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Personal Relationships , (2011). Printed in the United States of America.Copyright © 2011 IARR; DOI: 10.1111/j.1475-6811.2011.01371.x Extending the actor–partner interdependence modelfor binary outcomes: A multilevel logistic approach SETH M. SPAIN, a JOSHUA J. JACKSON, b AND GRANT W. EDMONDS ca University of Nebraska, Lincoln; b University of Illinois at Urbana-Champaign; c Oregon Research Institute Abstract Relationship researchers face many challenges when samples consist of dyads. The actor–partner interdependencemodel provides a framework for the analysis of dyadic data (D. A. Kashy & D. A. Kenny, 2000). Binary variableslike discrete health diagnoses present additional challenges that are not easily handled using existing models. Theanalysis was demonstrated using SAS PROC GLIMMIX and HLM6. An example is presented where couple’spersonality traits are used to predict discrete health outcomes. A series of Monte Carlo simulations were performedto compare PROC GLIMMIX to HLM. GLIMMIX performed acceptably compared to HLM for small dyad-levelsample sizes (100 or fewer), but HLM clearly outperformed in larger samples. Both had problems estimating thevariance of the random intercept and its standard error. Relationship researchers face many data ana-lytic challenges. The nature of the researchrequires the use of samples such as datingor married couples, in which the observa-tions are not necessarily statistically indepen-dent of one another and thus may violatethe assumptions of standard analysis practices.There are several frameworks to overcomethese difficulties and properly analyze dyadicdata, particularly the actor–partner interde-pendence model (APIM; e.g., Kenny, Kashy,& Cook, 2006). The APIM combines a con-ceptual framework for studying two-person Seth M. Spain, Department of Management, Universityof Nebraska, Lincoln; Joshua J. Jackson, Department of Psychology, University of Illinois at Urbana-Champaign;Grant W. Edmonds, Oregon Research Institute.SAS syntax for the Monte Carlo simulation run in thisstudy is available from S.M.S. The authors would liketo thank Serena Wee, Chris Nye, Paige Deckert, EmilyLove, and Brent Roberts for comments on an earlier draftof this article. The authors also would like to thank JacquiSmith for insight on and assistance with the HRS data.Correspondence should be addressed to Seth M.Spain, Institute for Innovative Leadership, University of Nebraska, Lincoln, 1240 R St., Lincoln, NE 68588-0497,e-mail:
[email protected]. interpersonal relationships and the statisti-cal machinery to analyze those relationships(Cook & Kenny, 2005). We can then accu-rately model the effects that both partners ina dyad have on each other’s outcomes.Additional complications arise when theoutcomes under study are binary and specificto a single person in the dyad. Outcomes suchas the incidence of disease diagnosis, death,retirement, or infidelity that can occur to oneor both partners without necessarily occurringto the other are difficult to analyze within cur-rent frameworks. As an example, the researchmay be interested in whether a participantengaged in a particular behavior. This articlepresents an extension of the APIM to dealwith the analysis of dichotomous outcomevariables such as these. Specifically, we eval-uate methods for addressing dichotomous out-comes where each dyad member’s outcomestate may be correlated with their partner’s,but does not depend on it entirely. The tech-niques given the most attention in this articleare like those listed above; we do not spendmuch time considering variables such asdivorce or childbirth that, by definition, occur1 2 S. M. Spain, J. J. Jackson, and G. W. Edmonds at the dyad level of analysis. To facilitateour discussion of how best to analyze binaryand member specific outcomes, we provide anexample where we predict the occurrence of diagnosis with cancer and psychiatric disorderfor each member in a married couple.First, we will outline the logic and themathematics of single and multilevel logisticmodels that allow for the extension of APIMfor dichotomous outcomes. An example of how to analyze such data will then be pro-vided using the SAS system with PROCGLIMMIX and the program HLM6. Wedemonstrate this example by examiningwhether the diagnosis of cancer or psychiatricproblems can be predicted from personalitytraits from both members of a married couple.Finally, we compare the parameter estimatesof these two approaches—PROC GLIMMIXand HLM6—in a Monte Carlo simulation todetermine approximately what sample sizesare necessary to provide accurate results. Wecompare these two methods, as they are prob-ably the most accessible routines available tothe widest number of relationship researchers.We expect that most relationship researcherswho will need to fit these models will befamiliar with one of these tools. There areseveral other statistical packages that can fitthese models, and we provide references inthe discussion.The goal is to provide the researcher witheffective and easy ways to analyze dichoto-mous outcomes such as mortality, retirement,mental health outcomes, specific disease diag-nosis, or infidelity, when data are nested indyads. In particular, we address the analysisof data where the dichotomous outcomes arenot time graded, for instance, when analyz-ing incidences of cancer when only the diag-noses have been recorded, but not the timewhen diagnoses occurred. Data that are dis-tinguished by time can be handled by meth-ods such as survival analysis (see Harrison,2002; Kenny et al., 2006). The methods pre-sented here may be particularly useful for theanalysis of archival data or when time dataare not available.We also focus on distinguishable dyads,such as opposite-sex couples, as opposedto indistinguishable dyads, such as same-sexroommates. For example, marriage has beenimplicated as a factor predicting reducedcriminal recidivism (Collins, 2010; Sampson,Laub, & Wimer, 2006). Recidivism is adichotomous outcome that either or bothpartners in a union could experience. Themodel we describe would allow researchersto account for possible dependency betweenpartners in the outcome, while evaluatingfeatures of individuals and their spouses onrecidivism. The model allows researchers toevaluate both actor effects and partner effects.That is, researchers can examine hypothesessuch as: Are more hostile individuals morelikely to experience recidivism (actor effect),and does his or her spouse’s hostility levelhave an additional effect on their recidivism,as well (partner effect)? Issues with dyadic data structures We will review the statistical problems withdyadic data briefly, as they are describedin many places (e.g., Campbell & Kashy,2002; Kashy & Kenny, 2000; McMahon,Pouget, & Tortu, 2006). The problems asso-ciated with dyadic data stem from the viola-tion of a fundamental assumption in standardstatistical analysis: that each observation isstatistically independent of the other observa-tions. This assumption is frequently violatedin relationships research, as relationship part-ners may select one another based partly onsimilar characteristics and also influence oneanother’s choices and behaviors over time.This may lead to relationship partners havinga greater likelihood to share outcomes, suchas developing cancer or suffering recidivism,than two randomly selected participants.To address research questions regardingclose relationships, a system for the designand analysis of studies with couples has beenproposed, the APIM (Kashy & Kenny, 2000;Kenny, 1996; Kenny & Cook, 1999). TheAPIM is a multilevel model, with the indi-vidual as Level 1 and the dyad as Level 2.The key features of the APIM is that theindividual-within-dyad is the unit of analysisand that an individual’s score on the inde-pendent variable affects not only his or herdependent variable score ( actor effect ) but Logistic models for dyads 3 also his or her partner’s dependent variablescore ( partner effect ; Campbell & Kashy,2002). Using the APIM, both actor and part-ner effects can be described. For instance, indescribing the relationship between personal-ity and subjective health, the actor’s person-ality may influence his or her health (actoreffect), whereas their partner’s personalitymay also impact the actor’s health (partnereffect; Roberts, Smith, Jackson, & Edmonds,2009). In addition, multiplicative and othernonlinear interaction terms can be modeled.For example, does the absolute differencebetween actor and partner on the trait matterover and above their actual trait standings? Werefer readers to Campbell and Kashy (2002)for an excellent overview of APIM for theanalysis of continuous dependent variables. Dichotomous dependent variables Complications arise when outcome variablesdo not follow a normal distribution such aswhen predicting discrete health outcomes likethe diagnosis of cancer. Standard linear mod-els fitted to such data will produce inaccurateresults. Many dependent variables of interestto relationship researchers fall into this cat-egory, such as retirement, health diagnoses,and committing infidelity. When participantsare not clustered into groups or dyads, the rec-ommendation for handling binary outcomesis to use logistic regression. Logistic regres-sion is an instance of the generalized linearmodel (Nelder & Wedderbrun, 1972). In gen-eralized linear models, the dependent variableis transformed using a link function, and thetransformed variable is linearly regressed on aset of predictors. The general linear model canbe formulated as a special case of the general-ized linear model, using the unit link function,which does not transform the linear predictorin any way.For binary dependent variables, the logit or log-odds link function is most frequentlyused. In the empirical example consideredin this article, we sought to predict discretehealth diagnoses (e.g., “Have you ever beendiagnosed with any type of cancer? Yes/No.”)using personality traits. Here the outcome iseach participant’s discrete diagnosis. In the 0.0 − 3 − 2 − 1 0 1 2 3 0.2 0.4 0.6p0.8 1.0 l o g i t ( p ) Figure 1. The logit function.logit transformation, this 1 or 0 outcome istransformed first into odds (Pr( Y = 1)/Pr( Y = 0)), and then the natural logarithm of theodds is calculated. The logit transformation isuseful for two reasons. First, although prob-abilities have a lower bound of 0 and anupper bound of 1, odds have no upper bound.Small increases in probability will translateinto large changes in odds. Taking the loga-rithm of the odds then eliminates the lowerbound, as odds between 0 and 1 will becomenegative. The logit function traces the curvepresented in Figure 1—notice that the proba-bilities range between 0 and 1, but the logit-transformed probabilities range from −∞ to ∞ . The probit function may be used as analternative (and very similar) transformation.Historically, lack of computational power pro-hibited general use of the probit function, butit is now a viable alternative.The logit transform effectively makes the1/0 outcome behave like a continuous vari-able, and we can linearly regress the log-oddson a set of predictor variables: η i = β 0 + β 1 x i (1)where η i is participant i ’s log-odds of theoutcome, β 0 and β 1 are the intercept andslope of the regression equation, x i is par-ticipant i ’s score on an independent vari-able predicting the outcome. β 0 is the valueof η i when the predictor x i is equal tozero. The usual residual term, r i , is omit-ted from Equation 2, as its variance followsdirectly from the success probability, suchthat σ 2 (r i ) = p i ( 1 − p i ) , and is assumed tobe fixed under the log-odds transformation 4 S. M. Spain, J. J. Jackson, and G. W. Edmonds (Snijders & Bosker, 1999). The error variancefor logistic regression has variance of π 2 /3,so it is fixed at approximately 3.29 (Aldrich& Nelson, 1984). It is possible for binomi-ally distributed data to show variance greateror less than that implied by the logistic func-tion (overdispersion or underdispersion; Gel-man & Hill, 2006, p. 116), and overdisper-sion is common when logistic regression isapplied to count data. Furthermore, overdis-persion may also occur in other generalizedlinear models (such as Poisson regression).Overdispersed models can be handled by esti-mating dispersion parameters using beta bino-mial and negative binomial distributions asthe link functions for logistic and Poissonregression models, respectively. Overdisper-sion is common for most count variables, inpractice (Lambert & Roeder, 1995). Lambertand Roeder (1995) provide a variety of testsfor overdispersion. Underdispersion is muchless common in practice. An additional optionfor handling overdispersion is to construct amultilevel model with a random effect at theunit level, such as the model presented inthis article. We also note that truly dichoto-mous dependent variables, in the absence of higher level nesting, can only have dispersionparameters equal to 1.0 (i.e., no overdisper-sion or underdispersion; Gelman & Hill, 2006,p. 302).The model of Equation 1 could be usedfor outcomes that affect both members of thedyad, such as divorce. In such a case, the sub-script i could be used to refer to the dyad.If using, say conscientiousness to predict theoutcome, we could add a predictor, say z i to Equation 1, with x i representing husbands’levels of conscientiousness, and z i represent-ing wives’ levels of conscientiousness (i.e., η i = β 0 + β 1 *Actor_conscientiousness + β 2 *Partner_conscientiousness). This approachwould then model the probability of a coupledivorcing, given both the husband’s and thewife’s standings on conscientiousness. 1 1. The model of Equation 1 can be fit with logisticregression methods, such as SAS PROC LOGISTIC. Combining multilevel and logistic approaches A multilevel approach is required if both part-ner’s diagnosis is used as an outcome. Thissection explains the multilevel logistic model,which is essentially the same as that presentedin the last section, but takes the dependencybetween dyad members into account. Thisapproach allows the data analyst to explic-itly model the correlation between the twomembers of a dyad while accounting for thedistributional properties of binary dependentvariables. This is accomplished by allowingfor random effects , or variability in modelparameters between dyads.We consider primarily the case of distin-guishable dyads, where there is a meaningfulvariable that differentiates the dyad members,such as sex in the case of opposite-sex mar-riages or supervisor–subordinate dyads. Themodel can be extended for indistinguishabledyads, such as same-sex friends or partnerswith equal status in a business venture, andwe will give some advice regarding this for-mulation later.As with the single-level case, we start withthe log-odds transformation, η ij , adding the j -subscript to account for dyad membership.In this formulation, i represents individual-within-dyad (e.g., 1 for husband and 2 forwife), and j represents dyad membership(e.g., 1 , 2 , 3 ,... ,J) . The multilevel structureof the data can be described using threeequations, starting with the basic level onelogistic regression as in Equation 1, but withthe added j subscript to indicate which dyadthe particular respondent belongs to: η ij = β 0 j + β 1 j x ij + β 2 j z ij (2)where β 0 j is the intercept term for dyad j and β 1 j is the slope of η ij on x ij and β 2 j is theslope of η ij on z ij for dyad j . If x ij is the indi-vidual’s score on X and z ij is their spouse’sscore on the same variable, we would referto β 1 j as the “actor effect” and β 2j as the“partner effect.” In the empirical example, thisis the regression of the log-odds of canceron self-neuroticism and spouse neuroticism,respectively.The intercept term from Equation 2 can beexpanded to include fixed and random effects Logistic models for dyads 5 using the Level 2 equation: β 0 j = γ 00 + U 0 j (3)where γ 00 is the average intercept over all J dyads and U 0 j is the Level 2 residual,conceptualized as the unique effect of dyad j on the intercept. This is the model we use inthe empirical example below, which containedno Level 2 predictors. We could includeLevel 2 predictors, which would be dyad-level variables, for example, the durationof the marriage, by incorporating them intoEquation 3 as β 0 j = γ 00 + γ 01 w j + U 0 .Models with both random intercepts andrandom slopes are generally underidentified inmultilevel modeling with small cluster sizes,such as the case with dyads. These modelshave too many parameters to estimate giventhe number of covariance elements in the data(Newsom, 2002). Therefore, we assume thatthe Level 2 slopes are fixed: β 1 j = γ 10 β 2 j = γ 20 (4)If we wanted to expand the model of Equation 2 to include other predictors, suchas preexisting health conditions, each addi-tional slope, B nj , would have a similar modelas β 1 j and β 2 j in Equation 4. We then substi-tute the Level 2 equations, using Equation 3for the intercept and Equation 4 for the slopes,into the Level 1 equation to yield the logisticmixed model: η ij = γ 00 + γ 10 x ij + γ 20 z i j + U 0 j (5)Only U 0 j is a random effect in this modelformulation, indicating that this model onlyhas a random intercept, while the slopesare all fixed effects. This means that theintercept term is allowed to vary betweendyads but that the slopes for x ij and z ij are not.Essentially, the regressions of the log-oddsonto the predictors are the same for all groups,but that the intercept, basically the “mean”odds for each dyad, can differ between groups.For instance, the weight of actor’s neuroticismfor predicting cancer risk must be the same forall groups, but an individual couple may havea higher baseline risk. The mixed model of Equation 5 is the basis for what will be fit todata using the programs SAS and HLM6. Estimating the parameters of the logisticmixed model Several procedures in the SAS system allowfor nonlinear models, such as PROCsNLMIXED, NLINMIXED, and GLIMMIX.The use of PROC NLMIXED for dyadic dataanalysis is detailed in McMahon and col-leagues (2006). NLMIXED is very flexibleand can be used to specify some nonstan-dard models. However, syntax for NLMIXEDis cumbersome and the procedure requiresstarting values for the model parameters,usually output from a generalized estimat-ing equations approach (GEE), such as SASPROC GENMOD or the GEE procedure inSPSS. GEEs are used to estimate mean effectswhen unknown dependency exists in thedependent variable (Hardin & Hilbe, 2003).GEEs may be useful when the researcher hasno interest in the form of the correlation inoutcomes for dyads, and simply wish to con-trol for nonindependence as a nuisance factor.PROC GLIMMIX and HLM6 can be used tospecify forms of nonindependence and can beused when the researcher wishes to model theassociation in a particular way. Empirical Example Sample and procedure We provide an example of multilevel logis-tic analysis using data from the Health andRetirement Study (HRS). The HRS is anationally representative longitudinal study of Americans age 50 or older and their spouses.The data analyzed here were collected in theeighth wave of assessment (collected in 2006).The HRS consists of respondents selectedusing a multistage probability design andtheir household. Within each household, thespouse or partner of the sample respondentwas interviewed regardless of the spouse orpartner’s age.For the eighth wave of assessment in thelongitudinal study, about half of the total HRSsample was visited for in-person interviews.
