Comparison of Nomothetic Versus Idiographic-Oriented Methods for Making Predictions About Distal Outcomes From Time Series Data

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1 Multivariate Behavioral Research, 48: , 2013 Copyright Taylor & Francis Group, LLC ISSN: print/ online DOI: / Comparison of Nomothetic Versus Idiographic-Oriented Methods for Making Predictions About Distal Outcomes From Time Series Data Laura Castro-Schilo and Emilio Ferrer University of California, Davis We illustrate the idiographic/nomothetic debate by comparing 3 approaches to using daily self-report data on affect for predicting relationship quality and breakup. The 3 approaches included (a) the first day in the series of daily data; (b) the mean and variability of the daily series; and (c) parameters from dynamic factor analysis, a statistical model that uses all measurement occasions to estimate the structure and dynamics of the data. Our results indicated that data from the first measurement occasion does not provide information about the couples relationship quality or breakup 1 to 2 years later. The mean and variability of the time series, however, were more informative: females average positive and negative affect across time was related to relationship quality, whereas males variability in negative affect across time was predictive of breakup. The dynamic factor analysis, in turn, allowed us to extract information central to the dyadic dynamics. This information proved useful to predict relationship quality but not breakup. The importance of examining intraindividual variability and couple dynamics is highlighted. In psychological research, it is common practice to administer a series of measures to a sample of individuals at a single timepoint. These data are then used to predict important outcomes, which typically are also measured on one instance. If the interest is on individual processes, however, the validity of findings derived from such data is questionable. The main assumption in these types of analyses Correspondence concerning this article should be addressed to Laura Castro-Schilo, Department of Psychology, University of California, One Shields Avenue, Davis, CA castro@ucdavis.edu 175

2 176 CASTRO-SCHILO AND FERRER is that a snapshot of behavior from a group of people at a particular timepoint is enough to characterize individual processes. There has been much discussion about the shortcomings of this so-called nomothetic approach in psychological research (Epstein, 1994; Hamaker, Dolan, & Molenaar, 2005; Molenaar, 2004; Nesselroade & Ford, 1985). Particularly, advocates of individual-level analyses have argued that for many of the most interesting aspects of humans, multivariate, multioccasion, multiperson measurement arrangements are likely to be essential (Nesselroade & Ford, 1985, p. 58). This argument rests on the fact that human behavior is complex, and thus, complex methods must be pursued to capture such complexity. Other researchers affirm that the trajectories of groups of individuals are not likely to characterize any particular pattern of intraindividual variability in the population. For example, Molenaar (2004) proved that the covariance structure of a one-factor model that has fixed loadings (i.e., same association between the factor and the corresponding manifest variable for each individual in the sample) is indistinguishable from that of a one-factor model with random loadings (i.e., loadings are freely estimated for each individual in the sample). This implies that the loadings from a one-factor model with fixed loadings might fall far from the loadings that any particular person in the sample would have, if they had been estimated. Supporters of an idiographic approach to psychological research recognize the challenge of generalizing the results from an individual-level analysis to a group of people, and thus, recommend repeated single-subject designs to gather information that applies to multiple individuals (Jones, 2007; Nesselroade & Ford, 1985). One way to investigate the potential benefits of analyses incorporating multivariate, multioccasion, and multiperson data to the study of psychological processes is to use empirical data to compare nomothetic versus idiographic-oriented methods. In doing so, one could identify the degree of information revealed from each approach. In this article we focus on models that incorporate data from multiple measurements at multiple timepoints for multiple individuals. The data for these models consist of daily self-reports from couples involved in romantic relationships. The core questions that we attempt to shed light on are as follows: Can we predict relationship quality or breakup equally from a one-timepoint assessment compared with a series of daily assessments? Is there an improvement in our understanding of affective processes and how these lead to relationship quality and breakup in relationships when we consider multiple timepoints of measurement? We begin with a historical discussion of the nomothetic and idiographic perspectives in psychological research followed by a brief literature review on relationship quality and dissolution. Then, we describe potential approaches for making predictions about relevant outcomes (in our example, relationship quality and breakup). We describe the Dynamic Autoregressive Factor Score

3 COMPARISON OF METHODS FOR MAKING PREDICTIONS 177 (DAFS) model, a statistical model tailored to the study of individual-level data. After detailing our methodology, we compare the results of three different prediction approaches: (a) using one-timepoint reports of positive and negative affect as predictors, (b) using the mean and standard deviation of positive and negative affect of all the available timepoints as predictors, and (c) using the resulting dynamic parameters from individual-level DAFS models with positive and negative affect as predictors. Although we apply these models to one unit of data (i.e., a dyad) at a time, we show an approach to extract the information from each DAFS model and submit it to a multiple regression analysis as a secondary step. We consider this third approach idiographic-oriented as it considers the dynamics of each unit across time while also providing group-level estimates of prediction. 1 Moreover, we discuss unresolved issues that arise when using information from individuals to make inferences about groups. We conclude the article highlighting the benefits and shortcomings of each approach. NOMOTHETIC AND IDIOGRAPHIC PERSPECTIVES IN PSYCHOLOGICAL RESEARCH In their quest for knowledge of reality, the empirical sciences either seek the general in the form of the law of nature or the particular in the form of the historically defined structure. On the one hand, they are concerned with the form which invariably remains constant. On the other hand, they are concerned with the unique, immanently defined content of the real event : : : scientific thought is nomothetic in the former case and idiographic in the latter case. (Windelband, 1894/1980, p. 175) In his rectorial address at the University of Strasbourg, Windelband (1894/1980), a German philosopher, first coined the terms nomothetic and idiographic to refer to the methodologies that different disciplines employed at the time. When discussing the place of psychology within these alternative perspectives, Windelband believed that it fell unambiguously within the nomothetic bounds. Stern (1911) reintroduced the nomothetic and idiographic terms in Germany in an attempt to organize psychological science around the individual. With a similar goal, Allport (1937) acquainted American psychologists with the nomothetic and idiographic terms to encourage the integration of idiographic methodology into psychological inquiry. The topic was controversial then and continues to 1 We purposely use the term idiographic-oriented to indicate that although this approach is based on modeling the individual unit of analysis, certain assumptions (described in the Discussion section) about these individual units are made to arrive at group-level estimates. Thus, this third approach is not idiographic in the purest sense of the word.

4 178 CASTRO-SCHILO AND FERRER be to the present day. Researchers have taken multiple stands on this issue, some arguing that idiographic approaches represent an antiscience point of view (Nunnally, 1967, p. 472), others arguing that findings from nomothetic methods are simply fundamentally inadequate (Lamiell, 1981, p. 276) to inform the understanding of individual experience, and yet others taking a less extreme position affirm that nomothetic-based knowledge can be used to craft hypotheses for idiographic research (Runyan, 1983). Kluckhohn and Murray s (1953, p. 53) classic statement Every man [sic] is in certain respects (a) like all other men, (b) like some other men, (c) like no other man was quoted by Runyan (1983) to suggest three levels of inquiry; he argued that personality psychology was charged with the task of understanding phenomena that apply to all individuals, to some individuals, and to only one individual. In this sense, nomothetic and idiographic approaches should be employed synergistically to advance psychological science. The nomothetic-idiographic debate is ongoing, as idiographic supporters believe that the focus in psychology is still primarily nomothetic (e.g., see Molenaar, 2004). Although idiographic supporters acknowledge the need for general laws, one pressing challenge is finding an optimal approach for combining information from individual-level analyses to make statements about groups of individuals or to arrive to general laws (however, see Runyan, 1983, who argues that idiographic research need not generalize to groups of people or universal laws). Several investigations have focused on this issue, some suggesting individual replications (Jones, 2007; Nesselroade & Ford, 1985); others have employed techniques in which several time series are stacked together and analyzed as one (e.g., Russell, Bryant, & Estrada, 1996) or have pooled time series by identifying equivalent variance-covariance matrices prior to conducting analyses and creating subgroups to which generalizations can be made (Nesselroade & Molenaar, 1999); yet others have opted for first carrying out the analyses and then identifying subgroups using cluster analysis (e.g., Hoeppner, Goodwin, Velicer, Mooney, & Hatsukami, 2008). In this article, we compare idiographic and nomothetic approaches. We conduct analyses at the dyad level (our unit of analysis) and take the results from these idiographic analyses to make nomothetic inferences. The analyses use information from couples daily affect and make predictions about future relationship quality and breakup. To put these analyses into context, we provide a brief review on romantic relationships quality and dissolution. AFFECT, RELATIONSHIP QUALITY, AND DISSOLUTION IN ROMANTIC RELATIONSHIPS Theoretical accounts of romantic relationships place great emphasis on the interdependence within dyads (e.g., interdependence theory; Kelley & Thibaut,

5 COMPARISON OF METHODS FOR MAKING PREDICTIONS ). However, a large body of literature on romantic relationships ignores the fundamental fact that romantic partners are a dynamic, interdependent system with mutual influences over time. Instead, some work in this area still treats partners as independent units by examining information from only one individual and relying on cross-sectional data (Karney & Bradbury, 1995). Moreover, findings from this body of work have been largely based on nomothetic methods. Although this methodology might be informative in some way, it disregards longitudinal processes that occur at the individual or dyadic level. The contributions of alternative methods (e.g., longitudinal, cross-sectional, nomothetic, or idiographic) for making predictions about romantic relationships have not been compared in the past. Specifically, the contribution of dynamic information which is gathered from complex models fit to longitudinal data has not been compared with other simpler approaches for predicting relevant outcomes such as relationship quality and dissolution. However, previous work using different methods does show that positive and negative affect are related to the quality and stability of romantic relationships (Caughlin, Huston, & Houts, 2000; Ferrer, Steele, & Hsieh, 2012; Kim, Martin, & Martin, 1989; Watson, Hubbard, & Wiese, 2000), that positive and negative affective dynamics in romantic couples are temporally interdependent (Steele & Ferrer, 2011), and that patterns of intraindividual variability are predictive of future breakup (Ferrer et al., 2012). In sum, these findings suggest that the dynamics of positive and negative affect in intimate relationships are predictive of relationship quality and dissolution. In the current study, we adopt an idiographic-oriented approach for testing this proposition. Our comparisons are intended to examine whether the time series of affect from dyads (using means and variability and, separately, using dynamic information from the time series) are predictive of relationship quality and/or breakup in the future. In addition, we examine the added predictive value of considering these longitudinal approaches over one timepoint of measurement. APPROACHES FOR PREDICTING DISTAL OUTCOMES When data are taken from multiple individuals at one timepoint, the statistical models for making predictions are easy to determine. Depending on the number of variables measured per individual, one might opt for a path analysis with or without latent variables, or simply a regression model. The results gathered from any of these statistical methods would represent a nomothetic approach for making predictions because one parameter would reflect the effect of the predictor on the outcome for every individual in the analysis. When data are highly dimensional, on the other hand, the options are not so clear. For example, Cattell (1946) introduced what he termed the covariation chart (also known as data box ), which consisted of a three-dimensional space with occasions

6 180 CASTRO-SCHILO AND FERRER (or time), variables, and persons (or organisms) in each axis (see Figure 1). He later expanded upon this structure by introducing the Basic Data Relation Matrix, which comprised a 10-dimensional space with persons, focal stimuli, response patterns, environmental backgrounds, observers, and five time variants from each of these (Cattell, 1966). For clarity, we limit our discussion to the original 3-dimensional covariation chart, as cross-sectional data which are most often used in psychological research consist of multiple variables from multiple individuals at one timepoint. The multiple dimensions of Cattell s (1966) data box illustrate the analytical options available for testing hypotheses. If one were to collect data with multiple measurements at multiple timepoints from multiple individuals (see Figure 1A), then a decision could be made about whether nomothetic or idiographic inferences are sought. To submit data like those from Figure 1A to traditional (nomothetic) statistical methods, one would have to ignore one of the three dimensions or aggregate across one dimension. Thus, depending on the dimensions that one chooses to maintain or aggregate across, the resulting data structure could entail multiple variables from one individual taken at multiple FIGURE 1 Alternative data structures. A) Three-dimensional data structure: multiple measurements from multiple people at multiple timepoints. B) Two-dimensional data structure: multiple measurements from one individual at multiple timepoints. C) Twodimensional data structure: multiple measurements from multiple people at one timepoint. D) Two-dimensional data structure: one measurement from multiple individuals at multiple timepoints.

7 COMPARISON OF METHODS FOR MAKING PREDICTIONS 181 timepoints (see Figure 1B), multiple variables taken from multiple people at one timepoint (see Figure 1C), or one variable from multiple people at multiple timepoints (see Figure 1D). Alternatively, multilevel models with time-varying covariates or second-order latent growth curve models could be applied to the three-dimensional data in Figure 1A. These analyses fall within nomothetical and idiographic standards because information about intraindividual and interindividual variability can be obtained. However, the individual trajectories derived from these models are pulled toward the group-based trajectory, which can obscure idiosyncratic patterns in the data. Data such as those depicted in Figure 1B are suited for idiographic methods, as these methods result in estimates that characterize one particular individual. One statistical technique apt for modeling the latter types of data is the DAFS, which is a time series model that uses latent variables. Other possible options include time series analysis without latent variables (Box & Jenkins, 1976) and dynamical systems models such as the damped linear oscillator (Nesselroade & Boker, 1994). As with any other analysis, the choice of model depends on the theoretical question to answer. Data such as those in Figure 1C are apt for nomothetic methods, with estimates that characterize a group of people. R-technique factor analysis, general linear models, and generalized linear models, among others, are techniques that can be applied to these data. Finally, data such as those shown in Figure 1D could, in theory, be submitted to nomothetic or idiographic analyses. Most often, these types of data are analyzed using repeated measures ANOVA or latent growth curve modeling, which are nomothetic approaches, even though the latter can yield information about both intraindividual changes and interindividual differences. Also, if only one person is selected, data such as those in Figure 1D could be analyzed in an idiographic manner, assuming that enough observations across time are available. We believe that a compromise between the nomothetic and idiographic approaches is possible by decomposing a three-dimensional data structure (as in Figure 1A) into multiple two-dimensional data structures (as in Figure 1B) and analyzing them with idiographic methods. This alternative is idiographicoriented in that dynamics unique to the individual are considered prior to aggregating information across people. In the next section we describe one statistical technique suited for idiographic analyses, the DAFS. DYNAMIC AUTOREGRESSIVE FACTOR SCORE MODEL In his endeavor to study the structure of intraindividual personality, Cattell suggested the application of the common factor model to time series data from one

8 182 CASTRO-SCHILO AND FERRER individual (Cattell, Cattell, & Rhymer, 1947). The resulting latent variables from this analysis represent intraindividual variation of the observed variables. A problem with this technique, however, is that it does not take into account the lagged relationships of both observed and latent variables (Anderson, 1963). Lagged relationships are important because they can indicate the degree to which one latent variable influences another latent variable or observed variables across time. Acknowledging the limitations of P-technique, the dynamic factor model was developed in which the lagged covariation among the various repeated measurements was incorporated into the model. One specification of the dynamic factor model is the DAFS (Browne & Nesselroade, 2005; Nesselroade, McArdle, Aggen, & Meyers, 2002). A lag-1 specification of the DAFS in matrix notation takes the following form: y t D F t C t t D B t 1 C d t ; t D 1; 2; : : :; T where y t is a p 1 vector of measurements on p variables at time t; F is a p q matrix of factor loadings that is invariant over time; t is a q 1 vector of factor scores at time t; t is a p 1 vector of unique factors at time t, assuming t N.0; D /; B is a q q matrix of regression weights indicating the influence of the lag-1 common factors on the current factors; and d t is a q 1 vector of residuals of t that could not be explained by the lagged effects, assuming d t N.0; /. In this model, common factors are assumed to be uncorrelated with unique factors. However, unique factors can have an autocorrelational structure. The t 1 term denotes this is a lag-1 model, but additional lagged effects can be included. Thus, this model shows that the influences of prior timepoints take place through the influence of prior factor scores on current factor scores, which, in turn, influence the current observed variables. Intercepts are typically not included in the equations because the DAFS is designed to model the covariations in the data. However, other specifications can account for trends and nonstationarity (Molenaar, De Gooijer, & Schmitz, 1992). The DAFS model is a combination of factor analysis and time series analysis. In the time series literature, there has been discussion about the difficulty entailed in identifying the appropriate model to fit to time series data (Velicer & Molenaar, 2013). However, lag-1 models have been advocated as appropriate for behavioral sciences data (Simonton, 1977), and simulation studies have indeed shown that a lag-1 model provides appropriate results for several different Auto Regressive Integrated Moving Average processes (Harrop & Velicer, 1985). In addition, previous analyses with intensive longitudinal data have shown a lag-1 model is appropriate to characterize daily affective processes (Ferrer & Nesselroade, 2003; Ferrer & Widaman, 2008). Although these findings suggest

9 COMPARISON OF METHODS FOR MAKING PREDICTIONS 183 a lag-1 DAFS model is appropriate for our analyses, we corroborated this assumption with examination of our empirical data, which we describe later. We chose to use the DAFS for our idiographic-oriented approach because of its particular usefulness with data that show consistent fluctuations over time (Browne & Nesselroade, 2005; Ferrer & Zhang, 2009; Song & Ferrer, 2009). Also, this model has been extended to examine affective processes in dyads over time, which allows for the simultaneous investigation of intraindividual and interindividual variability within the dyad (Ferrer & Nesselroade, 2003; Ferrer & Widaman, 2008; Song & Ferrer, 2009, 2012). Moreover, the autoregressive structure of the DAFS is specified at the latent level. That is, the effects of previous timepoints on future timepoints are modeled with variables that are free from measurement error. This feature of the DAFS should make this model preferable over a model that utilizes composite scores instead of latent variables. To verify this claim, we compared the DAFS with an autoregressive lag-1 (AR1) model that has observed (i.e., composite) scores instead of latent variables. Figure 2 shows a lag-1 dyadic DAFS model with one factor and three observed variables, for two individuals (person A and person B), across measurement occasions t and t C 1. In addition to the autoregressive coefficients (i.e., influences within the same person across time), the model depicts cross-lagged coefficients, representing the influence of one person on the other across time. The notion of an autoregressive effect, or lagged effect, has been labeled in the affective literature as inertia (Kuppens, Allen, & Sheeber, 2010; Suls, Green, & Hillis, 1998). That is, high inertia suggests a strong influence of a variable on itself from one timepoint to the next. The notion of a cross-lagged effect, on the other hand, has been conceptualized as reactivity (Suls et al., 1998), for it represents how reactive one variable is to the influence of another variable across time. Multilevel models have been employed with daily diary data (similar to the data used in this investigation) to study, for example, daily intimacy and disclosure in married couples (Laurenceau, Troy, & Carver, 2005) and to identify emotional contagion in couples under stress over time (Thompson & Bolger, 1999). These models have attractive features. For example, they do not require equally spaced data or an equal number of observations across individuals. Moreover, in a multilevel framework, between-level and within-level parameters are estimated simultaneously, describing a common trajectory for the sample and differences in the trajectory across individuals. This feature makes these models more appealing than repeated measures ANOVA, for which only mean trajectories are estimated. However, if one is interested in idiographic approaches, these multilevel models are more limited than models that place all their focus on the individual (or other unit of interest). The primary reason for this is multilevel models have assumptions about the variability across individuals; that is, they assume random effects are normally distributed. As a consequence, if one wanted to obtain empirical Bayes estimates, individuals trajectories would be pushed

10 184 CASTRO-SCHILO AND FERRER FIGURE 2 Lag-1 dyadic Dynamic Autoregressive Factor Score (DAFS) model with three observed variables and one factor. Double-headed arrows in the structural model denote correlations between the latent variable residuals. Although not depicted in this figure, covariances among unique factors across time and persons are specified.

11 COMPARISON OF METHODS FOR MAKING PREDICTIONS 185 toward the group-mean trajectory. The degree of shrinkage toward the fixed effects would depend on the number of data points for a particular individual in the sample and the relative difference in ordinary least squares estimates of such individual in comparison with others in the sample. Although one could avoid getting empirical Bayes estimates by using multilevel models directly in a one-stage procedure, the assumption of normality of the parameters is one that persists. On the other hand, purely idiographic models focus solely on the unit of analysis and its dynamics over time. Parameters from individually run DAFS models do not have distributional assumptions, and as such, we can expect more variability in the parameters, which might result in greater predictive value of distal outcomes. Furthermore, fitting a DAFS model per dyad allows for unique factorial structures across couples, something that could not be accommodated in one multilevel model. Given the aims of our study and the characteristics of our empirical data (i.e., dyadic multivariate time series data, described later), the dyadic DAFS seems ideal for delineating dyadic affective dynamics for each couple over time. We also investigate whether such dynamics carry unique information about each couple by using differences in the dynamics across dyads to predict relationship quality and stability at a later time. Participants and Procedures METHOD Our data are part of a longitudinal project about dyadic interactions. Participants are couples who began the study while they were in a premarital or marital relationship. Advertisements were placed in local newspapers and on the Internet. Individuals could participate only if their partners participated as well. During an intake appointment, participants gave informed consent and completed a questionnaire containing measures related to their relationship, affect, and demographic questions. Upon completion, participants received daily diary packets containing questions about their daily emotional experiences. They were instructed to complete one page each evening for up to 90 days. For our analyses, we considered those couples that had a minimum of 50 days of daily data (N D 197 couples). The couples in our subsample ranged in age from 17 to 74 years (M D 25:08, SD D 10:39) and reported having been in the relationship from 1 month to 54 years (M D 3:39, SD D 6:52). Measures Positive and negative affect. All participants were asked to complete individually a 20-item daily questionnaire about their positive and negative

12 186 CASTRO-SCHILO AND FERRER affect for up to 90 consecutive days (the Positive and Negative Affect Schedule [PANAS]; Watson, Clark, & Tellegen, 1988). Participants answered all items in response to the stem Indicate to what extent you have felt this way today. Each item was rated on a 5-point Likert scale (1 D very slightly or not at all and 5 D extremely). The reliability estimates (coefficient alpha) of the PA and NA subscales of the PANAS at the first occasion were.87 and.86 for males and.85 and.84 for females. Relationship quality and breakup. Between 1 and 2 years after the initial visit, couples returned to the laboratory for a follow-up interview about relationship quality and status. If participants indicated they were no longer together with their previous partners, they were considered broken up. To assess relationship quality, six items from the Perceived Relationship Quality Component Inventory (Fletcher, Simpson, & Thomas, 2000) were used. Sample items included How satisfied are you with your relationship? How committed are you with your relationship? and How intimate are you in your relationship? These items were rated on a 7-point Likert scale (1 D not at all and 7 D extremely) and were averaged into a composite score at the dyad level, resulting in one quality score per couple. The reliability estimates (coefficient alpha) of the relationship quality scale were.92 for females and.95 for males. Data Analysis We proposed three approaches for making predictions about relationship quality and breakup from the affective data. In the first approach, we used information from one measurement occasion. We computed composite scores of PA and NA based on the PANAS reports from the first day of data collection and used these as predictors of relationship quality and breakup. This approach consisted of just using the scores collected on the first measurement occasion for prediction, unlike the second and third approaches (described later), which used information from all measurement occasions, and as such, might capitalize on having higher reliability. 2 In an attempt to further evaluate the first approach in a way not biased by its potential lower reliability, we used the estimated internal consistency (coefficient alpha) of the two PANAS subscales, separately for males and females, to correct each predictor for the effects of measurement error. This correction was performed by fixing the residual variance of each first-occasion predictor at (1 reliability) predictor variance. 2 The interpretation of the first measurement occasion variables as having lower reliability is arguable (see Hertzog & Nesselroade, 1987) because in our case these variables are a measure of state affect, which might be highly variable across time (suggesting low test-retest reliability) but can have high internal consistency.

13 COMPARISON OF METHODS FOR MAKING PREDICTIONS 187 In the second approach, we incorporated data from the entire time series (from 50 to 90 days of measurement) for each couple by computing the mean PA and NA as well as the standard deviation of PA and NA across all daily reports. Then, we used the mean and standard deviation of each individual in the couple as predictors of relationship quality and breakup. The standard deviation was used to represent the variability in affect for each individual within the couple. Using the standard deviation for this purpose has been shown to yield meaningful results (e.g., Eid & Diener, 1999). In the third approach, we also considered all of the time series but we modeled the dynamics across time for each individual couple using a lag-1 dyadic DAFS model with two factors (i.e., PA and NA) per person. We chose a lag-1 model based on the time series literature mentioned before but also based on empirical evidence gathered from examination of the linearly detrended time series autocorrelation function (ACF) and partial autocorrelation function (PACF) plots. We selected a random sample of 20 dyads, computed composite scores of their positive and negative affect detrended time series, and plotted their ACF and PACF. 3 For the most part, the plots suggested a lag-1 or no-lag model. We chose a lag-1 model as an approximation model that could be fit to all dyads, although in very few instances (5 out of the 80 time series) the ACF and PACF plots pointed to a lag-2 model. Inspection of the ACF and PACF plots also suggested stationarity. To reduce the number of observed variables in the DAFS model and improve the psychometric properties of the factors, we grouped the PANAS items to create a total of six parcels (Kishton & Widaman, 1994), three representing positive affect (PA) and three for negative affect (NA). 4 We used these parcels as observed indicators of PA and NA in the dyadic DAFS. To identify the model, we set the variance (or residual variance in the case of endogenous factors) of the latent variables to unity. We carried out the DAFS analyses by running Mplus (Muthén & Muthén, 2010) in batch mode through R (R Development Core Team, 2010). That is, we ran 197 dyadic DAFS models in Mplus and extracted all the standardized dynamic parameters (i.e., the autoregressive and cross-lagged standardized estimates, 16 in total) from the models using R. These parameters represented all possible influences of PA and NA between the 2 individuals in the couples. For example, a female s PA on a particular day could have an influence on her own 3 ACF and PACF plots are available from Laura Castro-Schilo upon request. 4 The parcels for positive affect were created in the following fashion: Parcel 1 D enthuse, interest, strong. Parcel 2 D excited, determined, attentive. Parcel 3 D proud, inspired, alert, active. For negative affect: Parcel 1 D afraid, irritable, hostile. Parcel 2 D distress, nervous, ashamed. Parcel 3 D upset, scared, guilty, jittery. The assignment of items across parcels was based on the domain-representative method for parcel construction put forth by Kishton and Widaman (1994) and entailed factor analyses at the group level.

14 188 CASTRO-SCHILO AND FERRER PA the day after (i.e., lagged/autoregressive parameters) and on her own NA the day after (i.e., cross-lagged parameters within a partner). Similarly, her PA on a particular day could influence her male partner s PA and NA the following day (i.e., cross-lagged parameters across partners). The standardized estimates were saved in a separate data set, which also included the variables from the first and second approach, and information about the couple s status (together or broken up) and relationship quality (the average across both partners). Finally, we used these dynamic parameters from each couple in the third approach as predictors of relationship quality and breakup. Because our goal was to compare the information gathered by each of the three proposed approaches, we used the structural equation modeling framework to run regression models, place restrictions on parameters of interest, and conduct Wald chi-square tests. Models were run in Mplus (Muthén & Muthén, 2010) and missing data were handled with multiple imputation using the Bayesian estimator, which employs the Markov Chain Monte Carlo algorithm based on the Gibbs sampler (Asparouhov & Muthén, 2010a). Missing data were present in the outcome measures (see Table 1) and, to a lesser extent, in the predictors from the DAFS models (10 cases were missing). We chose multiple imputation over other approaches, such as Full Information Maximum Likelihood (FIML), TABLE 1 Descriptive Statistics for the First Timepoint Score, Mean and Variability of Time Series Variables N M SD Min Max First timepoint Males PA Females PA Males NA Females NA Mean and variability of time series Mean of males PA Mean of females PA Mean of males NA Mean of females NA SD of males PA SD of females PA SD of males NA SD of females NA Variables at follow-up Relationship quality Breakup Note. PA D positive affect; NA D negative affect. Each PANAS item was measured on a 5-point Likert scale where 1 D very slightly or not at all and 5 D extremely.

15 COMPARISON OF METHODS FOR MAKING PREDICTIONS 189 because FIML was more computationally intensive, particularly for the models with the categorical outcome (i.e., breakup). We imputed 10 data sets from an unrestricted variance covariance model using all relevant variables (i.e., all predictors and both outcomes). Parameter estimates from the models were averaged across replications, and standard errors computed according to Rubin s rules (1987). The Wald chi-square tests were performed using the estimated asymptotic variance of the parameters (for technical details see Asparouhov & Muthén, 2010b). In the first model, we specified the respective outcome (relationship quality or breakup) as a function of males and females PA and NA from the first measurement occasion. To assess the overall predictive value of these firstoccasion predictors, we compared a model in which all predictors had an effect on the outcome with one in which these predictions were fixed to zero. Following the same logic, we specified a second model with the first-occasion predictors together with the mean and variability of the time series as predictors (i.e., predictors from the second approach). Effects from this latter model were compared with an alternative model in which the effects of the second approach predictors on the outcome were fixed to zero. Finally, we assessed the overall predictive value of the dynamic parameters by fitting a model in which predictors from all three approaches were specified to relate to the corresponding outcome and compared it to a model with the dynamic parameters fixed to zero. Rescaling of variables. To facilitate interpretation of the intercept in our models, we rescaled the predictors. In the case of the first measurement occasion, we subtracted 1 from every value, resulting in values that ranged from 0 4 instead of the original 1 5. The mean and variability for the second approach were rescaled in two different ways: (a) the mean of males and females PA and NA were rescaled such that a value of zero represented the lowest possible level of PA and NA, and (b) the standard deviations of males and females PA and NA were centered such that a value of zero represented the average amount of variability across the sample. There was no need to rescale the dynamic predictors from the DAFS as a meaningful zero already existed in the data: a lack of affective influence. Descriptive Statistics RESULTS Table 1 lists the sample size, mean, standard deviation, minimum, and maximum of the PA and NA scores from the first day of data collection; the means and standard deviations of individuals PA and NA across all days of data collection;

16 190 CASTRO-SCHILO AND FERRER and descriptives of relationship quality and breakup. The means of the variables at the first timepoint show that couples reported experiencing higher levels of PA than NA on the first day of the study, and the standard deviation suggests more variability of PA than NA. The average of the entire time series shows a similar pattern, with slightly lower mean scores of PA and NA than on the first day. With regard to the outcomes at follow-up, relationship quality and breakup were negatively correlated, r pb.136/ D :30, p < :001, suggesting lower relationship quality for those who are more likely to break apart. Relationship quality data were missing from 58 couples and breakup data were missing from 44 couples. We obtained relationship quality data from some of the couples who reported breaking up, as 21 couples notified us of their breakup after having visited our lab for their follow-up, on which they filled out the relationship quality questionnaire. These data were included in all our analyses. As it would be expected, the incidence of breakup is not as high as that of staying together (35 couples reported breaking up), and relationship quality is higher than the middle of the scale (M D 5:88 in a 1 7 scale). Dynamic Factor Analysis Of the 197 DAFS models, 187 converged to a stable solution with estimates within the accepted boundary space. Descriptive statistics for the 16 dynamic parameters are listed in Table 2. As in previous work (Ferrer & Widaman, 2008), these parameters show ample variability across individuals. Estimates representing within-affect autoregressive parameters (e.g., males positive affect regressed on their own positive affect) are positive, with means ranging from.17 to.20, whereas means for other estimates are close to zero. Overall, the fit of the models is generally acceptable, with CFIs ranging from.61 to 1.00.M D :91/ and RMSEAs from 0 to.16.m D :07/. Zero-order correlations between our outcomes of interest and the predictors from all three approaches are presented in Table 3. Based on these coefficients, the mean of females NA, the variability in males and females NA, and the day-to-day influences of females NA to males and females NA, males PA to females PA and males NA to females PA are significantly related to relationship dissolution up to 2 years later. The mean of females PA and the day-to-day influences of females NA to females PA and males NA to females NA are significantly related to relationship quality. Model 1: First Measurement Occasion The predictors from the first regression model did not explain a significant amount of variability in relationship quality, R 2 D :02, p D :43. The Wald test

17 COMPARISON OF METHODS FOR MAKING PREDICTIONS 191 TABLE 2 Descriptive Statistics of Parameters From DAFS Models Variables M SD Min Max Fit indices CFI RMSEA Lagged relations Males PA! Males PA Males NA! Males NA Females PA! Females PA Females NA! Females NA Cross-lagged relations within partners Males NA! Males PA Males PA! Males NA Females NA! Females PA Females PA! Females NA Cross-lagged relations across partners Males PA! Females PA Males NA! Females PA Males PA! Females NA Males NA! Females NA Females PA! Males PA Females NA! Males PA Females PA! Males NA Females NA! Males NA Note. PA D positive affect; NA D negative affect. Each variable represents a vector of regression weights from the Direct Autoregressive Factor Score (DAFS) (e.g., Males NA! Males PA D males positive affect regressed on males negative affect). Descriptives are based on the standardized DAFS parameters. revealed that fixing the first measurement occasion predictors to zero did not worsen the model fit, 2.4/ D 0:72, p D :95. Thus, reports of PA and NA on the first day of the study were not predictive of couples future relationship quality. We followed the same procedure to investigate whether a single measurement of PA and NA was predictive of couples relationship status (i.e., together, breakup) in the future. Results from the first model for breakup suggested that the PA and NA from the first occasion of measurement did not significantly predict breakup, 2.4/ D 1:52, p D :82. We proceeded to run two additional models correcting the first occasion predictors for measurement error. The results from the latter analyses were nearly identical to the previous results. Thus, regardless of measurement error, the first assessment of affect was not predictive of relationship quality or breakup.

18 TABLE 3 Zero-Order Correlations Among First Occasion Scores, Means and Standard Deviation of Time Series, and Distal Outcomes Variable Breakup Relationship quality st time males PA st time males NA st time females PA st time females NA Mean of males PA Mean of males NA Mean of females PA Mean of females NA SD of males PA SD of males NA SD of females PA SD of females NA Note. PA D positive affect; NA D negative affect. Boldface indicates coefficients significant at the.05 alpha level. 192

19 TABLE 3 (Continued) Zero-Order Correlations Among First Occasion Scores, Means and Standard Deviation of Time Series, and Distal Outcomes Variable Males PA! Males PA Males NA! Males PA Males PA! Males NA Males NA! Males NA Females PA! Males PA Females NA! Males PA Females PA! Males NA Females NA! Males NA Females PA! Females PA Females NA! Females PA Females PA! Females NA Females NA! Females NA Males PA! Females PA Males NA! Females PA Males PA! Females NA Males NA! Females NA Note. PA D positive affect; NA D negative affect. Boldface indicates coefficients significant at the.05 alpha level. 193

20 TABLE 3 (Continued) Zero-Order Correlations Among First Occasion Scores, Means and Standard Deviation of Time Series, and Distal Outcomes Variable Males PA! Males PA Males NA! Males PA Males PA! Males NA Males NA! Males NA Females PA! Males PA Females NA! Males PA Females PA! Males NA Females NA! Males NA Females PA! Females PA Females NA! Females PA Females PA! Females NA Females NA! Females NA Males PA! Females PA Males NA! Females PA Males PA! Females NA Males NA! Females NA Note. PA D positive affect; NA D negative affect. Boldface indicates coefficients significant at the.05 alpha level. 194

21 COMPARISON OF METHODS FOR MAKING PREDICTIONS 195 Model 2: Mean and Variability of Time Series In the next step, we focused on the mean and standard deviation of PA and NA from each individual s entire time series. When these variables entered the prediction of relationship quality together with the first-time variables, the Wald chi-square test indicated an overall predictive value, 2.8/ D 26:19, p < :01. The first timepoint predictors, together with the mean and standard deviation of the time series, explained 15% of the variance in relationship quality.r 2 D :15; p < :01/. The regression coefficients from this analysis are presented in Table 4. The model s intercept indicates that the average level of relationship quality reported by those couples with the lowest levels of PA and NA in the first occasion and across the time series and with average variability of affect across their time series, was 5.56 out of a possible 7. This suggests that couples with low levels of PA and NA, and average variability in their PA and NA, reported a relatively high level of relationship quality. Relationship quality increased significantly, b D 0:54, SE D 0:15, p < :01, for every unit increase in the mean of the females PA time series and decreased significantly, b D 0:91, SE D 0:40, p < :05, for every unit increase in the mean of the females NA time series. These results suggest that females mean PA and mean NA of a 3-month span are related to relationship quality 1 to 2 years later. Analyses adjusting for measurement error in the first set of predictors were nearly identical. TABLE 4 Regression Coefficients From the Prediction of Relationship Quality Based on the First Occasion and the Mean and Variability of the Time Series Predictors of Relationship Quality b SE t P Intercept <.01 1st time males PA st time females PA st time males NA st time females NA Mean of males PA Mean of females PA <.01 Mean of males NA Mean of females NA <.05 SD of males PA SD of females PA SD of males NA SD of Females NA R 2 D :15 Note. PA D positive affect; NA D negative affect.

22 196 CASTRO-SCHILO AND FERRER Next, we investigated the unique contribution of the mean and variability of the time series to predict relationship breakup using logistic regression. As before, the model was specified so the predictors from the first and second approach had freely estimated predictive parameters, and a Wald test compared this model against one in which the predictors from the second approach were fixed to zero. The test indicated a significant unique contribution for the prediction of breakup from the mean and variability of the time series, 2.8/ D 20:06, p < :05. The resulting regression weights (in log-odds) and odds ratios are presented in Table 5. The single significant predictor of breakup was the variability in males NA across the time series, b D 5:66, SE D 2:27, p < :05. The odds ratio for this parameter suggests that, for every unit increase in the standard deviation of males NA across time, the odds of breaking up (vs. staying together) increase by a factor of 288. However, one unit increase in the standard deviation is not within the possible values of our data (the mean-centered variable in our sample has a maximum value of 0.53). Dividing the regression weight in half and exponentiating it results in the odds ratio for a half unit increase, which is 17. Thus, the odds of breaking up increase by a factor of 17 for those males who go from having an average amount of variability in NA to having the maximum amount of variability in NA in our sample. Results from the analysis correcting for measurement error in the first-occasion predictors were very similar. In sum, the second set of predictors had a significant contribution for predicting breakup. TABLE 5 Regression Coefficients and Odds Ratios From the Prediction of Breakup Based on the First Occasion and the Mean and Variability of the Time Series Predictors of Breakup b SE t p Odds Ratio Intercept st time males PA st time females PA st time males NA st time females NA Mean of males PA Mean of females PA Mean of males NA Mean of females NA SD of males PA SD of females PA SD of males NA < SD of Females NA Note. PA D positive affect; NA D negative affect. Dependent variable was coded 0 D staying together, 1 D breakup.