Dyadic Analysis and the Reciprocal One-with-Many Model: Extending the Study of Interpersonal Processes with Intensive Longitudinal Data

Abstract

Newly available data streams from experience sampling studies and social media are providing new opportunities to study individuals’ dyadic relations. The “one-with-many” (OWM) model (Kenny & Winquist, 2001; Kenny et al., 2006) was specifically constructed for and is used to examine features of multiple dyadic relationships that one set of focal persons (e.g., therapists, physicians) has with others (e.g., multiple clients, multiple patients). Originally, the OWM model was constructed for and applied to cross-sectional data. However, the model can be extended to accommodate and may be particularly useful for the analysis of intensive repeated measures data now being obtained through experience sampling and social media. This paper (a) provides a practical tutorial on fitting the OWM model, (b) describes how the OWM model is extended for analysis of repeated measures data, and (c) illustrates application of the OWM model using reports about interpersonal behavior and benefits individuals experienced in 64,111 social interactions during 9 weeks of study (N = 150). Our presentation highlights the utility of the OWM model for examining interpersonal processes in everyday life.

Data from experience sampling studies and social media are providing new opportunities to describe and study individuals’ dyadic relations. These intensive data, often collected from multiple individuals who each report on a variety of relationships, allow for the examination of interpersonal processes and how these processes may differ between individuals as well as between relational partners within individuals. The “one-with-many” (OWM) model (Kenny & Winquist, 2001; Kenny et al., 2006) was specifically constructed for and is used to examine features of multiple dyadic relationships that one set of focal persons (e.g., therapists, physicians) has with others (e.g., multiple clients, patients). While this model was originally constructed for and applied to cross-sectional data, the model can be extended to accommodate and may be particularly useful for the analysis of intensive repeated measures data now being obtained through experience sampling and social media.

Two types of multi-person models are prevalent in the literature. The standard dyadic model emerges from study designs where data are collected from both partners (e.g., husband, wife) of multiple dyads. As seen in Panel A of Figure 1, each person is paired with only one other person. In the top portion of the panel, the arrows connecting two circles indicate which individuals have a dyadic relationship, and in the bottom portion of the panel, the Xs indicate who is reporting on whom. For example, Person A reports only about their relationship with Person B, and Person B reports only about their relationship with Person A. In contrast, the social relations model emerges from study designs where data are collected from many individuals who are all connected to each other (e.g., a fully connected network). As shown in the top of Panel C in Figure 1, each person is paired with all other persons in their group, with the specific reporting indicated by the Xs in the bottom of the panel. For example, Person A reports about their relationships with Person B, Person C, and Person D. Similarly, Person D reports about their relationships with Person A, Person B, and Person C, and so on.

Figure 1.

Depictions of the standard dyadic model, one-with-many model, and social relations model, respectively. The circles represent people and the arrows (and Xs in the tables) indicate who interacts and reports on whom.

A third type of multi-person model, the OWM model, combines the standard dyadic model and the social relations model. Panel B of Figure 1 depicts a simple OWM model. In this design, Person A (the focal person) reports about their relationships with Person B, Person C, and Person D (the partners), while Persons B, C, and D only report about their relationship with Person A. The OWM model is used to examine dyadic relationships in which one person interacts with multiple persons, and has been successfully used in a variety of settings. For example, DePaulo and Kashy (1998) used the OWM model to examine whether participants (focal persons) lied to different types of interaction partners – family member, friend, etc. – at the same rate. Marcus and colleagues (Marcus et al., 2009, 2011) used the OWM model to examine whether therapists (focal persons) rate the alliance with all patients (partners) similarly, whether all patients rate the alliance with the therapist similarly, and whether the ratings of the alliance are unique to specific therapist-patient pairs. Similar applications can be formulated relatively easily, for example, to study how focal persons rate the behaviors/personalities of multiple friends or how focal persons interact with many friends and acquaintances on social media.

Despite its applicability to a wide range of research questions, the OWM model is not used frequently. In a non-exhaustive survey, only 16% of 75 articles on dyadic relationships used this design (Kenny et al., 2006). Looking forward, we believe the OWM model, especially when extended to accommodate repeated measures, is particularly useful for a variety of novel data streams, including experience sampling and social media data. For example, researchers have already begun to use the OWM model to examine differences in warmth of adolescents’ computer-mediated communication across multiple types of relationships (parents versus friends; Ackerman et al., 2019). Seeing the potential for many future applications of the model, the current paper (a) provides a practical tutorial on fitting the OWM model, (b) describes how the OWM model is extended for analysis of repeated measures data, and (c) illustrates how the OWM model can be used to study interpersonal processes in everyday life (e.g., Hopwood, 2018; Pincus et al., 2014) using reports about individuals’ behaviors and feelings that were each obtained after 64,111 social interactions during 9 weeks of study (N = 150). In sum, this paper provides a tutorial on implementing the OWM model, and to the best of our knowledge, is the first paper to develop a three-level OWM model for repeated measures data to examine both between-focal persons and within-focal person differences (for an exception using a structural equation modeling framework, see Ackerman et al., 2019).

OWM Model

Stated generally, the OWM model examines the associations between focal persons’ and/or partners’ features and a particular outcome (fixed effects), and how focal persons’ and/or partners’ features are (in)consistent across relationships and (in)consistent across focal persons (random effects). Before describing the technical aspects of the OWM model, we first define and distinguish some relevant terminology.

Focal Persons and Partners.

Focal persons are the persons of interest, and partners are their dyadic partners. Each focal person is the “one” who is connected “with many” partners. Focal persons can provide reports about partners and/or partners can provide reports about focal persons. Thus, the distinction between focal persons and partners is a substantive distinction driven by the specifics of the research question and not a direct function of the reporting structure. Indeed, there are several variations on the OWM model (see Table 14.1 in Kenny & Winquist, 2001 for an illustration) that differ on the number of focal persons, the number of partners, and whether there is reciprocity in reporting (i.e., both members of the dyad report on the other dyad member). An example of a one focal person reporting on many partners design is when a sample of adolescents are each asked to rate/describe the behaviors of three friends. An example of a many partners reporting on one focal person design is when each student in a classroom is asked to evaluate their teacher. In both of these examples, reports are only given by one side of the focal person-partner relationship. In a reciprocal OWM design, focal persons provide reports about multiple partners and the partners all provide reports about the focal person. For example, the teacher rates each of the students in their classroom, and each student rates their teacher. This tutorial focuses on the implementation of models for reciprocal OWM designs.

Dyads Nested Within Persons.

A dyad is a pair, two people who are linked in some way. In Figure 1, a dyad is designated by an arrow connecting two individuals. In the OWM model in particular, the arrows connect each focal person with multiple partners. In most dyadic analytic techniques, two individuals (e.g., husband–wife; parent-child; supervisor–supervisee) are nested within a dyad. In contrast, in the OWM model, multiple dyads are nested within each focal person. This “reversal” of nesting (dyads nested within focal persons, rather than persons nested within dyads) is an essential feature of the OWM model and governs how it is implemented within a multilevel modeling framework.

Distinguishability.

Distinguishability is when members of the dyad can be identified by their role in or another feature important to the relationship (Kenny et al., 2006). Distinguishable dyads include husband-wife pairs, parent-child pairs, and boss-employee pairs, whereas indistinguishable dyads include friend pairs, identical twins, co-worker pairs, and randomly paired experimental participants. In the OWM model, focal persons are by definition distinguishable from partners. We must consider, however, whether partners are distinguishable from each other. For example, when a person goes on multiple dates and reports on each date, the partners are treated as replicates and considered indistinguishable. In contrast, when a person reports on their dyadic relationships with their mother, father, best friend, and boss, the partners are not identical replicates and are considered distinguishable. In this tutorial, we demonstrate both cases. As will be described in more detail below, we use the OWM model to study indistinguishable partners in a 2-level (cross-sectional) OWM model and to study distinguishable partners in a 3-level (intensive longitudinal) OWM model. The general format of these reciprocal models will be described in the sections the “Standard Two-level Reciprocal OWM Model: Analysis of Between-Focal Persons Differences” and “Three-level Reciprocal OWM Model: Analysis of Within-Dyad Process and Between-Dyad Differences,” and these models will be implemented in the “Illustrative Empirical Example” section.

Data and Assumptions.

The data for the reciprocal OWM model consist of reports obtained from both focal persons and partners. Data can be cross-sectional or longitudinal, and can be collected in a variety of experimental contexts or naturalistic settings.

The key feature of OWM data is the patterning in whose reports are collected rather than the actual content or context in which the data are collected. The OWM model assumes that (1) focal persons are each associated with multiple partners, but not each other and (2) partners are not associated with each other, neither within a focal person nor across focal persons, when conditioning on the focal person – i.e., there is an assumption of local independence. These assumptions are illustrated in Panel B of Figure 1. Specifically, there are arrows indicating focal persons’ associations with multiple partners, but there are no arrows between a given focal person’s partners (i.e., the assumption is that partners are not associated with each other except through the focal person, and that partners do not influence the relationships the focal person has with other partners) and there are no arrows between focal persons that connect focal persons or the partners. While these assumptions are strong, their mathematical convenience does map to many types of relationships and situations. As usual, some care should be taken in interpretation of results and appropriate qualifications applied.

Standard Two-level Reciprocal OWM Model: Analysis of Between-Focal Persons Differences

With the terminology and assumptions of the OWM model in place, we now describe the statistical aspects of the model. The two-level OWM model (Kenny et al., 2006, Chapter 10) is most commonly used to examine how focal persons (e.g., therapists) differ in their relationships (e.g., perceptions of therapeutic alliance) with partners (e.g., multiple clients) using cross-sectional data (see Table 1 for example data). The two-level OWM model is used to answer a variety of research questions, including (1) how much variability in an outcome is attributable to differences within focal persons versus differences between focal persons?; how much variability in perceptions of therapeutic alliance is due to differences between therapists or to differences between clients within a therapist? (2) do focal persons view their relationships with their partners similarly?; do therapists perceive their therapeutic alliance with all of their clients similarly? (3) do partners differ in their relationship with the focal person?; do clients of the same therapist have similar perceptions of therapeutic alliance? (4) are there unique relationship effects between therapists and specific clients – i.e., are there components of the relationship not accounted for by how the therapist views all of their clients and how all of the clients view the therapist?

Table 1.

An example of the format of the data before conducting a two-level OWM analysis.

Focal Person ID	Interaction Partner Type	Communion	Extraversion	Focal Dummy	Partner Dummy
1	6	87	3.0	1	0
1	6	84	3.0	0	1
1	5	84	3.0	1	0
1	5	78	3.0	0	1
1	2	74	3.0	1	0
1	2	67	3.0	0	1
150	1	83	1.5	1	0
150	1	88	1.5	0	1
150	5	81	1.5	1	0
150	5	83	1.5	0	1
150	0	60	1.5	1	0
150	0	16	1.5	0	1

Mathematical Model.

The nesting of dyads within focal persons in the reciprocal OWM model is accommodated using a multilevel modeling framework. Dyad-level differences in the outcome of interest are modeled at Level 1, such that

$$Y_{Fdi}={\gamma }_{0Fi}+{\gamma }_{1Fi}{Z}_{di}+e_{Fdi}$$ (1)

$$Y_{Pdi}={\gamma }_{0Pi}+{\gamma }_{1Pi}{Z}_{di}+e_{Pdi}$$ (2)

where Y_Fdi and Y_Pdi are the outcome of interest for dyad d within focal person and partner i, respectively; γ_0Fi and γ_0Pi are the intercepts or expected values of the outcome variable (conditional on other predictors) for focal person and partner i, respectively; γ_1Fi and γ_1Pi are the associations between a the dyad-level variable Z and the outcome for focal person or partner i, respectively; and e_Fdi and e_Pdi are unexplained differences in focal person or partner i’s reports, respectively.

In practice, Equation 1 and 2 are then combined into a single model. In the multilevel modeling framework, this is done by collecting the focal persons’ reports and partners’ reports into a single outcome variable, Y_di, along with a pair of dummy variables, Focal_di and Partner_di, that indicate whether a particular observation of Y_di was provided by a focal person or a partner. These dummy variables act as a “switch” that either “turn on” or “turn off” specific parameters that are associated with the dyad member of interest (Bolger & Laurenceau, 2013; MacCallum, Kim, Malarkey, & Kiecolt-Glaser, 1997). The general form of the two-level reciprocal OWM model using this framework is

Level 1:

$$\begin{gathered} Y_{di}={\gamma }_{0Fi}Foca{l}_{di}+{\gamma }_{0Pi}Partne{r}_{di}+ \\ {\gamma }_{1Fi}Foca{l}_{di}{Z}_{di}+{\gamma }_{1Pi}Partne{r}_{di}{Z}_{di}+e_{Fdi}+e_{Pdi} \end{gathered}$$ (3)

where Y_di, the outcome of interest for dyad d within focal person i, is a “stacked” variable that includes all the multiple reports provided by each focal person (Y_Fdi) and the reports about each focal person provided by multiple partners (Y_Pdi), as is typical in the multivariate multilevel framework used in dyadic analysis (see e.g., Chapter 8 of Bolger & Laurenceau, 2013; MacCallum et al., 1997). No global intercept is estimated. Instead, the use of the dummy variables allows for estimation of separate intercepts for the focal persons and for the partners. Specifically, γ_0Fi is the intercept or expected value of the outcome variable (conditional on other predictors) for focal person i based upon all of focal person i’s reports about their partners (i.e., perceiver effect); γ_0Pi is the intercept or expected value of the outcome variable (conditional on other predictors) for all partners’ reports about focal person i (i.e., partner effect); γ_1Fi is the association between a focal person i’s reports and the dyad-level variable Z; γ_1Pi is the association between a focal person i’s partners’ reports and the dyad-level variable Z; and e_Fdi and e_Pdi are unexplained differences in focal person i’s reports and their partners’ reports, respectively, and capture unexplained within-dyad processes across different partners. The focal person (e_Fdi) and partner (e_Pdi) residuals are assumed independent, each with their own homogeneous variance, and may be correlated. Specifically, the within focal person (Level 1) covariance (i.e., error) structure is

$$\left[\begin{array}{cc}{\sigma }_{e_{Fdi}}^2& {\sigma }_{e_{Fdi}{e}_{Pdi}}\\ {\sigma }_{e_{Fdi}{e}_{Pdi}}& {\sigma }_{e_{Pdi}}^2\end{array}\right]$$ (4)

where ${\sigma }_{e_{Fdi}}^2$ is the residual focal person variance, ${\sigma }_{e_{Pdi}}^2$ is the residual partner variance, and ${\sigma }_{e_{Fdi}{e}_{Pdi}}$ is the covariance between the focal person and partner residuals. The focal person-specific parameters at Level 1 are then modeled as

Level 2:

$${\gamma }_{OFi}={\pi }_{00F}+{\pi }_{01F}{W}_i+v_{0Fi}$$ (5)

$${\gamma }_{OPi}={\pi }_{00P}+{\pi }_{01P}{W}_i+v_{0Pi}$$ (6)

$${\gamma }_{1Fi}={\pi }_{10F}+{\pi }_{11F}{W}_i+v_{1Fi}$$ (7)

$${\gamma }_{1Pi}={\pi }_{10P}+{\pi }_{11P}{W}_i+v_{1Pi}$$ (8)

where the π_00F and π_00P parameters are the expected value of the outcome variable for the focal person and partner, respectively, when the predictor variable W is zero; π_01F and π_01P indicate how those expected values are associated with the predictor variable for the focal persons and partners, respectively; and ν_0Fi and ν_0Pi represent the unexplained differences in intercepts for the focal persons and partners (e.g., how much a particular focal person-partner pair deviates from the mean of all focal person-partner pairs), respectively. In turn, π_10F and π_10P are the prototypical association between the focal person and partner outcome variables, respectively, and the dyad-level variable Z; π_11F and π_11P indicate how that association is moderated by the predictor variable W; and ν_1Fi and ν_1Pi represent unexplained differences in the dyad-level associations.

The between focal persons (Level 2) covariance matrix of random effects is

$$\left[\begin{array}{cccc}{\sigma }_{v_{0Fi}}^2& {\sigma }_{v_{0Fi}{v}_{0Pi}}& {\sigma }_{v_{0Fi}{v}_{1Fi}}& {\sigma }_{v_{0Fi}{v}_{1Pi}}\\ {\sigma }_{v_{0Fi}{v}_{0Pi}}& {\sigma }_{v_{0Pi}}^2& {\sigma }_{v_{0Pi}{v}_{1Fi}}& {\sigma }_{v_{1Pi}{v}_{1Fi}}\\ {\sigma }_{v_{0Fi}{v}_{1Fi}}& {\sigma }_{v_{0Pi}{v}_{1Fi}}& {\sigma }_{v_{1Fi}}^2& {\sigma }_{v_{1Fi}{v}_{1Pi}}\\ {\sigma }_{v_{0Fi}{v}_{1Pi}}& {\sigma }_{v_{1Pi}{v}_{1Fi}}& {\sigma }_{v_{1Fi}{v}_{1Pi}}& {\sigma }_{v_{1Pi}}^2\end{array}\right]$$ (9)

where ${\sigma }_{v_{0Fi}}^2$ is the variance between focal persons’ intercepts, ${\sigma }_{v_{0Pi}}^2$ is the variance between partners’ intercepts, ${\sigma }_{v_{1Fi}}^2$ is the variance in focal persons’ association between the dyad-level predictor and the outcome, ${\sigma }_{v_{1Pi}}^2$ is the variance in partners’ association between the dyad-level predictor and the outcome, and the off-diagonal elements are the covariances among the intercepts and slopes. As per usual in multilevel modeling, the residual covariance structures can be constrained to match substantive hypotheses (e.g., homogeneity of effects) or data constraints. In brief, the number and interpretation of parameters in the reciprocal OWM model can quickly become complex. To help reduce this complexity, we have compiled in Table S1 of the online supplemental material a brief description of each model parameter and an explanation of how each parameter maps onto both a general substantive research question and the specific substantive question examined in our illustrative example.

Centering of variables within this model has implications for model set-up and interpretation. The dyad-level predictors (Z) can be centered in various ways depending on the desired model interpretation and the associated measurement scale of the predictor variable (e.g., nominal vs. interval). For instance, when researchers are interested in the expected value of the outcome variable for the “average” level of predictor Z that is measured on an interval scale (e.g., the dyad’s reported relationship satisfaction), they should engage dyad-level mean centering. Alternatively, when researchers are interested in the expected value of the outcome variable for a specific level of a dyad-level predictor that is measured on a nominal scale (e.g., dyads composed of same-sex pairs in comparison to dyads of opposite sex pairs), they should code the predictor with a meaningful zero. Similar to the predictor variables in Level 1, the Level 2 predictor variables (W) can be centered to sample-level means or specific values to obtain the desired model interpretation (Bolger & Laurenceau, 2013).

As described above, the standard two-level OWM model provides for examination of if and how dyadic relationships with multiple partners differ between focal persons. However, this model assumes each dyadic relationship is static and does not allow for the description or comparison of within-dyad changes. As constructed, the model does not provide description of how the behaviors of clients and therapists change from week to week, or how those changes are associated with differences in therapeutic alliance. Examination of how the dyadic interactions differ across partners and across focal persons requires longitudinal data and a three-level OWM model. Next, we describe this extension of the model.

Three-level Reciprocal OWM Model: Analysis of Within-Dyad Process and Between-Dyad Differences

The two-level OWM model described above examined between-focal persons differences in dyadic relations using cross-sectional data. The three-level OWM model, however, can be quite useful to study the dynamics of interpersonal interactions over time and to quantify how within-dyad processes operate differently for different focal persons (see Table 2 for example data).

Table 2.

An example of the format of the data before conducting a three-level OWM analysis.

Focal Person ID	Interaction Number	Interaction Partner Type	Communion	Benefit	Extraversion	Focal Dummy	Partner Dummy
1	1	6	87	64	3	1	0
1	1	6	84	80	3	0	1
1	2	5	84	70	3	1	0
1	2	5	78	63	3	0	1
1	3	2	74	36	3	1	0
1	3	2	67	58	3	0	1
150	264	1	84	46	1.5	1	0
150	264	1	82	44	1.5	0	1
150	265	2	53	75	1.5	1	0
150	265	2	53	75	1.5	0	1
150	266	6	73	40	1.5	1	0
150	266	6	71	71	1.5	0	1

Note: Interaction number = social interaction number, such that 1 = the first interaction through N = the final interaction for each focal person.

The three-level OWM model might be especially useful when examining interactions on social media. For example, consider a researcher who wishes to examine how emotional contagion differs depending on someone’s interaction partner on social media. The researcher recruits 100 participants who each share the last twenty Facebook posts or messages they exchanged with their best friend, their spouse, and a coworker. In contrast to most of the current OWM model work, this researcher now has repeated measures (i.e., 20 messages) for each partner type for each participant (i.e., focal person). The three-level OWM model would allow the researcher to appropriately accommodate the nesting of multiple messages within dyad within focal person, as well as examine (1) the dynamics of how a dyad-level variable is related to an outcome (e.g., does a friend’s negative emotional expression prompt a focal person’s negative emotional expression?), (2) the differences between dyads within focal person (e.g., is emotional contagion stronger for friends than for co-workers?), (3) the association between a focal person variable and an outcome (e.g., is a focal person’s neuroticism associated with how their emotional expressions are perceived and they perceive others’ emotional expressions on social media?), and (4) trends across time within partner (e.g., is emotional contagion stronger as the dyadic relationship lengthens?).

Similar to the empirical example described later, we present example data in Figure 2 from two focal persons (top vs. bottom panel) who each had five social interactions with three different partners (as differentiated by the three colors in each plot). After each social interaction, the focal person and partner both rated their satisfaction with the interaction and the quality of support received during the interaction. Focal person satisfaction and support are depicted by the black squares and circles, respectively. Partner satisfaction and support are depicted by the white squares and circles, respectively. Looking horizontally across the red, blue, and green sections of each panel, we see the within-focal person, between-partner differences in both level and variability of satisfaction and support. Looking vertically across the upper and lower panels, we also see that there are systematic between-focal person differences in level and variability of satisfaction and support. Note that although reports on each focal person’s interactions with friends, spouses, and others are collected together in Figure 2 to highlight the between-partner differences in level and variability, the interactions did not actually occur sequentially (e.g., first all friends, then spouses), nor are they assumed sequential when using the OWM model.

Figure 2.

Example data from two focal persons (top vs. bottom plot) with three partners each (as differentiated by the three colors in each plot), and five interactions for each partner. After each social interaction, the focal person and partner each rated their satisfaction with the interaction and support received during the interaction. Focal person satisfaction and support are depicted by the black squares and circles, respectively. Partner satisfaction and support are depicted by the white squares and circles, respectively.

Mathematical Model.

The three-level OWM model addresses new kinds of research questions by expanding the model to accommodate repeated measures at Level 1. The set-up is similar to the two-level OWM model outlined earlier, except that the means and variances that were previously modeled at Level 1 in the two-level OWM model (i.e., the dyad-level) are now modeled at Level 2, and the means and variances that were previously modeled in Level 2 in the two-level OWM model (i.e., the focal person level) are now modeled at Level 3. Brief descriptions of each model parameter and an explanation of how each parameter maps onto both a general substantive research question and the specific substantive question examined in our illustrative example are collected together in Table S2 of the online supplemental material for easy reference.

Similar to the set-up of the two-level OWM model, the outcome variable for the focal persons and partners can be expressed separately (as in Equations 1 and 2), but are combined and estimated simultaneously in practice. Using the combined expression, we extend Equations 3 to 9 to three levels, as well as adding a time-varying covariate (TVC), to get

Level 1:

$$\begin{gathered} Y_{tdi}={\beta }_{0Fdi}Foca{l}_{tdi}+{\beta }_{0Pdi}Partne{r}_{tdi}+ \\ {\beta }_{1Fdi}Foca{l}_{tdi}TV{C}_{tdi}+{\beta }_{1Pdi}Partne{r}_{tdi}TV{C}_{tdi}+e_{Ftdi}+e_{Ptdi} \end{gathered}$$ (10)

where Y_tdi is a “stacked” variable of the outcome of interest at a specific time t for the focal person (Y_Ftdi) and partner (Y_PtFdi) in each dyad d (focal person – partner pair) within focal person i; β_0Fdi and β_0Pdi are intercepts for each unique focal person and each unique partner; β_1Fdi and β_1Pdi are the within-person associations between the time-varying variable, TVC_tdi, and the outcome, Y_tdi, for each unique focal person and each unique partner; and e_Ftdi and e_Ptdi are occasion-specific residual errors. Residual errors for focal persons and partners are assumed independent, normally distributed, and may be correlated. Specifically, the Level 1 covariance (i.e., error) structure is

$$\left[\begin{array}{cc}{\sigma }_{e_{Ftdi}}^2& {\sigma }_{e_{Ftdi}{e}_{Ptdi}}\\ {\sigma }_{e_{Ftdi}{e}_{Ptdi}}& {\sigma }_{e_{Ptdi}}^2\end{array}\right]$$ (11)

where ${\sigma }_{e_{Ftdi}}^2$ is the variance of the occasion-specific residuals for focal person reports, ${\sigma }_{e_{Ptdi}}^2$ is the variance of the occasion-specific residuals for partner reports, and ${\sigma }_{e_{Ftdi}{e}_{Ptdi}}$ is the covariance between those occasion-specific residuals (unexplained context-specific influences that are shared). The within-person intercepts and associations from Level 1 are then modeled as

Level 2:

$${\beta }_{0Fdi}={\gamma }_{00Fi}+{\gamma }_{01Fi}{Z}_{di}+u_{0Fdi}$$ (12)

$${\beta }_{0Pdi}={\gamma }_{00Pi}+{\gamma }_{01Pi}{Z}_{di}+u_{0Pdi}$$ (13)

$${\beta }_{1Fdi}={\gamma }_{10Fi}+{\gamma }_{11Fi}{Z}_{di}+u_{1Fdi}$$ (14)

$${\beta }_{1Pdi}={\gamma }_{10Pi}+{\gamma }_{11Pi}{Z}_{di}+u_{1Pdi}$$ (15)

where γ_00Fi and γ_00Pi are the expected values of the outcome of interest (conditional on predictors) for the focal person and partner in each focal person-partner dyad, γ_01Fi and γ_01Pi indicate how the outcome is associated with the dyad-level predictor, Z; γ_10Fi and γ_10Pi indicate the prototypical within-person association of the TVC with the outcome variable for each dyad; γ_11Fi and γ_11Pi indicate how those within-person associations are moderated by the dyad-level variable for both the focal persons and partners. The u_0Fdi and u_0Pdi are unexplained dyad-specific residuals in the intercepts for focal persons and partners (e.g., how much a particular focal person-partner pair deviates from the mean of all focal person-partner pairs within a focal person), and u_1Fdi and u_1Pdi are unexplained dyad-specific residuals for the within-person associations for focal persons and partners. The Level 2 covariance matrix of random effects is

$$\left[\begin{array}{cccc}{\sigma }_{u_{0Fdi}}^2& {\sigma }_{u_{0Fdi}{u}_{0Pdi}}& {\sigma }_{u_{0Fdi}{u}_{1Fdi}}& {\sigma }_{u_{0Fdi}{u}_{1Pdi}}\\ {\sigma }_{u_{0Fdi}{u}_{0Pdi}}& {\sigma }_{u_{0Pdi}}^2& {\sigma }_{u_{0Pdi}{u}_{1Fdi}}& {\sigma }_{u_{0Pdi}{u}_{1Pdi}}\\ {\sigma }_{u_{0Fdi}{u}_{1Fdi}}& {\sigma }_{u_{0Pdi}{u}_{1Fdi}}& {\sigma }_{u_{1Fdi}}^2& {\sigma }_{u_{1Fdi}{u}_{1Pdi}}\\ {\sigma }_{u_{0Fdi}{u}_{1Pdi}}& {\sigma }_{u_{0Pdi}{u}_{1Pdi}}& {\sigma }_{u_{1Fdi}{u}_{1Pdi}}& {\sigma }_{u_{1Pdi}}^2\end{array}\right]$$ (16)

where ${\sigma }_{u_{0Fdi}}^2$ is the residual variance in dyad-specific intercepts for focal person reports, ${\sigma }_{u_{0Pdi}}^2$ is the residual variance in dyad-specific intercepts for partner reports, ${\sigma }_{u_{1Fdi}}^2$ is the residual variance in the within-person associations for focal persons, ${\sigma }_{u_{1Pdi}}^2$ is the residual variance in the within-person associations for partners, and the off-diagonal elements are all possible covariances among those residuals.

Level 3:

$${\gamma }_{00Fi}={\pi }_{000F}+{\pi }_{001F}{W}_i+v_{00Fi}$$ (17)

$${\gamma }_{00Pi}={\pi }_{000P}+{\pi }_{001P}{W}_i+v_{00P_i}$$ (18)

$${\gamma }_{01Fi}={\pi }_{010F}+{\pi }_{011F}{W}_i$$ (19)

$${\gamma }_{01Pi}={\pi }_{010P}+{\pi }_{011P}{W}_i$$ (20)

$${\gamma }_{10Fi}={\pi }_{100F}+{\pi }_{101F}{W}_i+v_{10Fi}$$ (21)

$${\gamma }_{10Pi}={\pi }_{100P}+{\pi }_{101P}{W}_i+v_{10Pi}$$ (22)

$${\gamma }_{11Fi}={\pi }_{110F}+{\pi }_{111F}{W}_i$$ (23)

$${\gamma }_{11Pi}={\pi }_{110P}+{\pi }_{111P}{W}_i$$ (24)

where π_000F and π_000P are the expected value of the outcome variable (conditional on predictors – i.e., when W = 0) for the prototypical focal person and partner; π_001F and π_001P indicate how between-person differences in those intercepts are associated with between-focal person differences in predictor W; π_010F and π_010P represent the prototypical association between the focal person or partner outcome variable and the dyad-level predictor, Z; π_011F and π_011P represent how the focal person variable, W, moderates the association between the focal person or partner outcome variable and the dyad-level predictor; π_100F and π_100P represent the prototypical within-person association between the focal person or partner outcome variable and the TVC; π_101F and π_101P indicate how the within-dyad association of the outcome variable and the TVC is moderated by the focal person predictor W for focal persons and partners; π_110F and π_110P represent how the prototypical within-person association of the outcome variable and the TVC is moderated by the dyad-level variable for the focal persons and partners; π_111F and π_111P represent the effects of the interaction of the focal person and dyad-level variables on the prototypical within-person association of the outcome variable and the TVC for the focal persons and partners; ν_00Fi and ν_00Pi represent unexplained differences in intercepts for focal persons or partners; and ν_10Fi and ν_10Pi represent unexplained differences in the dyad-level associations. The inclusion of additional random effects (e.g., in Equations 19, 20, 23, and 24) is possible and depends upon the discretion of the researcher and the structure of the data, but we have chosen only to model unexplained differences in focal persons’ intercepts and in dyad-level associations.

Finally, the between focal person (Level 3) covariance matrix of random effects is

$$\left[\begin{array}{cccc}{\sigma }_{v_{00Fi}}^2& {\sigma }_{v_{00Fi}{v}_{00Pi}}& {\sigma }_{v_{00Fi}{v}_{10Fi}}& {\sigma }_{v_{00Fi}{v}_{10Pi}}\\ {\sigma }_{v_{00Fi}{v}_{00Pi}}& {\sigma }_{v_{00Pi}}^2& {\sigma }_{v_{00Pi}{v}_{10Fi}}& {\sigma }_{v_{00Pi}{v}_{10Pi}}\\ {\sigma }_{v_{00Fi}{v}_{10Fi}}& {\sigma }_{v_{00Pi}{v}_{10Fi}}& {\sigma }_{v_{10Fi}}^2& {\sigma }_{v_{10Fi}{v}_{10Pi}}\\ {\sigma }_{v_{00Fi}{v}_{10Pi}}& {\sigma }_{v_{00Pi}{v}_{10Pi}}& {\sigma }_{v_{10Fi}{v}_{10Pi}}& {\sigma }_{v_{10Pi}}^2\end{array}\right]$$ (25)

where ${\sigma }_{v_{00Fi}}^2$ is the variance between focal persons’ intercepts, ${\sigma }_{v_{00Pi}}^2$ is the variance between partners’ intercepts, ${\sigma }_{v_{10Fi}}^2$ is the variance in focal persons’ association between the dyad-level predictor and the outcome, ${\sigma }_{v_{10Pi}}^2$ is the variance in partners’ association between the dyad-level predictor and the outcome, and the off-diagonal elements are the covariances between the intercepts and slopes.

Similar to the two-level OWM model, the dyad-varying predictors in Level 2 and focal person predictors in Level 3 should be centered to obtain the desired model interpretation. The centering of TVCs in the three-level OWM model requires a few additional considerations beyond those discussed in the two-level OWM model. The disaggregation of between- and within-person effects that result when the TVCs are centered (e.g., at the sample-, focal person-, or dyad-level) influence the interpretation of the between- and within-dyad effects that are obtained. In most cases, within-person deviation scores will be calculated during data pre-processing by taking the difference between each observation and the selected mean-level in order to obtain the desired interpretation. Further discussion of the effects of TVC centering can be found in Wang and Maxwell (2015) and Hamaker and Grasman (2015) in the context of autoregressive TVCs.

Summary

The OWM model can be used to study a wide variety of dyadic relationships in which there is a set of focal persons that interacts or is associated with multiple partners, such as the many social interactions an individual has in daily life (e.g., DePaulo & Kashy, 1998) or the weekly interactions between therapists and their clients (e.g., Marcus et al., 2009, 2011). The OWM model allows for the examination of focal person effects (e.g., does the focal person behave similarly across partners?), partner effects (e.g., do partners behave similarly toward the focal person?), and relationship effects (e.g., are there unique aspects of a specific focal person-partner dyad unexplained by focal person or partner effects?). We have presented the conceptual and technical components of the two- and three-level OWM models to help researchers apply the model to their own research questions. To further that possibility, we illustrate how the two- and three-level OWM models can be used to examine interpersonal processes surrounding social interactions. Specifically, we use reports obtained in an intensive experience sampling study of everyday social interactions (Ram et al., 2014) to examine how interpersonal communion (i.e., interpersonal warmth on the interpersonal circumplex; Wiggins, 1979) is related to perceived benefit across multiple social interactions, how the association between communion and benefit differs across partner types, and how focal persons’ extraversion (which sits between warmth and status on the interpersonal circumplex; McCrae & Costa, 1989) moderates these associations.

Illustrative Empirical Examples

Data for our illustrations were drawn from the Intraindividual Study of Affect, Health, and Interpersonal Behavior (iSAHIB), a multiple time-scale experience sampling study supporting a wide variety of methodological and substantive inquiries (Ram et al., 2014; Pennsylvania State University Institutional Review Board protocol #33706). Several of the specific measures used here have been examined previously with a different method and purpose (e.g., Vogel et al., 2017) than we engage here. Of note, the iSAHIB study was not initially conceived as a reciprocal OWM design in that the focal person and partner reports described below were both obtained from the focal person (i.e., the focal persons report their own experiences and their perceptions of their partners’ experiences). However, for didactic purposes, we refer to and analyze the focal person and partner reports as if these reports were obtained from different people. While the current data set has its limitations with respect to lacking separate focal person and partner reports, there are two primary characteristics of these data that make them suitable for this illustrative example. First, we assume this data set has the necessary design for an OWM model such that (1) focal persons are the only connection between their partners (i.e., partners are locally independent of each other), (2) focal persons are not connected to each other (e.g., in a network), and (3) reports exist for both focal persons and partners. Second, the data set contains repeated interactions (i.e., reports) for each partner within each focal person, allowing us to construct a three-level OWM model and examine and compare the dynamics of interpersonal interactions. So, although we were able to “retro-fit” and present the data as if it were obtained using an OWM design, conclusions drawn from these analyses are solely meant for analytical illustration. Inferences about the underlying dyadic processes should be made cautiously because of the threat of single-source bias (Campbell & Fiske, 1959). We use the focal persons and partners terminology throughout to demonstrate how researchers would interpret results from a truly reciprocal OWM model with their own data.

Participants and Procedure

The iSAHIB sample consists of 150 adults from the Pennsylvania State University and local community, stratified by gender (50% female) and age across the adult life span (M_Age = 47.64, SD_Age = 18.85). Participants self-identified as Caucasian (91%), African American (4%), Asian American (1%), and Mixed or Other (4%) ethnicity, with the prototypical participant having 16.36 (SD = 3.90) years of formal education, yearly family income of ‘$50,000 – $74,999’ (Mode_Income = ‘$20,000 – $49,999’), and 1.5 children (SD = 1.41).

Data for this illustration are drawn from the baseline questionnaire and the three 3-week measurement bursts that were completed over the course of a year (bursts collected at approximately 4.5 month intervals). During each measurement burst, participants went about their normal daily lives, providing end-of-day reports about their feelings, thoughts, and behaviors and reporting on their social interactions throughout the day using a study-provided smartphone with a custom survey application. Specifically, after each social interaction that lasted at least five minutes, participants completed a 27-item questionnaire that asked about the social interaction partner, how long the interaction lasted, and thoughts and feelings the participant had about the social interaction and their partner. Across the three bursts (90.7% completion), participants reported on a total of 64,111 social interactions. On average, 150 focal persons reported on 59.58 (SD = 71.64) interactions they had with 6.59 (SD = 0.61) unique partners (distinguished by partner type).

Measures

Three sets of measures obtained from the baseline questionnaire and after each social interaction are used to illustrate and interpret results from two- and three-level OWM models.

Focal Person Level – Extraversion.

As part of the baseline survey, focal persons completed the short form Big 5 Inventory (BFI; Rammstedt & John, 2007). Our analyses specifically make use of responses to the items “I see myself as someone who is reserved” and “I see myself as someone who is outgoing, sociable.” Answers provided on a five-point Likert scale (disagree strongly to agree strongly) were averaged to obtain an index of each focal person’s level of extraversion (M = 3.17, SD = 1.02). The extraversion variable was centered at the sample mean (M = 0, SD = 1.02) so that the intercepts in the models represent effects for the prototypical participant.

Dyad Level –Partner Type.

After each social interaction, participants described the type of person they had interacted with, selecting from one of seven categories: stranger, friend, co-worker, romantic partner, family, service professional, and other. Of the 64,111 social interactions, 14,989 were with a friend (23.4%), 14,404 were with a romantic partner (22.5%), 13,285 were with a co-worker (20.7%), 11,345 were with a family member (17.7%), 4,073 were with a service professional (6.4%), 3,494 were with an “other” type of person (5.4%), and 2,345 were with a stranger (3.7%). Although participants interacted with multiple people within any given partner type category, this analytic demonstration treats all partners within a category as interchangeable (within focal person). In our models below, partner type was recast as a set of 6 dummy variables. The stranger partner type was used as the reference category, and the model parameters associated with the 6 dummy variables represent deviations for each partner type from the stranger partner type.

Repeated Measures Level – Focal Persons’ and Partners’ Ratings of Social Interaction.

Two types of ratings were obtained about focal persons’ and partners’ impressions of each social interaction.

Communion.

Level of partner’s and focal person’s communal behavior during each social interaction was assessed as ratings of how distant to friendly, indicated using a 0–100 slider, each person acted during the interaction (a modified version of the interpersonal grid; Moskowitz & Zuroff, 2005). The average rating of partner communion was 80.44 (SD = 15.96) and the average rating of focal person’s communion was 82.09 (SD = 14.78).

Benefit.

The extent to which the partner and the focal person benefited from the social interaction was rated as very costly to very beneficial on a −50 to +50 scale so that ratings greater than zero indicated that the focal person or partner found the interaction beneficial while ratings less than zero indicated that the focal or partner found the interaction costly. The average rating of partner’s benefit was 15.51 (SD = 19.13) and the average rating of focal person’s benefit was 14.11 (SD = 20.46). Benefit was centered at 0 for easy interpretation since zero is a meaningful point – i.e., a neutral interaction that was neither costly nor beneficial.

Two-Level One-with-Many Analyses

Our illustration of the two-level OWM analysis is structured as a cross-sectional analysis examining the association of level of communion and extraversion within everyday social interactions, and how this association differs between focal persons. In this demonstration, we use data from the first, and only the first, social interaction each of the N = 150 participants had with up to seven different partners (first stranger, first friend, etc.). In this section we describe the data preparation, model, and results of the two-level OWM analysis to examine the focal person, partner, and relationship effects on communion ratings of social interactions with multiple partners.

Data Preparation

To create cross-sectional data, the larger data were subset to the first social interaction for each partner type during the first burst of the study. The two-level data consist of five variables: two person-level measures – focal person id (i.e., participant number, i = 1 to 150) and focal person extraversion (centered; continuous) – and three repeated measures – interaction partner type (nominal with 7 categories), partner report of communion (continuous), focal person report of communion (continuous). The data are in a “long” format – there is a row for each partner within a focal person. The data set has 951 rows (average of 6.56 rows per focal person).

To set-up the data for analysis in a multivariate multilevel modeling framework, we first need a row for each focal person report and a row for each partner report. In the format our data are in now, the reports of communion for the partner and focal person are contained in the same row (because both reports have been obtained from the focal person). To be able to manipulate the statistical program later, we need this information in separate rows while maintaining the current column format. If we had two reports of extraversion (i.e., if we also had the partner’s extraversion), we would also need to put extraversion reports in their respective rows, but for this demonstration, we are only interested in the focal person’s extraversion and its association with reports of communion. Second, we create a focal person dummy variable that indicates whether a row contains information about the focal person. Finally, we create a partner dummy variable that indicates whether a row contains information about the partner. These dummy variables are primarily created as a switch that will turn on/off information (i.e., communion reports) about either the focal person or partner. Stated differently, the dummy coded variables direct the statistical program to obtain estimates for focal persons using only information about the focal person or obtain estimates for the partners using information from only the partners. An abridged depiction of the prepared data is contained in Table 1. Code for the data preparation makes use of the base and dplyr packages in R (R Core Team, 2016; Wickham, Francois, Henry, & Müller, 2017) and is available on the Penn State Quantitative Developmental Systems Group website (https://quantdev.ssri.psu.edu/).

Model

We next present a two-level OWM model for these data, built from the general layout given earlier in Equations 3 to 9. Specifically, we are interested in (1) whether there is variability in reports of communion across partners, (2) whether focal person extraversion is associated with reports of communion after an interaction, and (3) whether there is variability in the association between extraversion and communion across partners (i.e., is the strength of association between extraversion and communion higher/lower depending on the interaction partner?). In the two-level OWM model, we are unable to examine differences between partner types (i.e., we assume that partners are indistinguishable) because we do not have repeated measures for each partner type. To test these empirical questions, we set up the model as

Level 1:

$$Communio{n}_{di}={\gamma }_{0Fi}Foca{l}_{di}+{\gamma }_{0Pi}Partne{r}_{di}+e_{Fdi}+e_{Pdi}$$ (26)

Level 2:

$${\gamma }_{0Fi}={\pi }_{00F}+{\pi }_{01F}Extraversio{n}_i+v_{0Fi}$$ (27)

$${\gamma }_{0Pi}={\pi }_{00P}+{\pi }_{01P}Extraversio{n}_i+v_{0Pi}$$ (28)

where Communion_di indicates the report of communion for each member of each dyad d (focal person – partner pair) within each focal person i; γ_0Fi represents the expected communion rating for focal person i based upon all their reports; and γ_0Pi represents the expected communion rating for focal person i’s partners. In turn, π_00F and π_00P represent the expected communion rating provided by focal persons and partners when the focal person had a prototypical level of extraversion, and π_01F and π_01P indicate how those ratings are associated with between-focal person differences in extraversion. Unexplained differences at the focal-person level are captured by ν_0Fi and ν_0Pi, while differences between dyads within a focal person are captured by e_Fdi and e_Pdi. Code for this model makes use of the nlme package (Pinheiro et al., 2016) and is given in the online supplemental material. For researchers interested in fitting this model using a proprietary program, please refer to Planalp and colleagues (2017) for a great tutorial on fitting longitudinal dyadic models as a starting point to translate our code into their preferred program.

Results and Interpretation

Results of this two-level OWM model are shown in Table 3. For the prototypical person in the sample, with average extraversion, focal person reports of communion were, on average, $\widehat{{\pi }_{00F}}=81.72$(SE = 3.05) and partner reports of communion were, on average, $\widehat{{\pi }_{00P}}=79.27$(SE = 2.93). Extraversion was not associated with reports of communion for focal persons or partners ($\widehat{{\pi }_{01F}}=0.04$ and $\widehat{{\pi }_{01P}}=0.12$). The random effects suggest that there were substantial between-focal person differences in partner ratings ($\widehat{{\sigma }_{v_{0F\iota }}}=10.27$) and substantial between-partner differences in focal person ratings ($\widehat{{\sigma }_{v_{0P\iota }}}=10.83$), and these differences were highly correlated ($\widehat{r}=0.99$), indicating that focal persons reporting high levels of communion had partners that also reported high levels of communion (which is expected here given that reports were actually coming from a single source). There was also a substantial amount of dyad-specific variation ($\widehat{{\sigma }_{e_{Fd\iota }}}=17.14$ for focal persons and $\widehat{{\sigma }_{e_{Pd\iota }}}=14.12$ for partners). This residual dyad-level variance is often labeled a “relationship effect” – i.e., the uniqueness of each focal-person - partner relationship, some aspects of which were noted by both members of those dyads ($\widehat{r}=-.08$). In summary, this two-level OWM model allowed us to determine that (1) there was significant variability in reports of communion across partners, (2) focal person extraversion was not associated with focal person’s or partner’s reports of communion, and (3) there was a significant association between how focal persons rated the interactions and how their partners rated the interactions.

Table 3.

Results from Two-level One-with-many Analysis Examining the Effect of Focal Person Extraversion in Reported Levels of Communion

Fixed Effects	Estimate	SE
Intercepts
Focal Person, π00F	81.79*	0.96
Partner, π00P	79.74*	0.95
Extraversion
Focal Person, π01F	0.04	0.94
Partner, π01P	0.12	0.94
Random Effects	SD	Corr
Focal person, v0Fi	10.83
Partner, v0Pi	10.27	0.99
Focal person residual, eFdi	17.14
Partner residual, ePdi	14.12	−0.08

Note: N = 951 social interactions nested within 150 focal persons. SE = Standard Error. SD = Standard Deviation. Corr = Correlation.

^*= p < .05

Three-Level One-with-Many Analyses

We then extended the two-level OWM analysis to a three-level analysis that accommodates and makes use of the repeated measures data. In this section, we describe the data preparation, model, and results of the three-level OWM model. Specifically, we walk through the necessary steps to conduct a three-level OWM analysis to examine the focal person, partner, and relationship effects on communion during social interactions with different types of partners, whether communion varied in accordance with the perceived benefit of the interaction, and if and how the ratings and processes were associated with differences in focal person’s level of extraversion. This analysis makes use of the whole longitudinal data set originally described: focal person and partner reports about 64,111 social interactions nested within 7 types of dyads nested within 150 individuals (i.e., focal persons).

Data Preparation

Our data set consists of six variables: focal person id (i.e., participant in this study), social interaction number (represented with “interaction number” in Table 2, such that 1 = the first interaction through N = the final interaction for each focal person), partner type (nominal variable depicted here in parsimonious form as a single variable with 7 values, but actually invoked as 6 dummy variables and a reference category = stranger dyads), communion from the focal persons and partners (continuous variable), benefit from the focal persons and partners (continuous variable), and focal person extraversion (centered; continuous variable). Similar to the two-level OWM analysis, the data are in a “long” format and we must set-up the data in preparation for our analyses. We create (1) a row for each focal person report and a row for each partner report for every social interaction, (2) a focal person dummy variable that indicates whether a row contained information about the focal person, and (3) a partner dummy variable that indicates whether a row contains information about the partner. An abridged depiction of the prepared data is contained in Table 2. Additionally, Figure 3 depicts the first 100 social interactions for two focal persons (top vs. bottom plot) with seven partner types each (as differentiated by the seven colors in each plot). Each partner type has a varying number of social interactions within the focal person. After each social interaction, the focal person and partner each rated the communion and benefit of the interaction. Focal person reports of communion and benefit are depicted by the black solid and dashed lines (after using a smoother for plotting purposes), respectively. Partner reports of communion and benefit are depicted by the white solid and dashed lines (after using a smoother for plotting purposes), respectively.

Figure 3.

The first 100 social interactions from the first burst of data collection for two focal persons (top vs. bottom plot) with seven partner types each (as differentiated by the seven colors in each plot). Each partner type has a varying number of social interactions with the focal person. After each social interaction, the focal person and partner each rated the communion and benefit of the interaction. Focal person communion and benefit are depicted by the black solid and dashed lines, respectively. Partner communion and benefit are depicted by the white solid and dashed lines, respectively.

Model

Next, we developed a three-level OWM model for these data, built from the general layout given earlier in Equations 10 to 25. We allow for random intercepts in expected level of communion for focal persons and partners, and in the within-focal person associations between benefit and communion for focal persons and partners, both across dyads (Level 2) and across individuals (Level 3). The expanded model becomes

Level 1:

$$\begin{gathered} Communio{n}_{tdi}={\beta }_{\text{0}Fdi}Foca{l}_{tdi}+{\beta }_{oPdi}Partne{r}_{tdi}+ \\ {\beta }_{1Fdi}FocalBenefi{t}_{tdi}+{\beta }_{1pdi}PartnerBenefi{t}_{tdi}+e_{Ftdi}+e_{Ptdi} \end{gathered}$$ (29)

where Communion_tdi indicates the report of communion obtained from each person in each dyad d within each focal person i at a particular occasion t; β_0Fdi and β_0Pdi represent intercepts for each unique focal person and partner; β_1Fdi and β_1Pdi represent the within-person associations between the TVC, benefit, and communion for each unique focal person and partner. The dyad-specific parameters are then modeled as

Level 2:

$${\beta }_{0Fdi}={\gamma }_{00Fi}+{\gamma }_{01Fi}Frien{d}_{di}+{\gamma }_{02Fi}Coworke{r}_{di}+\cdots +{\gamma }_{06Fi}Othe{r}_{di}+u_{0Fi}$$ (30)

$${\beta }_{0Pdi}={\gamma }_{00Pi}+{\gamma }_{01Pi}Frien{d}_{di}+{\gamma }_{02Pi}Coworke{r}_{di}+\cdots +{\gamma }_{06Pi}Othe{r}_{di}+u_{0Pi}$$ (31)

$${\beta }_{1Fdi}={\gamma }_{10Fi}+{\gamma }_{11Fi}Frien{d}_{di}+{\gamma }_{12Fi}Coworke{r}_{di}+\cdots +{\gamma }_{16Fi}Othe{r}_{di}+u_{1Fi}$$ (32)

$${\beta }_{1Pdi}={\gamma }_{10Pi}+{\gamma }_{11Pi}Frien{d}_{di}+{\gamma }_{12Pi}Coworke{r}_{di}+\cdots +{\gamma }_{16Pi}Othe{r}_{di}+u_{1Pi}$$ (33)

where γ_00Fi and γ_00Pi represent the expected communion rating for focal person i based upon the prototypical focal person – stranger dyad (i.e., the reference category) from the focal person’s and partner’s perspective; γ_01Fi through γ_06Fi and γ_01Pi through γ_06Pi represent the expected difference in communion rating for different dyad types from the prototypical focal person – stranger dyad rating, from the focal person’s and partner’s perspective; γ_10Fi and γ_10Pi represent the within-dyad association of the TVC, benefit, with communion for the prototypical focal person – stranger dyad from the focal person’s and partner’s perspective; γ_11Fi through γ_16Fi and γ_11Pi through γ_16Pi represent how the within-dyad association of the benefit with communion for each type of dyad differs from focal person i’s prototypical focal person – stranger dyad, from the focal person’s and partner’s perspective; u_0Fi and u_0Pi represent the unexplained dyad-specific residuals in the intercepts for focal persons and partners; and u_1Fi and u_1Pi represent unexplained dyad-specific residuals for the within-person associations for focal persons and partners.

Level 3:

$${\gamma }_{00Fi}={\pi }_{000F}+{\pi }_{001F}Extraversio{n}_i+v_{00Fi}$$ (34)

$${\gamma }_{01Fi}={\pi }_{010F}+{\pi }_{011F}Extraversio{n}_i$$ (35)

$${\gamma }_{02Fi}={\pi }_{020F}+{\pi }_{021F}Extraversio{n}_i$$ (36)

⋯

$${\gamma }_{06Fi}={\pi }_{060F}+{\pi }_{061F}Extraversio{n}_i$$ (37)

$${\gamma }_{00Pi}={\pi }_{000P}+{\pi }_{001P}Extraversio{n}_i+v_{00Pi}$$ (38)

$${\gamma }_{01Pi}={\pi }_{010P}+{\pi }_{011P}Extraversio{n}_i$$ (39)

$${\gamma }_{02Pi}={\pi }_{020P}+{\pi }_{021P}Extraversio{n}_i$$ (40)

⋯

$${\gamma }_{06i}={\pi }_{060P}+{\pi }_{061P}Extraversio{n}_i$$ (41)

$${\gamma }_{10Fi}={\pi }_{100F}+{\pi }_{101F}Extraversio{n}_i+v_{10Fi}$$ (42)

$${\gamma }_{11Fi}={\pi }_{110F}+{\pi }_{111F}Extraversio{n}_i$$ (43)

$${\gamma }_{12Fi}={\pi }_{120F}+{\pi }_{121F}Extraversio{n}_i$$ (44)

⋯

$${\gamma }_{16Fi}={\pi }_{160F}+{\pi }_{161F}Extraversio{n}_i$$ (45)

$${\gamma }_{10Pi}={\pi }_{100P}+{\pi }_{101P}Extraversio{n}_i+v_{10Pi}$$ (46)

$${\gamma }_{11Pi}={\pi }_{110P}+{\pi }_{111P}Extraversio{n}_i$$ (47)

$${\gamma }_{12Pi}={\pi }_{120P}+{\pi }_{121PC}Extraversio{n}_i$$ (48)

⋯

$${\gamma }_{16Pi}={\pi }_{160P}+{\pi }_{161P}Extraversio{n}_i$$ (49)

where π_000F and π_000P represent the expected value of communion (conditional on focal person extraversion) for the prototypical focal person and partner in focal person – stranger dyads; π_010F through π_060F and π_010P through π_060P represent the difference in the expected value of communion (conditional on focal person extraversion) for each dyad type relative to the prototypical focal person – stranger dyad, for the prototypical focal person and partner; π_001F and π_001P indicate how the differences in level of communion for focal person – stranger dyads are related to differences in extraversion; π_011F through π_061F and π_011P through π_061P indicate how the dyad-type specific differences are moderated by differences in extraversion; π_100F and π_100P indicate how partner type moderates the expected association between communion and benefit for the prototypical focal person and partner in focal person – stranger dyads; π_110F through π_160F and π_110P through π_160P represent the differences in how partner type moderates the expected association between communion and benefit; π_101F and π_101P indicate how focal person extraversion moderates the interaction between partner type and benefit for focal person and partner reports about focal person – stranger dyads; π_111F through π_161F and π_111P through π_161P indicate how those interactions are moderated by focal person extraversion; ν_00Fi and ν_00Pi represent unexplained differences in expected communion for focal persons and partners; ν_10Fi and ν_10Pi represent unexplained differences in the association between benefit and communion for focal persons and partners; and e_Ftdi and e_Ptdi occasion-specific residual errors. Interpretation of each of the model parameters can be found in Table S2 of the online supplemental material. Code for this model makes use of the nlme package (Pinheiro et al., 2016) and is available at the aforementioned website and within the online supplemental material.

Results and Interpretation

The results of this three-level OWM model are presented in Table 4. Across focal persons having an average level of extraversion, we found that average reports of communion for strangers was 78.12 ($\widehat{{\pi }_{000F}}=78.12$, SE = 0.90) and was significantly higher for friends ($\widehat{{\pi }_{010F}}=3.13$, SE = 0.63) and family members ($\widehat{{\pi }_{040F}}=3.86$, SE = 0.66), but significantly lower for service professionals ($\widehat{{\pi }_{050F}}=-1.65$, SE = 0.68). Across partners conditional on the focal person having an average level of extraversion, we found the average reports of communion for strangers was 75.82 ($\widehat{{\pi }_{000P}}=75.82$, SE = 0.95), with friends ($\widehat{{\pi }_{010P}}=3.93$, SE = 0.79) and family members ($\widehat{{\pi }_{040P}}=2.16$, SE = 0.83) having significantly higher average reports of communion. Added value from the 3-level OWM model includes examination of within-dyad dynamics, operationalized here as the association between time-varying changes in benefit and individuals’ communion. Reports of benefit were positively associated with reports of communion for strangers from both the focal person ($\widehat{{\pi }_{100F}}=0.16$, SE = 0.02) and partner perspectives ($\widehat{{\pi }_{100P}}=0.16$, SE = 0.02; see left column of Figure 4). Romantic partners (from the focal person:$\widehat{{\pi }_{130F}}=0.12$, SE = 0.02; and partner perspective:$\widehat{{\pi }_{120P}}=0.16$, SE = 0.03) and family members (from the partner perspective: $\widehat{{\pi }_{140P}}=0.07$, SE = 0.03) had an even stronger association between reports of benefit and communion as compared to stranger dyads. It is worth noting, however, that the benefit variable was not centered (a choice we made because the benefit variable has a true zero), which affects the interpretation of benefit. Specifically, the interpretation of these results is the overall effect of benefit on communion; the between-person and within-person effects cannot be disaggregated with the uncentered variable, which is a blend of the two effects.

Table 4.

Results from Three-level One-with-many Analysis Examining the Association between Benefit and Communion with Partner Type and Extraversion Moderating this Association

Fixed Effects	Estimate	SE
Focal person Intercepts
Stranger, π000F	78.12*	0.90
Friend, π010F	3.13*	0.63
Co-worker, π020F	−0.12	0.65
Romantic partner, π030F	0.41	0.66
Family, π040F	3.86*	0.66
Service professional, π050F	−1.65*	0.68
Other, π060F	−0.10	0.69
Partner Intercepts
Stranger, π000P	75.82*	0.95
Friend, π010P	3.93*	0.79
Co-worker, π020P	0.31	0.81
Romantic partner, π030P	0.74	0.83
Family, π040P	2.16*	0.83
Service professional, π050P	0.64	0.83
Other, π060P	−1.22	0.87
Focal person - benefit
Stranger, π100F	0.16*	0.02
Friend, π110F	−0.03	0.02
Co-worker, π120F	−0.01	0.02
Romantic partner, π130F	0.12*	0.02
Family, π140F	0.03	0.02
Service professional, π150F	0.03	0.02
Other, π160F	0.00	0.02
Partner - benefit
Stranger, π100P	0.16*	0.02
Friend, π110P	−0.02	0.03
Co-worker, π120P	−0.04	0.03
Romantic partner, π130P	0.12*	0.03
Family, π140P	0.07*	0.03
Service professional, π150P	0.02	0.03
Other, π160P	−0.01	0.03
Focal person - extraversion
Stranger, π001F	3.13*	0.89
Friend, π011F	−1.27*	0.62
Co-worker, π021F	−1.29*	0.63
Romantic partner, π031F	−2.04*	0.64
Family, π041F	−1.73*	0.66
Service professional, π051F	−0.63	0.67
Other, π061F	−0.75	0.67
Partner - extraversion
Stranger, π001P	1.34	0.93
Friend, π011P	0.27	0.78
Co-worker, π021P	−0.15	0.79
Romantic partner, π031P	−0.32	0.80
Family, π041P	0.55	0.82
Service professional, π051P	−0.03	0.81
Other, π061P	0.47	0.84
Focal person - benefit X extraversion
Stranger, π101F	−0.03	0.02
Friend, π111F	0.03	0.02
Co-worker, π121F	0.03	0.02
Romantic partner, π131F	0.02	0.02
Family, π141F	0.01	0.02
Service professional, π151F	0.01	0.02
Other, π161F	0.01	0.02
Partner - benefit X extraversion
Stranger, π101P	0.07*	0.02
Friend, π111P	−0.07*	0.02
Co-worker, π121P	−0.04	0.02
Romantic partner, π131P	−0.07*	0.02
Family, π141P	−0.09*	0.03
Service professional, π151SpF	−0.03	0.03
Other, π161P	−0.05	0.03
Random Effects	SD	Corr
Person-level (Level 3)
Focal person, v00Fi	9.31
Partner, v00Pi	8.84	0.97
Focal person X benefit, v10Fi	0.11	−0.73	−0.68
Partner X benefit, v10Pi	0.10	−0.67	−0.67	−0.94
Dyad-level (Level 2)
Focal person, u0Fdi	4.50
Partner, u0Pdi	5.67	0.85
Focal person X benefit – between partners, u1Fdi	0.11	−0.76	−0.64
Partner X benefit – between partners, u1Pdi	0.14	−0.61	−0.73	0.81
Occasion-level (Level 1)
Focal person residual, eFtdi	10.35
Partner residual, ePtdi	12.06	0.00†

Note: N = 64,111 social interactions nested within 150 focal persons. SE = Standard Error. SD = Standard Deviation. Corr = Correlation.

^*= p < .05.

^†= Correlation between residual terms set to zero.

Figure 4.

Estimated association of benefit and communion by partner type and level of extraversion from the three-level OWM model. The upper left panel depicts the benefit-communion association by partner type from the focal person’s perspective for focal persons who have reported extraversion −1 standard deviation below the mean. The bottom left panel depicts the benefit-communion association by partner type from the focal person’s perspective for focal persons who have reported extraversion +1 standard deviation above the mean. The upper right panel depicts the benefit-communion association by partner type from the partner’s perspective for focal persons who have reported extraversion −1 standard deviation below the mean. The bottom right panel depicts the benefit-communion association by partner type from the partner’s perspective for focal persons who have reported extraversion +1 standard deviation above the mean.

Additionally, we found that focal person extraversion was positively associated with reports of communion for focal person – stranger social interactions from the focal person’s perspective ($\widehat{{\pi }_{001F}}=3.13$, SE = 0.89). Friends ($\widehat{{\pi }_{011F}}=-1.27$, SE = 0.62), co-workers ($\widehat{{\pi }_{021F}}=-1.29$, SE = 0.63), romantic partners ($\widehat{{\pi }_{031F}}=-2.04$, SE = 0.64), and family members ($\widehat{{\pi }_{041F}}=-1.73$, SE = 0.66) had a significantly weaker association between focal person extraversion and communion as compared to strangers. From a partner’s perspective, focal person extraversion was not associated with reports of communion for any partner types. Furthermore, we tested the interaction between reports of focal persons’ and partners’ benefit and focal person extraversion on reports of communion and found (1) from the focal person’s perspective, extraversion did not moderate the association between benefit and communion for any partner types (compare upper and lower panels in the left column of Figure 4) and (2) from the partner’s perspective, extraversion moderated the association between benefit and communion for strangers ($\widehat{{\pi }_{101P}}=0.07$, SE = 0.02) such that it made the association stronger, whereas friends ($\widehat{{\pi }_{111P}}=-0.07$, SE = 02), romantic partners ($\widehat{{\pi }_{131P}}=-0.07$, SE = 0.02), and family members ($\widehat{{\pi }_{141P}}=-0.09$, SE = 0.03) had a significantly weaker moderation effect compared to strangers (compare upper and lower panels in the right column of Figure 4).

The random effects were also of particular interest in this model. We found there were substantial between focal person differences in partner ratings $\left(\widehat{{\sigma }_{v_{0F\iota }}}=9.31\right)$ and focal person ratings $\left(\widehat{{\sigma }_{v_{0P\iota }}}=8.84\right)$, after accounting for partner type and extraversion. The focal person and partner random effects were highly correlated $(\widehat{r}=0.97)$, which indicates that focal persons in a dyad who reported higher levels of communion than average had partners who also reported higher levels of communion than average (which is expected here given that reports were coming from a single source). There were also significant differences in ratings within a focal person (across dyads), both from the focal person’s perspective $\left(\widehat{{\sigma }_{u_{\text{o}F\iota }}}=4.50\right)$ and the partner’s perspective $\left(\widehat{{\sigma }_{u_{\text{o}F\iota }}}=5.67\right)$. These random effects were also highly correlated $(\widehat{r}=0.85)$ indicating that within a focal person, focal persons in a dyad who reported higher levels of communion than average had partners who also reported higher levels of communion than average. Furthermore, there were differences in the association between benefit and communion (within-dyad dynamics) from the focal person’s and partner’s perspectives, both between focal persons ($\widehat{{\sigma }_{v_{1F\iota }}}=0.11$ and $\widehat{{\sigma }_{v_{1P\iota }}}=0.10$ ) and between partners ($\widehat{{\sigma }_{u_{1F\iota }}}=0.11$ and $\widehat{{\sigma }_{u_{1P\iota }}}=0.14$), that were not accounted for by extraversion or partner type. Finally, there was substantial occasion-specific variance ($\widehat{{\sigma }_{e_{Fd\iota }}}=10.35$ for focal persons and $\widehat{{\sigma }_{e_{Pd\iota }}}=12.06$ for partners), indicating there is something unique about each of the specific social interactions that was not shared by the participants in that situation (correlation between the focal person and partners’ residual error was 0.00).

In sum, this three-level OWM model allowed us to examine (1) the association between partner type and communion from the focal persons’ and partners’ perspective, (2) the association between extraversion and communion for each partner type, (3) the association between reported focal person/partner benefit and focal person/partner communion at each social interaction, (4) the interaction of focal persons’/partners’ benefit and focal person extraversion on reports of focal persons’/partners’ communion, (5) the focal person and partner random effects and their association, and (6) the focal person and partner random effects within focal persons.

Discussion

In this article, we presented the two- and three-level OWM models as being increasingly useful for testing and refining theories about interpersonal processes. Our hope is that as researchers in many disciplines making use of newly-arriving data streams may find this model particularly useful in their study of dyadic relations. To illustrate this model, we applied the two- and three-level OWM model to data obtained in an intensive experience sampling study of social interactions (Ram et al., 2014). We described how characteristics of the repeated interactions (i.e., benefit), dyads (i.e., partner type), and focal persons (i.e., extraversion) were associated with focal persons’ and partners’ perceptions of communion. While these results can help inform theory with regards to interpersonal perceptions, we caution that our dyadic analysis was actually based only on the focal persons’ perceptions of their and their partners’ experiences. Our purpose here was to provide a tutorial on the modeling process rather than form substantive conclusions about the specific dyadic processes captured in these data.

The OWM model is unique within the dyadic literature in that it is a blend of the standard design (pairs of people only connected to each other, such as spouses) and the social relations model (each person in a network being connected to every other person in the network, such as in a round robin design). While this model has primarily been used to examine situations in which the OWM model naturally forms (such as in therapist-client relationships), we see this model as particularly useful when complete network data are not or cannot be collected. For example, in studies of in-person and/or online social interactions, it is unlikely that researchers will be able to collect information about all the interactions that occur among individuals and their social partners over time – i.e., all interactions for the entire social network. Using the OWM model, researchers can still make good use of information about all the interactions the focal person has (e.g., from their Facebook stream, assuming that partners’ online interactions are not connected to each other in any meaningful way). Such data are already readily available (Eichstadt et al., 2018).

Limitations and Future Directions

The OWM model presented here has several limitations worth noting, and these limitations go hand-in-hand with future extensions of the model. First, the OWM models we presented assumed a linear association between the predictors and outcome. However, we believe this model can be easily extended to capture potential nonlinearities in the association between predictors and outcomes, for example, by adding quadratic effects. Second, we ignored time dependencies within our models. Specifically, (1) we did not model trends or reactivity effects, which can be accommodated by including time as a predictor in the model and (2) we did not accommodate the possibility that reports from the prior social interaction are associated with reports from the current social interaction, which can be modeled using an autoregressive error term in the model or error structure (Wang et al., 2012). We also assumed the timing between social interactions was irrelevant (i.e., that the social interactions were independent and identically distributed). Future extensions could account for the unequal time intervals or carry-over effects between social interactions through clever use of a timing variable and a continuous time modeling framework (van Montfort et al., 2018). Third, the OWM model assumes that the partners of any given focal person are not connected to each other. This may not be a realistic assumption when studying adolescents and their friends, employees and their co-workers, or teachers and their students. One way of accommodating this issue includes allowing correlations among partners’ error terms. Fourth, while we can make rough inferences based on what is generally known about multilevel modeling, simulation studies are needed to better understand how this model performs under different conditions, including different number of time points, partners, and overall sample size (e.g., Maas & Hox, 2005). Furthermore, this model does not necessarily have to be fit within a multilevel modeling framework; it can also be fit within a structural equation modeling framework (Ackerman et al., 2019). The usual considerations for running a model in a multilevel modeling versus structural equation modeling framework apply here (Grimm et al., 2016). For instance, a SEM model can be used when a measurement model is needed to represent the constructs of interest or when assessing model fit is the goal of the analysis. In our case, we chose to use a multilevel modeling framework because all the variables were either single items or established scales (thus no measurement model was needed) and all the participants had unique temporal designs (differing number and spacing of repeated measures), design characteristics easily accommodated in multilevel implementations. Finally, the OWM model examples we presented examined data that assumed there was reciprocity in reporting (i.e., both members of the dyad reported about each other). In cases when there is not reciprocity in the reporting of information, the equations would be modified by removing either the focal person or partner terms in the Level 1 equation of the multilevel model (and the subsequent, related terms that appear in the Level 2 and Level 3 equations). As the two-level and three-level OWM models get applied to the rich data streams flowing in from experience sampling studies and social media, we are certain that the modeling approach will be fine-tuned for additional complexities.

Conclusion

This paper has presented and extended the OWM model to incorporate repeated measures data. The OWM model exists at the intersection of standard dyadic designs and social relations models, and is well-suited to examine a variety of interpersonal phenomena (e.g., therapist-client relationships) and to examine data in which full network information is not available (e.g., data from new social media streams). We hope this paper facilitates understanding and use of the OWM model, while also shedding light on potential methodological directions to advance dyadic analytic techniques.