\mathrm{female},i} + \epsilon_{\mathrm{male},i} \\[2em] Y_{\mathrm{female},i} = b_{0,\mathrm{F}} + b_{1,\mathrm{F}} \times X_{\mathrm{female},i} + b_{2,\mathrm{F}} \times X_{\mathrm{male},i} + \epsilon_{\mathrm{female},i} \end{gathered} \]

Could we just use two separate models?

Male model

male_model <- glmmTMB(
  
  outcome_male ~ X_male + X_female,
  
  data = data
)

Female model

female_model <- glmmTMB(
  
  outcome_female ~ X_female + X_male,
  
  data = data
)

Could we just use two separate models?

Yes, we could! But then we:

can’t test whether the two actor effects or partner effects are equal
can’t pool equal paths into a shared estimate, which can increase power
ignore unmodelled sources of interdependence between partners
- joint modelling tells us how much interdependence remains after accounting for the predictors

(Iida et al., 2018, pp. 68–69; Kenny & Cook, 1999, pp. 439–440)

Residual interdependence in the distinguishable APIM

What remains after prediction?

Fitting the distinguishable APIM Interpretation →

Fitted distinguishable APIM for provided support and MVPA.

Ask participants to interpret

MVPA: minutes of average daily total MVPA across the 55 days

All associations are conditional on both partners’ provided support.
The intercepts, 35.28 for female partners and 31.24 for male partners, are the predicted daily MVPA when both partners’ average provided support equals the pooled sample mean (0.92 on the original scale).
Female actor effect, -4.61: for each one-unit higher support provided by a female partner, her own predicted daily MVPA is 4.61 minutes lower, holding her male partner’s support constant.
Female partner effect, 15.06: for each one-unit higher support provided by a male partner, the female partner’s predicted daily MVPA is 15.06 minutes higher, holding her own support constant.
Male actor effect, 17.92: for each one-unit higher support provided by a male partner, his own predicted daily MVPA is 17.92 minutes higher, holding his female partner’s support constant.
Male partner effect, -9.87: for each one-unit higher support provided by a female partner, the male partner’s predicted daily MVPA is 9.87 minutes lower, holding his own support constant.
The residual SDs, 21.40 minutes for female partners and 16.73 minutes for male partners, describe the remaining person-to-person spread around those predictions. An SD is a typical scale of residual deviation, not a variance and not the uncertainty of a coefficient.
The residual correlation, .78, indicates strong positive remaining within-couple interdependence: couples in which the female partner is above her prediction also tend to have a male partner above his prediction, and likewise for being below prediction. This remains after accounting for both support scores.

These are associations in the aggregated observational data, not causal effects.

The exchangeable APIM

Sometimes:

partners may not be theoretically meaningfully distinguishable
no evidence of difference in effects \(\rightarrow\) pool effects for greater power
we want to formally test distinguishability

\(\rightarrow\) Then, we may fit an exchangeable model that imposes symmetry

The exchangeable APIM

Conceptual exchangeable APIM with pooled actor and partner paths.

The exchangeable APIM

Exchangeability constraints

Distinguishable and exchangeable APIM equations, predictions, and residual constraints.

In this example, imagine we have a few couples with exactly \(X_F = 1\) and \(X_M = 1\) on the GM-centered scale. For these couples, the following would be true…

On the left:

intercepts, actor effects, partner effects, and residual standard deviations may differ by role.
Follow each prediction from its intercept through the actor contribution in blue and the partner contribution in purple.
The open circle is the conditional prediction.
- The nearby dots represent illustrative observed outcomes,
- and the arrow labelled \(\epsilon\) isolates one residual as the distance from the prediction to an observation.
The shaded area is supposed to visualize the residual variance

On the right:

the same intercept, actor effect, partner effect, and residual standard deviation are used for both arbitrary member positions.
Shared coefficients do not imply identical predictions!!: the actor and partner predictor values are exchanged between the two equations.
Equal residual variance is a constraint on the modeled residual distribution, not a claim that the two members have identical residual values.
In both models, partners’ residuals may still be correlated!

Fitting the exchangeable APIM Interpretation →

Fitted exchangeable APIM for provided support and MVPA.

Testing exchangeability

library(dyadMLM)

compare_nested_glmmTMB_models(
  exchangeable_apim,
  distinguishable_apim
)

Likelihood-ratio test for nested models fitted to equivalent data
Assumes mathematical nesting and an appropriate chi-squared reference distribution.

                     Df    AIC    BIC  logLik deviance  Chisq Chi Df Pr(>Chisq)
exchangeable_apim     5 617.55 628.94 -303.78   607.55                         
distinguishable_apim  9 611.59 632.08 -296.80   593.59 13.962      4   0.007418
                       
exchangeable_apim      
distinguishable_apim **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Conclusion (5% level): The likelihood-ratio test provides evidence that `distinguishable_apim` fits better than `exchangeable_apim` (p = 0.00742).

Level-and-difference models

Dyad-Individual Model & Dyadic Score Model

Shifting the analytical lens

Sometimes, actor and partner associations are not the primary interest.

Instead, we may want to understand:

how couples compare with other couples
how partners stand relative to one another within a couple

This shifts the focus to dyadic levels and within-dyad differences.

for exchangeable dyads, use the dyad-individual model (DIM)
for distinguishable dyads, use the dyadic score model (DSM)

Research Question 1b

RQ1b: How are couples’ average levels and partner differences in provided emotional support associated with the couple-average MVPA and partner differences in MVPA?

The dyad-individual model (DIM)

The DIM is:

A re-parametrization of the exchangeable APIM!
We decompose predictors into two variables:
- Dyad mean (GM-centered): \(\bar{X} - \mu_{X}\)
- Within-dyad member deviation: \(X_i - \bar{X}\)

In other words, these predictors express:

A couple’s average level compared with the average of all couples
A partner’s level compared with the average of their couple

DIM predictors are APIM predictors in disguise!

APIM coordinates

Actor, x_actor0.00 Partner, x_partner0.00

DIM coordinates

Dyad mean, x_mean0.00 Member deviation, x_dev0.00

x_mean = (x_actor + x_partner) / 2 x_dev = (x_actor − x_partner) / 2

APIM

DIM

b_mean = a + p

b_dev = a − p

Drag the dot or move either set of sliders. Both equations give the same fitted change in MVPA.

The DIM

Conceptual DIM with a between-dyad path from the dyad mean and a within-dyad path from the within-dyad member deviation.

The DIM

Conceptual DIM separating the between-dyad effect of the dyad mean from the within-dyad effect of the within-dyad member deviation.

Why are the mean and deviation residuals uncorrelated?

An elevator represents the dyad mean residual, which moves both partners together; a seesaw represents the within-dyad residual, which moves partners in opposite directions. Swapping arbitrary member labels reverses only the seesaw, requiring zero covariance under exchangeability.

Swap the arbitrary member labels: \(r_{\mathrm{M}} \mapsto r_{\mathrm{M}} \qquad r_{\mathrm{D}} \mapsto -r_{\mathrm{D}}\)

Exchangeability requires the same covariance after relabelling:

\[ \operatorname{Cov}(r_{\mathrm{M}},r_{\mathrm{D}}) = \operatorname{Cov}(r_{\mathrm{M}},-r_{\mathrm{D}}) = -\operatorname{Cov}(r_{\mathrm{M}},r_{\mathrm{D}}) \quad\Longrightarrow\quad \operatorname{Cov}(r_{\mathrm{M}},r_{\mathrm{D}})=0 \]

Zero does not mean independent partners: \(\operatorname{Cov}(e_1,e_2) =\operatorname{Var}(r_{\mathrm{M}})-\operatorname{Var}(r_{\mathrm{D}})\)

Their residual association is encoded by the relative elevator and seesaw variances.

POTENTIALLY SKIP FOR TIME

Elevator: once the couples’ mean value X has accounted for some variance in the couples’ outcome Y, there might still be some movement up or down that is unaccounted. \(\rightarrow\) residual

Seesaw: once a persons’ difference from the couple mean has accounted for some variance in the persons’ difference from the couple mean in Y, there might still be some additional deviation \(\rightarrow\) residual

In the DIM, the member positions on the seasaw are arbitrary. Swapping them leaves the elevator coordinate unchanged but reverses the sign of the seesaw coordinate.

You might wonder, why the elevator and seasaw residuals are not correlated.

A positive mean–deviation residual correlation would mean: - couples above their prediction on average also tend to tilt toward the arbitrarily named Member 1. - But Member 1 and Member 2 are assigned arbitrarily. Across couples, reversing these arbitrary assignments produces equally valid positive and negative signed deviations, which cancel in expectation. - Therefore, the exchangeable DIM therefore constrains the population correlation to zero.

But again, this does not make the partners independent, and this model IS exactly equivalent to the APIM

Fitting the DIM Interpretation →

Fitted DIM for provided support and MVPA.

DIM to exchangeable APIM coefficients

Once we fit an APIM, we have already fitted a DIM—and vice versa.
We can transform values from a fitted APIM into DIM values.

DIM to exchangeable APIM coefficients

The intercept is omitted; APIM predictors are grand-mean centered.

For a focal person, let \(X_{\mathrm{self}}=1,\quad X_{\mathrm{partner}}=1\),

so that in DIM coordinates: \(X_{\mathrm{mean}}=1,\quad X_{\mathrm{dev,self}}=0.\)

So,

\[ \hat Y_{\mathrm{self}}=1\times a+1\times p \]

implies

\[ \hat Y_{\mathrm{self}}=1\times b_{\mathrm{mean}}+0\times b_{\mathrm{dev}} =b_{\mathrm{mean}}. \]

Therefore,

\[ b_{\mathrm{mean}}=a+p. \]

DIM to exchangeable APIM coefficients

The intercept is omitted; APIM predictors are grand-mean centered.

For a focal person, let \(X_{\mathrm{self}}=1,\quad X_{\mathrm{partner}}=-1\),

so that in DIM coordinates: \(X_{\mathrm{mean}}=0,\quad X_{\mathrm{dev,self}}=1\).

So,

\[ \hat Y_{\mathrm{self}}=1\times a+(-1)\times p \]

implies:

\[ \hat Y_{\mathrm{self}}=0\times b_{\mathrm{mean}}+1\times b_{\mathrm{dev}} =b_{\mathrm{dev}}. \]

Therefore,

\[ b_{\mathrm{dev}}=a-p. \]

DIM to exchangeable APIM coefficients

APIM coordinates

Actor, x_actor0.00 Partner, x_partner0.00

DIM coordinates

Dyad mean, x_mean0.00 Member deviation, x_dev0.00

x_mean = (x_actor + x_partner) / 2 x_dev = (x_actor − x_partner) / 2

APIM

DIM

b_mean = a + p

b_dev = a − p

Drag the dot or move either set of sliders. Both equations give the same fitted change in MVPA.

The dyadic score model (DSM)

What if we want DIM-style interpretation but have distinguishable dyads?

The DSM is:

the unconstrained distinguishable version of the DIM
a re-parametrization of the distinguishable APIM

DIM: deviation from the couple mean \(X_i - \bar{X}\)
- opposite for each partner
DSM: meaningful directed differences \(X_\mathrm{female} - X_\mathrm{male}\)
- repeated for both partners

The DSM

Conceptual DSM with paths between predictor and outcome scores.

The DSM

Conceptual DSM linking predictor mean and difference to outcome mean and difference.

Why may the DSM mean and difference residuals correlate?

An elevator represents the dyad mean residual, which moves both partners together; a seesaw with female and male seats represents the directed outcome-difference residual. Because these substantive labels cannot be swapped arbitrarily, the two residual components may correlate.

Substantive labels: \(D_Y=Y_{\mathrm{F}}-Y_{\mathrm{M}}\)

Swapping Female and Male changes the meaning and sign of the difference, so label-swapping is not a symmetry of the DSM.

\[ \operatorname{Cov}(r_{\mathrm{M}},r_{\mathrm{D}}) \quad\text{is not constrained to zero.} \]

A positive \(\rho_{r_\mathrm{M}r_\mathrm{D}}\) means: couples whose outcomes are higher than predicted on average also tend to have a larger female–male difference than predicted.

Fitting the DSM Interpretation →

Fitted DSM for provided support and MVPA.

-> students interpret results, including correlation.

Intercepts: For a couple with an average support mean and no female–male support difference:

a10: predicted couple-mean MVPA is 33.26 minutes,
a20: the female is predicted to be 4.04 min more active than the male.

Holding the couple-difference score at zero, a 1 unit higher couple-mean support score is associated with:

a11: 9.25 additional minutes of couple-mean MVPA,
a21: a 2.4-minute larger female-male MVPA difference (The female is 2.4 minutes + 4.04 minutes more active than the male).

Holding the couple-mean score at the sample-average, a one-unit larger female-male support difference is associated with:

a12: 11.86 fewer minutes of couple-mean MVPA
a22: a 4.06-minute larger female-male MVPA difference (The female is 4.06 minutes + 4.04 minutes more active than the male)

Residual correlations

After accounting for the support mean and female–male support difference, couples whose average MVPA is higher than predicted also tend to have a more positive female–male MVPA difference than predicted.

In other words, their unexplained MVPA tends to tilt toward the female partner.

DSM to distinguishable APIM coefficients

Once we fit a DSM, we have already fitted a distinguishable APIM—and vice versa.
We can back-transform the DSM fixed effects into APIM coefficients:

\[ \begin{aligned} b_{0,\mathrm F} &=a_{10}+\tfrac12a_{20}, & b_{0,\mathrm M} &=a_{10}-\tfrac12a_{20},\\[0.4em] b_{1,\mathrm F} &=\tfrac12a_{11}+a_{12}+\tfrac14a_{21}+\tfrac12a_{22}, & b_{1,\mathrm M} &=\tfrac12a_{11}-a_{12}-\tfrac14a_{21}+\tfrac12a_{22},\\[0.4em] b_{2,\mathrm F} &=\tfrac12a_{11}-a_{12}+\tfrac14a_{21}-\tfrac12a_{22}, & b_{2,\mathrm M} &=\tfrac12a_{11}+a_{12}-\tfrac14a_{21}-\tfrac12a_{22}. \end{aligned} \]

Here, \(b_1\) is the actor effect and \(b_2\) is the partner effect. The APIM predictors and DSM couple mean now use the same pooled grand-mean reference, so no additional centering adjustment is needed for the intercepts.

Common Fate Model

Modeling Dyadic Constructs

When the construct is dyadic

Sometimes, a construct describes the couple as a unit, rather than either partner individually.

Both partners report on the same dyadic phenomenon.
Differences between their reports are not the substantive focus.
Interest lies in the couple’s underlying dyadic level.

For our data-example:

Collaborative planning as a dyadic predictor
Joint physical activity as a dyadic outcome

Research Question 3

RQ3: How is collaborative planning associated with joint physical activity in romantic couples?

Measurement - CFM predictor

Dyadic predictor: collaborative planning

Rough english translation of the original German item
“We have already planned precisely for tomorrow when, where, and how we will be physically active together.” Adapted from Scholz et al. (2008)

0 = not at all true today
to 5 = completely true today

We aggregate each person’s ratings into their average daily collaborative planning across the 55 days.

Do partners report similar dyadic levels?

Partner-agreement plots for collaborative planning and joint MVPA, with female reports on the horizontal axis, male reports on the vertical axis, and diagonal identity lines.

The distinguishable CFM

Conceptual distinguishable Common Fate Model with female and male reports as indicators of shared latent levels.

The distinguishable CFM

Conceptual distinguishable Common Fate Model with female and male indicator intercepts, shared latent predictor and outcome levels, and symbolic standard deviations for all four indicator residuals.

Fitted distinguishable CFM

Model constraint: The female and male joint-MVPA residual variances were constrained equal to obtain admissible fit.

Fitted Common Fate Model for collaborative planning and joint MVPA, including four observed indicator intercepts and unstandardized standard deviations for all four indicator residuals.

When should we model a CFM as exchangeable?

Partner positions are arbitrary, or theory and the data support treating the measurement process as invariant across roles.
Swapping the partner labels should then leave the model unchanged.
We constrain corresponding partner parameters to be equal:
- intercepts: the same expected report level
- residual variances: the same amount of indicator-specific variation
- same-member residual associations: the same residual predictor–outcome association for both partners

Ledermann & Macho (2009); Peugh et al. (2013)

Fitted CFM with partner-role equality constraints

Fitted exchangeable Common Fate Model for collaborative planning and joint MVPA. Female and male indicator intercepts, corresponding residual variances, and same-member residual covariances are constrained to equality.

Likelihood-ratio test of the remaining partner-role equality constraints: \(\Delta\chi^2(4) = 0.65\), \(p = .958\); no detectable loss of fit.

Conclusion

RQ3: In this small observational sample, a one-unit higher latent collaborative-planning level was associated with 12.30 more minutes of joint MVPA per day at the couple level (\(R^2 = 0.39\)).

Other interesting dyadic models to consider

Other dyadic models and designs

Truth and Bias model: Separates accuracy in judging a partner from directional bias and assumed similarity (West & Kenny, 2011).
Mutual-influence model: Tests whether partners’ outcomes directly and reciprocally influence one another (Kenny, 1996).
APIM mediation model: Traces whether actor and partner associations operate through either partner’s mediator (Ledermann et al., 2011).
Dyadic response surface analysis: Tests whether partner similarity, discrepancy, or particular combinations of scores relate to an outcome (Schönbrodt et al., 2018).
One-with-many and Social Relations models: Extend dyadic analysis to repeated relationships with several partners or round-robin groups, separating person and relationship components (Kenny et al., 2006; Kenny & La Voie, 1984).
Network models: Explain the formation and evolution of ties within an interconnected group, including reciprocity, clustering, and actor-driven network change (Holland & Leinhardt, 1981; Snijders, 2001).

The Truth and Bias model (Kenny & Acitelli, 2001; West & Kenny, 2011)

Dyadic Truth and Bias model in which female and male self-reports predict both partner judgments. Same-perceiver paths represent assumed-similarity bias, cross-partner paths represent direct accuracy, intercepts represent directional bias, self-report covariance represents actual similarity, and judgment residuals covary.

Direct accuracy tracks the target’s report; assumed similarity tracks the perceiver’s own report.
The target’s report is the operational “truth,” not necessarily objective truth; after centering, \(b_0\) is directional bias.

Has the structure of a distinguishable APIM.
The subscript on each path coefficient identifies the perceiver.
The blue actor paths are assumed-similarity bias: each person’s own report predicts their judgment of their partner.
The purple partner paths are direct accuracy: the target’s report predicts the perceiver’s judgment of that target, controlling for assumed similarity.
The correlation between self-reports is actual similarity. Together with assumed similarity, it can produce indirect accuracy even if direct accuracy is zero.
Bias is systematic information other than the truth criterion, not random error and not necessarily inaccuracy.
Self-reports and judgments must use the same scale and be centered on the relevant truth-criterion mean for the intercept to represent adjusted directional bias.
In the basic observed-variable model, the target’s self-report is treated as an error-free criterion for estimation. This defines accuracy relative to the target’s report; it does not establish objective truth or perfect self-knowledge.
The paths are predictive associations, not necessarily causal effects.
Direct accuracy is the truth association remaining after the measured bias variable has been controlled.
Separate standardization by partner would make the role-specific coefficients difficult to compare; the model is interpreted using unstandardized paths on a common metric.

Dyadic response surface analysis (Schönbrodt et al., 2018)

Two hypothetical response surfaces from one dyadic model. The first shows Partner A's predicted satisfaction and the second Partner B's predicted satisfaction, each as a function of both partners' support scores.

One joint model: Both partner outcomes are estimated with correlated residuals.
Two surfaces: Each visualizes one partner-specific coefficient set.

These are deliberately hypothetical surfaces, not results from today’s data.
Both surfaces come from the same conceptual joint model: the two polynomial outcome equations are estimated simultaneously and their residuals correlate.
The horizontal axes stay in the same orientation in both plots. Only the outcome on the vertical axis and its regression coefficients change.
Dyadic response surface analysis can test equality constraints across the two partners’ surface parameters.
Along the line of congruence, both partners have the same predictor score; the first surface illustrates a rising high-high congruence pattern.
The second surface deliberately differs to illustrate that the same predictor combination may relate differently to the two partners’ outcomes.
A response surface is more informative than a difference score because the same difference can arise from substantively different combinations of the two partners’ levels.

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Appendix

Interpreting the distinguishable APIM ← Results

Red estimates are not statistically distinguishable from zero at \(\alpha = .05\). Their numerical translations are shown only to demonstrate how APIM coefficients are interpreted; they should not be treated as evidence of a nonzero effect.

Intercepts: When both partners’ support scores equal the pooled sample mean:

\(b_{0,\mathrm{F}}\): Predicted female MVPA is 35.28 minutes.
\(b_{0,\mathrm{M}}\): Predicted male MVPA is 31.24 minutes.

Female MVPA, holding the male partner’s support constant:

\(a_{\mathrm{F}}\): One-unit higher female support is associated with 4.61 fewer minutes of her own MVPA.
\(p_{\mathrm{F}}\): One-unit higher male support is associated with 15.06 additional minutes of female MVPA.

Male MVPA, holding the female partner’s support constant:

\(a_{\mathrm{M}}\): One-unit higher male support is associated with 17.92 additional minutes of his own MVPA.
\(p_{\mathrm{M}}\): One-unit higher female support is associated with 9.87 fewer minutes of male MVPA.

Residual correlation (\(\rho = .78\)): When the female’s MVPA is higher than predicted, the male’s MVPA also tends to be higher than predicted, and likewise for being lower. This is the remaining within-couple interdependence after accounting for both support scores.

Interpreting the exchangeable APIM ← Results

The intercept, actor effect, partner effect, and residual SD are pooled across members.

Pooled intercept: When both partners’ support scores equal the pooled sample mean:

\(b_0\): Predicted MVPA is 31.21 minutes for either member position.

Pooled effects: For either member, holding the other support score constant:

\(a\): One-unit higher own support is associated with 7.75 additional minutes of own MVPA.
\(p\): One-unit higher partner support is associated with 1.97 additional minutes of own MVPA.

Pooling means that the same coefficients are used for both member positions. It does not require the two members to have identical predictions because their own and partner support scores may differ.

Residual correlation (\(\rho = .77\)): When one member’s MVPA is higher than predicted, the partner’s MVPA also tends to be higher than predicted, and likewise for being lower. The residual correlation remains freely estimated even though the other parameters are pooled.

Interpreting the fitted DIM ← Results

Red estimates are not statistically distinguishable from zero at \(\alpha = .05\). Their numerical translations are shown only to demonstrate how DIM coefficients are interpreted; they should not be treated as evidence of a nonzero effect.

Intercepts: For a couple with an average support mean:

\(b_0\): Predicted couple-mean MVPA is 31.21 minutes.
The member-deviation intercept is constrained to zero: a member with no support deviation has no predicted MVPA deviation from the couple mean.

Holding the members’ support deviations fixed, a one-unit higher couple-mean support score is associated with:

\(b_{\mathrm{mean}}\): 9.72 additional minutes of couple-mean MVPA.

Holding couple-mean support at the sample average, a one-unit larger member support deviation from the couple mean is associated with:

\(b_{\mathrm{dev}}\): A 5.78-minute larger member MVPA deviation from the couple mean. Thus, a member whose support is one unit above the couple mean is predicted to have MVPA 5.78 minutes above the couple mean.

Residual correlation: Exchangeability constrains the correlation between the residual couple mean and residual member deviation to zero. Thus, couples whose MVPA is higher than predicted on average are not expected to tilt systematically toward either arbitrarily labeled member in residual MVPA. This is a model constraint, not an empirical finding, and it does not imply that the two partners’ residuals are independent.

Interpreting the fitted DSM ← Results

Red estimates are not statistically distinguishable from zero at \(\alpha = .05\). Their numerical translations are shown only to demonstrate how DSM coefficients are interpreted; they should not be treated as evidence of a nonzero effect.

Intercepts: For a couple with an average support mean and no female–male support difference:

\(a_{10}\): Predicted couple-mean MVPA is 33.26 minutes.
\(a_{20}\): The female is predicted to be 4.04 minutes more active than the male.

Holding the female–male support difference at zero, a one-unit higher couple-mean support score is associated with:

\(a_{11}\): 9.25 additional minutes of couple-mean MVPA.
\(a_{21}\): A 2.40-minute larger female–male MVPA difference. Thus, when couple-mean support is one unit above the sample average, the predicted difference is \(4.04 + 2.40 = 6.44\) minutes.

Holding couple-mean support at the sample average, a one-unit larger female–male support difference is associated with:

\(a_{12}\): 11.86 fewer minutes of couple-mean MVPA.
\(a_{22}\): A 4.06-minute larger female–male MVPA difference. Thus, when the female–male support difference is one unit, the predicted difference is \(4.04 + 4.06 = 8.10\) minutes.

Residual correlation (\(\rho = .37\)): Couples with higher-than-predicted mean MVPA also tend to show a more positive-than-predicted female–male MVPA difference; residual MVPA therefore tilts toward the female.

Exchangeable CFM: constraints and degrees of freedom

Positions 1 and 2 are arbitrary. The model must be unchanged when they are swapped (Ledermann & Macho, 2009; Peugh et al., 2013).

\[ \begin{aligned} \nu_{X_1} &= \nu_{X_2}, & \nu_{Y_1} &= \nu_{Y_2} \\ \sigma^2_{\epsilon_{X_1}} &= \sigma^2_{\epsilon_{X_2}}, & \sigma^2_{\epsilon_{Y_1}} &= \sigma^2_{\epsilon_{Y_2}} \\ \operatorname{Cov}(\epsilon_{X_1},\epsilon_{Y_1}) &= \operatorname{Cov}(\epsilon_{X_2},\epsilon_{Y_2}) \end{aligned} \]

All four loadings remain fixed to 1.
Ordinary wide-data df = 6; df = 0 against the exchangeable saturated model.
Usual exchangeability test: \(\Delta df = 5\); here: \(\Delta df = 4\) (\(\sigma^2_{\epsilon_Y}\) is already equal).

Equality is imposed on the two same-member residual covariances. Because the corresponding residual variances are also equal, the residual correlations are equal as a consequence.
lavaan reports 6 degrees of freedom because its default saturated reference model is fully unrestricted. Six observed-moment restrictions are required for exchangeability.
The exchangeable CFM reproduces the exchangeable saturated model, so it has zero degrees of freedom relative to that appropriate reference model. Its ordinary chi-square is therefore a test of exchangeability, not an additional test of the basic CFM structure.
An unconstrained distinguishable CFM has df = 1, giving the usual five-degree- of-freedom exchangeability test. Our distinguishable model already equates the two outcome residual variances and has df = 2, so the comparison tests the four remaining equality constraints.

Distinguishable APIM: residual covariance

Within-dyad residual covariance block (shared across dyads):

\[ \operatorname{Cov} \begin{pmatrix} \epsilon_{Fi} \\ \epsilon_{Mi} \end{pmatrix} = \boldsymbol{\Sigma}_{\epsilon} = \begin{bmatrix} \sigma_{\epsilon_F}^{2} & \rho_{\epsilon_F\epsilon_M}\sigma_{\epsilon_F}\sigma_{\epsilon_M} \\ \rho_{\epsilon_F\epsilon_M}\sigma_{\epsilon_F}\sigma_{\epsilon_M} & \sigma_{\epsilon_M}^{2} \end{bmatrix} \]

Residual covariance matrix for all dyads (first three shown):

\[ \boldsymbol{\Sigma}_{\mathrm{model}} = \begin{bmatrix} \boldsymbol{\Sigma}_{\epsilon} & \boldsymbol{0} & \boldsymbol{0} & \cdots \\ \boldsymbol{0} & \boldsymbol{\Sigma}_{\epsilon} & \boldsymbol{0} & \cdots \\ \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{\Sigma}_{\epsilon} & \cdots \\ \vdots & \vdots & \vdots & \ddots \end{bmatrix} \]

Exchangeable APIM: residual covariance

For arbitrary member positions 1 and 2, the residual variances must be equal:

\[ \operatorname{Cov} \begin{pmatrix} \epsilon_{1i} \\ \epsilon_{2i} \end{pmatrix} = \boldsymbol{\Sigma}_{\epsilon,\mathrm{exch}} = \begin{bmatrix} \sigma_{\epsilon}^{2} & \rho_{\epsilon_1\epsilon_2}\sigma_{\epsilon}^{2} \\ \rho_{\epsilon_1\epsilon_2}\sigma_{\epsilon}^{2} & \sigma_{\epsilon}^{2} \end{bmatrix} \]

The equal diagonal elements express exchangeability.
The off-diagonal element still allows residual interdependence.
Across dyads, these blocks form the same block-diagonal model matrix as on the previous slide.

DIM: residual covariance

\[ \begin{aligned} r_{\mathrm{M}i} &= \bar{\epsilon}_i = \frac{\epsilon_{1i}+\epsilon_{2i}}{2}, \\ r_{\mathrm{D},mi} &= \epsilon_{mi}-r_{\mathrm{M}i} \quad (m=1,2), \qquad r_{\mathrm{D},1i} = \frac{\epsilon_{1i}-\epsilon_{2i}}{2} = -r_{\mathrm{D},2i}. \end{aligned} \]

Using the arbitrary orientation \(r_{\mathrm{D}i}=r_{\mathrm{D},1i}\), exchangeability gives:

\[ \operatorname{Cov} \begin{pmatrix} r_{\mathrm{M}i} \\ r_{\mathrm{D}i} \end{pmatrix} = \begin{bmatrix} \sigma_{r_\mathrm{M}}^{2} & 0 \\ 0 & \sigma_{r_\mathrm{D}}^{2} \end{bmatrix}. \]

Back-transform to the exchangeable APIM: partner covariance need not be zero.

\[ \begin{bmatrix} \color{#1769AA}{\sigma_{r_\mathrm{M}}^{2}+\sigma_{r_\mathrm{D}}^{2}} & \color{#7B3FA1}{\sigma_{r_\mathrm{M}}^{2}-\sigma_{r_\mathrm{D}}^{2}} \\[0.35em] \color{#7B3FA1}{\sigma_{r_\mathrm{M}}^{2}-\sigma_{r_\mathrm{D}}^{2}} & \color{#1769AA}{\sigma_{r_\mathrm{M}}^{2}+\sigma_{r_\mathrm{D}}^{2}} \end{bmatrix} = \begin{bmatrix} \color{#1769AA}{\sigma_{\epsilon}^{2}} & \color{#7B3FA1}{\rho_{\epsilon_1\epsilon_2}\sigma_{\epsilon}^{2}} \\[0.35em] \color{#7B3FA1}{\rho_{\epsilon_1\epsilon_2}\sigma_{\epsilon}^{2}} & \color{#1769AA}{\sigma_{\epsilon}^{2}} \end{bmatrix}. \]

DSM: residual covariance

With distinguishable roles, the directed difference may covary with the mean:

\[ r_{\mathrm{M}i}=\frac{\epsilon_{Fi}+\epsilon_{Mi}}{2}, \qquad r_{\mathrm{D}i}=\epsilon_{Fi}-\epsilon_{Mi}. \]

\[ \operatorname{Cov} \begin{pmatrix} r_{\mathrm{M}i} \\ r_{\mathrm{D}i} \end{pmatrix} = \begin{bmatrix} \sigma_{r_\mathrm{M}}^{2} & \rho_{r_\mathrm{M}r_\mathrm{D}} \sigma_{r_\mathrm{M}}\sigma_{r_\mathrm{D}} \\ \rho_{r_\mathrm{M}r_\mathrm{D}} \sigma_{r_\mathrm{M}}\sigma_{r_\mathrm{D}} & \sigma_{r_\mathrm{D}}^{2} \end{bmatrix}. \]

Back-transform to the distinguishable APIM, with \(c_{\mathrm{MD}} =\rho_{r_\mathrm{M}r_\mathrm{D}} \sigma_{r_\mathrm{M}}\sigma_{r_\mathrm{D}}\).

\[ \begin{bmatrix} \color{#1769AA}{ \sigma_{r_\mathrm{M}}^{2}+\frac{1}{4}\sigma_{r_\mathrm{D}}^{2}+c_{\mathrm{MD}}} & \color{#7B3FA1}{ \sigma_{r_\mathrm{M}}^{2}-\frac{1}{4}\sigma_{r_\mathrm{D}}^{2}} \\[0.35em] \color{#7B3FA1}{ \sigma_{r_\mathrm{M}}^{2}-\frac{1}{4}\sigma_{r_\mathrm{D}}^{2}} & \color{#B45309}{ \sigma_{r_\mathrm{M}}^{2}+\frac{1}{4}\sigma_{r_\mathrm{D}}^{2}-c_{\mathrm{MD}}} \end{bmatrix} = \begin{bmatrix} \color{#1769AA}{\sigma_{\epsilon_F}^{2}} & \color{#7B3FA1}{ \rho_{\epsilon_F\epsilon_M}\sigma_{\epsilon_F}\sigma_{\epsilon_M}} \\[0.35em] \color{#7B3FA1}{ \rho_{\epsilon_F\epsilon_M}\sigma_{\epsilon_F}\sigma_{\epsilon_M}} & \color{#B45309}{\sigma_{\epsilon_M}^{2}} \end{bmatrix}. \]