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University of Groningen
Generalizations of linear modelling in the biomedical sciencesGill, Nazia
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Chapter 2
The association between positiveand negative affect at the inter- andintra-individual level
Nazia Parveen Gill1, Elisabeth H Bos2, Ernst C. Wit1,and Peter de Jonge2
1 Johann Bernoulli Institute of Mathematics and Computer Science,University of Groningen, The Netherlands
2 University of Groningen, University Medical Center Groningen,Interdisciplinary Center Psychopathology and Emotion regulation, The
Netherlands
ISubmitted for publication
AbstractBackground: It is unclear whether positive affect (PA) and nega-
tive affect (NA) represent a single dimension. The present study aimedto find unbiased estimates of the between- and within-subjects asso-ciation between PA and NA.Methods: 85 participants recorded their PA and NA daily by meansof an electronic diary (mean 38 observations, range 2 to 56). A linearmixed-effects model was applied. The covariances between the ran-dom effects at the person- and measurement level were used to si-multaneously estimate the between- and within-subjects correlationbetween PA and NA.Results: The within- and between-subjects correlation between PAand NA were large: r=-0.56 (95% CI -0.58 to -0.54) and r=-0.52 (95% CI-0.54 to -0.50), respectively. The difference between the correlationswas small but significant (Fisher Z=-3.42; P<0.001). In participants whocompleted ≥ 80% of the measurements (n=46), the within- andbetween-subjects correlation were r=-0.59 (95% CI -0.61 to -0.57) andr=-0.50 (95% CI -0.52 to -0.48), respectively (Z=-6.61; P<0.001).Conclusion: Our study suggests that the correlation between PA andNA is large, especially at the intra-individual level. Within individuals,fluctuations in positive and negative affect are so strongly correlatedthat they may come close to two ends of a single continuum.
Keywords: mixed-effects models, affect, diary, longitudinal, within-subject and between-subject effects.
32 Applications of LMM for PA and NA
2.1 IntroductionMost affective states in humans can be characterized as either pos-
itive or negative [29]. Positive affect (PA) reflects an individual’s levelof pleasurable engagement with the environment. Persons high onPA may feel enthusiastic, energetic, and alert [31]. Conversely, neg-ative affect (NA) has been conceptualized as a dimension of subjectivedistress and unpleasant engagement. It consists of a variety of moodstates including fear, anger, guilt, and distress [31]. Negative affect isstrongly correlated with stress and mental health complaints [30] andplays a role in several of the DSM-defined psychiatric disorders, de-pression in particular (American Psychiatric Association) [1]. There isan ongoing debate whether positive and negative affect are indepen-dent affect dimensions or represent two opposites of a single affectdimension. This debate has not been resolved despite many attempts:several studies have reported high correlations between the two affectstates whereas others found low or even no correlations [19, 32].
A reason for the inconsistency in findingsmay lie in the fact that PAand NA differ not only between individuals but also within individualsover time [6, 28]. Affect is clearly not a constant but may fluctuate, andin fact these temporal affect dynamics have been considered impor-tant in understandingwhy some persons develop psychiatric disordersand others do not [5]. It is therefore important to disaggregate thesetwo sources of variation, because results at the inter-individual levelare not necessarily translatable to the intra-individual level. Between-subjects and within-subjects correlations may be of different magni-tudes or even different directions [6, 8, 33]. In other areas of researchimportant differences have been reported when correlations are inves-tigated at the individual versus the group-aggregate level, leading toquite different conclusions [14, 17, 23].
Theoretically, it may be hypothesized that within a given individ-ual, a change in positive affect will often be accompanied by a changein the opposite direction in negative affect, and vice versa. This doesnot mean, however, that all individuals scoring high on positive af-fect will have low scores on negative affect as well: some individualscan be characterized as having little feelings in general whereas othersmay have many positive as well as many negative feelings. From thisperspective, a difference in correlation between PA and NA at the two
2.2 Method 33
different levels of investigation seems plausible, with stronger (nega-tive) coupling within individuals than between individuals. Only withrepeated assessments of PA and NA in a sample of multiple individ-uals, i.e. with both a sufficient sample size and a sufficient number ofassessment points, we are able to disentangle these two effects. Unfor-tunately, the vast majority of studies on the association between PAand NA have only examined the between-subjects correlation [9]. Al-though some authors did argue that focusing only on inter-individualvariance will lead to false conclusions [11, 20], this practice remainsextremely dominant. To date, only a few studies investigated the PAand NA association at both the between- and within-subjects level[6, 12, 13, 27]. In these studies, estimates for the correlation betweenPA and NA were reported of -0.54 to -0.15 within-subjects and −0.23to 0.26 between-subjects, in daily diary studies of varying sample sizesand assessment points. Unfortunately, in these studies the correlationswere estimated in two separate analyses using product-moment cor-relations, and potential serial dependency was not accounted for. As aresult the reported correlations may be biased. Also, the sample sizesand/or number of assessments in these studies were rather small.
Recent statistical advances have made it possible to disentanglewithin- and between-subject variance in three-dimensional data in asingle model [10, 14, 26]. Three-dimensional data are characterizedby containing persons, variables and data points in a single matrix.Mixed-effect models have become popular to analyze data in whichthe data points are nested in higher-order variables [7]. In a longi-tudinal setting, repeated measurements are nested in individuals [4].In the present study, we have analyzed the association between in-dividuals’ PA and NA in a three-dimensional data set, using a linearmixed-effect model. Our aim was to derive unbiased estimates of thewithin-subjects and between-subjects correlation between the two af-fect measures, and to evaluate whether there would be differences be-tween these two estimates. Specifically, we hypothesized that strongercorrelations between PA and NA would be present when analyzingwithin-subjects variance than between-subjects variance.2.2 Method
Participants and ProcedureThe data were collected during a Mindfulness-Based Stress Reduc-
34 Applications of LMM for PA and NA
tion (MBSR) program in the period January 2010 until June 2012, inTilburg, The Netherlands. Participants were recruited by means of awebsite and advertisements in local newspapers. The MBSR programwas advertised as a group training program aimed at reducing men-tal distress and increasing wellbeing through cultivation of mindful-ness. Participants of the MBSR program were offered a 10% discount iftheywere willing to participate in the diary study. An online screeningwas used to check exclusion criteria for the study. Exclusion criteriawere severe psychiatric disorders and insufficient reading and writingskills. A total of 187 individuals participated in the MBSR program, ofwhom 85 (45%) agreed to fill out the daily diary. These 85 participantswere used in the present analysis. All participants provided writteninformed consent.
The MBSR program followed the manual of the group program de-veloped by Jon Kabat-Zinn [16]. It consisted of 8 sessions of 2.5 hours,over a period of 8 to 9 weeks, a silent retreat of 6 hours, and dailyhomework practice. Participants filled out an online daily question-naire during the entire MBSR program, each day after 5pm. They werepermitted to miss only one entry per week. Before the start of theintervention participants filled out an online pre-treatment question-naire. The diary study period ranged from 3-62 days, with an averageof 46.1 days (SD=12.4). The participants completed on average 38.2measurements (SD=12.9, range 2-56). A total of 3240 observations forPA and NA was included in the present study; 17.1% of the observa-tions wasmissing. These observationsweremissing either because theparticipants did not fill out the diary or filled it out before 5pm or after4am the next day.
Because the diary series of some participants were either very shortperiod or had a lot ofmissing values, we also estimated the correlationsin a sample of the most compliant patients. For this compliant samplewe selected those individuals whose series covered at least 45 days andhad filled out at least 80% of the measurements (n=46, total number ofobservations=2160).2.3 Measures
The pre-treatment questionnaire consisted of questions on demo-graphic and disease-related information, as well as psychological well-being and mindfulness. The daily diary questionnaire consisted of 30
2.4 Statistical analysis 35
items on momentary affect, mindfulness, and stress. For the daily as-sessment of positive and negative affect the Dutch version of the Pro-file of Mood States short form (POMS-SF) was used [25]. We selected13 items from this questionnaire, to keep the diary short [24]. Weselected items with high factor loadings on the corresponding sub-scale [2] and applicability in daily life. Negative affect was measuredwith eight items, two for each of the Negative Affect subscales of thePOMS-SF: blue and miserable (Depression), nervous and tense (Ten-sion), tired and fatigued (Fatigue), peeved and angry (Anger). Positiveaffect was measured with five items, three from the Vigor subscale ofthe POMS-SF (energetic, lively, cheerful), and two extra items (relaxedand happy).
The items were rated on a scale ranging from 1 to 5, where 1 rep-resents not at all and 5 extremely. PA and NA affect scores were cal-culated as the average score for the eight NA items and five PA items,respectively. The PA and NA scores can range from 1 to 5 and describehow the participants feel at the time of reporting. The internal consis-tency of the scales at the first measurement day was 0.84 for NA and0.88 for PA. The selected PA and NA items correlated highly with thetotal NA scale (r=.95) and PA scale (r=.96) of the POMS-SF as assessedat baseline.2.4 Statistical analysis
A three-level linear mixed model was used with affect scores PAand NA (level 1) nested in time (level 2), and time nested in patients(level 3). Because trends in the variables over time may induce spuri-ous correlations among the variables [21], we first removed any timetrends in the diary data. We did so by modeling the trend over time inPA and NA and saving the residuals of this model. To this end, mixedmodelswere fittedwith the affect score as the dependent variable and adummy variable indicating the affect valence (NA vs PA), time, and theinteraction between affect valence and time as independent variables.We tested models with linear, quadratic, and cubic time trends, as wellasmodels with different error-covariance structures. Time trendswereallowed to vary among persons by modelling random effects at theperson x time interaction level. The models were fitted with restrictedmaximum likelihood estimation (REML) using lme (nlme package) inR. The best model was chosen on the basis of the Akaike Information
36 Applications of LMM for PA and NA
Criterion (AIC).Next, the residuals of the above model were used as outcome vari-
able in the estimation of the between- and within-subject correlationbetween PA and NA. This model was fitted with REML using lmer(lme4 package) in R. The between- and within-subject correlation be-tween PA and NA was derived from the variance-covariance matrixof the random effects corresponding to the affect dummy variable atthe person- and person x time level, respectively. These variances andcovariances were estimated freely; thus, no structure was imposed onthemodel. The 95% confidence intervals for the correlationswere com-puted using Fisher’s z-transformation. We tested whether the differ-ence between the within- and the between-subjects correlation be-tween PA and NA was statistically significant by means of the FisherZ statistic.2.5 Results
The baseline characteristics of the 85 participants of the diary studyare shown in Table 2.1. The majority of the participants was female(70.2%) and had a high educational level. Mean age was 40.5 years (SD= 10.1). Themajority of the participants was employed, either full-timeor part- time (77.4%), was married or living together (60.7%), and hadreceived treatment in the past for psychological problems (63.1%).
Table 2.1: Baseline characteristics of the participants
Variable Mean (SD) / percentage
Female 70.2%Age, mean 40.5 (10.1)EducationLower vocational/ primary school 2.4%Vocational/ secondary school 36.9%Higher post-education/ college 60.7%Employed, full-time or part-time 77.4%Married or living together 60.7%Chronic somatic disease 27.4%Past treatment for psychological problems 63.1%Current use of anti-depressant 16.7%
2.5 Results 37
Mean NA scores over the entire diary period were 1.8 (SD = 0.7,range1 − 5). Mean PA scores were 2.6 (SD = 0.9, range 1 − 5). Weestimated how much of the total variance in PA and NA in our samplewas attributable to within- and between-subject variation. With re-spect to NA, 38% of the variance was attributable to within-subjectsvariation and 62% to between-subject variation. With regard to PA,55% of the variance was attributable to within-subjects variation and45% between-subjects variance.2.5.1 Curve estimation
The mixed-effect models aimed at curve estimation showed that amodel with a linear time trend and an autoregressive covariance struc-ture of the second order had the best fit according to the AIC. NAscores decreased somewhat over time (B = −0.14, p = .02), whilePA scores slightly – although not significantly – increased (B = 0.05,p = 0.58), on average. There was significant heterogeneity in the ini-tial PA and NA scores across individuals (SD = 0.48), but very littlevariation in the individual trends (SD < 0.001). The autoregressivecoefficients were 0.33 and 0.36 for the AR(1) and AR(2) parameters,respectively. We saved the residuals of this model for use in the nextmodel.2.5.2 Between- andwithin-subject associations betweenPAand
NAThe associations between PA andNAwere estimated using the ran-
dom effects approach described above. The within-subjects correla-tion between PA and NA was r= -0.561 (95% CI -0.579 to -0.542). Thebetween-subjects correlation between PA and NA was r= -0.518 (95%CI -0.538 to -0.497). There was a significant difference between thesetwo correlations (Fisher Z = −3.42; P< 0.001).2.5.3 Compliant sample
The results for the compliant sample were comparable to those inthe full sample, but the discrepancy between the within-subjects andthe between-subjects correlation between PA and NA was stronger.The within-subjects correlation was r= -0.59 (95% CI -0.61 to -0.57).The between-subjects correlation was r= -0.50 (95% CI -0.52 to -0.48).The difference between the two correlations was significant (FisherZ=-6.61; P<0.001).
38 Applications of LMM for PA and NA
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−2 −1 0 1 2
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NA random effect
PA r
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m e
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−2 −1 0 1 2
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Figure 2.1: Predicted within-subjects (a) and between-subjects correlation (b) be-tween PA and NA based on BLUP
We calculated BLUPs (best linear unbiased predictors) [3] of therandom effects of the fitted model. Based on these BLUPS, we plot-ted the scatter plots for the within-subjects and the between-subjectscorrelation in the compliant sample (Figure 2.1a and 2.1b). From thesefigures it is clear that the within-subject variation is larger than thebetween subject variation. In Figure 2.2 we plotted a scatter plot forthe detrended raw scores, thus before the shrinkage by the BLUP es-timates.
2.6 Discussion 39
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−1 0 1 2
−2
−1
01
2
standardized Negative Affect (NA)
stan
dard
ized
Pos
itive
Affe
ct (
PA)
Figure 2.2: Detrended raw scores with individual regression lines
2.6 DiscussionThe aim of this study has been to derive unbiased estimates of the
within- and between-subject association between PA and NA. Whatour study adds to the existing literature is that we analyzed a three-dimensional dataset with 85 participants and up to 62 repeated assess-ments by means of a random-effects approach that allowed us to dis-entangle within- and between-subject effects. We found that the pro-portion of total variance of PA and NA attributable to within-subjectsdifferences was about 50% larger than the proportion attributable tobetween-subjects fluctuations.
Although both the within-subjects and between-subjects correla-tion between PA and NA were large and negative, the strongest asso-ciation was observed within-subjects. This was especially true whenthe number of completed observations was large, as in our compli-ant sample. This shows the importance of collecting intensive time-series data with a substantial number of repeated observations, as thewithin-subject effects can be estimated most reliably if the number ofdata points within individuals is large. For smaller number of obser-vations, this correlation cannot be estimated accurately and is shrunktowards zero. This result is also in line with previous literature, in
40 Applications of LMM for PA and NA
which the size of the within-subjects correlation seems to be larger instudies with a greater number of assessment points [12, 13, 27].
To date, therewas no firm evidence in the literaturewhether PA andNA represent two independent mood dimensions or two opposites ofa single dimension [22, 32]. Independence of PA and NA implies thatan increase in the one affect dimension does not necessary lead toa decrease in the other affect, and therefore it is possible that peo-ple can be high on both affect dimensions at the same time [18, 19].Conversely, the bipolar view assumes that positive and negative af-fect are two opposite ends of a single continuum, which implies thatpeople cannot be happy and sad at the same time [22]. Our resultslargely support the latter view, as the within-subjects correlation wassubstantial, especially in the compliant sample. Our between-subjectscorrelation estimate was somewhat smaller, but still rather large, andalso larger than found in earlier studies [6, 12, 13, 29]. Taken together,our study is more consonant with the bipolar than with the indepen-dence view, but also suggests that the answer partly depends on thekind of variance that has been examined.
The main strength of this study was the use of three-dimensionalaffect data in combination with the random-effects approach for thedecomposition of the within- and between-subjects variance in theassociation between PA and NA. A limitation may be that our sam-ple consisted of individuals entering a mindfulness-based interven-tion, hampering our ability to generalize to the general population.Future studies should address this issue, using a sample that is morerepresentative for the general population or persons with specific at-tributes, i.e. a mental disorder.
Another limitation may be our selection of the PA and NA items. Itis known that the size of the correlation between PA and NA dependson the specific items used in the questionnaire [12, 32]. The size ofthe correlation may also be dependent on the variance of the scoresand the time frame that the questions cover: the lower the varianceand the longer the time frame, the lower the correlation [12]. Theseaspects should be kept in mind when correlations found in differentstudies are compared. A final limitation may be that the random-effectapproach applied in this study yields a single estimate for the within-subjects association, possibly ignoring the fact that this association
2.6 Discussion 41
may be different in different individuals (as the regression lines inFigure 1c suggest). An alternative approach would be to analyze thewithin-subjects association for each individual separately, using tech-niques for the analysis of individual time series. There is, however, atrade-off here: if the time series are short, individual estimates maybe unreliable. In that case, a group-based approach may be preferable.Nevertheless, a unit-by-unit approach may be the method of choicewhen the number of time points is very high and between-subject het-erogeneity is large [15].
As a more general remark we suggest that future studies in whichthe association between two fluctuating variables is evaluated shouldtake both sources of variance into account in their design and analy-sis. In our study, about one third of variance would have been missedwhen a single assessment of subjects’ PA and NA would have beendone. Moreover, the conclusions drawn on the between-subjects vari-ance would not have been translatable in a straightforward way to theindividuals in the sample. Neither a repeated assessments design in asingle subject nor a group-based cross-sectional design would be ableto capture the full three-dimensional data matrix and runs the risk toproduce biased effect estimates.
To conclude, this study presented amultilevel approach for estimat-ing the between- and within-subjects correlation between PA and NAin one single model. The results largely support the view that PA andNA are two opposite ends of a single affect dimension, although thestrength of the correlation partly depends on which kind of variancehas been examined.
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