Content uploaded by Nicholas S. Perry

Author content

All content in this area was uploaded by Nicholas S. Perry on Mar 07, 2017

Content may be subject to copyright.

RUNNING HEAD: GRAPHIC METHODS FOR RM-APIM

Graphic Methods for Interpreting Longitudinal Dyadic Patterns from Repeated-Measures

Actor-Partner Interdependence Models

Nicholas S. Perry1, Katherine J. W. Baucom1, Stacia Bourne1, Jonathan Butner1,

Alexander O. Crenshaw1, Jasara N. Hogan1, Zac Imel2,3,Travis J. Wiltshire1, and Brian

R.W. Baucom1

1Department of Psychology, University of Utah

2Department of Educational Psychology, University of Utah

3Department of Psychiatry, University of Utah

Corresponding author:

Nicholas Perry

Department of Psychology

University of Utah,

Room 502

380 South 1530 East

Salt Lake City, UT, 84112

E-mail: Nicholas.perry@psych.utah.edu

Phone: 781-526-5179

Fax: (801) 581-5841

GRAPHIC METHODS FOR RM-APIM

Abstract

Researchers commonly use repeated-measures Actor-Partner Interdependence models

(RM-APIM) to understand how romantic partners change in relation to one another over

time. However, traditional interpretations of the results of these models do not fully or

correctly capture the dyadic temporal patterns estimated in RM-APIM. Interpretation of

results from these models largely focuses on the meaning of single parameter estimates in

isolation from all the others. However, considering individual coefficients separately

impedes the understanding of how these associations combine to produce an

interdependent pattern that emerges over time. Additionally, positive within-person, or

actor, effects are commonly misinterpreted as indicating growth from one time point to

the next when they actually represent decline. We suggest that change-as-outcome RM-

APIMs and vector field diagrams (VFDs) can be used to improve the understanding and

presentation of dyadic patterns of association described by standard RM-APIMs. The

current paper briefly reviews the conceptual foundations of RM-APIMs, demonstrates

how change-as-outcome RM-APIMs and VFDs can aid interpretation of standard RM-

APIMs, and provides a tutorial in making VFDs using multilevel modeling.

Keywords: Actor-Partner Interdependence Models; vector field diagrams; longitudinal;

quantitative methods

GRAPHIC METHODS FOR RM-APIM

1

Graphic Methods for Interpreting Longitudinal Dyadic Patterns from Repeated-Measures

Actor-Partner Interdependence Models

Researchers studying family and relationship processes, such as attachment,

communication, empathy, power, or emotional adjustment, are often interested in questions

about patterns of dyadic influence, or how one individual’s characteristics influence their

romantic partner, and how these patterns of influence unfold over time. There are numerous

ways to estimate longitudinal patterns of dyadic influence, including dyadic growth models (e.g.,

Raudenbush, Brennan, & Barnett, 1995), common fate growth models (Ledermann & Macho,

2014), latent bivariate change models (McArdle, 2009), and coupled oscillator models (e.g.,

Boker & Laurenceau, 2005), among others.

Among these techniques, the standard repeated-measures Actor-Partner Interdependence

Model (RM-APIM) is one commonly used approach for modeling dyadic processes over time as

it allows for extending the popular cross-sectional APIM to examining within-dyad effects over

time (e.g., Cook & Kenny, 2005; Kenny, Kashy, & Cook, 2006). Despite their popularity for

estimating dyadic patterns of association over time, the longitudinal patterns represented by

standard RM-APIMs are commonly misinterpreted and the reciprocal dyadic nature of these

patterns is often incompletely interpreted. These interpretive errors likely stem from

misunderstandings about the conceptual foundations of standard RM-APIMs, as well as the lack

of a simple method for plotting the results. Supplementing standard RM-APIM results with

change-as-outcome RM-APIM results and using vector field diagrams (VFDs) provides a more

intuitive representation of longitudinal patterns of association between romantic partners. These

methods can be used to aid complete and correct interpretation. The current paper reviews how

longitudinal dyadic patterns are described in RM-APIMs, discusses how change-as-outcome

GRAPHIC METHODS FOR RM-APIM

2

RM-APIMs and VFDs can be used to better understand these patterns, and provides a tutorial on

VFDs using R code provided in the Appendix.

Brief Introduction to Repeated-Measures Actor-Partner Interdependence Models (RM-

APIMs)

Relationship researchers have commonly used standard RM-APIMs to understand how

the states of an individual and his or her partner measured at one point in time are associated

with both partners’ states measured at a later point in time (e.g., Cook & Kenny, 2005). Figure 1

displays a standard RM-APIM that models four separate paths between husband and wife

relationship satisfaction at Time 1 and husband and wife relationship satisfaction at Time 2.

These four paths include two actor paths (from husband relationship satisfaction at Time 1 to

husband relationship satisfaction at Time 2 [Path 1] and from wife relationship satisfaction at

Time 1 to wife relationship satisfaction at Time 2 [Path 4]) and two partner paths (from husband

relationship satisfaction at Time 1 to wife relationship satisfaction at Time 2 [Path 2] and from

wife relationship satisfaction at Time 1 to husband relationship satisfaction at Time 2 [Path 3]).

In addition to these four paths that describe associations between relationship satisfaction

measured at two different points in time, covariances between spouses’ variables at Time 1 and

between residual errors at Time 2 (double-headed curved arrow) are also estimated.

Such a model is well suited to answer research questions focusing on how the states of an

individual and his or her partner measured at one point in time are associated with both partners’

states measured at a later point in time because of the level of specificity with which actor and

partner paths can be interpreted. Because both partners’ states are included as predictors at Time

1 and a covariance is included between partners’ residual error terms at Time 2, actor paths can

be interpreted as the unique association between that partner’s state at Time 1 and Time 2 while

GRAPHIC METHODS FOR RM-APIM

3

controlling for all four cross-partner associations (i.e., the two covariances and the two partner

paths). A similar interpretation can be made for the statistical uniqueness of the partner paths.

Despite the wide applicability of the standard RM-APIM for longitudinal dyadic

research, misinterpretations of the results are frequent. The widespread interpretive mistakes of

RM-APIMs are: 1) describing actor and partner paths with the same sign as representing the

same form of longitudinal association and 2) interpreting each path in isolation. The first

interpretive error is most easily observed in cases where the actor and partner effects for one

spouse are both significant and positive. For example, suppose that the unstandardized parameter

estimate for Path 1 (husband actor effect) in Figure 1 is B = 0.5, p < .05, and for Path 3 (husband

partner effect) is B = 0.4, p < .05

1

. This pattern of associations is frequently interpreted as,

“higher levels of husband and wife satisfaction at Time 1 are both uniquely associated with

higher levels of husband satisfaction at Time 2.” While this interpretation is correct with respect

to variability about the mean, it is incorrect with respect to variability over time.

The error in the temporal aspect of this interpretation is that, within a repeated-measures

framework, Path 1 and Path 3 represent conceptually different temporal patterns. Path 1 (an actor

effect) represents stability in husbands’ satisfaction over time (e.g., Cook & Kenny, 2005),

whereas Path 3 (a partner effect) represents prediction of change in husband satisfaction from

Time 1 to Time 2. Path 1 represents stability because it is what is known as an autoregressive

term (i.e., a variable measured at a later time point is regressed onto that same variable measured

at an earlier time point; Cronbach & Furby, 1970). The basis of numerous regressed and latent

change models comes from two central ideas (see McArdle, 2009 for a review). These are that

the unweighted covariance between the same variable measured at two or more points in time

1

This pattern of associations is represented in equation form as:

Y(Husband satisfactiont2) = Intercept + 0.5*Husband satisfactiont1 + 0.4* Wife satisfactiont1 + e1

GRAPHIC METHODS FOR RM-APIM

4

represents stability or lack of change, and variance unexplained by an autoregressive term

represents a combination of change in that variable and measurement error. Despite the

widespread acceptance and use of regressed and latent change models, the standard RM-APIM is

rarely recognized as belonging to this class of models of change.

In addition to representing conceptually different temporal patterns (i.e., stability vs.

change), the positive signs of Path 1 and Path 3 may represent different directions of change in

husband satisfaction from Time 1 to Time 2. To illustrate this point, assume that the example

coefficients for Paths 1 and 3 described above are the result of a model where Time 1

relationship satisfaction scores are grand-mean centered.

2

Suppose that grand-mean centered

husband and wife relationship satisfaction are both equal to 1 at Time 1 (i.e., both husband and

wife relationship satisfaction are one unit above the mean). Solving for the estimate of stability

in husband satisfaction from Time 1 to Time 2 (i.e., the actor effect; Path 1) yields a value of 1 x

.5 = .5. This solution indicates that husbands’ relationship satisfaction is closer to husbands’

mean relationship satisfaction at Time 2 (.5 units above the mean) than it is at Time 1 (1 unit

above the mean), even though Path 1 has a positive coefficient. Actor effects that are smaller in

value than positive one represent a longitudinal pattern where the individual returns to mean

levels, whereas actor effects that are larger in value than positive one represent a longitudinal

pattern where the individual departs from mean level.

In contrast, solving the equation for the estimate of prediction of change (i.e., the partner

effect; Path 3) in husband relationship satisfaction from Time 1 to Time 2 yields a value of 1 x .4

= .4. Because Path 3 represents prediction of change over time, this solution indicates that

husband relationship satisfaction at Time 2 is .4 units higher than it would otherwise be. Positive

2

Grand mean centering refers to subtracting the sample mean of relationship satisfaction at Time 1 from each

relationship satisfaction score at Time 1.

GRAPHIC METHODS FOR RM-APIM

5

partner effects represent a longitudinal pattern of departure from the mean. In contrast, negative

partner effects represent a longitudinal pattern where the affected spouse returns to the mean.

As has been noted for other models of dyadic change (e.g., Boker & Laurenceau, 2005),

actor and partner effects in RM-APIMs must be considered jointly to correctly interpret the

longitudinal patterns they describe. The second error common in RM-APIM interpretation,

interpreting paths in isolation, can be seen when combining these two example solutions. Though

both Path 1 and Path 3 have positive parameter values in this example, solving for husband

relationship satisfaction at Time 2 generates a longitudinal pattern of return to the mean that

varies in rate depending on the combination of husband and wife values at Time 1. When

husband and wife values have the same sign at Time 1, husband relationship satisfaction is

slightly lower at Time 2. In the example above, the total estimate for Time 2 husband

relationship satisfaction would be .9 units above the mean (.5 + .4 = .9) compared with the Time

1 value of 1 unit above the mean. When husband and wife values have opposite signs at Time 1,

husband relationship satisfaction returns more rapidly to the mean. For example, if husband

relationship satisfaction was 1 at Time 1 and wife relationship satisfaction was -1, the total

estimate for husband relationship satisfaction at Time 2 is .1 units above the mean (.5 - .4 = .1).

Interpretive Aids for RM-APIM

The change-as-outcome RM-APIM (i.e., a RM-APIM where the dependent variable is a

change score from Time 1 to Time 2) and VFDs are methods that can be used to aid the

interpretation of longitudinal dyadic patterns described by standard RM-APIMs. The primary

advantage of these methods is that they directly describe change over time. VFDs hold particular

promise for helping researchers correctly interpret standard RM-APIMs because they depict

multivariate data over time in a single intuitive visualization.

GRAPHIC METHODS FOR RM-APIM

6

The current paper provides a step-by-step tutorial in using change-as-outcome RM-

APIMs and VFDs to interpret results of standard RM-APIMs. First, we estimate standard RM-

APIMs for three different couples using structural equation modeling (SEM) to estimate each

dyad separately as examples of these methods. Second, we describe how to estimate and interpret

change-as-outcome RM-APIMs for these examples. Third, we introduce the steps involved in

creating VFDs from the change-as-outcome RM-APIM results and provide detailed

interpretations of the VFDs. We then describe how to conduct this same set of analyses to model

longitudinal dyadic patterns for multiple couples in the same model using multilevel models

(MLMs). Finally, we discuss benefits of depicting dyadic change over time using these

supplemental methods to aid in accurate interpretation. In doing so, we take a broad approach to

defining repeated-measures designs as those that generate at least two measurements of the same

variable per person. As is true of any specific method, even though the methods described in this

manuscript can be used with a wide range of sampling designs the appropriate sampling design

for any given research question must be determined on a case-by-case basis.

Illustrative Single Case Examples

We estimate separate RM-APIMs for three couples (Cases A, B, and C) using separate

SEMs to maintain our emphasis on the interpretation of longitudinal dyadic patterns while

minimizing the additional modeling complexities that arise with estimating these models for

multiple couples at once. We address issues specific to MLM for multiple couples in the MLM

example below. The data preparation steps and interpretive principles described here extend

directly to those used in the MLM example. Indeed, they can be applied to any estimation

method that generates an unstandardized solution for the two-intercept parameterization of the

standard RM-APIM (e.g., time-series panel models, Bayesian estimation, etc.).

GRAPHIC METHODS FOR RM-APIM

7

Data for single-dyad Cases A, B, and C come from a larger study of emotion and

behavior in romantic relationships. As part of their participation, spouses wore Actiheart

biosensors (CamNTech Inc., 2014) that continuously recorded their heart rate (HR) during all

waking hours over 7 consecutive days. The electrocardiogram waveform was sampled at 128 Hz,

and these data were used to generate HR values for each 1-minute interval for each spouse. We

selected data from three different couples that vary in the length of the time series, as well as in

the longitudinal dyadic patterns they generate. The data for Case A are n = 210 total minutes of

HR data per spouse (N = 420 data points), the data for Case B are n = 60 total minutes of HR

data per spouse (N = 120 data points), and the data for Case C are n = 120 total minutes of HR

data per spouse (N = 240 data points).

Standard RM-APIM Data Preparation

In the following examples, we implement the two-intercept parameterization of RM-

APIMs. The two-intercept parameterization generates separate, simultaneous estimates of all

parameters for each partner. Although the results of the two-intercept RM-APIM are statistically

identical to the cross-level interaction RM-APIM

3

, we selected the two-intercept RM-APIM

because it clearly shows the actor and partner estimates separately for both spouses. This quality

of the two-intercept RM-APIM is advantageous because it eases interpretation. We note that the

two-intercept RM-APIM implicitly assumes that dyad members are distinguishable; however,

this should be directly tested before proceeding with the two-intercept parameterization (see

Kenny, Kashy, & Cook, 2006 for tests of distinguishability and Olsen & Kenny, 2006 for further

discussion of distinguishability).

As mentioned, standard RM-APIMs model data in an autoregressive manner where the

3

The cross-level interaction parameterization of APIMs involves creating interaction terms between a variable

denoting the spouse and the intercept, actor, and partner effects (Kenny, Kashy, & Cook, 2006, pp.173-178).

GRAPHIC METHODS FOR RM-APIM

8

outcome at Time 2 is regressed onto that same variable at Time 1 as illustrated by Equation 1:

Y(HRT2) = Husband*[B1*(Intercept) + B2*(HRT1_ActHH) + B3 *(HRT1_PartHW)] + Wife*[B4

*(Intercept) + B5*(HRT1_ActWW) + B6*(HRT1_PartWH)] + Error (1)

In Equation 1, the husband actor effect (HRT1_ActHH)

4

represents the association between his own

HR at time 2 and his HR at Time 1. HRT1_PartHW represents the husband’s partner effect, which is

the association between his wife’s HR at Time 1 and his HR at Time 2. Similarly, wife’s actor

effect (HRT1_ActWW) represents the association between her own HR at Time 1 and her HR at

Time 2. HRT1_PartWH represents the wife’s partner effect, which is the association between

husband’s HR at Time 1 and her HR at Time 2.

In the SEM implementation of this equation, the two-intercept standard RM-APIM is

estimated using a dataset structured such that there is one row for each dyad per time point, with

separate variables for each spouse’s dependent variable at each time point and separate variables

for each spouse’s independent variables at each time point. In our example dataset (see Table 1),

the variables included are Dyad, a variable that identifies specific couples, and Time, a variable

that distinguishes time points for each couple. In addition to these variables, criterion (HRh and

HRw for husband and wife HR respectively) and predictor variables are in the same row, first for

the wife (HRActWW, the wife’s effect on her own heart rate, and HRPartWH, the husband’s

effect on wife’s heart rate) and then for the husband (HRActHH, the husband’s effect on his own

heart rate, and HRPartHW, the wife’s effect on husband’s heart rate).

In addition to constructing separate versions of each predictor variable for each partner,

consideration needs to be given to whether to use any form of centering on the predictors prior to

4

The naming convention used for variables is HR=heart rate; H = husband, W = wife; Act = actor effect, Part =

partner effect; subscripts indicate the direction of effect (e.g., HRActHH is husband’s effect on himself, HRPartWH

is wife’s effect on husband).

GRAPHIC METHODS FOR RM-APIM

9

estimating the standard RM-APIM. Centering refers to shifting the zero value from the original

scale of a variable and is typically used to ensure a meaningful value of the intercept. We mean

centered the actor and partner variables by subtracting each spouse’s mean across time from his

or her own raw scores (i.e., spouse-centered) for the actor and partner predictor variables for two

reasons. First, with spouse mean-centering, the husband and wife intercepts in the standard RM-

APIM represent husband’s and wife’s average HR when all other variables in the model equal 0.

Given that the other variables in the model are the husband’s and wife’s previous HRs, the

intercepts would be conceptually uninterpretable if husband’s and wife’s previous HR were left

in the original metric because it is biologically implausible for HR to be 0 in this situation.

Choosing to center the predictor variables at each spouse’s mean has an additional

advantage for understanding change over time because the intercept represents the average value

in the outcome variable when there is no change in the actor or partner effects relative to the

mean. The point at which there is no change is referred to as an attractor within the dynamical

system literature (see Abraham & Shaw, 1992; Butner et al, 2015 for further discussion).

5

An

attractor is understood as a point to which the system is drawn over time or its natural “set

point”. Alternatively, within dynamical systems literature, in the absence of perturbations (i.e.,

influences in the system that affect the dyad’s trajectory), it is the most likely value of the dyadic

system. This conceptualization of an attractor is similar to the definition of the intercept in a

model with centered predictors. Finally, the concept of an attractor also incorporates the idea that

this specific set point emerges over time within the dyadic system.

Table 1 illustrates the variables needed to run standard RM-APIMs with the spouse-

5

An attractor is one of several topological features that can occur in a dynamic system. A thorough review of

topological features in dynamical systems is beyond the scope of this manuscript. Interested readers are directed to

Butner, et al. 2015 and Abraham & Shaw, 1992 for accessible overviews and introductions.

GRAPHIC METHODS FOR RM-APIM

10

centered predictors. HR represents the given spouse’s heart rate during that minute; HRActHHsc

and HRPartHWsc represent the husband’s centered actor and partner variables respectively; and,

HRctWWsc and HRPartWHsc represent the wife’s centered actor and partner variables. The first

two rows of data for the actor and partner variables for each couple are missing because the data

is lagged to shift the data down by one time point. Lagging is done because they are the first

observations in the series and there is therefore no previously occurring data available to predict

the outcome at Time 1. The data presented in this example dataset (Table 1) is simulated to

closely reproduce the results of the models run for our examples, which use real data.

6

The top left panel of Table 2 presents results of standard RM-APIMs run separately for

Case A, Case B, and Case C using SEM. Here we focus on Case A for descriptive purposes and

then discuss it in comparison to Cases B and C. In Case A, the Husband intercept is 92.24, and

the Husband actor effect is significant and positive (B = 0.73, p < .001). Similarly, the Wife

intercept is 112.16, and the Wife actor effect is also significant and positive (B = 0.58, p < .001).

The intercepts represent each spouse’s average HR when no change is occurring. The significant,

positive actor effects that are both less than 1 indicate that there is stability in husband and wife

HR from Time 1 to Time 2 and that husband and wife HR return to the mean over time. Both

partner effects are non-significant (B = 0.06, p = .43; B = 0.08, p = .20, for the husband and wife

respectively). These non-significant partner associations indicate that wife’s HR during the

previous minute is not significantly associated with change in husband’s HR from one minute to

the next and, similarly, that husband’s HR during the previous minute is not significantly

associated with change in wife’s HR from one minute to the next.

Alterations to RM-APIM for Building VFDs

6

The complete simulated data sets are available to be downloaded from http://systems.psych.utah.edu.

GRAPHIC METHODS FOR RM-APIM

11

To translate the standard RM-APIM into a format that can be graphed and interpreted,

some changes need to be made to the data and to the parameterization of the standard RM-

APIM. First, the outcome variable needs to be a change score in HR from Time 1 to Time 2

rather than the actual value of HR at Time 1, as illustrated in Table 1. This change score is

created by subtracting HRtime1 from HRtime2 (i.e., HRchange = HRtime2 – HRtime1). Second, the

original, uncentered versions of the actor and partner variables need to be used as predictors so

that the VFDs reproduce the results of the spouse-centered standard RM-APIM.

The reason that uncentered versions of the actor and partner variables need to be used in

this case is to ensure that the intercepts in this model are conceptually consistent with the

intercepts produced in the centered standard RM-APIM. Because the actor and partner effects

were spouse centered in the standard RM-APIM, the intercept in that model represents the

average HR when there is no relative change from the previous time point. In standard RM-

APIM models, relative change from the previous time point is represented by the variables on the

right side of the equation (i.e., the predictors). Using the change score for HR as the dependent

variable in the change-as-outcome RM-APIM means that the relative change from the previous

time point is now represented on the left side of the equation (i.e., the outcome). In the change-

as-outcome RM-APIM, the change score (which is the criterion) equals 0 when there is no

change from one time point to the next, and the point at which no change occurs is each spouse’s

average HR. Put differently, in the change-as-outcome RM-APIM, the outcome of the equation

equals 0 when the value of the spouse’s previous HR is equal to their average HR, whereas in the

standard RM-APIM the outcome of the equation equals the spouse’s average HR when the value

of their previous HR is equal to 0 in centered metric. Both models are describing the same dyadic

pattern, but that information is represented by different terms in the two models.

GRAPHIC METHODS FOR RM-APIM

12

To illustrate the changes to the data necessary to generate the VFD, Table 1 also shows

the variables needed to run the change-as-outcome RM-APIM on the right-hand side. HRchngh

and HRchngw represents the change in each spouse’s heart rate from the previous minute, with

“h” indicating the change in heart rate variable for husband and “w” indicating the change in

heart rate variable for wife. HRActHH and HRPartHW represent the husband’s actor and partner

variables in original units of measurement; and, HRActWW and HRPartWH represent the wife’s

actor and partner effects in original units of measurement. These variables are then used to

parameterize the change-as-outcome RM-APIM as illustrated by Equation 2:

Y(HRT2 – HRT1) = Husband*[B1*( Intercept) + B2 *(HRT1_ActHH) + B3 *(HRT1_PartHW) + Wife*

[B 4 (Intercept) + B 5 (HRT1_ActWW) + B 6 *( HRT1_PartWH)] + Error (2)

Given the modification of the model criterion to a change score, the interpretation of the

longitudinal dyadic pattern based on actor and partner effects also changes. In the change-as-

outcome RM-APIM, both actor and partner effects predict change, whereas in the standard RM-

APIM actor effects predict stability and partner effects predict change. Therefore, the substantive

interpretation of the actor effects is different across the two models, whereas the meaning of the

partner effects remains the same.

The right panel of Table 2 displays the results of the change-as-outcome RM-APIM.

Comparing these parameter estimates to those produced by the respective standard RM-APIMs

demonstrates that the values of the actor and partner effects in the two models represent the

conceptual differences described above. The magnitude of the actor effects in the change-as-

outcome RM-APIMs is 1 unit lower than the corresponding actor effects in the standard RM-

APIMs. This consistency in the relative difference of the actor effects across the models is not

GRAPHIC METHODS FOR RM-APIM

13

coincidental; actor effects in the two types of models will always differ by 1.

7

Conceptually, the

reason for this difference in the magnitudes of actor path estimates can be understood by

comparing the values of the actor effects that represent perfect stability/no change in the two

models: an unstandardized actor path with a value of 1 in a standard RM-APIM and an

unstandardized actor path with a value of 0 in a change-as-outcome RM-APIM. Turning to the

partner effects, not only do partner effects in the two models both represent prediction of change,

but they also show that the parameter values of the partner effects are identical in the two

models. This equivalence in the partner effects occurs because even though the criterion changes

across the two models, the standard RM-APIM and the change-as-outcome RM-APIM describe

the same longitudinal dyadic pattern; the extent to which the partner’s previous HR predicts

change in the spouse’s HR from Time 1 to Time 2 must be identical across both models.

This representation of the longitudinal dyadic pattern in the change-as-outcome RM-

APIM is particularly helpful because both actor and partner effects represent the same

longitudinal pattern (i.e., prediction of change from Time 1 to Time 2) and can be directly

compared to each other. If the actor and partner effects share the same sign in the change-as-

outcome RM-APIM, they indicate the same type of longitudinal change for the spouse. This

makes it considerably simpler to identify the dyadic pattern of change over time in the change-

as-outcome RM-APIM relative to the standard RM-APIM.

Generating and Understanding VFDs

Another benefit of the change-as-outcome RM-APIM is that the estimates can be used to

generate VFDs. There are numerous methods that can be used to generate a VFD (Abraham &

Shaw, 1992; Boker & McArdle, 1995; Butner, Gagnon, Geuss, Lessard, & Story, 2015). Here,

7

The algebraic proof of this mathematical equivalence is available from the first author.

GRAPHIC METHODS FOR RM-APIM

14

we use R code (provided in the online supplemental appendix) that makes use of the pracma

package (Borchers, 2016) to perform this step.

Conceptually, a VFD is analogous to a scatterplot of raw values, but the metric is one of

change (Butner et al., 2015). A VFD depicts the amount of change in the outcome variable for

both husbands and wives that would be expected given observed values during the previous

observation. Figure 2 displays the VFDs for Cases A, B, and C. In these diagrams the vector

arrows represent the amount and direction of change in husband’s and wife’s HR given

husband’s and wife’s HR during the previous minute. These diagrams depict husband’s HR

along the X-axis and wife’s HR along the Y-axis. Finally, there is a single attractor in each

diagram. This attractor is the pair of HR values (one for husband, one for wife) that the vectors

move toward over time.

Beyond the location of the attractor, other elements of the diagram, such as the length of

the arrows and the arrows’ orientation, convey other pieces of useful information about the

longitudinal dyadic pattern. The length of the arrows conveys the rate of change in HR, where

longer arrows indicate greater change from one minute to the next. In the examples provided in

Figure 2, the arrows are larger at the edges of the diagram, indicating greater change, but they

then become shorter as spouses move toward the attractor indicating slower change.

Finally, the orientation of the arrows provides a visual depiction of the contribution of

actor and partner effects. This is most apparent in the diagrams when examining the arrows that

are in the space adjacent to the attractor. Arrows oriented straight vertically or straight

horizontally indicate actor effects (i.e., husbands exerting stability on their own HR over time

and wives exerting stability on their own HR over time). In contrast, swirled lines indicate

partner effects (i.e., husband exerting change on wife’s HR over time and wife exerting change

GRAPHIC METHODS FOR RM-APIM

15

on husband’s HR over time).

In summary, researchers can model a standard RM-APIM to estimate effects of actor and

partner variables on an outcome over time. To generate a VFD to visualize these patterns and

ensure correct interpretations, researchers can run a change-as-outcome RM-APIM using

uncentered predictors and enter the coefficients from the change-as-outcome RM-APIM into our

supplemental R code to generate the VFD.

Identifying Dyadic Patterns in VFDs

Using our single-dyad cases, we identified several examples of common dyadic patterns,

including actor-oriented, couple-oriented (where actor effects are approximately equal to partner

effects), and, what Kenny and Cook, termed a social comparison model (where actor and partner

effects are approximately equal, but opposite signs and, thus, opposite effects on the spouse)

(Kenny & Cook, 1999). We do not include an example of a fourth pattern, the partner-oriented

dyadic pattern (where actor effects are non-significant), because this pattern did not emerge in

the data set used for the illustrative examples. The likely reason this pattern does not emerge is

that the cardiovascular system is a self-regulating, homeostatic system, meaning that significant

actor effects are nearly always present. This omission should not be understood as an indication

that partner-oriented dyadic effects are unlikely to occur in RM-APIMs; they could certainly

emerge when measuring other constructs.

Prior to examining each of these three patterns, it is important to recognize that when

applying labels of patterns created to describe cross-sectional dyadic effects (e.g., Kenny &

Cook, 1999) to RM-APIMs the pattern labels are most consistent with the change-as-outcome

RM-APIM. This is because, in the change-as-outcome RM-APIM, the actor and partner effects

both represent predictions of change. In the standard RM-APIM, the actor effects represent

GRAPHIC METHODS FOR RM-APIM

16

stability and the partner effects represent change (controlling for stability). This difference in

what the actor and partner effects conceptually measure in the standard RM-APIM complicates

translating results from them into commonly identified cross-sectional dyadic patterns.

In Case A, both husband and wife actor effects in the change-as-outcome model are

statistically significant and negative, with no statistically significant partner effects for either

spouse. Conceptually, this set of effects is like an actor-oriented dyadic pattern (Kenny & Cook,

1999). The combination of negative, significant actor effects and non-significant partner effects

creates the largely straight horizontal and vertical lines that converge on the attractor.

In contrast, in Case B, both husband’s and wife’s actor effects and partner effects in the

change-as-outcome model are significant and negative; this pattern of results is consistent with a

couple-oriented dyadic pattern (Kenny & Cook, 1999). This combination of two significant,

negative partner effects creates a much stronger swirled orientation of the arrows, which can be

seen where both husband’s (horizontal) and wife’s (vertical) arrows begin to swirl closer to the

edge of the diagram. In this case, husband’s arrows (horizontal) are drawn vertically by the

wife’s partner effect toward the attractor, whereas the wife’s arrows (vertical) are drawn

horizontally by the husband’s partner effect. This creates a much more angled orientation of the

swirled arrows for Case B when compared to the VFD for Case A because both partner effects

are significant for Case B, but neither is significant for Case A. When considered jointly, the

negative partner effects speed up the return to the set point.

Finally, Case C has significant partner and actor effects as well. However, these effects

are working opposite one another in a push-pull dynamic, consistent with Kenny’s social

comparison pattern (Kenny & Cook, 1999). Here, the actor effects are both negative, which

indicates these effects draw the spouses back to their set point. However, partner effects are both

GRAPHIC METHODS FOR RM-APIM

17

positive, which indicates they are moving each spouse away from their set point. In conjunction,

the actor effects are pulling each spouse, whereas the partner effects are pushing each spouse,

and the overall pattern is determined by the strength of the actor effects relative to the partner

effects. Because of this push-pull dynamic, a less prominent swirling of the arrows is seen in

Case C as compared to Case B.

VFDs can be used to visually map the effects of actor and partner effects relative to one

another, with a stronger swirled orientation indicating a stronger resultant combination of actor

and partner effects. This type of graphic depiction could also be used as a visual adjunct to other

quantitative means of estimating the relative influence of actor and partner effects (e.g., Kenny &

Ledermann, 2010). Additionally, the VFD can be used to identify dyadic patterns of effects

through a different means than examining model results, which could be simpler and more

intuitive for readers.

Correcting Conceptual Understanding of Temporal Change using VFDs

We now focus on three important contributions of using change-as-outcome RM-APIM

and VFDs to model change over time. These models help highlight 1) the conceptual meaning of

the attractor (i.e., where are spouses drawn), 2) the rate of change, and 3) the contribution of

actor and partner effects. Across couples, the nature of change over time could vary by any one

of, or combination of, these factors.

The attractor of the diagram depicts the importance of the intercept in determining how

both partners change over time. Visually, this demonstrates that both partners are drawn together

toward a common point, which emphasizes the dyadic nature of their movement over time

toward a shared point where no change occurs (i.e., a set point). Additionally, different couples

could change at identical rates, but be drawn to a different place over time, which could be

GRAPHIC METHODS FOR RM-APIM

18

observed in couples moving towards different attractors when comparing their VFDs. VFD

results that show this pattern would indicate that different couples have different natural set

points over time in the construct of interest (e.g., heart rate in our examples). Conceptually, this

pattern of results is the visual representation of a characteristic of a dyad that might shift their

inherent set point, such as relationship satisfaction. Thus, being able to see the attractor point

within the VFD orients the reader to the dyad-level end-point (i.e., that the attractor is about the

couple, not each partner individually) and allows the reader to examine if multiple attractors

might exist within a sample because of characteristics that distinguish groups of dyads.

The rate of change toward the attractor is a separate piece of information provided by the

VFD. Couples in a sample could be drawn to the same place over time (i.e., have the same

intercept), but move toward that attractor point at different rates over time. This difference would

be seen visually in the VFD in variation in the size of the arrows across two couples’ VFDs.

Conceptually, some couples might return to their attractor point faster or slower than others.

Additionally, the VFD can offer more detailed information about the rate of change with respect

to how the rate of change for the couple occurs as a dyadic pattern. For example, within Case A

of Figure 2, larger arrows are seen in the lower right corner at the edge of the VFD. In Case B,

larger arrows are observed in both corners of the VFD along the edges. In Case A, when one

partner is higher on HR and the other partner is lower (i.e., mismatched), change is much greater

(as signified by larger arrows). In contrast, in Case B, when both partners match (i.e., both

partners are lower in HR or higher in HR), change is much greater. Finally, in Case C, compared

to Case B, the relatively shorter arrows indicate that the combination of actor and partner effects

in Case C results in the couple returning to their set point more slowly. In sum, the main

conceptual idea is that some couples may return to their attractor point more quickly when both

GRAPHIC METHODS FOR RM-APIM

19

are coming from the same place (i.e., matched, Case B), whereas others might return more

quickly when the difference between the partners is greater (Case A) or when the combination of

effects generally slows their return (Case C). The VFD offers a helpful visualization that the rate

of change over time (i.e., the size of the coefficients) is a distinct component of understanding

dyadic longitudinal patterns. Further, the VFD depicts information about the rate of change with

respect to both partners in a dyadic context.

Finally, couples could also vary in their relative contribution of actor and partner effects.

The VFD can be used to visually interpret the relative contribution of each type of effect. This is

an important issue because the strength of these effects necessarily influences the couples’ rate of

change over time (i.e., the size of the arrows). We point to the panels in Figure 2 of different

models to emphasize the orientation of the arrows (i.e., straight versus swirled) changes

noticeably with the addition of significant partner effects. A straight, rather than swirled,

orientation of the arrows indicates that significant partner effects are not present in a model.

When comparing Case A versus Case B, the rate of change increases (as demonstrated by a

greater number of larger arrows along the edges of the VFDs) because of the difference in

significant partner effects. In these examples, partner effects are drawing spouse’s closer to the

attractor (i.e. having a stabilizing effect), resulting in an increased rate of change, which is then

shown in the VFD by the greater number of large arrows. However, in Case C, where the actor

and partner effects are working in opposite directions on the spouses’ return to their mean, the

arrows are much more uniform in size.

The VFD allows for a rapid visual assessment of the relative influence of actor and

partner effects. While quantitative approaches exist for this as well (e.g., Kenny & Ledermann,

2010), the ease with which the effects are identifiable from the orientation of the arrows in the

GRAPHIC METHODS FOR RM-APIM

20

diagram allows the reader to understand the direct relationship between the strength of the actor

and partner effects and the rate at which the couple moves toward their attractor by a visual

means. Further, the visual depiction allows readers to understand how different elements of the

dyadic pattern relate to one another, which is a nuance that could be missed when examining

output or results in a table.

Illustrative Multilevel Modeling Example

Our final case example models change over time for multiple couples simultaneously

using the same set of analyses as for our first case example, but estimates these models using

MLM. As in our previous examples, data come from the same larger study of emotion and

behavior in romantic relationships and are time series of HR data for both spouses. The data in

the MLM case were generated in a similar manner to those earlier case examples. For the data

used in this example, spouses wore wireless Bionomadix transmitters (Biopac Systems, 2016)

while having a 7-minute conversation about their courtship and early relationship history. The

electrocardiogram waveform was initially sampled at 1000 Hz, and these data were used to

generate HR values for each 1-minute interval for each spouse, similar to our earlier case

examples. We selected the data to illustrate flexibility of these modeling and graphical

techniques for application with time series of differing lengths (n = 7 observations per spouse)

and for differing numbers of couples (n = 59 couples; N = 826 HR measurements).

In the multilevel modeling implementation of this equation, the dataset is structured in a

pairwise format (see online appendix for Table 4). Here, there is one row for each spouse’s

dependent variable per time point and there are separate variables for all actor and partner effects

(two for husbands and two for wives). In our example dataset, the variables included are Dyad, a

variable that identifies specific couples, Person, a variable that distinguishes spouses within the

GRAPHIC METHODS FOR RM-APIM

21

couple, Husband, a dummy coded variable that indicates rows where husband is the criterion,

and Wife, a dummy coded variable that indicates rows where wife is the criterion. In addition to

these variables, criterion (HR) and predictor variables are in the same row, for the wife

(HRActW and HRPartWH, the wife’s actor and partner variables respectively) and for the

husband (HRActHH and HRPartHW, the husband’s actor and partner variables, respectively).

The results from a standard multilevel RM-APIM with sex-mean centered

8

predictors is

presented in Table 3. The results of the change-as-outcome multilevel RM-APIM with

uncentered predictors are provided on the bottom right side of the table. Both MLMs were run

using restricted maximum likelihood estimation and a compound symmetry variance-covariance

matrix; results are reported with robust standard errors. As with the single dyad case examples, a

principal consideration in estimating standard multilevel RM-APIMs and change-as-outcome

multilevel RM-APIMs is if and how to center the actor and partner effects (see Enders &

Tofighi, 2007 for further discussion of centering in MLM). To permit examination of the

multitude of ways that longitudinal dyadic patterns can differ, as discussed in the preceding

section (i.e., between-couple differences in the location of the attractor, the rate of change, and

the contribution of actor and partner effects), we sex-mean centered the actor and partner

variables in the standard multilevel RM-APIMs and did not center the actor and partner variables

in the change-as-outcome multilevel RM-APIMs. Sex-mean centering predictors in the standard

multilevel RM-APIM and leaving predictors uncentered in the change-as-outcome multilevel

RM-APIM result in the actor and partner effects in both models representing total effects, which

are a combination of within- and between-couple associations (see Raudenbush & Bryk, 2002 for

8

Sex-mean centering refers to centering each variable by the grand mean for that spouse. For example, sex-mean

centering the husband actor variable refers to subtracting the mean of all husband’s actor effects for each individual

husband actor variable value.

GRAPHIC METHODS FOR RM-APIM

22

additional discussion). It is necessary that the actor and partner effects in the standard multilevel

RM-APIM and change-as-outcome multilevel RM-APIM represent the same level of association

(i.e., both representing total effects in our case examples) for the VFD based on the change-as-

outcome multilevel RM-APIM results to accurately reflect the same longitudinal dyadic pattern

described by the standard multilevel RM-APIM.

Because MLMs are regression-based approaches, the interpretation of the standard

multilevel RM-APIM and the change-as-outcome multilevel RM-APIM follows the identical

interpretive principles as the previous case examples. Accordingly, the intercept in the standard

multilevel RM-APIM represents the point where no change is occurring. As can be seen on the

left of Table 3, both spouses’ intercepts are approximately 70 beats per minute, which is the

average HR for husbands and wives when no change is occurring. There is also a strong

significant actor effect for the husband, which is less than 1, indicating a return to mean levels

over time. Additionally, there is a significant partner effect of husbands’ HR on change in wives’

HR. Finally, there is a marginally significant partner effect of wives’ HR on change in husbands’

HR from one time point to the next. As can be seen in the change-as-outcome multilevel RM-

APIM (right side of Table 3), both the wife’s actor effect and the husband’s effect on her change

in HR share the same sign. Exactly like the single-dyad case example of the change-as outcome

RM-APIM, this denotes that these effects are both working together to move the wife along the

same longitudinal trajectory.

Consistent with this interpretation, there is a clear attractor in the VFD for these results

(Figure 3), which suggests a natural average set point for couples in the sample. Additionally, the

strong horizontal orientation of most of the arrows demonstrates the husbands’ actor effect shifts

his trajectory back toward his mean value (i.e., indicating stability in his HR over time). Finally,

GRAPHIC METHODS FOR RM-APIM

23

the swirled arrows suggest the presence of partner effects. The direction of those arrows

(counter-clockwise) visually demonstrates husbands’ influence on wives where wives’ vertical

arrows are shifted horizontally over time. Taken together, these results embedded in the VFD

visually depict the aggregate pattern of change for the full sample of couples.

Extensions and Additional Considerations

The case examples presented demonstrate the value of change-as-outcome RM-APIMs

and VFDs for interpreting the longitudinal dyadic patterns described by standard RM-APIMs.

These methods can also be flexibly applied to analyzing one couple at a time or multiple couples

at a time. In addition to aiding detailed interpretation, the examples also describe how VFDs can

be used to efficiently communicate the longitudinal dyadic patterns described by standard RM-

APIMs in a single, intuitive visual representation. We selected certain common patterns of

effects (e.g., actor-oriented, couple-oriented, “social comparison”) to demonstrate the value of

VFDs and note that other types of dyadic patterns represented in standard RM-APIMs can be

visually depicted in VFDs (see Kenny & Cook, 1999; Kenny & Ledermann, 2010 for discussion

of dyadic patterns). We now consider some areas of future work in the use of these diagrams.

The current paper makes use of MLM to estimate standard RM-APIMs and change-as-

outcome RM-APIMs for a sample of multiple couples. We elected to use MLM over other

possible estimation methods (e.g., multilevel SEM) because the sample size in the multiple-dyad

case is considerably smaller than is typically considered acceptable for SEM. As noted above,

other statistical approaches could be used for this type of analysis. Regardless of the specific

modeling approach used, the same concerns that apply to regression-based models (e.g.,

sufficient variability in the predictor and outcome variables, appropriate sample size) continue to

apply. For example, estimating these RM-APIMs in SEM can be done with a sufficiently large

GRAPHIC METHODS FOR RM-APIM

24

sample size of couples, with samples larger than 150 being a general rule of thumb for stable

estimation of model parameters (see Snijders & Bosker, 2012 for discussions of sample size for

SEM). Similarly, for sample sizes smaller than what can be stably estimated in MLM (i.e., less

than 20-30; Raudenbush & Bryk, 2002), time series panel analyses or other small sample MLM

approaches can be employed (e.g., Shadish, Kyse, & Rindskopf, 2013).

We chose to describe the use of VFDs with RM-APIMs with simple actor and partner

effects both to reduce the interpretive complexity associated with more elaborate models and

because of the widespread use of standard RM-APIMs with simple actor and partner effects.

Here we briefly describe how VFDs can be used to plot standard RM-APIMs with interactions,

which can be used to test whether the aggregate pattern of change in a sample of couples varies

based on the level (or presence) of another variable (for further discussion of moderation in

APIMs see Garcia, Kenny, & Ledermann, 2015).

In much the same way that simple slopes can be used to depict interactions, simple

equations can be used to generate VFDs at different values of a moderator (Cohen, Cohen, West,

& Aiken, 2013). Simple equations refer to generating a point estimate of the intercept and

regression coefficients at a specific level of the moderator. In keeping with convention for

estimating simple slopes in regression models, we suggest that simple equations be estimated at

one standard deviation above and below the mean of continuous moderators; simple equations

for categorical moderators should be estimated in a manner consistent with the coding scheme

used for the categorical moderators. In doing so, researchers can then interpret the VFDs that

represent the simple effects under different conditions by directly comparing and contrasting the

two figures (i.e., when the moderator is high/present or low/absent).

Finally, we note that our paper focused on producing and interpreting VFDs for

GRAPHIC METHODS FOR RM-APIM

25

distinguishable dyads. In cases where indistinguishable dyads are sampled, researchers could

generate a diagram for actor effects plotted against partner effects. We should point out that, like

the standard indistinguishable RM-APIM, this approach would assume that the actor and partner

effects are the same for both partners (see Olsen & Kenny, 2006 for further discussion of dyadic

modeling with indistinguishable dyads).

Conclusions

Our goal within the current paper has been to demonstrate the value of using change-as-

outcome RM-APIMs and VFDs to aid and expand interpretation of standard RM-APIMs by

visually capturing the complex dynamics of change over time that occur within close

relationships. In contrasting how results are depicted across standard RM-APIMs, change-as-

outcome RM-APIMs, and VFDs, several meaningful steps of interpretation that are often

overlooked in standard RM-APIMs emerged as being well represented by change-as-outcome

RM-APIMs and VFDs. We suggest that VFDs in particular offer an innovative method of

extending the interpretation of standard RM-APIMs that more fully captures these interesting

dynamics of change. Finally, as can be seen in our case examples, VFDs offer a clear and direct

corollary to standard RM-APIMs that we believe will be familiar to relationship scientists, as

they stem directly from standard RM-APIM results. Researchers are often taught and encouraged

to organize statistical results into tables and graph or plot data as a means of grasping the larger

picture of their findings and VFDs provide a novel means of doing so for the standard RM-

APIM. We hope the simple elegance of these diagrams will be appealing to dyadic researchers

and will help the field advance our understanding of complicated, interdependent processes of

change that unfold in close relationships.

GRAPHIC METHODS FOR RM-APIM

26

References

Abraham, R. & Shaw, C. D. (1992). Dynamics: The geometry of behavior (2nd ed.). Redwood

City, CA: Addison-Wiley.

ACQKnowledge (version 4) [Software]. Biopac Systems, Inc.

Boker, S. M., & McArdle, J. J. (1995). Statistical vector field analysis applied to mixed cross-

sectional and longitudinal data. Experimental Aging Research, 21, 77-93. doi:

10.1080/03610739508254269

Boker, S. M. & Laurenceau, J. P. (2006). Dynamical systems modeling: An application to the

regulation of intimacy and disclosure in marriage. In T. A. Walls & J. L. Schafer (Eds.).

Models for intensive longitudinal data. New York, New York: Oxford University Press.

Borchers, H. W. (2016). Package ‘pracma’. Retrieved from: https://cran.r-

project.org/web/packages/pracma/pracma.pdf

Butner, J. E., Gagnon, K. T., Geuss, M. N., Lessard, D. A., & Story, T. N. (2015). Utilizing

topology to generate and test theories of change. Psychological methods, 20(1), 1-25. doi:

10.1037/a0037802

Actiheart [Software]. CamNtech Inc.

Cohen, J., Cohen, P., West, S. G., & Aiken, L. S. (2013). Applied Multiple

Regression/Correlation Analysis for the Behavioral Sciences. Mahwah, NJ: Routledge.

Cook, W. L., & Kenny, D. A. (2005). The Actor–Partner Interdependence Model: A model of

bidirectional effects in developmental studies. International Journal of Behavioral

Development, 29, 101–109. doi: 10.1080/01650250444000405

Cronbach, L. J., & Furby, L. (1970). How we should measure “change”: Or should we?

Psychological Bulletin, 74, 68–80. doi: 10.1037/h0029382

GRAPHIC METHODS FOR RM-APIM

27

Enders, C. K., & Tofighi, D. (2007). Centering predictor variables in cross-sectional multilevel

models: a new look at an old issue. Psychological methods, 12, 121-138. doi: 10.1037/1082-

989X.12.2.121

Garcia, R. L., Kenny, D. A., & Ledermann, T. (2015). Moderation in the actor–partner

interdependence model. Personal Relationships, 22, 8-29. doi: 10.1111/pere.12060

Kenny, D.A., Kashy, D.A., & Cook, W.L. (2006). Dyadic Data Analysis. New York: Guilford.

Kenny, D. A., & Ledermann, T. (2010). Detecting, measuring, and testing dyadic patterns in the

actor–partner interdependence model. Journal of Family Psychology, 24, 359-366. doi:

10.1037/a0019651

Ledermann, T., & Macho, S. (2014). Analyzing change at the dyadic level: The common fate

growth model. Journal of Family Psychology, 28, 204-213. doi: 10.1037/a0036051

McArdle, J.J. (2009). Latent variable modeling of differences and changes with longitudinal data.

Annual Review of Psychology, 60, 577-605. doi: 10.1146/annurev.psych.60.110707.163612

Raudenbush, S. W., Brennan, R. T., & Barnett, R. C. (1995). A multivariate hierarchical model

for studying psychological change within married couples. Journal of Family Psychology, 9,

161-174. doi: 10.1037/0893-3200.9.2.161

Raudenbush, S. W. & Bryk, A. S. (2002). Hierarchical linear models: Applications and data

analysis methods. Thousand Oaks, California: Sage Publishers.

Shadish, W. R., Kyse, E. N., & Rindskopf, D. M. (2013). Analyzing data from single-case designs

using multilevel models: New applications and some agenda items for future research.

Psychological methods, 18, 385-405. doi: 10.1037/a0032964

Snijders, T. A. B.& Bosker, R. J. (2012). Multilevel analysis: An introduction to basic and

advanced multilevel modeling. London, England: Sage Publishers.

GRAPHIC METHODS FOR RM-APIM

28

Table 1

Simulated dyad dataset for Repeated Measures APIM and APIM for VFD with structural equation modeling

Variables used in RM APIM

Variables used in change-as-outcome RM-APIM

Dyad

Time

HRh

HRw

HRActHHsc

HRPartHWsc

HRActWWsc

HRPartWHsc

HRchngh

HRchngw

HRActHH

HRPartHW

HRActWW

HRPartHW

1

1

104.00

111.75

---

---

---

---

---

---

---

---

---

1

2

105.50

108.99

11.68

-0.41

-0.41

11.68

1.51

-2.76

104.00

111.75

111.75

104.00

1

3

104.39

105.80

13.18

-3.17

-3.17

13.18

-1.11

-3.2

105.5

108.99

108.99

105.50

1

4

103.33

108.33

12.07

-6.37

-6.37

12.07

-1.06

2.53

104.39

105.80

105.80

104.39

…

…

…

…

…

…

…

…

…

…

…

…

2

1

80.68

95.50

---

---

---

---

---

---

---

---

---

2

2

82.50

96.75

-4.76

-6.47

-6.47

-4.76

1.82

1.26

80.68

95.50

95.5

80.68

2

3

81.75

91.50

-2.94

-5.21

-5.21

-2.94

-0.75

-5.25

82.50

96.75

96.75

82.50

2

4

79.75

95.50

-3.69

-10.46

-10.46

-3.69

-2.01

4.00

81.75

91.50

91.5

81.75

…

…

…

…

…

…

…

…

…

…

…

…

3

1

99.01

117.40

---

---

---

---

---

---

---

---

---

3

2

91.00

121.66

-3.12

8.19

8.19

-3.12

-8.00

4.26

99.01

117.4

117.4

99.01

3

3

91.75

116.75

-11.12

12.44

12.44

-11.12

0.74

-4.92

91.00

121.66

121.66

91.00

3

4

96.50

113.75

-10.37

7.53

7.53

-10.37

4.76

-3.00

91.75

116.75

116.75

91.75

…

…

…

…

…

…

…

…

…

…

…

…

Note: sc = Spouse-mean centered, consistent with our centering choice

GRAPHIC METHODS FOR RM-APIM

29

Table 2.

Repeated measures Actor-Partner Interdependence Model results of change in heart rate for single-dyad case examples1

1 Case A represents the results from one couple, Case B represents the results from a second couple, Case C represents the results from

a third couple.

Standard RM-APIM (SEM)

Change-as-outcome RM-APIM (SEM)

Case A

B

SE B

p-value

95% CI

B

SE B

p-value

95% CI

Husband

Intercept

92.24

0.47

<.001

[91.31, 93.17]

17.68

9.37

0.06

[-0.79, 36.15]

Actor effect

0.73

0.06

<.001

[0.61, 0.85]

-0.27

0.06

<.001

[-0.39, -0.15]

Partner effect

0.06

0.08

0.43

[-0.93, 0.22]

0.06

0.08

0.43

[-0.90, 0.22]

Wife

Intercept

112.16

0.47

<.001

[111.23, 113.09]

39.57

9.37

<.001

[21.09, 58.03]

Actor effect

0.58

0.08

<.001

[0.43, 0.74]

-0.42

0.08

<.001

[-0.57, -0.26]

Partner effect

0.08

0.06

0.2

[-0.04, 0.19]

0.08

0.06

0.2

[-0.04, 0.19]

Case B

Husband

Intercept

102.14

0.59

<.001

[100.98, 103.3]

60.57

15.26

<.001

[30.46, 90.86]

Actor effect

0.64

0.07

<.001

[0.51, 0.78]

-0.36

0.07

<.001

[-0.49, -0.22]

Partner effect

-0.22

0.10

.02

[-0.41, -0.03]

-0.22

0.10

0.02

[-0.41, -0.03]

Wife

Intercept

109.04

0.59

<.001

[107.88, 110.2]

48.88

15.26

<.001

[18.77, 78.99]

Actor effect

0.68

0.10

<.001

[0.49, 0.87]

-0.32

0.10

<.001

[-0.51, -0.12]

Partner effect

-0.14

0.07

.04

[-0.27, -0.10]

-0.14

0.07

0.04

[-0.27, -0.07]

Case C

Husband

Intercept

57.51

0.49

<.001

[56.54, 58.48]

13.33

1.066

<.001

[11.24, 15.42]

Actor effect

0.81

0.01

<.001

[0.79, 0.93]

-0.19

0.01

<.001

[-0.21, -0.17]

Partner effect

0.05

0.01

<.001

[0.03, 0.07]

0.05

0.01

<.001

[0.03, 0.07]

Wife

Intercept

51.33

0.49

<.001

[50.36, 52.30]

9.28

0.01

<.001

[7.19, 11.37]

Actor effect

0.80

0.01

<.001

[0.78, 0.82]

-0.20

0.01

<.001

[-0.22, -.18]

Partner effect

0.08

0.01

<.001

[0.06, 0.11]

0.08

0.01

<.001

[0.06, 0.11]

GRAPHIC METHODS FOR RM-APIM

30

Table 3.

Repeated measures Actor-Partner Interdependence Model results of change in heart rate using multilevel modeling (N=59 dyads)

Husband

Intercept

71.68

1.65

<.001

[68.50, 74.86]

89.97

4.96

<.001

[80.25, 99.69]

Actor effect

-0.19

.04

<.001

[-0.27, -0.11]

-1.19

.04

<.001

[-1.27, -1.11]

Partner effect

-0.09

.05

.083

[-0.19, 0.01]

-0.09

0.05

.083

[-0.19, 0.01]

Wife

Intercept

74.61

1.72

<.001

[71.24, 77.98]

85.53

7.84

<.001

[70.16, 100.90]

Actor effect

-0.04

.09

.683

[-0.22, 0.14]

-1.04

0.09

<.001

[-1.22, -0.86]

Partner effect

-0.12

.05

<.05

[-0.22, -0.02]

-0.12

0.05

<0.05

[-0.22, -0.02]

Standard RM-APIM (MLM)

Change-as-outcome RM-APIM (MLM)

B

SE B

p-value

95% CI

B

SE B

p-value

95% CI

GRAPHIC METHODS FOR RM-APIM

31

Figure 1. Path diagram representation of RM-APIM. The solid lines represent actor effects and the dotted lines represent partner

effects. Single-headed arrows represent regression paths and double-headed arrows represent covariance. eH and eW represent residual

errors for husband and wife. Satisfaction T1 = Satisfaction at Time 1. Satisfaction T2 = Satisfaction at Time 2.

Husband SatisfactionT1

Wife SatisfactionT1

Wife SatisfactionT2

Husband SatisfactionT2

Path 1

eH

ew

Path 4

GRAPHIC METHODS FOR RM-APIM

32

Figure 2. Vector field diagrams of single-dyad repeated measures APIMs in SEM

Case B

Case C

Case A

GRAPHIC METHODS FOR RM-APIM

33

Figure 3. Vector field diagram of multilevel repeated measures APIM (N=59 couples)

Husband Heart Rate

Wife Heart Rate

GRAPHIC METHODS FOR RM-APIM

34

Table 4

Simulated pairwise dataset for Repeated Measures APIM and APIM for VFD with multilevel modeling

Variables used in RM APIM

Variables used in change-as-outcome RM-APIM

Dyad

Person

Husband

Wife

Time

HR

HRActHHsc1

HRPartHWsc

HRActWWsc

HRPartWHsc

HRchng

HRActHH

HRPartHW

HRActWW

HRPartWH

1

1

0

1

1

111.75

---

---

---

---

---

---

---

---

---

1

2

1

0

1

104.00

---

---

---

---

---

---

---

---

---

1

1

0

1

2

108.99

0

0

-0.41

11.68

-2.76

0

0

111.75

104.00

1

2

1

0

2

105.50

11.68

-0.41

0

0

1.51

104.00

111.75

0

0

1

1

0

1

3

105.80

0

0

-3.17

13.18

-3.2

0

0

108.99

105.50

1

2

1

0

3

104.39

13.18

-3.17

0

0

-1.11

105.5

108.99

0

0

1

1

0

1

4

108.33

0

0

-6.37

12.07

2.53

0

0

105.80

104.39

1

2

1

0

4

103.33

12.07

-6.37

0

0

-1.06

104.39

105.80

0

0

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

2

1

0

1

1

95.50

---

---

---

---

---

---

---

---

---

2

2

1

0

1

80.68

---

---

---

---

---

---

---

---

---

2

1

0

1

2

96.75

0

0

-6.47

-4.76

1.26

0

0

95.5

80.68

2

2

1

0

2

82.50

-4.76

-6.47

0

0

1.82

80.68

95.50

0

0

2

1

0

1

3

91.50

0

0

-5.21

-2.94

-5.25

0

0

96.75

82.50

2

2

1

0

3

81.75

-2.94

-5.21

0

0

-0.75

82.50

96.75

0

0

2

1

0

1

4

95.50

0

0

-10.46

-3.69

4

0

0

91.5

81.75

2

2

1

0

4

79.75

-3.69

-10.46

0

0

-2.01

81.75

91.50

0

0

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

3

1

0

1

1

117.40

---

---

---

---

---

---

---

---

---

3

2

1

0

1

99.01

---

---

---

---

---

---

---

---

---

3

1

0

1

2

121.66

0

0

8.19

-3.12

4.26

0

0

117.4

99.01

3

2

1

0

2

91.00

-3.12

8.19

0

0

-8

99.01

117.4

0

0

3

1

0

1

3

116.75

0

0

12.44

-11.12

-4.92

0

0

121.66

91.00

3

2

1

0

3

91.75

-11.12

12.44

0

0

0.74

91.00

121.66

0

0

3

1

0

1

4

113.75

0

0

7.53

-10.37

-3

0

0

116.75

91.75

3

2

1

0

4

96.50

-10.37

7.53

0

0

4.76

91.75

116.75

0

0

…

…

…

…

…

…

…

…

…

…

…

…

…

…

…

1 Note: sc = Sex-mean centered, consistent with our centering choice for our multilevel example

GRAPHIC METHODS FOR RM-APIM

35

Appendix: R code to generate Vector Field Diagrams using the pracma package

# Load Packages -----------------------------------------------------------

#Generation of the vectorfield plots requires installation and loading of the pracma package in R.

#After the first installation, the require function must be used each time R is restarted.

install.packages("pracma")

require(pracma)

# APIM Model Coefficients -------------------------------------------------

#This part of the code creates variables for the model coeficients based on APIM output

#husband and wife are the intercept values

#h_act and h_part are the husband's actor and partner effects

#w_act and w_part are the wife's actor and partner effects

husband<-17.68388

h_act<--.268861

h_part<-.062908

wife<-39.56347

w_act<--.4164373

w_part<-.0773451

# Generate Vectorfield Plots ----------------------------------------------

#xRange and yRange are variables that determine the possible range of values for X and Y

#The first and second values in the parentheses represent the min and max values

#The third value in the parentheses is the number of values of x or y to iterate by

#Changing the third value allows for increasing/decreasing the density of vectors in the plots

xRange <- seq(40,160,10)

yRange<-seq(40,160,10)

#temp generates two matrices using the values specified in xRange and yRange

temp <- meshgrid(xRange,yRange)

#u and v create the predicted values for APIM equations using each of the specified x and y

values

u <- husband + h_act*temp$X + h_part*temp$Y

v <- wife + w_act*temp$Y + w_part*temp$X

#Generates a blank plot with the specified x and y range and appropriate axis labels

plot(range(xRange),range(yRange),type="n",xlab="Husband",ylab="Wife")

GRAPHIC METHODS FOR RM-APIM

36

#Populates the plot with vectors based on the values in the x, y range as well as those generated

for u and v

#scale (adjusts the length of arrows), length (adjust length of arrow edged)

#and angle(adjusts angle between shaft and edge of arrow) can all be modified

quiver(temp$X,temp$Y,u,v,scale=0.5,length=0.05,angle=1)

GRAPHIC METHODS FOR RM-APIM

37

A preview of this full-text is provided by American Psychological Association.

Content available from Journal of Family Psychology

This content is subject to copyright. Terms and conditions apply.