Abstract
This work describes an application of the actor–partner interdependence model (APIM) that allows researchers to test hypotheses in terms of interdependence theory (IT). The authors’ goal is to move beyond the obvious similarities of these two frameworks by providing a detailed conceptual integration. This analysis demonstrates that aspects of APIM analysis reveal a useful perspective on interdependence not explicitly articulated by IT. They also expand on ideas presented by Kenny and Ledermann by exploring the relationship between their ratio parameter k and IT, and introducing two additional ratios (h and c) also suggested by IT. A complete worked example of APIM analysis from the perspective of IT, along with a SAS MACRO that produces confidence intervals for k, h, and c, is provided.
Interdependence theory (IT; Kelley et al., 2003; Kelley & Thibaut, 1978; Rusbult & Buunk, 1993; Rusbult & Van Lange, 2003; Thibaut & Kelley, 1959) has long been recognized as a powerful framework for understanding the dynamics of dyadic interaction. Over time, the theory has evolved beyond a general explanation of the terms and conditions of interdependent relationships to describe the dispositional and contextual factors leading to specific patterns of interdependence (Kelley et al., 2003). Although Kelley and colleagues (Kelley et al., 2003; Kelley & Thibaut, 1978; Thibaut & Kelley, 1959) provided considerable justification for the presence of interdependent processes in close relationships, specific procedures for testing aspects of IT in the form of a data-analytic model have not been described in detail. As a result, researchers adopting an IT perspective have utilized a piecemeal approach to test only components of interdependent processes.
Since its introduction in the mid-1990s, the actor–partner interdependence model (APIM; Kenny, 1996; Kenny, Kashy, & Cook, 2006) has become the de facto method for analyzing dyadic data. In the APIM, the association between a predictor and outcome variable for members of a dyad is decomposed into two distinct parts. The unique effect of a person’s own predictor on his or her own outcome is known as the actor effect, and the unique effect of that person’s predictor on their dyadic partner’s outcome is known as the partner effect. A number of prior works describe procedures for estimating regression coefficients for actor and partner effects, using a variety of statistical frameworks (Campbell & Kashy, 2002; Kenny, 1996; Kenny & Cook, 1999; Kenny et al., 2006). This article demonstrates how the APIM can be used to directly examine hypotheses in terms of IT (Kelley et al., 2003; Kelley & Thibaut, 1978; Thibaut & Kelley, 1959). Our goal is to move beyond the obvious similarities of these two frameworks by providing a detailed conceptual integration. We also illustrate how some aspects of APIM analysis reveal a useful perspective on interdependence not explicitly articulated by IT.
At least two prior works have suggested links between IT and the APIM. For instance, when discussing the implications of partner effects, Kenny and Cook (1999) made a brief reference to IT by directing the reader to Kelley and Thibaut (1978) for a “parallel formulation” (p. 435). More recently, Kenny and Ledermann (2010) described an application of the APIM that focused on interpreting the ratio of partner to actor regression coefficients (b Partner:b Actor), known as k. The authors presented a technique for estimating k, along with a discussion of how different values of k are associated with different forms of interdependence. Although these contributions provided insight into many aspects of dyadic data analysis, theoretical integration was not the primary focus of either, and a number of important connections between APIM and IT remained unexplored.
This article does not question the validity of prior contributions in any way. Instead, our intention is to demonstrate a specific procedure for using APIM to evaluate hypotheses derived from IT. Given the complexity and broad scope of IT, it would be impossible to incorporate every aspect of the theory into our presentation. As a result, we concentrate on the core aspects of IT that are most relevant to dyadic data analysis using the APIM. Some of the more complex aspects of IT (e.g., transition lists, transformations) are discussed as potential avenues for future work. In addition, we focus specifically on distinguishable dyads, which possess a categorical variable that differentiates between the two members of the dyad (Kenny, 1996). Examples of distinguishable dyads include heterosexual couples whose members may be distinguished on the basis of gender (i.e., male vs. female) and nontwin siblings who may be classified according to birth order (i.e., older vs. younger). In contrast, indistinguishable dyads possess no attribute or characteristic that differentiates between the members. Examples of indistinguishable dyads include homosexual couples and monozygotic twins. Although many of the ideas discussed here also apply to indistinguishable dyads, for the sake of simplicity, a detailed exploration of the link between IT and the APIM for indistinguishable dyads is left to future work.
We begin with a thorough review of IT and the APIM, followed by a comparison and contrast of their central features. Emphasis is placed on understanding how the statistical parameters estimated by the APIM (i.e., intercepts, actor and partner slopes, variances, and covariances) relate to the concepts described by IT. Aspects of IT that are not typically addressed through APIM analysis, but that may be “recovered” through model specification and follow-up procedures, are discussed. We also expand on the core ideas presented by Kenny and Ledermann (2010) by exploring further the associations between k and IT, and introducing two additional ratios (c and h) also suggested by IT. A complete worked example of APIM analysis from the perspective of IT, along with a SAS MACRO that produces confidence intervals (CIs) for k, c, and h, is provided. The article concludes with a discussion of the theoretical and analytic implications of this synthesis, as well as some advanced aspects of IT and the associated research designs.
IT
The central idea of IT (Kelley et al., 2003; Kelley & Thibaut, 1978; Thibaut & Kelley, 1959) is that individuals belonging to the same dyad affect one another in complex ways. IT maintains that in addition to the mutual (direct) influence that dyad members exert on each other, the joint combination of decisions or attributes exhibited by dyad members also plays a role in determining the outcome for each individual. As a result, changes in the attributes or behavioral decisions of either dyad member may affect the outcome for both dyad members. Given its utility, the popularity of IT over the past six decades is hardly surprising. Indeed, a number of theories on personal relationships trace their lineage to IT, including models of ideal standards (Fletcher, Simpson, Thomas, & Giles, 1999), intimacy (Reis & Shaver, 1988), investment (Rusbult, 1980), responsiveness (Lemay, Clark, & Feeney, 2007), risk regulation (Murray, Holmes, & Collins, 2006), and self-expansion (Aron & Aron, 1986).
IT is also a very flexible framework for understanding dyadic interaction, which allows it to be combined with other theoretical perspectives. For example, an IT perspective on adult romantic attachment (Bowlby, 1973; Hazan & Shaver, 1987) suggests that an individual’s relationship satisfaction should be related to one’s own attachment anxiety, the attachment anxiety of their romantic partner, as well as the unique combination of both members’ attachment anxiety. Although the predictor in this example (attachment anxiety) is a continuous variable, the original sources for IT (Kelley et al., 2003; Kelley & Thibaut, 1978; Thibaut & Kelley, 1959) typically used discrete predictors in their examples to reflect situations where dyad members decided between one of two possible options (e.g., clean the apartment vs. not clean the apartment). We contend that the fundamental aspects of IT are unchanged by broadening the theory’s focus to accommodate continuous predictors. 1 In fact, given that many of the questions relevant to contemporary dyadic research involve attributes measured on interval scales, this generalization allows IT to be applied to a wider range of questions. Next, we review the core components of IT used to describe the nature of interdependent processes.
The Outcome Matrix
Concrete examples of IT are best expressed in the form of a matrix, borrowed from game theory research (Kelley et al., 2003). This 2 × 2 outcome matrix describes each member’s standing on some outcome variable as a function of the decisions or attributes of each dyad member on the predictor variable (Kelley et al., 2003; Kelley & Thibaut, 1978; Thibaut & Kelley, 1959). The values in the outcome matrix represent hypothetical losses or gains along some hedonic dimension of a positively or negatively valenced outcome, with more positive values reflecting better outcomes (Kelley et al., 2003). Figure 1A illustrates an example outcome matrix describing the association between attachment anxiety (the predictor) and relationship satisfaction (the outcome) among heterosexual couples. Note that each cell contains two values. The numbers in the upper right corner within each cell of the matrix reflect outcome values for men, and those in the bottom left corner correspond to outcomes for women. According to Kelley and colleagues (2003), an outcome matrix is said to be symmetric when the values are the same for both dyad members within all cells. In contrast, an outcome matrix is asymmetric when any of the cells contain different values.
Outcome matrix and distribution
In our example, the left vertical axis of the matrix lists values of attachment anxiety for the woman, whereas the top horizontal axis lists values for the man. Examining the possible combinations of attachment anxiety, cells along the main diagonal (upper left and lower right) represent situations in which dyad members “match” on the predictor (e.g., both partners are high or low in attachment anxiety). In this example, we observe high satisfaction for the man, but low satisfaction for the woman, when both dyad members report low attachment anxiety. In contrast, we observe low satisfaction for both dyad members when the man and woman are high in attachment anxiety. Turning to the remaining cells in which dyad members mismatch (e.g., one partner is low attachment anxiety and the other is high) on the predictor, we observe moderate satisfaction for both dyad members when the man is low in attachment anxiety and the woman is high. Finally, we observe high satisfaction for the woman and moderate satisfaction for the man when the man is high in attachment anxiety and the woman is low.
According to Kelley and colleagues (2003), the outcome matrix is conceptually similar to a 2 × 2 table of means from a traditional ANOVA. In ANOVA terms, each dyad member represents a separate independent variable, or factor, and the possible decisions available to each member reflect levels within the factor. Later sections expand on the similarities between ANOVA and the outcome matrix, an association that helps to connect IT and the APIM. Next, we discuss how the outcome matrix may be reformatted, decomposed, and analyzed in ways that provide insight into the dyadic processes described by IT.
The Outcome Distribution
The outcome matrix can also be expressed as a distribution of expected outcome values, allowing researchers to illustrate the degree and form of interdependence for a given scenario. Kelley and colleagues (Kelley et al., 2003; Kelley & Thibaut, 1978) developed the outcome distribution as a visual method for providing insight into the pattern of values described by the outcome matrix. Creating an outcome distribution involves plotting each cell of the outcome matrix along a two-dimensional plane as in Figure 1B. Each dyad member is assigned an axis, and the values for each cell of the outcome matrix are used to plot coordinates on this plane. These points are connected to form the outcome distribution. Each point reflects the outcome values (of relationship satisfaction, for example) for the man and woman in one of the four cells of the outcome matrix. The labels refer to the cell in the outcome matrix from which the values are drawn. For example, the point located at +8.5 along the horizontal (man) axis, and −5 along the vertical (woman) axis, is labeled “ML, WL” because it represents the cell in which both dyad members are low in attachment anxiety.
Range and levels
Features of the outcome matrix and distribution convey important information about the interdependent processes at work. According to Kelley and colleagues (Kelley et al., 2003; Kelley & Thibaut, 1978), the range of possible values observed for each dyad member reflects the amount of variability in the outcome across the four cells. Small estimates of range suggest limited variability in the outcome of interest (i.e., the outcome value changes little across the four cells), which indicates a weak association between the predictor and the outcome. The range is computed by subtracting the minimum outcome value from the maximum for each dyad member. For this example, the observed ranges were 8.5 − (−3.5) = 12 for the man and 13 − (−5) = 18 for the woman. This means that across the four possible combinations of attachment anxiety, there is greater variability in relationship satisfaction for the woman (relative to the man). The average value of the outcome variables for each dyad member, known as the level, is also of interest. The levels for both dyad members are located near the center of the outcome distribution. In this example, the average outcome level for relationship satisfaction was 4 for the man and 1 for the woman. Given that the outcome variable in this example does not possess a natural, readily interpretable scale, the levels have no intrinsic value when considered in isolation. However, relative differences between levels for dyad members may be of theoretical interest and lead to future research questions (e.g., Why do men report higher average levels of satisfaction?).
Covariation
The covariation in an outcome matrix refers to the degree of correspondence or correlation between dyad member outcome values (after subtracting each member’s level), much like a Pearson correlation coefficient. Positive covariation occurs when dyad members exhibit similar values within each cell of the outcome matrix, and negative covariation arises when dyad member outcomes are inversely related. According to Kelley and colleagues, positive covariation reflects corresponding interests between dyad members (i.e., what is good for you is also good for me), whereas negative covariation reflects conflicting interests among dyad members (i.e., what is good for you is bad for me, or vice versa). Zero covariance suggests that there is no association between dyad member outcomes (i.e., what is good for you may be either good or bad for me). The figure formed by connecting the points in the outcome distribution indicates the strength and direction of the covariance, with greater spread among the points (a “wider” shape) indicating a weaker covariance (i.e., closer to zero). In contrast, a pattern of points that is more linear (a “narrower” shape) suggests a stronger covariance.
The outcome distribution in Figure 1B follows a positive trend (corresponding interests), and the breadth of the shape is consistent with covariation that is moderate in magnitude. In terms of IT, this pattern of covariation suggests that across the four possible combinations of attachment anxiety (i.e., man high/woman high, man high/woman low, man low/woman high, man low/woman low), when the man has higher values of relationship satisfaction, the woman also has higher levels of satisfaction. Together, the range, levels, and covariation provide a template for understanding the pattern of observed values in the outcome matrix. These aspects of the outcome matrix and distribution can be further deconstructed to provide researchers with a comprehensive understanding of the amount and nature of interdependence present.
Source Matrices
The outcome matrix and distribution are useful for detecting and describing the nature of interdependence, but they do not provide information about how dyad members arrive at a particular outcome pattern (i.e., who is influencing whom). However, Kelley and colleagues (Kelley et al., 2003; Kelley & Thibaut, 1978; Thibaut & Kelley, 1959) developed a procedure for determining the sources of variability among the values in the outcome matrix. Based on the logic of ANOVA, the outcome controls, or source matrices, are computed by applying the same procedure used to solve for the main effects and interaction for a 2 × 2 table of cell means.
Figure 2 presents the source matrices for our attachment anxiety and relationship satisfaction example. Notice that the single matrix to the left of the equal sign is the outcome matrix from Figure 1A. The intercept matrix contains the average value of the outcome for each dyad member. The bilateral actor control (AC) component describes the extent to which an individual’s standing on the predictor (attachment anxiety) influences his or her own outcome (relationship satisfaction). The mutual partner control (PC) matrix reflects the extent to which an individual’s outcome is uniquely influenced by his or her partner. Finally, the mutual joint control (JC) component captures the extent to which the specific configurations of dyad members’ predictors influence the outcome, otherwise known as the interaction effect.
Outcome and source matrices
The amount of change in the outcome attributable to that component—the control value—is listed below each matrix. Each control value is reflected as a deviation score in the corresponding source matrix (e.g., AC effect for the woman of −6 corresponds to values of −3 and +3 in adjacent cells). Computing the control value for the intercept involves taking the mean outcome value across all cells for each dyad member (e.g., woman’s intercept = [−5 + 13 −5 + 1] / 4 = 4 / 4 = 1). The remaining component matrices reflect the change in outcome values attributable to each source component. 2
In this example, the AC effect for the woman was computed by subtracting the row marginal means of the outcome matrix (e.g., WHigh − WLow = [1 −5] / 2 − [13 −5] / 2 = [−2] − [4] = −6) to reflect the expected change attributable to one’s own attachment anxiety (i.e., moving from low to high attachment anxiety). Specifically, a woman who is high (vs. low) in attachment anxiety is associated with relationship satisfaction outcomes that are 6 units lower. The PC effect for the woman was computed by subtracting the column marginal means (high − low) because these marginal means reflect the effect of having a partner who is high (vs. low) in attachment anxiety (e.g., MHigh − MLow = [(13 −5) / 2 − (−5 + 1) / 2] = [8] − [−2] = +6). Finally, the JC component for women may be computed by subtracting the means of the diagonal elements of the outcome matrix (e.g., WMinor − WMajor = [(–5 −5) / 2 − (13 + 1) / 2] = [−5] − [7] = −12). We can verify our arithmetic by comparing the sum of the source matrices for a given element with the corresponding element in the outcome matrix.
Combinations
Kelley and colleagues (Kelley et al., 2003; Kelley & Thibaut, 1978) went to great lengths to describe the theoretical importance of the ratios among the component matrices (i.e., AC:PC, AC:JC, PC:JC). These combination ratios describe the specific form or type of interdependence in a given outcome matrix. The sign of the ratio indicates the congruence—the degree to which the given effects undermine or bolster one another. Specifically, ratios are concordant when the sign of the effects is the same (e.g., positive AC and positive PC control value), indicating that the sources bolster or reinforce each other. In contrast, discordant ratios arise when the sources have different signs (e.g., positive AC control value and negative PC), indicating that the sources undermine or contradict each other. Furthermore, the size of the ratio (e.g., AC:PC < 1 vs. AC:PC > 1) indicates the relative strength of the two sources. For example, an AC:PC ratio greater than 1 indicates that the AC control value is larger than the PC control value, and a ratio less than 1 suggests that the PC control value is larger than the AC control value.
Naming the game
To provide context for these ratios, Kelley and colleagues described a three-dimensional “globe of interdependence,” although this figure is actually cube shaped. Each axis of this three-dimensional figure corresponds to a combination ratio, which serves to locate a given outcome matrix within the globe of interdependence. Many of the locations associated with these ratios correspond to paradigms from game theory research (as illustrated throughout chap. 4 of Kelley et al., 2003). These “named locations” may be useful in describing the underlying motives that generated the pattern of interdependence. In our example, the observed AC:PC ratio of −1 corresponds to the “threat game” scenario described by Kelley and colleagues (2003, Figure 4.1). In this threat game, men experience the most positive outcome when they are high in attachment anxiety and their partner is low, and women experience the most positive outcome when they and their partner are low in attachment anxiety. However, men and women experience their least positive outcome when they are both high in attachment anxiety.
The remaining combinations (i.e., AC:JC and PC:JC) also correspond to locations on the globe of interdependence, described by “maps” provided by Kelley and colleagues (2003, Figures 4.1-4.3). A detailed discussion of each named scenario described by IT is beyond the scope of this article (see Kelley et al., 2003, for a comprehensive treatment of this topic). At present, Kelley and colleagues have identified only a handful (<10) of combination ratios with named patterns, so in practice, the majority of observed ratios will not correspond to locations with existing names (e.g., threat game, prisoners’ dilemma). One way to deal with this potential problem involves using nearby named patterns (described in Kelley et al., 2003) as interdependence “landmarks” and interpreting the observed combination in reference to them.
Combinations simplified
Relying solely on the named scenarios may limit a researcher’s ability to describe the results of a study concisely. In many instances, a simple and concrete interpretation of combinations may be useful. Although the AC component certainly answers substantive and relevant questions (e.g., Does one’s own attachment anxiety predict one’s relationship satisfaction?), it does not reflect interdependent processes. In contrast, the PC and JC sources describe the influence of one’s partner, either directly (PC) or as a function of both dyad members’ predictors (JC). This means that AC:PC and AC:JC ratios between 0 and 1 reflect stronger interdependence, relative to independence (because the PC or JC component is larger). In addition, AC:PC and AC:JC ratios greater than 1 indicate less interdependence, relative to independence (because the AC component is larger). Kelley and colleagues (2003) distinguished between two basic types of interdependence determined by the relative strength of PC versus JC effects. Interdependence based on coordination occurs when the JC component is stronger (relative to PC) because the interdependence arises when dyad members match or mismatch on the predictor. In contrast, interdependence based on mutual exchange occurs when the PC component is stronger (relative to JC) because the interdependence stems from the direct influence of one’s partner, as opposed to the coordinated (matching) effect reflected in the JC component.
Another important aspect of combinations involves concordance or discordance reflected in the sign of the ratio. In our example, the AC:PC combination is concordant for the man (−6 / −6 = +1) but discordant for the woman (−6 / 6 = −1), and the absolute magnitude of the AC:PC ratio is the same for both. The concordant ratio for the man suggests that the AC and PC sources complement or reinforce each other, whereas the discordant ratio for the woman suggests that these sources work in opposite directions. We also know that AC and PC effects play an equally important role in determining the outcome matrix, because the magnitude of the AC:PC ratio is 1 for the man and woman. The AC:JC ratio is concordant for the man (−6 / −3 = +2) and the woman (−6 / −12 = +0.5). The ratio AC:JC falls between 0 and 1 for the woman, suggesting that the JC effect is stronger than the AC effect, but the AC:JC ratio is greater than 1 for the man, indicating that the AC effect is stronger than the JC effect. Finally, the PC:JC ratio is concordant for the man (−6 / −3 = +2) and discordant for the woman (+6 / −12 = −0.5). The concordant ratio for the man suggests that these effects complement each other, and the discordant ratio for the woman suggests that these processes work in opposite directions. For the man, interdependence is based more on mutual exchange, as the PC:JC ratio is greater than 1. However, because the PC:JC ratio is between 0 and 1, we know that interdependence is based more on coordination for the woman.
Given the breadth and depth of IT, it is not practical to review every aspect of the theory. Instead, we have sought to familiarize the reader with the fundamental components of IT. In particular, we emphasized the aspects of IT that are applicable to methods and statistical techniques commonly applied to dyads (e.g., APIM). Accordingly, other aspects of IT (e.g., transition lists, transformations) that are not as relevant to these research designs are addressed in later sections. Next, we review relevant aspects of the APIM, followed by a discussion of what is similar and dissimilar between these theoretical frameworks.
Dyadic Data Analysis and the APIM
A number of models for analyzing dyadic data have been proposed (see Kashy & Grotevant, 1999, for a review). Most of these approaches incorporate some form of partner effect to model the extent to which one’s outcome is uniquely associated with his or her partner’s predictor. Many of these procedures require the researcher to transform correlation coefficients manually, disaggregate dyad members by role, or aggregate across member roles (ignoring nonindependence). In contrast, the APIM is specified as a multivariate regression model, with the parameters estimated by structural equations modeling (SEM) or mixed effects modeling (MEM) software. As we will see, complex forms of interdependence suggested by IT (i.e., coordination) require interaction terms, which are available only in the modeling frameworks associated with APIM.3,4 The following section reviews many of the core issues associated with dyadic data, with an emphasis on the components of the APIM that directly relate to IT.
Dyadic Nonindependence
Researchers have long recognized that collecting responses from both members of a dyad introduces nonindependence among observations. Nonindependent observations violate a core assumption of general linear model (i.e., ordinary least squares [OLS] regression, ANOVA), resulting in biased significance tests (Kenny, 1996; Kenny & Judd, 1986). 5 Fortunately, two methods exist that allow researchers to account for nonindependence between distinguishable dyad members and conduct unbiased significance tests (Kenny et al., 2006). One procedure involves estimating the correlation or covariance between the outcomes of dyad members, with higher absolute values reflecting greater nonindependence. Alternatively, the researcher may partition the total variability in dyad member outcomes into between- and within-dyad components using a random intercept model. In a random intercept model, a greater proportion of random intercept (between-dyad) variance relative to residual (within-dyad) variance indicates greater nonindependence (Kashy & Snyder, 1995; Kenny, 1995; Raudenbush, Brennan, & Barnett, 1995). 6
Variables in dyadic data fall into three basic categories. A variable is classified as varying between dyads when it reflects an attribute that is shared by both members of the dyad (Kenny et al., 2006). Scores on between-dyad variables are the same for both members of the same dyad (e.g., relationship length, marital status, number of children, household income). In contrast, a variable is classified as within dyads when the scores are different for members belonging to the same dyad (e.g., gender, actual proportion of housework completed) but are the same over all dyads when the values from the same dyad are averaged. Within variables often serve as distinguishing variables that define a respondent’s role. Within- and between-dyad variables contain no variance that is specific or unique to a given dyad member because one dyad member’s standing on the variable is perfectly predicted by his or her partner’s standing on the variable. Mixed variables reflect attributes specific to individual, as well as the dyad. These variables contain shared (within- or between-dyad) variance and individual variability. For example, relationship satisfaction is a mixed variable because members of the same dyad may have different appraisals of relationship quality. However, the attributes or behaviors of one dyad member may influence the other dyad member’s satisfaction, leading to a correlation between dyad member outcomes (see Kenny, 1996, for a discussion of the sources of nonindependence). Finally, a variable is considered independent when it contains only individual variability and no shared variance.
Typical Implementation of APIM
We conducted a survey of research articles to understand how the APIM is typically employed. PsycInfo was used to search for papers citing any three of the most frequently cited technical presentations of APIM (i.e., Campbell & Kashy, 2002; Kenny, 1996; Kenny et al., 2006). As of February 2010, more than 200 independent research articles were identified. The initial pool of articles was reduced by retaining only papers that estimated partner effects and included outcomes for both dyad members. A pool of 154 research articles containing 162 dyadic samples remained. From these articles, we identified two often-neglected aspects of APIM analysis, which have important theoretical and analytic consequences.
First, a number of important interactions among actor, partner, and role predictors were often not considered. As will be shown, these interactions play a critical role in linking APIM and IT, and their omission may result in biased or incomplete results. Table 1 presents the proportion of studies that reported testing various forms of interactions. As recommended by prior work (Campbell & Kashy, 2002; Kenny et al., 2006), a large proportion of studies, 75%, included Actor × Role (A × R) and Partner × Role (P × R) interactions to examine the equality of actor and partner effects across dyad member role (e.g., gender). However, only 21% of articles reported testing Actor × Partner (A × P) interactions, despite being described by prior work (Campbell & Kashy, 2002; Kenny et al., 2006). In addition, only one fourth of the articles that tested A × P interactions (7% overall) went on to examine the extent to which the A × P interaction varied as a function of partner role (A × P × R). As we will see, the estimation of these effects is essential to fitting a form of APIM that is consistent with IT.
Interactions in Studies Using APIM
Note: APIM = actor–partner interdependence model; A × R = actor-by-role; P × R = partner-by-role; M × M = mixed-by-mixed (actor × actor, partner × partner); A × P = actor-by-partner; A × P × R = actor-by-partner-by-role; A × B = actor-by-between; P × B = partner-by-between; B x R = between-by-role. Percentage values based on N = 162 studies, unless otherwise noted.
Only applicable to 131 studies using distinguishable dyads.
Second, although researchers often disclosed the method used to account for nonindependence (i.e., random intercept vs. covariance), the values describing the direction and degree of nonindependence were rarely reported. These parameters (variances and covariances) are important because they provide insight into how well the predictors modeling interdependent processes explain nonindependence among dyad member outcomes. The following section describes a number of conceptual and technical extensions to the basic APIM that play an important role in understanding the complex interplay between APIM and IT.
Beyond the Typical APIM
To provide a better understanding of our perspective on the APIM, we review the single predictor APIM for distinguishable dyads. The APIM may be expressed as either a bivariate (two-equation) or univariate (one-equation) model. Understanding both formulations is important because some modeling frameworks use the bivariate (i.e., SEM, the “two-intercept” MEM; Kenny et al., 2006; Raudenbush et al., 1995) and others use the univariate (i.e., univariate MEM; Campbell & Kashy, 2002; Kenny et al., 2006) formulation. The bivariate expression bears some resemblance to the IT matrices described in the previous section:
where Y is the outcome variable, subscripts “m” and “w” refer to men and women, respectively, Xm and Xw are predictors, and Xm × Xw is the multiplicative product of Xm and X w. In addition, i represents the regression intercept, a refers to actor slopes, p to the partner slopes, a × p to the actor-by-partner interaction, and e to the error terms (residuals). These equations contain role-specific subscripts for each estimated parameter. In terms of the APIM, this means that actor, partner, and A × P coefficients are estimated uniquely for each dyad member (i.e., they are not pooled). Specifying separate equations for each dyad member type (e.g., men, women) makes the distinction between dyad member roles explicit.
The univariate formulation is slightly more complicated because the distinction between dyad member roles is no longer expressed through separate equations. When the APIM is represented as a univariate equation, the potential differences between dyad member types must be accounted for by including a separate predictor variable identifying the role of the respondent. Interaction terms between the role predictor and the remaining predictor variables must also be included to allow for unique actor, partner, and A × P coefficients for each dyad member. This univariate formulation describes the exact same model as the bivariate formulation (i.e., Equations 1 and 2) using different terms:
where Y is the outcome for person p in dyad j, Genderp is the predictor variable designating the role of person p and may be dummy or effect coded. X pj is the predictor for person p in dyad j, and X′ is the predictor for the dyadic partner of person p in dyad j. The regression intercept i is the expected value of the outcome when all predictor variables are zero. The regression coefficient g represents the expected difference in the outcome across dyad member roles, and the regression coefficients for the first-order actor and partner effects are represented by the a and p terms. The a × g and p × g coefficients describe the difference in first-order actor and partner effects as a function of dyad member role (i.e., gender), and the a × p coefficient describes the interaction between dyad member predictors. The a × p × g coefficient expresses the degree of difference in the lower order a × p interaction as a function of dyad member role. Finally, epj represents the person-specific residual or the difference between the observed and expected values of Ypj.
The A × R, P × R, and A × P × R interactions mentioned earlier allow the researcher to estimate and examine differences between role-specific coefficients. However, the procedure for obtaining and interpreting these interactions depends on how the model is specified. When fitting a bivariate model using SEM software (e.g., MPlus, AMOS, EQS, SAS PROC CALIS) or a “two-intercept” model using MEM (Raudenbush et al., 1995), these interactions are obtained by freely estimating (vs. constraining) all actor, partner, and A × P coefficients across dyad members, as seen in Equations 1 and 2. However, when specifying the univariate model (Campbell & Kashy, 2002) using MEM software (e.g., SAS PROC MIXED, SPSS MIXED, HLM), these interactions are obtained by including interactions among the role predictor (Genderp) and the actor (Xpj), partner (X′pj), and actor-by-partner (Xpj × X′pj) predictor variables.
Furthermore, the interpretation of all model parameters (i, a, p, a × p, a × g, p × g, a × p × g) is determined by the coding of the role predictor (Genderp). When effect coding is used, the lower order terms that do not contain the role predictor (i, a, p, a × p) are interpreted as pooled coefficients that reflect the average effect across both dyad member types. In addition, the coefficients for interactions containing the role predictor (a × g, p × g, a × p × g) reflect role-specific deviations from the lower order pooled estimate. When dummy coding is used, lower order terms are specific to the dyad member type that is coded as the reference group (i.e., 0). For example, if Genderp is dummy coded such that men = 0 and women = 1, the lower order coefficients (i, a, p, a × p) are specific to men, and g reflects the expected difference in the outcome for women (relative to men). Similarly, the interaction terms containing the role predictor (a × g, p × g, a × p × g) represent the difference in the lower order coefficients (i, a, p, a × p) for women (relative to men).
Terms representing the intercept(s) and regression slopes (a, p, a × p, etc.) are known as fixed effects because they are estimates of a single population parameter (slope or intercept). Practically, the fixed effects describe the relationship between the predictor and outcome variables. The final term in each equation (e m, e m, or e pi) represents the residuals or errors in prediction. Whereas each fixed effect has a value that applies to every observation (e.g., the actor slope for men across all dyads, a m = 0.4), residuals are random effects, which vary across observations (e.g., for one dyad e m = −1.2, for another dyad e m = +0.5); consequently they must be described in terms of variances and covariances. In the APIM (and most linear models), residuals are assumed to be normally distributed with means of 0, variances (σ2 M and σ2 W), and a nonzero covariance (σM,W). Our survey of research articles using the APIM revealed a notable lack of consistency in how fixed and random effects are specified. The following sections discuss this issue in more detail.
Fixed effects
Although the majority of studies in our sample allowed actor and partner effects to vary across member role (A × R and P × R interactions), only a small percentage included the A × P term. Excluding these effects is equivalent to fixing the a × p and coefficient(s) to zero, which means that the variance potentially explained by the A × P predictor is relegated to the error term. If the A × P effect is significant, then excluding the coefficients distorts measures of effect size (e.g., proportion of variance explained) and inflates standard error estimates, which may result in increased Type II error rates for significance tests of first-order (main effect) actor and partner coefficients. Excluding the A × P interaction also has the potential to bias first-order actor and partner regression coefficients. Furthermore, constraining the magnitude of A × P effects to be equal for both dyad members, by excluding the A × P × R interaction, is also problematic. For example, if the correct value for the A × P effect is positive for men (i.e., a × p m = +0.5) and negative for women (i.e., a × p w = −0.5), but the A × P × R interaction is not included, then the pooled A × P effect would be estimated at 0 (i.e., a × p m = a × p w = 0).
Random effects
In linear modeling, random effects refer to variance and covariance parameters estimated by the model. Concretely, these random effects describe the errors in prediction (variances) and the association between errors for different variables (covariances). When applied to dyadic data, the variances often pertain to a variable for a specific dyad member (e.g., men’s relationship satisfaction, women’s relationship satisfaction), and the covariances often represent nonindependence among dyad members (e.g., covariance between men and women’s relationship satisfaction). Prior work (Kenny, 1996; Kenny et al., 2006) suggests that nonindependence may arise from at least one of four possible dyadic processes. In the common-fate model, dyad members exhibit nonindependence due to the influence of a common factor or shared experience. In the mutual influence model, dyad member outcomes influence each other in the form of a “feedback loop” or nonrecursive (Bollen, 1989) process. Assortative mating or compositional effects occur when individuals enter into a relationship because of preexisting similarities on a given attribute (i.e., “Birds of a feather flock together”). Finally, in a partner effects model, the “predictor” attributes of one dyad member influence the outcome of the other dyad member.
When using the APIM, the researcher makes a theoretical assumption that the partner effects model describes the process responsible for the nonindependence. However, examining how the covariance among dyad members is explained (using path tracing or covariance algebra), reveals that actor, partner, and A × P effects are also responsible for nonindependence among dyad members. Practically, this means that adding predictors to the model will reduce the degree of individual variance and nonindependence among dyad member outcomes (assuming one of the coefficients is nonzero), relative to a model containing no predictors. Moreover, noting the change in the covariance between dyad member outcomes (or random intercept variance) from a model with no predictors to a model with predictors allows the researcher to determine the amount of the nonindependence explained. Similarly, the change in variance parameters after adding predictors provides a measure of the amount of individual variability in the outcome that is explained for each dyad member. Researchers rarely report and interpret these aspects of the APIM analysis, which is unfortunate because they play an important role in describing dyadic interdependence. Later sections provide a detailed description of this procedure.
Integration
This section describes the correspondence between the central features of IT and the APIM.
Source Matrices Are Fixed Effects
The most straightforward link between APIM and IT involves the source matrices described by IT and the fixed effects estimated by the APIM. In both cases, the actor effect (AC matrix) reflects the extent to which an individual’s predictor is associated with his or her own outcome. Partner effects (PC matrix) and A × P (JC matrix) interactions describe the extent to which one member’s outcome is contingent on the attributes of the other, as well as the manner in which specific combinations of dyad member predictor values have unique effects on the outcome, respectively. In addition, interaction terms that include the distinguishing “role” variable (e.g., gender) are important because they correspond to the notion of asymmetry in IT. Forcing dyad members to have common (pooled) estimates for actor, partner, and A × P coefficients is equivalent to imposing symmetry on the source components. If any of the sources is asymmetrical across dyad members, then imposing symmetry (excluding significant A × R, P × R, or A × P × R interaction) biases the resulting outcome matrix and outcome distribution, and ultimately distorts our interpretation of the pattern of interdependence present in a given sample.
The Outcome Matrix Is a Simple-Slope Analysis
Careful examination of the IT outcome matrix reveals that it is comparable with a plot of expected means, as described by Aiken and West’s (1991) “simple-slope” formulation. By decomposing a fully factorial APIM using the simple-slope procedure (where “low” and “high” levels of the factor are defined by values of the predictor ±1 SD from the mean), the researcher is able to reproduce a comparable expected outcome matrix. In addition, the outcome matrix derived from a simple-slope analysis allows the researcher to create an outcome distribution, which may be used to describe the range, covariation, and levels for a given predictor and outcome.
Using the APIM to provide estimates for IT source controls leads to a slightly different interpretation of the outcome matrix relative to the original presentation (Kelley et al., 2003). Specifically, in the classic IT matrix formulation, the range reflected the difference between the minimum and the maximum values in the outcome matrix for a single dyad member. However, applying IT based on APIM analysis, the outcome matrix comprises means derived from the entire sample. Thus, constructing an outcome matrix and distribution using the APIM, the minimum and the maximum values are defined by representative (±1 SD from the predictor mean) values. Finally, the APIM intercept parameters in Equations 1 and 2 correspond to the concept of levels in the IT outcome distribution because they reflect the average level of the outcome variable.
Combination Ratios Are the Parameters k, h, and c
Kenny and Ledermann (2010) described a parameter, k, that is obtained by computing the ratio of partner to actor effects. 7 Although not explicitly discussed in prior work, k directly corresponds to the AC:PC combination described by IT. Furthermore, the additional source ratios suggested by IT (i.e., the AC:JC and PC:JC ratios) may be estimated and interpreted within the context of APIM. The former, which we label h, describes the ratio of actor effect (AC) to interaction effect (JC). The ratio of partner (PC) to the A × P (JC) effect, which we term c, describes the relative strength of the partner effect compared with the interaction component. Because the actor, partner, and A × P coefficients from an APIM analysis correspond to the three sources of outcome variance in IT, APIM estimates may be used to identify the pattern of interdependence as described by IT.
Making predictions about the sign and limits of these ratios can be simplified into a series of “if–then” rules. 8 First, if the process is expected to be largely independent (i.e., characterized by strong actor effect), then the lower CI for k (AC:PC) and h (AC:JC) ratios should be >1. Second, if the process is expected to be largely interdependent, and the partner effect is expected to be stronger than the interaction, then the lower CI for c (PC:JC) should be >1. Finally, effect coefficients that complement each other (e.g., positive actor and positive partner) should have concordant (positive) ratios, and those that contradict each other (e.g., positive actor and negative partner) should have discordant (negative) ratios. Positive ratios emerge when a predictor is consistently related to an outcome across sources (e.g., my own and my partner’s attachment anxiety is negatively related to my satisfaction). Negative ratios emerge when a predictor is inconsistently related to an outcome across sources (e.g., my own attachment anxiety is negatively related to my satisfaction, but my partner’s attachment anxiety is positively related to my satisfaction). Alternatively, researchers could generate hypotheses by locating the most appropriate outcome matrices using Kelley and colleagues’ IT atlas and decomposing the combined outcome matrix into source matrices. Regardless, these procedures should encourage researchers to approach their construct-specific hypotheses from the broader, more fundamental, perspective of IT.
Interdependence is nonindependence explained
Earlier, we described how actor, partner, and A × P effects explain nonindependence between dyad members, and how the effects responsible for nonindependence in APIM (i.e., actor, partner, A × P) and the components of interdependence in IT (i.e., source matrices) are functionally equivalent. Specifically, the predictors in an APIM (actor, partner, and A × P) reflect interdependent processes that explain nonindependence between dyad members. Moreover, the nonindependence explained in an APIM analysis is synonymous with the concept of interdependence described by IT. Significant residual nonindependence (covariance or random intercept variance) in an APIM analysis suggests that dyad member outcomes are related, even after accounting for the effect of actor, partner, and A × P effects. Therefore, the pattern of interdependence reflected in the predictors is only one component of the total interdependence picture for a given outcome variable. Additional constructs or higher order (polynomial) interactions may be responsible for the remaining nonindependence (unexplained interdependence) among dyad member outcomes. Along the same lines, a residual covariance parameter that is near zero suggests that the nonindependence among dyad member outcomes has been explained by the interdependent processes (fixed effects).
Implications
The previous sections described a number of common features between a major theory in social psychology (IT) and a popular data-analytic model for dyads (APIM). We believe that this insight will allow researchers to combine IT with other theoretical frameworks by using APIM in this manner. Unlike the majority of psychological theories, IT does not stipulate specific predictor or outcome variables or, as stated by Rusbult and Van Lange (2003), “interdependence theory does not identify an overarching need or drive that fuels interpersonal behavior (e.g., reproduction, security, mastery)” (p. 354). Instead, IT addresses a fundamentally different aspect of any theoretical question, specifically, how people engage, interact, and influence one another. This focus is in contrast to most other theories of close relationships that posit specific causal mechanisms and consequences in an attempt to explain why a given outcome arises. As a result, IT does not encroach on or contradict hypotheses derived from other theories. In fact, the full utility of IT may be realized only by adopting IT as a perspective, or lens, through which hypotheses derived from other theories may be examined.
Calculational Example
The practical utility of an IT approach to APIM analysis can be demonstrated through didactic example. Our primary goal is to show how conceptualizing APIM analysis in terms of IT components, and extending IT logic to unique elements of APIM, provides researchers with a more comprehensive and precise understanding of the phenomenon than would otherwise be obtained. The constructs of interest in our example are from work on need satisfaction in romantic relationships (NSR; Knee, Lonsbary, Canevello, & Patrick, 2005; Patrick, Knee, Canevello, & Lonsbary, 2007). Specifically, Knee and colleagues found that the fulfillment of one’s basic psychological needs for autonomy, competence, and relatedness (Deci & Ryan, 2000) was associated with more positive relationship outcomes. Across multiple studies, results suggested that higher levels of need fulfillment (especially relatedness) were associated with greater relationship satisfaction and commitment, as well as less defensive responses to relationship conflict (Knee et al., 2005; Patrick et al., 2007).
Although APIM analysis was conducted in Knee et al. (2005) and Patrick et al. (2007), an interpretation of the results in terms of IT was not provided. Using new data, we demonstrate how applying and interpreting the APIM in terms of IT reveals an interesting and highly descriptive pattern of results. In the present example, we attempted to replicate the dyadic “matching” (i.e., A × P interaction) that was observed between the relatedness component of NSR and relationship satisfaction (Patrick et al., 2007, Study 2). In terms of IT, this pattern of results would be reflected in significant AC, PC, and JC effect coefficients, all in the positive direction, resulting in concordant IT combinations.
Variables and Sample
A total of 71 dating and married couples drawn from the participant pool at a large Southwestern University completed a packet of self-report measures. Participants responded to three items from the Relatedness subscale of the Need Satisfaction in Relationships scale (La Guardia, Ryan, Couchman, & Deci, 2000; for example, “When I am with my romantic partner, I feel loved and cared about”), using a 7-point Likert-type scale with endpoint labels strongly disagree and strongly agree. Participants also completed the five-item Satisfaction subscale of Rusbult’s Investment Model scale (Rusbult, Martz, & Agnew, 1998; for example, “I feel satisfied with our relationship”), using a 9-point Likert-type scale ranging from 0 = do not agree at all to 8 = agree completely, with a midpoint label (agree somewhat). Scale items were averaged to form composites and, consistent with the recommendation of Aiken and West (1991), the predictors were grand mean centered (ignoring gender) prior to computing the A × P term.
Analysis Strategy
Researchers can use SEM or MEM software to estimate the APIM, and existing work describes the basic procedure for each method in detail (Kenny et al., 2006). In this example, we used MEM, specifically SAS PROC MIXED (SAS Institute, 2008) with restricted maximum likelihood estimation. Nonindependence was modeled by specifying a covariance parameter between dyad member outcomes. In PROC MIXED, this involves using a REPEATED statement with TYPE = UN to allow for a covariance term and separate variance estimates for each dyad member. A sample data set, as well as SAS and SPSS syntax for all models, is provided in the appendix.
Using MEM software requires that multiple models be estimated. The researcher should begin by fitting a null model that contains only an effect-coded variable that distinguishes between dyad member roles. For the present example, dyad members may be distinguished on the basis of gender, so our distinguishing variable is coded +1 for men and −1 for women. The variance–covariance parameters from the null model will serve as baseline (unconditional) estimates. Next, actor and partner predictors are added, along with all possible interactions, resulting in an initial fully saturated (conditional) model. For this first conditional model, the researcher should retain the effect-coded role variable (“gender” in the present example) so that the lower order coefficients will reflect pooled estimates averaging across men and women. This configuration provides an informative “first pass” for the model of interest, as well as an evaluation of the asymmetry (role) interactions. In addition, the residual variance and covariance estimates from the initial conditional model may be used (along with the unconditional estimates) to compute the proportion of variance and covariance (nonindependence) explained by the predictors.
The follow-up models should use dummy-coded versions of the role predictor (gender), combined with the simple-slope procedure described by Aiken and West (1991). Role-specific simple-slope equations are computed for each dyad member by forming two versions of the dummy-coded role predictor that differ only in the dyad member assigned to the reference category (i.e., 0). For example, women-specific slopes may be obtained by including a dummy-coded predictor in which women = 0 and men = +1. These models provide role-specific estimates for the actor, partner, and A × P coefficients, as well as predicted means that can be used to construct a simple-slope plot, outcome matrix, and distribution. Finally, point estimates for k, c, and h can be computed by taking the ratio of role-specific actor, partner, and A × P coefficients. A SAS MACRO program (supplementary file available online at http://pspr.sagepub.com/supplemental), which uses a nonparametric bootstrapping procedure, is available to provide unbiased CIs for the k, c, and h ratios. Bootstrapped CIs are necessary because the sampling distributions for these ratios are nonnormal rendering standard significance tests (e.g., t test) unreliable.
IT–APIM analysis proceeds somewhat differently when the researcher specifies a bivariate model using SEM software or a two-intercept MEM (Raudenbush et al., 1995). When using SEM, freely estimating the paths leading from all predictors to each dyad member’s outcomes allows for asymmetry in effects, and computing simple slopes follows the same general procedure described by Aiken and West (1991). Although most SEM packages provide estimates of variance explained by the predictors (R2), we are aware of no software that provides estimates of covariance explained. As a result, the researcher may need to fit a null model in which actor, partner, and A × P paths are fixed to 0 to obtain an unconditional estimate of nonindependence. Point estimates and bootstrap CIs for c and h may be obtained by modifying the SEM procedure described by Kenny and Ledermann (2010). 9 At present, only HLM (Raudenbush, Bryk, & Congdon, 2004) and MLwiN (Rasbash, Charlton, Browne, Healey, & Cameron, 2009) support the two-intercept MEM described by Raudenbush and colleagues (1995). As with the SEM framework, the two-intercept MEM provides role-specific estimates for actor, partner, and A × P coefficients, and simple slopes should also be computed using the procedure described by Aiken and West (1991). Estimates of variance explained by the predictors may be obtained using the same procedure as the univariate MEM (i.e., comparing unconditional to conditional variances and covariances). Unfortunately, a procedure does not currently exist that provides bootstrap CIs for ratio parameters.
Results
The three models needed to conduct a complete IT–APIM analysis using MEM are presented below. The null model provides unconditional estimates of the variance–covariance parameters, which may be compared with the residual variance–covariance estimates provided by the initial model to determine the proportion of nonindependence and individual variability explained by the predictors. Finally, the follow-up model uses dummy coding and the simple-slope analysis to provide role-specific actor, partner, and A × P estimates, which can be used to create an outcome matrix and distribution.
Unconditional model
To obtain estimates of unconditional variances and the covariance between dyad member outcomes (relationship satisfaction), a model was specified with gender as the only predictor. The unconditional variance was 2.75 for men, and 1.37 for women, suggesting that men tend to be more variable in relationship satisfaction than women. The unconditional covariance was estimated at 0.836. A standardized estimate of nonindependence can be computed by applying the standard formula for a correlation (i.e.,
Initial conditional model
As stated previously, an effect coding scheme was used for the gender predictor (i.e., women = −1, men = +1), and consistent with our recommendations, all possible interactions were included. Because all possible interactions are estimated, effect coding the distinguishing variable results in actor, partner, and A × P coefficients that are pooled (averaged) across dyad member roles. Unstandardized coefficients are listed in Table 2. The gender coefficient suggests that men report slightly higher levels of satisfaction, relative to women, but this difference is not significant (p > .30). A significant first-order actor effect (p < .01) suggests that relatedness is positively associated with one’s own relationship satisfaction. Because gender was effect coded, the A × G interaction term represents the difference between the pooled actor slope and each dyad member’s gender-specific slope. In this example, men exhibit a slightly stronger actor effect, bMen Actor = (0.77 + [+1 × 0.06]) = 0.83, relative to women, bWomen Actor = (0.77 + [−1 × 0.06]) = 0.71; however, this difference is not statistically significant (p = .51). A significant first-order partner effect also emerged (p < .05), indicating that the level of relatedness reported by one’s partner is positively associated with the respondent’s satisfaction, but as before, this effect does not differ across member role (p = .89). Although the lower order A × P interaction term was nonsignificant (p = .56), the A × P × G interaction was statistically significant (p < .01), suggesting that the magnitude of the underlying A × P interaction is different for men and women. Although it is permissible to drop the nonsignificant A × G and P × G terms, we choose to retain them because the power of these tests is often low (Kenny & Ledermann, 2010).
Parameter Estimates (SE) for Initial Model of Relatedness Satisfaction
Note: A = actor; P = partner; G = gender predictor. The G is effect coded (men = +1, women = −1).
p < .05. **p < .01.
Follow-up model
The follow-up model uses dummy coding for the gender predictor. This is useful because it yields significance tests for role-specific slopes and allows us to compute predicted means that may be used to construct an outcome matrix. Specifically, trading dummy for effect codes takes advantage of a well-known property of interactions in regression analysis, that is, lower order terms in the regression equation reflect conditional effects when all other predictors with which the lower order term is crossed are zero (Aiken & West, 1991).
Figure 3 provides estimates based on the dummy-coded analyses. As in the initial model, actor effects for men and women are statistically significant and comparable in magnitude. Although the gender-specific partner effects are positive and of similar magnitude, the standard error for this effect is notably larger for women leading to a nonsignificant association. Turning to the A × P interaction effects, we see that women exhibit a marginally significant (p < .10) interaction in the positive direction, suggesting that increases in their partner’s level of relatedness translate into a more positive effect of their own relatedness. In contrast, men exhibit a significant (p < .05) A × P interaction in the negative direction, suggesting that increases in their partner’s level of relatedness result in a weaker (more negative) effect of her own relatedness.
Path diagram for relatedness-satisfaction example ** p < .01. * p < .05. †p < .10.
The asymmetrical JC effects suggest differences in the combination of dyadic NSR relatedness necessary to optimize satisfaction. We can explore these effects in more detail by computing conditional actor effects (simple slopes) for men and women when partners report relatively low and high levels of relatedness (Aiken & West, 1991). Table 3 contains estimates of conditional intercepts and simple slopes, and Figure 4 provides a plot of slopes and predicted means. For men, as their partner’s relatedness increases, their average level of satisfaction increases, and their own level of relatedness becomes less important in determining their satisfaction. Women’s level of satisfaction also increases along with their partner’s relatedness, and their own level of relatedness becomes more important as the partner increases in relatedness.
Unstandardized Simple Slopes for Relatedness-Satisfaction Example
Note: All intercepts and slopes are significant at p < .01. Low and high refer to partner relatedness at ±1 SD relative to the mean.
Simple-slope plot for relatedness-satisfaction example
IT Interpretation
Until this point, our presentation of the example has been consistent with an “APIM-only” interpretation of model parameters. To demonstrate the utility of our suggestions, the following section explores the observed results in terms of IT.
Outcome matrix and distribution
Consistent with our recommendations, predicted means derived from the simple-slope analysis were used to construct an outcome matrix and distribution. The first step required to compute the outcome matrix involves choosing reasonable values for the predictor variables to complete each simple-slope equation. Consistent with the recommendations of Aiken and West (1991), we computed the standard deviation of the predictor (s relatedness = 1.08). Referring to Table 3, we see that the simple regression equation for men with a partner low in relatedness is
Providing a value corresponding to −1 SD on relatedness (i.e., −1.08) yields the predicted outcome value (5.22) for men reporting low levels of relatedness, who also have partners reporting low levels of relatedness. Similarly, for women we use the comparable simple regression equation:
which yields the expected value, or mean, of relationship satisfaction for a woman reporting low relatedness, with a partner who is also low in relatedness. These expected values can be used to fill in the first quadrant of an outcome matrix for relationship satisfaction. Completing the remaining cells involves computing expected values as a function of the remaining simple equations and relevant values of the predictor (i.e., +1.08). The final outcome matrix and distribution are illustrated in Figure 5.
Outcome matrix and distribution for relatedness-satisfaction example
In terms of IT components, our analysis suggests that men and women exhibit similar range values in relationship satisfaction (men = 7.52 − 5.22 = 2.3, women = 7.70 − 5.65 = 2.15). In addition, the role-specific intercepts reflect the average level of the outcome for each dyad member. These values may be computed by averaging across the four cells of the outcome matrix for each dyad member (e.g., [5.22 + 7.52 + 6.19 + 7.49] / 4 = 6.6 for men), or using the overall intercept (5.56) and gender coefficient (.04) to compute role-specific deviations (e.g., 6.56 + .04 × +1 = 6.6 for men). The outcome distribution reveals a positive covariation between dyad members’ satisfaction (with respect to relatedness), consistent with the IT notion of corresponding interests. However, scrutiny of the outcome distribution reveals an interesting finding as a function of dyad member role. Specifically, men who report higher relatedness tend to experience high satisfaction regardless of their partner’s level of relatedness. In contrast, women who report higher relatedness experience greater satisfaction when their partner also reports higher relatedness.
Combination ratios
The role-specific estimates for actor, partner, and A × P coefficients produced by the dummy-coded follow-up models can be used to compute the combination ratios described by Kelley and colleagues (2003), which are provided in Table 4. The sampling variability for the ratio parameters as estimated by the SAS MACRO is considerable, but this is not surprising given the relatively small sample size and modest effect size of the individual coefficients in our example. However, as observed in Kenny and Ledermann (2010), the sampling variability of these ratios decreases with larger sample sizes and stronger effects. Although all of the 95% (and all but one of the 90%) CIs included the critical values (e.g., |1|), we provide interpretation of the point estimates in terms of IT to maintain continuity with earlier sections.
Point Estimates and Nonparametric Bootstrap CI Estimates for k, h, and c Ratios for NSR Relatedness- and Relationship-Satisfaction Example
Note: AC = actor control; PC: partner control; JC = joint control; CI = confidence interval; NSR = need satisfaction in romantic relationships. CIs based on 1,000 bootstrap samples.
The point estimate for k (AC:PC) ratio is greater than 1 and positive for men (+3.83) and women (+2.88), which suggests that the association between NSR relatedness and relationship satisfaction is complementary across the two sources (i.e., actor and partner effects are in the same direction). In terms of “named” IT combinations, k values of this magnitude are similar to the “Co-Op” pattern described by IT (Kelley et al., 2003, Figure 4.1) but with a stronger AC influence. The Co-Op pattern suggests that the interplay between relatedness and relationship satisfaction represents a cooperation contingency such that being higher in relatedness benefits the person and his or her partner. In the present example, the actor (AC) component is considerably stronger (3 and 4 times the magnitude of partner), suggesting a stronger direct benefit to the actor. The h (AC:JC) ratio is greater than 1 for both partners indicating stronger actor effects relative to A × P effects. However, because the direction of the interaction is different for men and women, the ratio is negative for men (−3.90) and positive for women (5.14). This pattern of combinations is consistent with a conflicting interests version of the “Martyr” pattern, indicating that for both genders, actor effects are much stronger than A × P effects. Whereas the A × P effects undermine or contradict the actor effect among men, the A × P bolsters or augments the actor effect among women. Finally, the c (PC:JC) ratio is close to 1, and negative for men (−1.02), but positive and greater than 1 for women (1.79). This pattern of ratios corresponds to a conflicting interests form of the “Turn Taking” situation in which the observed interdependence is equally apportioned between partner and A × P sources for men but may be more partner based for women. In addition, the negative ratio for men suggests that the partner and A × P effects contradict each other, whereas the positive ratio for women indicates that the effects complement each other.
(Co)Variance explained
In a fully saturated model, the residual covariance between dyad member outcomes was .105. We can compute the proportion of nonindependence explained by subtracting the ratio of residual covariance to unconditional covariance from 1. This means that approximately 85% (1 − [.105 / .863] = .847) of the nonindependence, or interdependence among dyad member relationship satisfaction, is explained by NSR relatedness. Whereas the unconditional covariance is significantly different from zero (p < .01), the residual covariance is not (p = .39), suggesting that additional predictors are not necessary to explain the interdependence among dyad member relationship satisfaction. Using a similar procedure, we can compute the proportion of independent variance explained for each dyad member by subtracting the ratio of residual variance to unconditional variance from 1. After accounting for the effect of NSR relatedness predictors, the variance estimates drop to 1.14 for men and 0.85 for women. This means that NSR relatedness explains approximately 59% (1 − [1.14 / 2.75] = .585) of the variance in men’s and 38% (1 − [0.85 / 1.37] = .847) of the variance in women’s relationship satisfaction.
Conclusion
The worked example demonstrates that much is to be gained by applying the logic of IT to APIM analysis. We found that the association between relatedness and relationship satisfaction is not a simple additive function of the individual’s or the partner’s relatedness, but rather it is a function of the specific configuration of both dyad members. Furthermore, the pattern of results produced by the A × P and A × P × G interactions is compelling because it suggests an interpersonal process model that more closely resembles our implicit and intuitive beliefs about relationships. One of these assumptions is that the implications of one’s own attributes or actions vary depending on the presence or absence of those same traits or behaviors in one’s partner. In terms of IT, we found that men and women exhibit some degree of corresponding interests with respect to relatedness and relationship satisfaction, as evidenced by the positive covariance illustrated in the outcome distribution. Men and women also exhibit comparable ranges and levels of relationship satisfaction. The k, h, and c ratios in our example were discussed in terms of named patterns described by Kelley and colleagues (2003). In addition, differences in the magnitude and sign of these ratios across dyad members suggest that different configurations of relatedness maximize satisfaction for men and women.
This example focused on illustrating the correspondence between APIM parameters (as well as their ratios) and central features of the IT matrix formulation, outcome distributions, and combinations of source matrices. As a result, we avoided making concrete recommendations regarding model selection procedures based on tests of significance. This decision was driven by our desire to provide as flexible a presentation of our central thesis as possible. In addition, the inferences about the combination ratios become difficult when the associated coefficients are near zero (due to the presence of extreme ratios; Kenny & Ledermann, 2010) or the sample size is modest (due to greater variability in the sampling distribution), both of which were true in the present example. Larger samples and stronger effect coefficients (particularly partner and A × P) will result in narrower CIs for the combination ratios, as observed by Kenny and Ledermann (2010).
Limitations and Implications for Future Work
The breadth and complexity of the IT and the APIM lead to a number of interesting directions for future integration.
Transformations and Higher Order Interactions
Kelley and colleagues (Kelley et al., 2003; Kelley & Thibaut, 1978) described a process by which a given outcome matrix may be transformed into a new, “effective” outcome matrix. Specifically, Kelley and colleagues (2003) suggested that a number of variables may influence how an observed pattern of interdependence may differ across dyads, such as individual “social-person” factors (p. 75), dyad-level attributes, and known features of the situation. For example, assume that the outcome matrix shown in Figure 1A represents a given matrix reflecting the typical interdependence configuration. If an additional construct (e.g., neuroticism) moderates the association between the man’s attachment anxiety and the woman’s relationship satisfaction (i.e., the PC matrix for women), the outcome matrix may be transformed as a function of this moderating variable. This transformation hypothesis may be examined by adding a new product term that reflects the interaction between the “partner predictor” for women (e.g., man’s anxious attachment) and the predictor for women’s neuroticism. A broader, albeit less theoretically driven, test of all possible transformation sources could be obtained by crossing the primary APIM–IT predictors (i.e., actor, partner, A × P) with the moderating variable that describes the transformational process. However, the power to detect and reliably estimate these effects is sensitive to sample size, as well as to the distribution of the potential moderator.
Interdependence Over Time
Many aspects of IT are relevant when dyad members are observed on multiple occasions. For example, the idea of transition lists as described by IT refers to situations in which individuals negotiate a series of repeated exchanges. The central idea conveyed by transition lists is that the magnitude of the source matrices can vary across these engagements, as a function of time, or as a function of the decisions made by dyad members on previous trials. The tenability of a potential transition list can be examined by specifying a series of interactions between the basic actor, partner, and A × P effects, and additional variables that describe aspects of the current (or previous) situation. Similarly, the notion of transformations (discussed earlier) takes on an additional layer of complexity, leading to a more ecologically valid representation in the context of a repeated measures framework. Specifically, transformations could be conceptualized as a between-persons or between-dyads process in which the outcome matrices for individuals or dyads, averaging across all measurement occasions, vary as a function of some fixed aspect of the person or dyad. Alternatively, transformations could also be driven by the specific situational context or even aspects of prior situations or “trials.” Researchers have long acknowledged the utility of event sampling and diary designs to assess between- and within-dyadic processes (Bolger, Davis, & Rafaeli, 2003), yet we are unaware of work that has discussed the analysis of experience sampling data for dyads in terms of IT. However, the complexity of these models makes them sensitive to the overall sample size, as well as the ratio of necessary predictors to sample size.
Final Remarks
Many theories of personal relationships strive to answer questions about the dynamic processes that characterize close relationships by focusing on the associations between specific attributes or cognitions and relationship outcomes. Although the content of these attributes/cognitions may pertain to the individual, the relationship, or the partner, the tenets of these theories do not speak to the emergent processes that arise from individuals interacting within the shared reality of the relationship. In contrast, IT provides a template for describing how the configuration of individual attributes within the dyad give rise to distinct individual outcomes. Instead of focusing on specific attributes or cognitions, IT deals with the fundamental processes that govern interpersonal interaction serving as a conduit for more specific theories. In this way, IT “plays well” with other theories because it speaks to a process that underscores other theories of interpersonal behavior. As a result, IT may serve as a basic framework through which to view interpersonal relationships.
This article provides a method for conducting the APIM analysis that makes it possible to extract components of IT from dyadic data generated by standard between-subjects designs. This feature allows the precise predictions put forth by specific theories (e.g., attachment, self-determination, trust) to be mapped onto a general IT perspective. The integration presented here also generalizes to domains of research in which zero- or limited-acquaintance social interactions are of interest (e.g., negotiation/conflict resolution, cooperation). Our analysis also demonstrated how aspects of APIM provide insight into aspects of interdependence not originally anticipated by IT. In this light, IT may be seen as a common thread that ties theories together. In fact, in his call for a more theoretically integrated approach to research in personal relationships, Reis (2007) suggested that if properly applied, IT may serve as the unifying theory of personal relationships. When applied in the manner described in this article, the APIM explicitly imposes an IT perspective on the hypotheses of interest. By extension, then, the APIM may be the method that leads to such unification.
Footnotes
Appendix
Sample Data
| cid | Gender | a_men_effect | a_men_dum | a_women_dum | a_nsr_rel | p_nsr_rel | a_satis |
|---|---|---|---|---|---|---|---|
| 1 | Man | 1 | 1 | 0 | 1.33 | 0.67 | 6.5 |
| 1 | Woman | −1 | 0 | 1 | 0.67 | 1.33 | 5.0 |
| 2 | Man | 1 | 1 | 0 | −1.67 | −1.00 | 4.5 |
| 2 | Woman | −1 | 0 | 1 | −1.00 | −1.67 | 4.0 |
| 3 | Man | 1 | 1 | 0 | 2.00 | 1.00 | 7.5 |
| 3 | Woman | −1 | 0 | 1 | 1.00 | 2.00 | 6.0 |
Acknowledgements
We would especially like to thank Kristen Capuozzo, Benjamin Hadden, Bennett Porter, Lindsey Rodriguez, and Kelty Wickham for their helpful comments on earlier versions of this article. We would also like to thank Amy Bush for the use of her data.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.
Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.
Notes
References
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