1. Segment 1: Actor-Partner Models vs. Common-Fate Models
Considering the Unit of Analysis in Dyadic Methods
Erin Kramer Holmes, Adam M. Galovan, and Christine M. Proulx

2. Gonzalez and Griffin (2012) note that in research about relationships:
“Interdependence is not treated as a nuisance that needs to be corrected but rather as one of the key psychological parameters to model” (p. 439).
This presentation is grounded in this assumption: that interdependence is something worth of study and we need to find methods that allow us to study it well.

3. Actor/partner Interdependence Model (APIM)
Mother Conflict
Mother Marital Quality
This is one way of modeling dyadic/family level data. It is probably the most common. It identifies actor effects and partner effects, reflecting potential reciprocity in relationships and reflecting the way individuals might influence their partner’s outcomes when they are in a close relationship.
Father Conflict
Father Marital Quality

4. Theory behind Studying Dyadic/Family Phenomena
Interpersonal interaction is a fundamental component of human life worth exploring.
Human thoughts, emotions, and actions are impacted not only by the individual displaying these outcomes, but also by those connected to the individual in his or her life (e.g. romantic partner, friend, child, etc).
Processes within the group can operate in direct and indirect ways. These direct and indirect processes represent unique group-level contexts (e.g. common expectations) which may impact other group or individual level phenomena.
Individual functioning is related not only to the individuals themselves, but also to the complex system of behaviors between members of the system (e.g. family rules, family goals, roles within the system, etc.)
The first two assumptions can be tested with the APIM. But the third brings up a few more considerations.
These processes at the group level may include implicit or explicit rules for all members of the system, roles that are assigned and filled within that unique system, behaviors experienced and perpetuated within the system, or expectations shared by multiple members of the group. As such, researchers still need to account for non-independence in the system, assuming that group-level phenomena specific to that system may impact other group-level phenomena, and/or other individual level phenomena that also occur within that system.

5. Two Methodological Issues to think about
Multiple individuals in the same group may share similar responses about the same group-level phenomenon (i.e. shared variance and non-independence of data)
Properties of interdependent relationships may represent either individual-level or group-level phenomena
“I am happy with my relationship;” “My partner listens to me.”
“It is a real zoo in our home”; “We can usually find things when we need them”; “In our relationship, we talk about things that make us angry.”

6. Conceptual Representation of Handling of Shared variance in the APIM, CFM, and Hybrid Models
Mother
Mother
Father
Father
Shared
Shared
[This slide features several animation sequences.]
As conceptually represented here, in the APIM the shared variance in actor and partner scores is modeled as correlated error/residual error [SHOWN IN PURPLE HERE], with paths considering only the unique effect of each partner on the unique variance in self and partner outcomes [SHOWN IN RED AND BLUE HERE].
Conversely, in the CFM the shared variance [IN PURPLE] is of the most interest and the remaining variance in each individual’s score is treated as error in the model. Paths consider how the shared variance for the independent variables is related to the shared variance for the dependent variables.
Hybrid Models consider either how the shared variance is related to variance in individual outcomes [first set of Red Arrows after the Purple arrow] OR how variance in individual outcomes is related to variance in shared outcomes [second set of Red Arrows after Purple arrow].
[Red and Blue Arrows disappear and Purple Arrow appears on click.]

7. USING the Common Fate Model
“The common-fate conception implies that two dyad members are similar to one another on a given variable due to the influence of a shared or dyadic latent variable” (Ledermann & Kenny, 2012, p. 141).
Measure (or items) with wording representing group-level rather than individual- level process
“It is a real zoo in our home”; “We can usually find things when we need them”; “In our relationship, we talk about things that make us angry”
Assessment of similarity between members (when r > .2, or factor loadings on common latent variable are > .4 see Ledermann & Kenny, 2012; or Ledermann & Macho 2009)
If in an APIM the actor and partner effects have the same sign, and there is positive non-independence in both variable pairs, the CFM should yield reliable estimates.
Ex. Because we share the same relationship the common fate model assumes that the two of us will report similar experiences. It models a variable that represents our shared variance, putting our unique variance in the error terms.

8. What Types of Research Questions are Answered by the CFM models?
Many questions we have in family research, can be conceptualized as dyad- or family-level questions
How are couples’ shared perceptions of conflict and coparenting quality related to their shared view of marital quality?
More simply: How is couple conflict and coparenting quality related to marital quality at the couple-level?
How is positive coparenting related to child behavior outcomes?
How is family chaos related to couple conflict resolution?

9. Building your common fate model
Mother Conflict
Father Conflict
Mother Marital Quality
Father Marital Quality
Couple Conflict
Couple Marital Quality 1
Discuss briefly Structural Equation Models for those who aren’t as familiar.
Parameters: One intercept for each indicator, one variance for each error term, one variance for the exogenous latent variable and one for the disturbance variance for the endogenous latent variable, two covariances between the error terms, and one direct path at the dyad-level. Remember, both of the paths from your latent variable to your observed variables need to be fixed to unity (i.e., the path weight is set equal to 1; see Macho & Ledermann, 2012).
As relations between family- or couple-level constructs may be inflated due to correlations between each partner’s individual reports—an idea referred to as intrapersonal dyadic dependence (see Peugh, DiLillo, & Panuzio, 2013)—disturbance terms for each partner’s report of one family- or couple-level variable were correlated with their own report of other family- or couple-level variables.
Peugh, DiLillo, & Panuzio (2013) set only one of the indicator path weights to 1. However, this often results in unidentified models. If you want to estimate a path for one of the indicator variables, you can also estimate the latent mean for the common fate variable and set the intercepts for each indicator to zero.
In terms of the structural model, the paths between common fate variables are usually not substantively different. We use Macho & Ledermann’s (2012) method in all of our examples. 1 1 1
Create your latent variables (representing how the observed variables are related to the latent group-level constructs).

10. Example with Generated Data
The data were generated in Mplus by using summary statistics such as means, standard deviations, correlations, reliabilities, and factor loadings either from data sets managed by the authors or from published articles
Busby et al (1995)
Feinberg, Brown, & Kan (2012)
After generating a dataset with the proper covariance structure, data were rescaled to fit a set metric
Coparenting items range from 0 to 6
Conflict items range from 1 to 5
The Consensus, Satisfaction, Cohesion, and BMI items were not rescaled
Higher coparenting scores denote more effective coparenting.
Higher conflict scores denote more conflict.
Higher Consensus, Satisfaction, and Cohesion scores indicate more positive Marital Adjustment.

11. Example with the Generated Data
Mother Conflict
Father Conflict
Mother Mar Adj
Father Mar Adj
Couple
Conflict
Couple Marital Adjustment
.68
.51
–.15
.71
Mother Coparenting
Father Coparenting
Coparenting
.74
.12
.03
–.10
–.54
.58
–.04
–.09
.13
For this example, we constructed the CFM variables as second-order latent variables, with each indicator to the latent shared variable also being a latent variable.
After a click, the structural paths change to show that the effect of Conflict on Marital Adjustment is fully mediated by Couple Coparenting. Bootstrap analyses showed this indirect effect was significant (p < .05).
.00
–.56
.68
N = Model Fit Statistics: χ2 (393) = , ns; CFI = .999; TLI = .999; RMSEA = .006.

12. Chaos, conflict Resolution, and Child Behavior Problems
SUBSTANTIVE EXAMPLE
Chaos, conflict Resolution, and Child Behavior Problems

13. Substantive Example: sample
732 couples and a target child
All participants in the Study of Early Child Care and Youth Development (SECCYD)
83% White, non-Hispanic
7.4% African American
5.2% Hispanic
4.4% “Other” Ethnicity
373 male children, 359 female children
Data here represent 3 periods in time: 3rd grade, 5th grade, and age 15

14. hypotheses
We hypothesize that greater chaos in the home will result in a less positive emotional tone following conflict.
Less positive resolution of conflict between parents will also be associated with child behavior problems.
We further hypothesize that associations between chaos in the home and child behavior problems are likely mediated by the impact of chaos on parental conflict resolution.

15. Measures Sample CHAOS items: You can’t hear yourself think in our home
Chaos in the Home
Confusion, Hubbub, and Disorder Scale (CHAOS), (Matheny, Wachs, & Phillips, 1995)
15 items assess routine, noise, and confusion.
Higher scores represent a more chaotic home environment
Conflict Resolution
Conflict Resolution Scale, (Kerig, 1996)
13 items designed to assess the “emotional tone” following conflict
Higher scores represent a more positive emotional tone following conflict.
Children’s Problem Behavior
Child Behavior Check List (Achenbach, 1991)
118 items (internalizing, externalizing, and other thought and behavioral problems)
Scores on the CBCL are standardized and reported as T-Scores with higher scores indicating more problem behavior.
Sample CHAOS items:
You can’t hear yourself think in our home
It’s a real “zoo” in our home
We are usually able to stay on top of things
α = .78 for fathers and α = .81 for mothers
Sample Conflict Resolution items:
We feel closer to one another than before the fight
We don’t resolve the issue
We continue to hold grudges
We each give in a little bit to the other
α = .87 for fathers and α = .88 for mothers.
CBCL
α = .95 for fathers and α = .94 for mothers

16. Mediation Model with Three Reports on the CBCL
Mother Chaos
Father Chaos
Mother CBCL
Father CBCL
Family Chaos
CBCL
.68
.72
.26
.51
.48
3rd Grade
Age 15
Mother Conflict Resolution
Father Conflict Resolution
Couple Conflict Resolution
.70
.75
6th Grade
–.21
–.23
–.33
–.28
–.15
–.06
Youth CBCL
.61
.38
.15
As is evident with the CBCL variable, the CFM model is not limited to only dyadic data. CFM variables can estimated using reports from 3 or more family members.
Once the child is added to the CBCL, mother and father scores can be correlated, as they are likely to share variance beyond what is shared with the child.
N = 732. Control variables include Father is Partner, Income-to-Needs ratio, Mother’s Education, Child’s Sex, Child is Firstborn, Child is Ethnic Minority, and Mother’s Age.
Model Fit Statistics: χ2 (36) = , ns; CFI = .980; TLI = .943; RMSEA = .034.

17. Discussion of Substantive Example
It is noteworthy that the reports of chaos in third grade were associated with outcomes in both fifth and grade and age 15.
According to family systems theory, once families establish patterns of interaction these patterns are maintained by the system.
Any effort to change these patterns would be met with resistance, concepts known as homeostasis and negative feedback (S. Minuchin, 1974).
By the time a child is in third grade, family interaction patterns are well- established. Thus, patterns of chaos would be established and likely to persist up through the later grades.

18. Methodological extensions
Dyer, Day and Harper (2014) and Gustavson et al (2012) argue that CFM models allow the researcher to explore both shared and unique perceptions
“In addition, these errors almost certainly contain a component of random error” (Dyer et al, 2014, p. 4).
Unique Perceptions
Thus, when predicting both unique and shared perceptions in a CFM model, an additional error term is required.
References
Dyer, W. J., Day, R. D., & Harper, J. M. (2014, July 7). Father Involvement: Identifying and Predicting Family Members’ Shared and Unique Perceptions. Journal of Family Psychology. Advance online publication.
Gustavson, K., Røysamb, E., von Soest, T., Helland, M., Karevold, E., & Mathiesen, K. (2012). Reciprocal longitudinal associations between depressive symptoms and romantic partners’ synchronized view of relationship quality. Journal of Social and Personal Relationships, 29(6), doi: /
Father
Score
Mother Score
Shared Perceptions

19. Models of Shared and Unique Perceptions
To explore predictors of both shared and unique perceptions of father involvement, Dyer et al. used CFM techniques and then saved the factor scores and residuals for use in regression models
Saving latent variable scores often results in decreased variance, which may lead to biased parameter estimates for analyses that use these parameters (Hoshino & Bentler, 2013)
An alternative to saving factor scores, would be to estimate the unique and shared perceptions within a latent variable framework
Gustavson et al (2012) employed this method to explore longitudinal relations among partners synchronized (shared) and unique views of relationship quality and each spouses’ depressive symptoms
To explore predictors of both shared and unique perceptions of father involvement, Dyer et al. used CFM techniques and then saved the factor scores and residuals for use in regression models.
Hoshino and Bentler (2013) note, however, that saving latent variable scores often results in decreased variance, which may lead to biased parameter estimates for analyses that use these parameters. An alternative to saving factor scores, would be to estimate the unique and shared perceptions within a latent variable framework.
Gustavson et al (2012) employed this method to explore longitudinal relations among partners synchronized (shared) and unique views of relationship quality and each spouses’ depressive symptoms.
Reference
Hoshino, T., & Bentler, P. M. (2013). Bias in Factor Score Regression and a Simple Solution. In A. R. de Leon & K. C. Chough (Eds.) Analysis of mixed data: Methods and application (pp ). New York: Chapman and Hall.

20. Challenges in Models of Shared and Unique Perceptions
It is often not possible to simultaneously estimate predictors of both individual and shared perceptions, as such models are unidentified. We propose a three- step process.
CFM model predicting only the shared perceptions is estimated.
Parameter estimates for the paths predicting the CFM outcome are set to the (unstandardized) values obtained in step one, and paths predicting individual perceptions are estimated.
To verify that stability of estimates, the paths predicting the individual perceptions are set to the values obtained in step two, and the paths predicting the shared perceptions are freely estimated.
The estimates for these freed paths should be identical to those obtained in step one.
It is not possible, however, to simultaneously estimate both predictors of both individual and shared perceptions, as such models are unidentified. Gustavson et al (2012) did not indicate how they handled these model identification issues. We propose a three-step process.
First, a CFM model predicting the shared perceptions is estimated.
Next, parameter estimates for the paths predicting the CFM outcome are set to the values obtained in step one, and paths predicting individual perceptions are estimated.
Finally, to verify that stability of estimates, the paths predicting the individual perceptions are set to the values obtained in step two, and the paths predicting the shared perceptions are freely estimated. The estimates for these freed paths should be identical to those obtained in step one.
This process essentially follows the procedure outlined by Dyer et al (2014)—as saving the residual scores and using them as dependent variables in later analyses assumes that the original regression paths predicting the shared perceptions is constant—but retains the advantage of latent variable models.

21. Mediation Model with Shared and Unique Perceptions
Mother Chaos
Father Chaos
Mother CBCL
Father CBCL
Family Chaos
CBCL
.69
.73
.28
.47
3rd Grade
Age 15
Mother Conflict Resolution
Father Conflict Resolution
Couple Conflict Resolution
.70
.74
6th Grade
–.22
–.27
–.33
–.17
–.02
Youth CBCL
.64
.40
.16
.13
Youth Unique CBCL
Mom Unique CBCL
Dad Unique CBCL
Unique CBCL predictor paths:
Family Chaos  Mom Unique CBCL = .03, ns Family Chaos  Dad Unique CBCL = .00, ns
Family Chaos  Youth Unique CBCL = –.05, ns
Couple Conflict Resolution  Mom Unique CBCL = –.03, ns
Couple Conflict Resolution  Dad Unique CBCL = .01, ns
Couple Conflict Resolution  Youth Unique CBCL = .05, ns
N = 732. Control variables include Father is Partner, Income-to-Needs ratio, Mother’s Education, Child’s Sex, Child is Firstborn, Child is Ethnic Minority, and Mother’s Age.
Model Fit Statistics: χ2 (32) = , p < .01; CFI = .979; TLI = .931; RMSEA = .038.
Family Chaos  Mom Unique CBCL = .03, ns Family Chaos  Dad Unique CBCL = .00, ns
Family Chaos  Youth Unique CBCL = –.05, ns
Couple Conflict Resolution  Mom Unique CBCL = –.03, ns
Couple Conflict Resolution  Dad Unique CBCL = .01, ns
Couple Conflict Resolution  Youth Unique CBCL = .05, ns

22. Limitations Estimation of CFM can sometimes be difficult
There can be issues with non-identification
Theoretically, items should be at a group- or family-level rather than the individual level
For example, it doesn’t make a lot of sense to model husband and wife depression as “couple depression”
As a field, we have many names for these types of models
Common Fate Model
Multi-informant model
Multi-level dyadic structural equation model
What is the difference between each?
OTHER LIMITATIONS???

23. Implications of CFM Modeling
As a family studies field, we often ask questions about processes at the family- or dyad-level.
Such questions may often be answered more correctly with analytical techniques that assess dyadic or family variables and processes, such as the CFM approach.
One of the challenges in using the CFM may be finding measures that are suited to assessing dyadic and family-level constructs.
In some cases, it might be appropriate to modify existing measures to ask questions using plural pronouns (e.g., we, our) rather than singular pronouns (e.g., I, my).
These measures may then better assess the dyadic- or family-construct rather than an individual’s perception. In other cases, modifying pronoun usage may not be sufficient.
Interdependence or systems frameworks may suggest that constructs scholars have traditionally considered at the individual-level deserve reconceptualization at the dyad- or family-level.
In some cases, it may be acceptable to use the CFM even when constructs are not assessed with plural language if the language seems to be clearly directed at assessing a dyadic- or family-level construct and the reporter scores are sufficiently correlated.
For example, partners may report on their participation in housework by rating their own contribution in comparison to their partner. Even though partners may be reporting their own contribution, they are reporting on how housework is shared—a dyadic-level construct—and thus each partner’s score could be coded and used as a CFM indicator.

24. Conclusion
As a field focused on studying families and close relationships, we should more often assess constructs at the dyad- and family-level and employ suitable methods for analyzing such data, such as the CFM.
The use of these methods will better align with our theoretical assumptions and add greater understanding to the base of research in many substantive areas of family studies.

25. Questions?

26. Acknowledgements
We are grateful to the Eunice Kennedy Shriver National Institute of Child Health and Human Development Early Child Care Research Network for designing and carrying out the data collection for the example in this project. The NICHD Study of Early Child Care is a study directed by a Steering Committee and supported by NICHD through a cooperative agreement that calls for scientific collaboration between the grantees and the NICHD staff. The content of this project is solely the responsibility of the named authors and does not represent the official views of the Eunice Kennedy Shriver National Institute of Child Health and Human Development, the National Institute of Health, or individual members of the Network.