Dyadic Score Model (DSM)

Pascal Küng

library(interdep)
has_glmmTMB <- requireNamespace("glmmTMB", quietly = TRUE)
dsm_fitted_alt <- "Fitted DSM diagram unavailable."

This vignette focuses on the Dyadic Score Model (DSM) for distinguishable dyads and its relationship to the distinguishable Actor-Partner Interdependence Model (APIM). The DSM expresses associations in terms of the dyad’s shared level and the directional difference between partners (Iida et al. 2018).

For the broader package workflow and an overview of the available model-specific vignettes, including the Actor-Partner Interdependence Model, Mixed-Composition APIM, and Dyad-Individual Model, see the Overview.

Preparing DSM Data

A DSM requires an explicitly declared direction. This direction should be substantively meaningful when the directional coefficients are interpreted. Here, c("female", "male") defines every difference as female minus male.

cross_dsm_data <- prepare_interdep_data(
  example_dyadic_crosssectional,
  group = coupleID,
  member = personID,
  role = gender,
  predictors = communication,
  model_type = "dsm",
  dsm_role_order = c("female", "male")
)

print(cross_dsm_data, n = 4)
#> # interdep data
#> # Rows: 190 | Dyads: 95 | Intensive longitudinal: no
#> # Structure: group = coupleID, member = personID, role = gender
#> # DSM direction: female - male
#> #
#> # Dyad compositions:
#> # female_x_male distinguishable 95 dyads
#> #
#> # Added columns:
#> #   .i_composition              inferred dyad composition
#> #   .i_composition_role         composition-specific member role
#> #   .i_is_{comp-role}           composition-role indicator columns
#> #   .i_dsm_role_contrast        DSM role contrast: +0.5 for the first declared
#> #                               role and -0.5 for the second declared role
#> #   .i_{pred}_dyad_mean_gmc     dyad-mean predictor: dyad's average predictor
#> #                               level, grand-mean centered
#> #   .i_{pred}_within_dyad_diff  DSM signed predictor difference: first declared
#> #                               role minus second declared role
#> #
#> # A tibble: 190 × 12
#>   personID coupleID gender communication satisfaction .i_composition
#>      <int>    <int> <fct>          <dbl>        <dbl> <fct>         
#> 1        1        1 female          4.79         4.37 female_x_male 
#> 2        2        1 male            3.80         2.34 female_x_male 
#> 3        3        2 female          2.91         2.44 female_x_male 
#> 4        4        2 male            6.51         6.08 female_x_male 
#> # ℹ 186 more rows
#> # ℹ 6 more variables: .i_composition_role <fct>,
#> #   .i_is_female_x_male_female <dbl>, .i_is_female_x_male_male <dbl>,
#> #   .i_dsm_role_contrast <dbl>, .i_communication_dyad_mean_gmc <dbl>,
#> #   .i_communication_within_dyad_diff <dbl>

For predictor values $X_{\mathrm{female}}$ and $X_{\mathrm{male}}$, interdep then creates:

• .i_communication_dyad_mean_gmc $= \frac{X_{\mathrm{female}} + X_{\mathrm{male}}}{2} - \mu_X$ (with $\mu_X$ representing the sample grand mean of the dyad-level predictor means)

• .i_communication_within_dyad_diff $= X_{\mathrm{female}} - X_{\mathrm{male}}$

• .i_dsm_role_contrast $= +0.5$ for female and $-0.5$ for male.

The dyad mean and signed difference are repeated on both member rows. The outcome remains unchanged and no transformation is needed.

Cross-Sectional Gaussian DSM

For a cross-sectional Gaussian DSM, a correlated dyad random intercept and role-contrast slope represent unexplained outcome-level and outcome-difference variation.

Conceptual cross-sectional DSM. Predictor mean and predictor difference each predict both outcome scores.

The same model can be displayed in the individual-member rows used by the long-format multilevel model.

Individual-level representation of the cross-sectional DSM used for the long-format multilevel model. The centered predictor mean and female-minus-male predictor difference appear on both member rows. For the female outcome, the intercept and slopes add one-half of the corresponding outcome-difference parameters. For the male outcome, they subtract one-half. The female and male residuals may have different variances and covary.

The path labels correspond directly to the terms in the model below:

dsm_model <- glmmTMB::glmmTMB(
  satisfaction ~

    # Outcome-level intercept
    1 +

    # Predictor level -> outcome level (a11)
    .i_communication_dyad_mean_gmc +

    # Predictor difference -> outcome level (a12)
    .i_communication_within_dyad_diff +

    # Outcome-difference intercept (a20)
    .i_dsm_role_contrast +

    # Predictor level -> outcome difference (a21)
    .i_communication_dyad_mean_gmc:.i_dsm_role_contrast +

    # Predictor difference -> outcome difference (a22)
    .i_communication_within_dyad_diff:.i_dsm_role_contrast +

    # Outcome-level and outcome-difference residual variances and their covariance
    us(1 + .i_dsm_role_contrast | coupleID),
  dispformula = ~ 0,
  family = gaussian(),
  data = cross_dsm_data
)

summary(dsm_model)
#>  Family: gaussian  ( identity )
#> Formula:          
#> satisfaction ~ 1 + .i_communication_dyad_mean_gmc + .i_communication_within_dyad_diff +  
#>     .i_dsm_role_contrast + .i_communication_dyad_mean_gmc:.i_dsm_role_contrast +  
#>     .i_communication_within_dyad_diff:.i_dsm_role_contrast +  
#>     us(1 + .i_dsm_role_contrast | coupleID)
#> Dispersion:                    ~0
#> Data: cross_dsm_data
#> 
#>       AIC       BIC    logLik -2*log(L)  df.resid 
#>     589.5     618.0    -285.7     571.5       167 
#> 
#> Random effects:
#> 
#> Conditional model:
#>  Groups   Name                 Variance Std.Dev. Corr  
#>  coupleID (Intercept)          0.632    0.795          
#>           .i_dsm_role_contrast 3.678    1.918    -0.16 
#> Number of obs: 176, groups:  coupleID, 88
#> 
#> Conditional model:
#>                                                        Estimate Std. Error
#> (Intercept)                                             5.04256    0.08481
#> .i_communication_dyad_mean_gmc                          1.99024    0.07834
#> .i_communication_within_dyad_diff                      -0.03201    0.05372
#> .i_dsm_role_contrast                                    0.95644    0.20459
#> .i_communication_dyad_mean_gmc:.i_dsm_role_contrast    -0.13847    0.18899
#> .i_communication_within_dyad_diff:.i_dsm_role_contrast  1.48641    0.12959
#>                                                        z value Pr(>|z|)    
#> (Intercept)                                              59.46  < 2e-16 ***
#> .i_communication_dyad_mean_gmc                           25.41  < 2e-16 ***
#> .i_communication_within_dyad_diff                        -0.60    0.551    
#> .i_dsm_role_contrast                                      4.67 2.94e-06 ***
#> .i_communication_dyad_mean_gmc:.i_dsm_role_contrast      -0.73    0.464    
#> .i_communication_within_dyad_diff:.i_dsm_role_contrast   11.47  < 2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Interpreting the DSM paths

For the outcomes given the predictors, the long-format model estimates the same paths as the conventional score-based DSM (Iida et al. 2018).

In the conventional score-based representation, the predictors are decomposed in the same way:

$$X_{\mathrm{mean}} = \frac{X_{\mathrm{female}} + X_{\mathrm{male}}}{2} - \mu_X, \qquad X_{\mathrm{diff}} = X_{\mathrm{female}} - X_{\mathrm{male}},$$

where $\mu_X$ is the sample grand mean of the dyad-level predictor means.

The outcomes are also decomposed:

$$Y_{\mathrm{mean}} = \frac{Y_{\mathrm{female}} + Y_{\mathrm{male}}}{2}, \qquad Y_{\mathrm{diff}} = Y_{\mathrm{female}} - Y_{\mathrm{male}}.$$

The long-format model fitted here does not create $Y_{\mathrm{mean}}$ and $Y_{\mathrm{diff}}$ as observed variables. Instead, it uses the member-level outcome directly. With complete outcome pairs, this is an equivalent parameterization of the same conditional outcome regressions.

Consider the conceptual SEM formulas:

$$\widehat{Y_{\mathrm{mean}}} = a_{10} + a_{11}X_{\mathrm{mean}} + a_{12}X_{\mathrm{diff}},$$

$$\widehat{Y_{\mathrm{diff}}} = a_{20} + a_{21}X_{\mathrm{mean}} + a_{22}X_{\mathrm{diff}}.$$

The fitted paths for this example are:

Fitted cross-sectional DSM for the example data. The nodes identify the mean and difference scores; edge and intercept labels show the estimated DSM coefficients, and the residual labels show the estimated score-component standard deviations and correlation.

The fixed effects from our MLM model map directly to these paths as such:

Long-format fixed effect	DSM SEM path and interpretation
Intercept	$a_{10}$: expected dyad-average satisfaction at the sample-average communication level and no female-male communication difference
Communication dyad mean	$a_{11}$: predictor level $\rightarrow$ outcome level
Communication difference	$a_{12}$: predictor difference $\rightarrow$ outcome level
DSM role contrast	$a_{20}$: expected female-minus-male outcome difference at the predictor reference values
Dyad mean $\times$ role contrast	$a_{21}$: predictor level $\rightarrow$ outcome difference
Communication difference $\times$ role contrast	$a_{22}$: predictor difference $\rightarrow$ outcome difference

Thus, for example, $a_{12}$ is the change in dyad-average satisfaction associated with a one-unit larger female-minus-male communication difference, holding communication level constant. In contrast, $a_{22}$ is the change in the female-minus-male satisfaction difference associated with that same one-unit larger communication difference, holding communication level constant. The $a_{12}$ and $a_{21}$ coefficients are the DSM cross-paths. They are omitted from the reduced DSM but are needed for the full model.

The random intercept variance is unexplained variation in outcome level, and the random role-contrast slope variance is unexplained variation in the full directional outcome difference. Their covariance indicates whether unexplained outcome level and unexplained outcome difference are associated.

The curved arrow $\rho_{r_m r_d}$ is the scale-free correlation between the outcome-mean residual and the outcome-difference residual. It is not the female-male residual correlation $\rho_{\epsilon_F\epsilon_M}$ from the APIM.

The DSM uses the full female-minus-male difference:

$$e_{\mathrm{F}} = r_{\mathrm{m}} + \frac{1}{2}r_{\mathrm{d}}, \qquad e_{\mathrm{M}} = r_{\mathrm{m}} - \frac{1}{2}r_{\mathrm{d}}.$$

Therefore, this relationship applies:

$$\operatorname{Cov}(r_{\mathrm{m}},r_{\mathrm{d}}) = \frac{\operatorname{Var}(e_{\mathrm{F}})-\operatorname{Var}(e_{\mathrm{M}})}{2}.$$

For this reason, a nonzero $\rho_{r_m r_d}$ indicates that the two roles have different residual variances. The remaining covariance between the partners’ residuals is

$$\operatorname{Cov}(e_{\mathrm{F}},e_{\mathrm{M}}) = \operatorname{Var}(r_{\mathrm{m}})-\frac{1}{4}\operatorname{Var}(r_{\mathrm{d}}).$$

Reversing the coding

Instead of computing $X_{\mathrm{female}} - X_{\mathrm{male}}$, we can reverse the direction and compute $X_{\mathrm{male}} - X_{\mathrm{female}}$. This changes the direction of the differences, but not the substantive model.

cross_dsm_data_inverted <- prepare_interdep_data(
  example_dyadic_crosssectional,
  group = coupleID,
  member = personID,
  role = gender,
  predictors = communication,
  # Request APIM columns too for comparison below.
  model_type = c("dsm", "apim"),
  dsm_role_order = c("male", "female")
)

dsm_model_inverted <- glmmTMB::glmmTMB(
  satisfaction ~
    .i_communication_dyad_mean_gmc +
    .i_communication_within_dyad_diff +
    .i_dsm_role_contrast +
    .i_communication_dyad_mean_gmc:.i_dsm_role_contrast +
    .i_communication_within_dyad_diff:.i_dsm_role_contrast +
    us(1 + .i_dsm_role_contrast | coupleID),
  dispformula = ~ 0,
  family = gaussian(),
  data = cross_dsm_data_inverted
)

female_minus_male <- glmmTMB::fixef(dsm_model)$cond
male_minus_female <- glmmTMB::fixef(dsm_model_inverted)$cond

knitr::kable(
  data.frame(
    `model term` = names(female_minus_male),
    `female - male` = unname(female_minus_male),
    `male - female` = unname(male_minus_female),
    check.names = FALSE
  ),
  digits = 3,
  align = c("l", "r", "r")
)

model term	female - male	male - female
(Intercept)	5.043	5.043
.i_communication_dyad_mean_gmc	1.990	1.990
.i_communication_within_dyad_diff	-0.032	0.032
.i_dsm_role_contrast	0.956	-0.956
.i_communication_dyad_mean_gmc:.i_dsm_role_contrast	-0.138	0.138
.i_communication_within_dyad_diff:.i_dsm_role_contrast	1.486	1.486

The two models have identical fitted values and model fit:

• .i_communication_within_dyad_diff reverses because the predictor difference reverses.

• .i_dsm_role_contrast reverses because the represented outcome difference reverses.

• .i_communication_dyad_mean_gmc:.i_dsm_role_contrast reverses because only the outcome difference reverses.

• .i_communication_within_dyad_diff:.i_dsm_role_contrast remains unchanged because both differences reverse.

The intercept and .i_communication_dyad_mean_gmc also remain unchanged. For the random effects, the variances of (Intercept) and .i_dsm_role_contrast remain unchanged, whereas their covariance reverses sign.

Relationship to the APIM and DIM

For distinguishable dyads, the full DSM and an unconstrained distinguishable APIM are alternative parameterizations of the same fixed associations (Iida et al. 2018).

Let’s fit the equivalent distinguishable APIM:

apim_model <- glmmTMB::glmmTMB(
  satisfaction ~
    # Role-specific intercepts
    0 +
    .i_is_female_x_male_female +
    .i_is_female_x_male_male +

    # Role-specific actor effects
    .i_is_female_x_male_female:.i_communication_actor +
    .i_is_female_x_male_male:.i_communication_actor +

    # Role-specific partner effects
    .i_is_female_x_male_female:.i_communication_partner +
    .i_is_female_x_male_male:.i_communication_partner +

    # Role-specific Gaussian residual covariance structure
    us(0 +
         .i_is_female_x_male_female +
         .i_is_female_x_male_male
       | coupleID),
  dispformula = ~ 0,
  family = gaussian(),
  data = cross_dsm_data_inverted
)

The two DSM directions and the APIM have identical fit statistics:

data.frame(
  model = c("DSM: female - male", "DSM: male - female", "APIM"),
  AIC = round(c(AIC(dsm_model), AIC(dsm_model_inverted), AIC(apim_model)), 3),
  BIC = round(c(BIC(dsm_model), BIC(dsm_model_inverted), BIC(apim_model)), 3),
  logLik = round(c(
    as.numeric(logLik(dsm_model)),
    as.numeric(logLik(dsm_model_inverted)),
    as.numeric(logLik(apim_model))
  ), 3)
)
#>                model     AIC     BIC   logLik
#> 1 DSM: female - male 589.491 618.026 -285.746
#> 2 DSM: male - female 589.491 618.026 -285.746
#> 3               APIM 589.491 618.026 -285.746

Fixed-effect transformation

Let $b_{\mathrm{actor},\mathrm{female}}$ and $b_{\mathrm{actor},\mathrm{male}}$ denote the actor effects on the female and male outcomes. The corresponding partner effects are $b_{\mathrm{partner},\mathrm{female}}$ and $b_{\mathrm{partner},\mathrm{male}}$, and the APIM intercepts are $b_{0,\mathrm{female}}$ and $b_{0,\mathrm{male}}$. Fixed APIM coefficients use $b$ and write out their effect and outcome role. The numbered paths $a_{10}$ through $a_{22}$ retain the published DSM notation.

The slope transformation can be understood in two steps. First, for each outcome role $r \in \{\mathrm{female},\mathrm{male}\}$, form the actor-plus-partner and actor-minus-partner combinations:

$$\begin{aligned} b_{\mathrm{sum},r} &= b_{\mathrm{actor},r} + b_{\mathrm{partner},r}, \\ b_{\mathrm{difference},r} &= b_{\mathrm{actor},r} - b_{\mathrm{partner},r}. \end{aligned}$$

The DSM slopes then follow by combining the female- and male-outcome effects:

$$\begin{aligned} a_{11} &= \frac{b_{\mathrm{sum},\mathrm{female}} + b_{\mathrm{sum},\mathrm{male}}}{2}, & a_{21} &= b_{\mathrm{sum},\mathrm{female}} - b_{\mathrm{sum},\mathrm{male}}, \\ a_{12} &= \frac{b_{\mathrm{difference},\mathrm{female}} - b_{\mathrm{difference},\mathrm{male}}}{4}, & a_{22} &= \frac{b_{\mathrm{difference},\mathrm{female}} + b_{\mathrm{difference},\mathrm{male}}}{2}. \end{aligned}$$

The APIM predictors retain their original scale, whereas the DSM predictor level is grand-mean centered. Keeping the raw APIM predictors on their original scale preserves their reference values; the centering difference is handled explicitly in the intercept transformation. If $\mu_X$ is the grand mean subtracted from the DSM predictor level, the intercepts transform as

$$a_{10} = \frac{b_{0,\mathrm{female}} + b_{0,\mathrm{male}}}{2} + \mu_X a_{11}, \qquad a_{20} = b_{0,\mathrm{female}} - b_{0,\mathrm{male}} + \mu_X a_{21}.$$

For the reverse slope transformation, first recover the role-specific actor-plus-partner and actor-minus-partner combinations:

$$\begin{aligned} b_{\mathrm{sum},\mathrm{female}} &= a_{11} + \frac{a_{21}}{2}, & b_{\mathrm{sum},\mathrm{male}} &= a_{11} - \frac{a_{21}}{2}, \\ b_{\mathrm{difference},\mathrm{female}} &= a_{22} + 2a_{12}, & b_{\mathrm{difference},\mathrm{male}} &= a_{22} - 2a_{12}. \end{aligned}$$

Then, for each outcome role $r$,

$$b_{\mathrm{actor},r} = \frac{b_{\mathrm{sum},r} + b_{\mathrm{difference},r}}{2}, \qquad b_{\mathrm{partner},r} = \frac{b_{\mathrm{sum},r} - b_{\mathrm{difference},r}}{2}.$$

The intercepts transform back as

$$b_{0,\mathrm{female}} = a_{10} + \frac{a_{20}}{2} - \mu_X\left(a_{11} + \frac{a_{21}}{2}\right),$$

$$b_{0,\mathrm{male}} = a_{10} - \frac{a_{20}}{2} - \mu_X\left(a_{11} - \frac{a_{21}}{2}\right).$$

The following comparison applies the APIM-to-DSM transformation to all six fixed effects:

APIM-to-DSM fixed-effect transformation (centering constant = 5.148).

DSM path	From APIM transformation	From DSM model
a10	5.043	5.043
a11	1.990	1.990
a12	-0.032	-0.032
a20	0.956	0.956
a21	-0.138	-0.138
a22	1.486	1.486

Random-effect transformation

Let $u_{\mathrm{female}}$ and $u_{\mathrm{male}}$ be the APIM random effects for the female and male outcomes. The DSM outcome-level and outcome-difference residuals are the same random effects expressed in different coordinates:

$$\begin{pmatrix} r_{Y,\mathrm{mean}} \\ r_{Y,\mathrm{diff}} \end{pmatrix} = \begin{pmatrix} 1/2 & 1/2 \\ 1 & -1 \end{pmatrix} \begin{pmatrix} u_{\mathrm{female}} \\ u_{\mathrm{male}} \end{pmatrix}.$$

Applying this rotation to the APIM covariance matrix reproduces the DSM random intercept variance, intercept-slope covariance, and role-slope variance:

apim_vcov <- as.matrix(glmmTMB::VarCorr(apim_model)$cond$coupleID)
dsm_vcov <- as.matrix(glmmTMB::VarCorr(dsm_model)$cond$coupleID)

rotation <- rbind(
  outcome_level = c(0.5, 0.5),
  outcome_difference = c(1, -1)
)
apim_to_dsm_vcov <- rotation %*% apim_vcov %*% t(rotation)

data.frame(
  parameter = c(
    "Var(outcome mean)",
    "Cov(outcome mean, outcome diff)",
    "Var(outcome diff)"
  ),
  from_DSM = round(c(
    dsm_vcov[1, 1],
    dsm_vcov[1, 2],
    dsm_vcov[2, 2]
  ), 3),
  from_APIM_transformation = round(c(
    apim_to_dsm_vcov[1, 1],
    apim_to_dsm_vcov[1, 2],
    apim_to_dsm_vcov[2, 2]
  ), 3)
)
#>                         parameter from_DSM from_APIM_transformation
#> 1               Var(outcome mean)    0.632                    0.632
#> 2 Cov(outcome mean, outcome diff)   -0.240                   -0.240
#> 3               Var(outcome diff)    3.678                    3.678

For exchangeable dyads, the direction of a member difference is arbitrary. The directional intercept, both cross-paths, and the covariance between outcome level and outcome difference must then be zero. The remaining reduced, label-invariant DSM is algebraically the Gaussian Dyad-Individual Model (DIM). In interdep, use model_type = "dim" for this exchangeable model and reserve model_type = "dsm" for distinguishable dyads.

Because the Gaussian DIM is the exchangeability-constrained version of the full DSM, exchangeability can also be tested by comparing these nested models. This is equivalent to the comparison shown in Testing distinguishability in the APIM vignette.

Intensive longitudinal DSM

The following sections are under construction and are coming soon! This extension is equivalent to the extension from the cross-sectional DIM to the ILD DIM. Please refer to that section.

Interpretation of concurrent ILD DSM coefficients

Equivalence of APIM and DSM in ILD

Including Random Slopes

Transforming DSM random slopes to APIM slopes

Dynamic ILD DSM example

The ILD models above do not model residual serial dependence. One way to model dynamics or to account for temporal dependency is to include lagged outcomes as predictors.

Note: Dynamic models, especially with small time series, are subject to bias. This, and the choice between raw and within-person-centered outcome lags, are addressed in the APIM vignette’s discussion of dynamic models.

---

Return to the Actor-Partner Interdependence Model vignette, see the Mixed-Composition APIM vignette or the Dyad-Individual Model vignette for related model specifications, or return to the Overview.

References

Iida, Masumi, Gwendolyn Seidman, and Patrick E. Shrout. 2018. “Models of Interdependent Individuals Versus Dyadic Processes in Relationship Research.” Journal of Social and Personal Relationships 35 (1): 59–88. https://doi.org/10.1177/0265407517725407.