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)

Measurement - Predictor

Emotional support provided

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"
  )

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

MVPA by partner role

One model dominates dyadic research

Many models to choose from

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

• Actor-partner interdependence model (APIM): interdependent individuals
• Dyad-individual model (DIM) and dyadic score model (DSM): within- and between dyad associations
• Common fate model (CFM): latent dyadic process

Many models to choose from

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

• Actor-partner interdependence model (APIM): interdependent individuals
• Dyad-individual model (DIM) and dyadic score model (DSM): within- and between dyad associations
• Common fate model (CFM): latent dyadic process

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).

The distinguishable APIM

The distinguishable APIM

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?

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

Residual interdependence in the distinguishable APIM

What remains after prediction?

Fitting the distinguishable APIM

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

The exchangeable APIM

Exchangeability constraints

Fitting the exchangeable APIM

Testing exchangeability

library(interdep)

compare_interdep_models(
  exchangeable_apim,
  distinguishable_apim
)

Likelihood-ratio test for nested models fitted to equivalent interdep 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).

The DIM

• A re-parametrization of the 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!

The DIM

The DIM

Why are the DIM residuals uncorrelated?

Fitting the DIM

DIM to 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.

Suppose \(X_i=1\) and \(X_j=1\),

then our DIM coordinates are:

\[ X_{\mathrm{mean}}=1,\qquad X_{\mathrm{dev},i}=0. \]

So,

\[ \hat Y_i=1\times a+1\times p \]

implies

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

Therefore,

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

DIM to APIM coefficients

Now suppose \(X_i=1\) and \(X_j=-1\) (imagine centered values),

then the DIM coordinates are:

\[ X_{\mathrm{mean}}=0,\qquad X_{\mathrm{dev},i}=1 \]

(\(i\) is 1 higher than the mean and 2 higher than partner \(j\)).

So,

\[ \hat Y_i=1\times a+(-1)\times p \]

implies for the higher partner:

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

Therefore,

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

DIM to APIM coefficients

The DSM

The dyadic score model (DSM) is:

• the unconstrained distinguishable version of the DIM

• a re-parametrization of the distinguishable APIM

• Just like the DIM, it estimates how predictor dyad mean and differences

• While the DIM uses deviation from the couple mean which is opposite for each partners \(X_i - \bar{X}\), the DSM uses meaningful directed differences, e.g., \(X_\mathrm{female} - X_\mathrm{male}\).

  • This allows the estimation of cross-paths, since both partners’ values do not cancel out like in the DIM$

The DSM

The DSM

In the DSM, the seats are labelled

Fitting the DSM

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.

Collaborative planning by partner role

Joint MVPA by partner role

The CFM

The CFM

Fitted distinguishable CFM

The exchangeable CFM

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).

Fitted exchangeable CFM

Honorable mention: The truth and bias model

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.

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 \(c_{\mathrm{MD}} =\rho_{r_\mathrm{M}r_\mathrm{D}} \sigma_{r_\mathrm{M}}\sigma_{r_\mathrm{D}}\).