Internship Research Paper
Diving into the Depths of Emotions: The
Relation Between Momentary and
Day-to-Day Emotial Inertia
Author: Jolanda J. Kossakowski Supervisor: Prof. Dr. Denny Borsboom Supervisor: Dr. Ellen L. Hamaker July 18, 2014
Contents Introduction 4 Model Formulation 6 Empirical Analysis 16 Exploratory Analyses 21 Discussion 23 References 26
Emotional inertia, the degree to which emotions resist change, is considered an impor-tant aspect of (healthy) emotion regulation. By regulating their emotions, humans can adapt to social situations. In this paper we expand a study conducted by Hamaker and Kuppens (2014) and aim to fit a multilevel model in which different levels of emotional inertia are related. By means of the experience sampling method (ESM), 79 participants rated ten different emotions ten to fifteen times a day for six to twenty-one consecutive days. We fitted a three-level multilevel model to see whether momentary inertia – the autocorrelation between an observation and the previous observation – and day-to-day inertia – the autocorrelation between a day and the previous day – are related. A sim-ulation study showed that overall emotion, momentary inertia and day-to-day inertia are estimated accurately, but that correlations between these parameters are estimated poorly. Results from an empirical analysis did not demonstrate that a correlation be-tween momentary and day-to-day inertia is present. We did find a negative correlation between mean positive affect and momentary inertia and a positive correlation between mean negative affect and day inertia. We conclude that momentary and day-to-day inertia are two independent emotion regulation processes; processes used to regulate emotions.
Introduction
Some people linger in their emotions, whereas others wake up every morning thinking it is “a new day”. This ability to bounce back from positive or negative experiences – called emotion regulation – is a two-way street: emotions can regulate our responses in situations (Gross & Muñoz, 1995) or we can change our emotions in response to situ-ations (Ekman, 1992; Keltner & Kring, 1998). Numerous studies have been conducted on emotion regulation and on the question of how emotions and psychological wellbeing interact. Kashdan and Rottenberg (2010) for instance, demonstrated that major de-pressive disorder (MDD) coincides with reduced psychological flexibility; the ability to respond to various situations.
A specific example of psychological inflexibility is emotional inertia, the degree to which emotions resist change (Kuppens, Allen, & Sheeber, 2010). Kuppens, Oravecz, and Tuerlinckx (2010) argue that an individual has a baseline level of emotion; internal and external situations result in a deviation from this baseline. Emotional inertia represents how fast an individual returns to his or her baseline level of emotion after he or she deviates. High emotional inertia means that an individual has trouble returning to his or her baseline level of emotion, whereas low emotional inertia means that an individual easily returns to his or her baseline level of emotion.
Several studies have related emotional inertia to personality traits and psychological disorders. For example, Suls, Green, and Hillis (1998) found that emotional inertia and neuroticism were positively related; higher emotional inertia coincided with increased neuroticism. Kuppens, Allen, and Sheeber (2010) demonstrated a negative relation be-tween self-esteem and emotional inertia; higher self-esteem was associated with lower emotional inertia. Kuppens, Allen, and Sheeber, together with Koval, Pe, Meers, and Kuppens (2013), also found that emotional inertia was positively related with depression: participants with depression displayed higher emotional inertia in comparison to partic-ipants without depression. Also, Koval, Kuppens, Allen, and Sheeber (2012) showed that emotional inertia and rumination (repetitive thoughts about one’s own depression
symptoms, their causes, and their effects) are positively related; higher rumination was associated with higher emotional inertia. Thus, it is shown that higher emotional inertia is related to increased neuroticism, lower self-esteem, higher rumination and depression. Kuppens, Allen, and Sheeber (2010) noted that it might be possible that emotion regulation processes that underlie emotional inertia might be different processes at dif-ferent time scales. Emotional inertia within days (momentary inertia) might be another emotion regulation process than emotional inertia between days (day-to-day inertia). Hamaker and Kuppens (2014) investigated this but failed to find a correlation between momentary and day-to-day inertia. The absence of a correlation in Hamaker and Kup-pens’s study might be due to power issues because the measurement period in which the data were collected was quite small.
Our aim in this study is to expand Hamaker and Kuppens’s study by using data with a larger measurement period as well as by working in a Bayesian framework. For example, it is possible that only momentary inertia is related to neuroticism or depres-sion, but that this trait and psychological disorder are not related to not day-to-day inertia. Investigating the relation between momentary inertia and day-to-day inertia is thus important to gain a better insight into the dynamics of emotions and their relation to personality traits or psychological disorders. The key question in this study that we aim to answer is what the dynamics of emotional inertia are. More specifically, we want to know whether and, if so, how the different levels of emotional inertia are related.
We expect a positive correlation between momentary and day-to-day inertia. Kuppens, Oravecz, and Tuerlinckx (2010) argue that individuals who lack emotion regulation pro-cesses have trouble returning to their baseline level of emotion. This lack of emotion regulation processes implies a high emotional inertia. If emotion regulation processes within and between days are similar and working together, this would imply that, when momentary inertia increases, day-to-day inertia also increases and vice versa. Thus, we hypothesise that we can accurately estimate the correlation between momentary and day-to-day inertia (1) and, in the case that hypothesis 1 can be confirmed, that this
correlation is positive (2). It is beyond the scope of this study to compare healthy and depressed participants, so we only included healthy participants.
Model Formulation
For this study we obtained data from an emotion study by Pe and Kuppens (2012). A total of 79 participants were recruited for this study (mean age = 23.52, SD = 7.82, 63.29% female). By means of ESM (Csikszentmihalyi & Larson, 1987), participants rated emotions at a signalling moment. With the use of a palmtop or a mobile app, participants are signalled ten to fifteen times a day to rate various emotions that indi-viduals experience at that moment. To randomise the moments of signalling, a stratified random-interval scheme is set up whereby a participants’ waking hours is divided by the amount of tests that a researcher wants participants to take, called intervals. Within these intervals, the signalling moment is randomised. Participants used a palmtop or mobile app for six to twenty-one consecutive days. Positive emotions (relaxed, happy, satisfied and excited) and negative emotions (anger, stress, depressed, irritation, anx-ious and sadness) were averaged to obtain individual composite scores of positive affect (PA) and negative affect (NA). Next to emotion rating, sociodemographic information was obtained (age, sex, education, self esteem, life satisfaction) as well as scores on the NEO-PI-R. More information on participant recruitment and the procedure can be found in Pe and Kuppens (2012).
We aim to model emotional inertia at different time scales, namely within days (mo-mentary inertia) and between days (day-to-day inertia). The autocorrelation, the corre-lation of a variable with itself on the previous time point, is the direct operationalisation of emotional inertia. To model momentary inertia, we predict PA or NA at a given time (m) from the previous measurement of PA or NA (m-1). Day-to-day inertia is modelled by predicting PA or NA at a given day (d ) from the measurement of PA or NA at the previous day (d -1).
0 50 100 150 0 10 20 30 40 50 60 70 High High da ys of person 14 βi = 0.454 φi = 0.595 0 20 40 60 80 100 0 10 20 30 40 50 60 70 Lo w da ys of person 28 βi = 0.037 φi = 0.687 0 10 20 30 40 50 60 0 10 20 30 40 Lo w da ys of person 6 βi = 0.482 φi = 0.045 0 20 40 60 80 100 120 140 0 5 10 15 20 25 30 βi φi da ys of person 75 βi = 0.052 φi = 0.092 Figur e 1 . : Vi sualization of emoti onal inert ia on p osi tiv e affect (P A) on differen t time scale s. Scores on x-axis represen t da ys that a partici pan t ra ted his or her P A. Scores on y-axis represen t P A. βi = da y-to-da y inertia. φi = momen tary inerti a. BLA CK D ASHED L INE = base line P A. BLA CK SOLID LINE = mean P A p er da y . GREY D ASHED LINE = momen tary inertia. GR EY SOLID LINE = da y-to-da y iner tia.
0 50 100 150 0 10 20 30 40 50 60 70 High High da ys of person 14 βi = 0.454 φi = 0.595 0 20 40 60 80 0 10 20 30 40 Lo w da ys of person 1 βi = 0.004 φi = 0.522 0 20 40 60 80 100 120 140 0 5 10 15 20 Lo w da ys of person 72 βi = 0.52 φi = 0.05 0 20 40 60 80 100 120 0 10 20 30 40 βi φi da ys of person 60 βi = 0.071 φi = 0.049 Figur e 2 . : Visualization of emo tional iner tia on negativ e a ff ect (NA) on differen t time scales. Scores on x-axis represen t da ys th at a participan t rated his or her NA. Scor es on y-axis repre sen t NA. βi = da y -to-da y inertia. φi = momen tary inertia. BLA CK D ASHED LINE = baseline NA. BLA CK SOLID LINE = mean NA p er da y . GREY D ASHED LINE = momen tary inertia. GREY SOLID LINE = da y-to-da y iner tia.
To illustrate, figure 1 and 2 visualise emotional inertia on different time scales. The upper left plot shows an example of a participant with both high momentary and day-to-day inertia. It can be seen that, when PA/NA increases, it tends to increase over observations and days. The upper right plot shows an example of a participant with high momentary inertia and low day-to-day inertia. When PA/NA increases within an observation, it tends to increase over observations, whereas the daily PA/NA ratings vary more. The lower left plot shows an example of a participant with low momentary inertia and high day-to-day inertia. PA/NA ratings over observations vary, whereas a pattern emerges in the PA/NA rating between days. The lower right plot shows an example of a participant with both low momentary and day-to-day inertia. PA varies between observations and between days. Note that momentary and day-to-day inertia as depicted in each individual plot is only an indication of momentary and day-to-day inertia; an indication of the true momentary and day-to-day inertia that are to be estimated by our model.
To examine emotional inertia at different time scales, we estimated three-level mul-tilevel models. We started out with estimating an empty model; a three-level mulmul-tilevel model without inertia parameters
Empty Model
Level 1 : ymdi= µdi+ emdi (1a)
Level 2 : µdi = µi+ r0di (1b)
Level 3 : µi = γ000+ u00i (1c)
where ymdi is the observation of participant i at day d and moment m. At level one, µdi
represents the mean emotion of participant i at day d (also called the baseline parameter
of emotion), with emdi as its corresponding error term. The mean µdi at level two is
term r0di. At level 3, µi consists of γ000, the grand mean and it corresponding error term
u00i.
The above-explained model will be used to determine how much variance exists on each level. Variance on each level is needed to fit our second model that is described below. Model two is a three-level multilevel model with parameters for momentary and day-to-day inertia. This model is written as follows:
AR(1)-Model
Level 1 : ymdi = µdi+ φi(ym−1,di− µdi) + emdi (2a)
Level 2 : µdi = µi+ βi(µd−1,i− µi) + r0di (2b)
Level 3 : µi = γ000+ u00i (2c)
βi = γ010+ u01i (2d)
φi = γ100+ u10i (2e)
where φi is the momentary inertia parameter. Day-to-day inertia is denoted by βi. The
parameters γ010and γ100represent fixed effects (between-subjects means) for momentary
and day-to-day inertia and u01i and u10i are its corresponding residual error terms. The
parameters u00i, u01i, u10i, r0di and emdi are assumed to be normally distributed in the
population. To test whether momentary and day-to-day inertia are related, u01i
(day-to-day inertia) and u10i (momentary inertia) will be correlated within the model. In
conclusion, the empty model and the AR(1)-model differ in the sense that, in model one, only the mean emotion is modelled, whereas in model two, emotional inertia at different levels is modelled as well. Note that all residual error terms are formulated in terms of standard deviations (SD).
All analyses are performed using JAGS (Plummer, 2003), the R statistical software version 3.0.3 (R Development Core Team, 2013), the R-package R2jags version 0.04-01 (Su & Yajima, 2014) and the R-package brew version 1.0-6 (Horner, 2011). Code files
2e+05 4e+05 6e+05 8e+05 1e+06 −0.5 0.0 0.5 traceplotcorrelation(βi, φi) iteration correlation ( βi , φi )
Figure 3. : Traceplot of the correlation between βi and φi under the AR(1)-model with
inertia correlation. Stationarity is reached when the traceplot appears as one think
horizontal strip. βi = day-to-day inertia. φi = momentary inertia.
from all analyses are available from the first author.
We use Bayesian statistics to estimate our multilevel models. The idea of Bayesian statistics is that data (likelihood) are combined with prior knowledge about the popu-lation values of the parameters (prior probability), resulting in an updated probability (posterior probability in Lesaffre & Lawson, 2012). The prior can be chosen in such a way that it expresses a certain belief about the data (informative prior), or that it contains as little information as possible (uninformative prior). In this study we chose to select uninformative priors. More information on our prior selection can be found in appendix I. We use the Deviance Information Criterion (DIC) for model selection since it is the only available information criterion in the R-package R2jags. We also use the pD for model selection because the pD gives an estimation of the complexity of the model; the higher the pD, the more complex the model is (Spiegelhalter, Best, Carlin, & Van Der Linde, 2002).
When applying Bayesian statistics to a model, it is essential to check the convergence of parameter estimates. Convergence checking entails checking how close the chains of estimations are to the true posterior distribution (Lesaffre & Lawson, 2012). Convergence is checked by investigating the stationarity of parameter chains and the accuracy of parameter estimates. To enhance the possibility of reaching stationarity, we use 100.000
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● 0 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 correlation(β, φ) lag A utocorrelation
Figure 4. : Autocorrelation plot of the correlation between βi and φi under the
AR(1)-model with inertia correlation. The closer the autocorrelation is to zero, the better the
convergence is. βi = day-to-day inertia. φi = momentary inertia.
iterations to estimate our parameters. The first 10.000 iterations are discarded – called the burn-in period – because these first iterations are unreliable. To check stationarity of parameters we plotted all individual estimates. An example of this can be found in figure
3, where individual iterations of the correlation between momentary (φi) and day-to-day
inertia (βi) are plotted, called a traceplot. Stationarity is reached when this trace plot
appears as one thick horizontal strip, as is the case in figure 3. We also plotted the autocorrelations between iterations to check for stationarity. An example of this can be
found in figure 4, where the autocorrelations of the correlation between momentary (φi)
and day-to-day inertia (βi) are plotted. When stationarity is reached, autocorrelations
decreases fast and remain small, close to zero. We see in figure 4 that this is the case. We start our analysis with the empty model. Results from the empty model give insight into the amount of variance that exists at each level. At each level, the parameters
µi, µdiand ymdiare estimated using a normal distribution with the parameter estimate of
the previous level as the mean and the precision (inverse of the variance) of that specific level. After analysing the empty model, we analysed the AR(1)-model. This procedure is split up into two parts. First, the AR(1)-model is fitted without estimating a correlation
between inertia parameters. Standard deviations of the inertia parameters (u01i for
define the Inverse-Wishart distribution (IW), which is the prior distribution for level 3
parameters µi, βi and φi. The second part of this procedure entails the analysis of the
AR(1)-model including the estimation of a correlation between inertia parameters.
Table 1:: Parameter Estimates for Simulated Data
Simulated data without inertia correlation Simulated data with inertia correlation Model Parameter True Value θ (SD) 95% CI True Value θ (SD) 95% CI 1 γ000 - 4.336 (0.270) 3.841 - 4.881 - 4.778 (0.261) 4.263 - 5.296 sd(emdi) - 1.035 (0.006) 1.022 - 1.047 - 1.029 (0.006) 1.016 - 1.041 sd(r0di) - 5.124 (0.126) 4.890 - 5.372 - 5.012 (0.127) 4.777 - 5.277 sd(u00i) - 1.875 (0.270) 1.364 - 2.423 - 1.814 (0.250) 1.362 - 2.325 DIC 41815 41622 pD 891 879 2 γ000 5 4.363 (0.274) 3.809 - 4.879 - 4.704 (0.269) 4.182 - 5.250 γ010 0.3 0.288 (0.043) 0.203 - 0.370 - 0.300 (0.044) 0.213 - 0.383 γ100 0.3 0.283 (0.016) 0.252 - 0.313 - 0.314 (0.013) 0.289 - 0.340 sd(emdi) 1 1.009 (0.007) 0.996 - 1.022 - 1.000 (0.007) 0.986 - 1.013 sd(r0di) 5 5.064 (0.136) 4.794 - 5.340 - 4.946 (0.131) 4.695 - 5.210 sd(u00i) 1.41 0.775 (0.468) 0.071 - 1.780 - 0.955 (0.482) 0.097 - 1.829 sd(u01i) 0.1 0.150 (0.061) 0.010 - 0.253 - 0.101 (0.054) 0.009 - 0.205 sd(u10i) 0.1 0.101 (0.014) 0.076 - 0.130 - 0.081 (0.014) 0.053 - 0.109 DIC 37712 37658 pD 1087 1251 3 γ000 - 4.354 (0.266) 3.841 - 4.869 5 4.717 (0.265) 4.188 - 5.255 γ010 - 0.297 (0.040) 0.221 - 0.378 0.3 0.306 (0.040) 0.223 - 0.382 γ100 - 0.283 (0.015) 0.255 - 0.312 0.3 0.315 (0.013) 0.288 - 0.342 sd(emdi) - 1.009 (0.007) 0.996 - 1.022 1 1.000 (0.006) 0.987 - 1.013 sd(r0di) - 5.083 (0.132) 4.835 - 5.345 5 4.950 (0.126) 4.710 - 5.195 sd(u00i) - 0.629 (0.231) 0.296 - 1.183 1.41 0.910 (0.329) 0.408 - 1.660 sd(u01i) - 0.133 (0.043) 0.065 - 0.229 0.1 0.098 (0.041) 0.042 - 0.195 sd(u10i) - 0.097 (0.003) 0.072 - 0.124 0.1 0.076 (0.014) 0.049 - 0.103 CorMuBeta - -0.019 (0.454) -0.797 - 0.780 0.4 0.354 (0.447) -0.653 - 0.913 CorMuPhi - 0.120 (0.411) -0.706 - 0.806 0.4 0.341 (0.341) 0.419 - 0.840 CorBetaPhi - 0.300 (0.329) -0.392 - 0.816 0.4 0.122 (0.427) -0.726 - 0.817 DIC 37756 37619 pD 1128 1204
CI = Credible Interval. θ = posterior mean. Model 1 = empty model. Model 2 =
AR(1)-model without inertia correlation. Model 3 = AR(1)-AR(1)-model with inertia correlation. µi
= mean emotion. βi = day-to-day inertia. φi = momentary inertia. CorMuBeta =
correlation between µi and βi. CorMuPhi = correlation between µi and φi. CorBetaPhi
= correlation between βi and φi.
Data were initially simulated to test the accuracy of all models and parameter esti-mates. Traceplots and autocorrelation plots (not shown for clarity of presentation) show that every parameter reached stationarity. Figures 3 and 4 show examples of the
corre-lation between βi and φi that reached stationarity. Table 1 shows parameter estimates
for all individual models. It can be seen that the empty model accurately estimates
the parameters γ000, sd(emdi), sd(r0di) and sd(u00i). Parameters within the
AR(1)-model without inertia correlation are also accurately estimated. Parameter estimates of
sd(u00i), sd(u01i) and sd(u10i), estimated in the AR(1)-model without inertia correlation,
are inserted into the JAGS-code of the AR(1)-model with inertia correlation. It can be seen in table 1 that this model also accurately estimates the parameters. Based on these results, we can conclude that convergence is reached for all parameters.
We extended the initial simulation to gain a deeper insight into the accuracy of our models. A hundred datasets were simulated under the AR(1)-model with an inertia correlation. After this simulation, these datasets were inserted into all models. Mean
estimates and standard deviations of level 3 parameters γ000, γ010 and γ100 as well as
correlation parameters were saved in order to investigate how biased our estimates are. First, we calculated the mean estimate of parameters over a hundred simulations and compared this with its true value. Second, the mean standard deviation over a hundred simulations was calculated and compared with the standard deviation calculated with the mean estimates of a hundred simulations and with the true standard deviation that was used to simulate data. Thirdly, 95% credible intervals for each parameter and simulation was computed and investigated whether the true value fell within the interval. We also calculated the width of each interval for the correlation parameters and whether zero fell within the intervals of the correlation parameters. Fourthly, we calculated the difference in DIC between the AR(1)-model with inertia correlation and the AR(1)-model with inertia correlation in order to see how many times which model is preferred.
Mean estimates found in the simulation study and their deviation from the parame-ters’ true value can be found in table 3. It can be seen that level 3 parameters deviated slightly from the mean estimate, which indicates that the true value is estimated accu-rately. However, mean estimates from the correlation parameters deviated more, which indicates that their estimate is less accurate. Thus, mean estimates of level 3 parameters
are estimated accurately, whereas this is not the case for the correlation parameters. Table 2 shows results of the comparison of the mean standard deviation (SD) from all simulated parameters and the SD from the mean estimates. Results show that the mean SD and the SD of the mean estimates only differ slightly. Both the mean SD and
the SD of the mean estimates for µi deviate the most from its true value (1.41;
√ 2), but only by one point. We conclude that the mean of the SDs and the SD of the mean estimates are similar.
Table 2:: Results Simulation Study: Standard Deviations
θ SD mean ∆ θ - ∆ θ - ∆ SD[mean(estimate)]
-(mean) (SD) SD(mean) mean(SD) mean(SD)
Model Parameter 1 µi 1.41 0.272 0.293 1.143 1.122 -0.021 2 µi 1.41 0.270 0.299 1.144 1.116 -0.029 βi 0.1 0.038 0.044 0.062 0.056 -0.006 φi 0.1 0.018 0.015 0.082 0.085 0.003 3 µi 1.41 0.321 0.294 1.094 1.120 0.027 βi 0.1 0.039 0.041 0.061 0.059 -0.002 φi 0.1 0.016 0.015 0.084 0.085 0.001
Model 1 = empty model. Model 2 = AR(1)-model without inertia correlation. Model 3 = AR(1)-model with inertia correlation. θ = true value. SD = standard deviation.
SD(mean) = SD of the mean estimates. mean(SD) = mean of the SDs. µi = mean
PA/NA. βi = day-to-day inertia for PA/NA. φi = momentary inertia for PA/NA.
Cor-MuBeta = correlation between µi and βi. CorMuPhi = correlation between µi and φi.
CorBetaPhi = correlation between βi and φi.
Results of the calculated 95% credible intervals (CIs) can be found in table 3. Small CIs signify that a parameter is estimated accurately, whereas a wide CI implies a poorly estimated parameter. It shows that level 3 parameters are estimated accurately, whereas the correlation parameters are estimated poorly. The mean width of the 95% CIs of the correlation parameters is more than one, which is very large for a correlation that runs between -1 and 1 and can have a CI with a maximum width of two. This indicates that estimated correlations between level 3 parameters are not accurately estimated. Note
Table 3:: Results Simulation Study: Mean Estimates and 95% Credible Intervals
Mean Deviation Mean width Percentage 95% CIs Percentage 95% CIs Model Parameter θ estimate from θ 95% CIs containing θ containing zero 1 µi 5.00 5.054 -0.054 1.139 96% -2 µi 5.00 5.048 -0.048 1.163 98% -βi 0.30 0.294 0.006 0.176 98% -φi 0.30 0.300 0.0003 0.059 90% -3 µi 5.00 4.927 0.071 1.154 92% -βi 0.30 0.303 -0.003 0.162 96% -φi 0.30 0.298 0.002 0.058 93% -CorBetaPhi 0.40 0.241 0.159 1.381 99% 95% CorMuBeta 0.40 0.193 0.207 1.451 100% 98% CorMuPhi 0.40 0.373 0.027 1.034 97% 75%
Model 1 = empty model. Model 2 = AR(1)-model without inertia correlation. Model
3 = AR(1)-model with inertia correlation. θ = true value. CI = credible interval µi =
mean PA/NA. βi = day-to-day inertia for PA/NA. φi = momentary inertia for PA/NA.
CorMuBeta = correlation between µi and βi. CorMuPhi = correlation between µi and
φi. CorBetaPhi = correlation between βi and φi.
correlation but an estimate of overall emotion, we consider this a small mean width of the 95% CIs. We conclude that the 95% CIs of level 3 parameters are small, whereas the 95% CIs of the correlation parameters are very wide and thus poorly estimated.
For both the DIC and the pD we calculated how many times the AR(1)-model without inertia correlation is chosen as the right model and how many times the AR(1)-model with inertia correlation is chosen as the right model. Results show that, for the DIC, the model without inertia correlation is chosen 53 out of 100 times, and the AR(1)-model with inertia correlation 47 out of 100 times. For the pD, results show that the model without inertia correlation is chosen 49 out of 100 times, and the AR(1)-model with inertia correlation 51 out of 100 times. We conclude that the DIC and the pD are indifferent when it comes to selecting a model: neither the AR(1)-model without an inertia correlation nor the AR(1)-model with an inertia correlation is preferred.
Empirical Analysis
After checking convergence for all models with simulated data, we proceeded with the analysis of the models with empirical data. Table 4 shows parameter estimates for
all models. The entire analysis was performed separately for PA and NA. We checked convergence for parameters in all three models by means of traceplots and autocorrelation plots. Stationarity was reached for all parameters with the exception of parameters in the model with inertia correlation for both PA and NA. We re-ran the AR(1)-model with inertia correlation using 1.000.000 iterations, a burn-in period of 100.000 and a thinning factor of ten. The latter means that we used every tenth iteration to estimate parameters. Convergence was reached with this number of iterations, burn-in period and thinning factor.
Parameters deviate from zero when zero does not fall within its 95% credible interval. Results show that, for every model and for PA and NA, all inertia parameters and
its standard deviations deviate from zero. Thus, fixed parameters γ000, γ010 and γ100
deviate from zero in all models for both PA and NA. All random parameters sd(emdi),
sd(r0di), sd(u00i), sd(u01i) and sd(u10i) also deviate from zero; they signify individual
differences in both PA and NA. This means that enough variance exists on each level of the empty model to fit the AR(1)-model. Individual parameter estimates and their corresponding 95% credible intervals are plotted for the AR(1)-model without inertia correlation for both PA (figure 5) and NA (figure 6), as well as for the AR(1)-model with inertia correlation for PA (figure 7) and NA (figure 8). These figures show that parameter estimates are more accurate for momentary inertia than for day-to-day inertia
and that fixed effects γ010 and γ100 are estimated more accurately than βi and φi.
However, results show that the correlation between momentary inertia (sd(u10i))
and day-to day inertia (sd(u01i)) does not deviate from zero. Although the correlation
between momentary and day-to-day inertia is positive for both PA and NA as hypothe-sised, it can be seen in table 4 that the 95% credible interval of the correlation between
(sd(u01i)) and (sd(u10i)) for both PA and NA contains zero. Thus we cannot
demon-strate that a correlation between momentary and day-to-day inertia is present. This implies that momentary and day-to-day inertia are two separate coping strategies used to regulate emotions.
Table 4:: Parameter Estimates for Empirical Data PA NA Model Parameter θ (SD) 95% CI θ (SD) 95% CI 1 γ000 58.713 (1.447) 55.854 - 61.486 11.255 (0.925) 9.520 - 12.967 sd(emdi) 13.701 (0.110) 13.497 - 13.924 9.016 (0.071) 8.873 - 9.153 sd(r0di) 7.885 (0.234) 7.424 - 8.366 5.337 (0.159) 5.014 - 5.649 sd(u00i) 12.441 (1.062) 10.632 - 14.690 8.044 (0.674) 6.894 - 9.529 DIC 76965 69022 pD 1447 1381 2 γ000 58.447 (1.484) 55.630 - 61.387 10.959 (0.927) 9.162 - 12.658 γ010 0.568 (0.066) 0.431 - 0.695 0.333 (0.073) 0.184 - 0.471 γ100 0.286 (0.020) 0.246 - 0.324 0.253 (0.022) 0.210 - 0.296 sd(emdi) 13.380 (0.111) 13.151 - 13.597 8.928 (0.073) 8.788 - 9.066 sd(r0di) 5.711 (0.320) 5.051 - 6.310 3.937 (0.228) 3.496 - 4.399 sd(u00i) 12.031 (1.080) 10.214 - 14.319 7.497 (0.777) 6.050 - 9.093 sd(u01i) 0.116 (0.065) 0.007 - 0.241 0.289 (0.057) 0.179 - 0.395 sd(u10i) 0.133 (0.018) 0.102 - 0.169 0.140 (0.018) 0.108 - 0.180 DIC 73464 66207 pD 1866 2299 3 γ000 58.235 (1.610) 54.921 - 61.118 11.094 (0.903) 9.339 - 12.888 γ010 0.565 (0.060) 0.452 - 0.677 0.382 (0.076) 0.232 - 0.532 γ100 0.286 (0.020) 0.247 - 0.325 0.253 (0.021) 0.210 - 0.294 sd(emdi) 13.387 (0.108) 13.178 - 13.600 8.967 (0.076) 8.818 - 9.117 sd(r0di) 5.676 (0.319) 5.041 - 6.293 3.712 (0.260) 3.193 - 4.209 sd(u00i) 11.891 (1.220) 9.781 - 14.542 7.549 (0.746) 6.191 - 9.115 sd(u01i) 0.164 (0.033) 0.105 - 0.234 0.234 (0.048) 0.147 - 0.336 sd(u10i) 0.127 (0.018) 0.094 - 0.165 0.134 (0.017) 0.102 - 0.170 CorBetaPhi 0.293 (0.289) -0.316 - 0.765 0.202 (0.290) -0.368 - 0.726 CorMuBeta -0.422 (0.256) -0.812 - 0.147 0.564 (0.190) 0.120 - 0.847 CorMuPhi -0.425 (0.165) -0.704 - -0.062 0.328 (0.160) -0.006 - 0.615 DIC 73434 66624 pD 1822 2644
CI = Credible Interval. θ = posterior mean. Model 1 = empty model. Model 2 = AR(1)-model without inertia correlation. Model 3 = AR(1)-AR(1)-model with inertia correlation. PA
= Positive Affect. NA = Negative Affect. µi = mean PA/NA. βi = day-to-day inertia
for PA/NA. φi = momentary inertia for PA/NA. CorMuBeta = correlation between µi
and βi. CorMuPhi = correlation between µi and φi. CorBetaPhi = correlation between
● ●●●● ● ● ●● ● ●●●● ● ●● ● ● ● ●●● ● ●● ●●● ● ● ● ● ●●● ● ●● ● ● ● ●● ● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●● ● ● ● ● ● ●●● −1.0 −0.5 0.0 0.5 1.0 γ010 1 12 23 34 45 56 67 78 ppn βi ● ● ●● ● ● ●● ●● ● ● ● ●●● ●● ● ● ● ● ● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●●● ● ● ●● ● ● ●● ● ● ● ● ●● ●●● ●● ● ● ● ● ●● ● ● ● ● ● −1.0 −0.5 0.0 0.5 1.0 γ100 1 12 23 34 45 56 67 78 ppn φi
Figure 5. : Visualisation of βi and φi in the AR(1)-model without inertia correlation on
PA. Mean estimates per individual and their corresponding 95% credible intervals are
plotted together with γ010, the fixed effect of day-to-day inertia (βi), and γ100, the fixed
effect of momentary inertia (φi).
● ●● ● ● ● ●●● ●●●●● ● ● ● ● ● ● ● ● ●● ● ●●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ●●●●● ●● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ● ●● ● ● ● ● ●●●● −1.0 −0.5 0.0 0.5 1.0 γ010 1 12 23 34 45 56 67 78 ppn βi ● ● ● ●● ● ●● ● ●● ●●● ●● ● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ●● ● ● ●●●● ● ● ● ● ● ●●●●● ● ● ● ● ● ●● ●●● ● ●● ● ●● ● ● ●● ● ● ● ● ●●● ● −1.0 −0.5 0.0 0.5 1.0 γ100 1 12 23 34 45 56 67 78 ppn φi
Figure 6. : Visualisation of βi and φi in the AR(1)-model without inertia correlation on
NA. Mean estimates per individual and their corresponding 95% credible intervals are
plotted together with γ010, the fixed effect of day-to-day inertia (βi), and γ100, the fixed
● ●●● ● ● ●● ●● ●●● ●●●●●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ●● ● ●●● ● ● ●● ● ● ● ●● ● ● ●● ● ●● ●● ● ●● ● ●● ● ● −1.0 −0.5 0.0 0.5 1.0 γ010 1 12 23 34 45 56 67 78 ppn βi ● ● ●● ● ● ● ● ●● ● ● ●● ●● ● ● ● ● ● ● ●●● ●● ● ●●● ● ● ● ● ● ● ● ● ● ●● ●● ● ● ● ●●● ● ●●● ● ● ●● ● ● ● ● ●● ●●● ●● ● ● ● ● ●●● ● ● ●● −1.0 −0.5 0.0 0.5 1.0 γ100 1 12 23 34 45 56 67 78 ppn φi
Figure 7. : Visualisation of βi and φi in the AR(1)-model with inertia correlation on PA.
Mean estimates per individual and their corresponding 95% credible intervals are plotted
together with γ010, the fixed effect of day-to-day inertia (βi), and γ100, the fixed effect of
momentary inertia (φi). ● ●● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●●●●● ● ● ●● ● ●●●●● ● ● ● ● ● ●● ● ●● ● ● ● ● ●● ●● −1.0 −0.5 0.0 0.5 1.0 γ010 1 12 23 34 45 56 67 78 ppn βi ● ● ● ●● ● ●●● ●● ●●● ●● ● ● ● ● ● ● ●● ●●● ● ● ● ● ●● ● ●● ● ● ● ●●● ●● ● ●●●●●●● ● ● ● ● ● ●● ●●● ● ● ● ● ● ● ● ● ●● ● ●● ● ●●● ● −1.0 −0.5 0.0 0.5 1.0 γ100 1 12 23 34 45 56 67 78 ppn φi
Figure 8. : Visualisation of βi and φi in the AR(1)-model with inertia correlation on NA.
Mean estimates per individual and their corresponding 95% credible intervals are plotted
together with γ010, the fixed effect of day-to-day inertia (βi), and γ100, the fixed effect of
Model selection is based on the DIC (Lesaffre & Lawson, 2012). When a difference of at least two points exists between DICs of different models, the model with fewer param-eters is preferred. Table 4 shows that, for PA, the AR(1)-model with inertia correlation is preferred over the AR(1)-model without inertia correlation. However, the reverse is found for NA, where the AR(1)-model without inertia correlation is preferred over the AR(1)-model with inertia correlation. We also investigated the pD as a second method for model selection. The pD gives an estimation of the complexity of the model; the higher the pD, the more complex the model (Spiegelhalter et al., 2002). Given that the pD reflects the complexity of a model, we expect higher values for the AR(1)-model with inertia correlation because more parameters are estimated. Table 4 shows that, for PA and NA, the more complex model is the AR(1)-model without inertia correlation. This can be explained by the fact that the AR(1)-model with inertia correlation also estimates correlations between level 3 parameters. Because these parameters share variance, the model is considered less complex.
Exploratory Analyses
After the main analyses and the simulation study, we performed exploratory analyses.
Next to the correlation between day-to-day inertia (sd(u01i)) and momentary inertia
(sd(u10i)), we investigated the correlation between overall PA/NA (sd(u00i)) and
day-to-day inertia (sd(u01i)) and overall PA/NA (sd(u00i)) and momentary inertia (sd(u10i)).
Figures 9 and 10 display the correlation between all level 3 parameters on the off-diagonals and their densities on the diagonals. Densities show that the level 3 parameters did not suffer from restriction of range. Note that the displayed correlations are based on PA/NA observations that are actually measured; missing values are not taken into account. In our empirical analysis observations that were not measured (for example, a participant forgot to fill out the questionnaire at the signalling moment) were also taken into account; these missing values altered the correlation between momentary inertia and day-to-day inertia.
µi 46 50 55 60 65 0.00 0.05 0.10 0.15 0.20 0.25 µi βi φi ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.1 0.2 0.3 0.4 0.5 0.6 0.7 40 50 60 70 80 βi 0.3 0.4 0.5 0.6 0.7 0.8 0 1 2 3 4 5 6 7 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● −0.15 0 0.15 0.3 0.45 0.65 40 50 60 70 80 φi ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● −0.15 0 0.15 0.3 0.45 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.19 0.23 0.27 0.31 0.35 0 5 10 15 20
Figure 9. : Panelplot showing the density of parameters µi, βi and φi on PA on the
diagonals and correlations between these parameters on the off diagonals. µi = mean
PA. βi = day-to-day inertia for PA. φi = momentary inertia for PA.
µi 7 9 11 13 15 0.0 0.1 0.2 0.3 0.4 µi βi φi ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● 0.1 0.2 0.3 0.4 0.5 0.6 0.7 40 50 60 70 80 βi 0.05 0.20 0.35 0.50 0.65 0 1 2 3 4 5 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● −0.15 0 0.15 0.3 0.45 0.6 40 50 60 70 80 φi ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● −0.15 0 0.15 0.3 0.45 0.6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.15 0.20 0.25 0.30 0.35 0 5 10 15
Figure 10. : Panelplot showing the density of parameters µi, βi and φi on NA on the
diagonals and correlations between these parameters on the off diagonals. µi = mean
Table 4 shows the correlation between parameters sd(u00i) and sd(u01i) and sd(u00i)
and sd(u10i). Results show that, for PA, a negative correlation between sd(u00i) and
sd(u10i) exists and that it deviates from zero; a negative correlation between sd(u00i and
sd(u01i was found but does not deviate from zero. This means that when overall PA
increases, momentary and day-to-day inertia decreases, although we could not demon-strate that the negative correlation between overall PA and day-to-day inertia exists.
For NA, results show that a positive correlation between sd(u00i) and sd(u01i) exists and
that it deviates from zero; a positive correlation was also found between sd(u00i) and
sd(u10i) but it does not deviate from zero. This means that, when overall NA increases,
momentary and day-to-day inertia also increases, although we could not demonstrate that the positive correlation between overall NA and momentary inertia exists.
Discussion
In this study we aimed to estimate the correlation between momentary and day-to-day inertia. Results from the empirical analysis did not demonstrate the presence of a correlation between momentary and day-to-day inertia, although results did show the positive correlation that was expected. We conclude that emotion regulation processes on different time scales that underly momentary and day-to-day inertia are two separate processes that work independently.
We performed a simulation study and found that level 3 parameters are estimated accurately by all models. Mean estimates deviated only slightly from their true values, the standard deviation of the mean estimates and the mean of the standard deviations were very similar and the 95% credible intervals were small. We could not conclude this for the correlation parameters. Mean estimates deviated and their 95% credible intervals were very wide, resulting in a poor estimation of the correlation parameters. Using the DIC, two models were selected for PA –the AR(1)-model with inertia correlation– and NA –the AR(1)-model without inertia correlation–. Finally, exploratory analyses found a negative correlation between a person’s individual PA and momentary inertia and a
positive correlation between a person’s individual NA and day-to-day inertia.
In this study we estimated momentary inertia – the autocorrelation between a sig-nalling moment and the previous one– and day-to-day inertia – the autocorrelation be-tween a day and the previous one. Participants rated emotions bebe-tween six and twenty-one consecutive days. This led to five to twenty autocorrelations; the minimum and maximum amount of days that participants rated emotions. On average, participants rated emotions for sixteen consecutive days. But with only fifteen autocorrelations that can be used to estimate day-to-day inertia, the estimate is not reliable enough. Ideally, day-to-day inertia is estimated with at least 25 days of ratings, so that the estimate becomes more reliable. Second, when estimating momentary inertia, we did not take the intervals between signalling moments into account. Signalling moments occurred via a stratified random-interval scheme; moments could be either five minutes apart or eighty minutes apart. Because we did not take this disparity of interval lengths into account, momentary inertia could not be estimated accurately. These two reasons could together explain the absence of a correlation between momentary and day-to-day inertia. Future research should try to run an ESM study that runs longer than sixteen consecutive days. Another reason for not finding a correlation between momentary and day-to-day iner-tia can be found in the results of the simulation study. Our simulation study found that correlations between level 3 parameters overall PA/NA, day-to-day inertia and momen-tary inertia were estimated poorly. We found inaccurate estimates of the correlations parameters as well as wide 95% credible intervals; a finding that can explain the in-accurate estimates. It is possible that this inin-accurate estimate caused the absence of a correlation between momentary and day-to-day inertia in our empirical analysis. Al-though this simulation study was on an exploratory basis, it did show that the estimation of the correlation parameters can be improved. Future research should focus on extend-ing the simulation study in which the cause of these inaccurate estimates is found and improving the model. This study could then be replicated to investigate whether there is in fact a correlation between momentary and day-to-day inertia.
Interestingly, although the correlation between momentary and day-to-day inertia did not deviate from zero for both PA and NA, model selection based on the DIC suggests that the AR(1)-model with this correlation is the best model for PA. Also, model selection based on the DIC points out that, for NA, the AR(1)-model without correlation is the best model. This contradiction is due to a different method of calculation that is used to calculate the DIC and the 95% credible interval. Future research should focus on conducting a simulation study and comparing model selections based on the 95% credible intervals and the DICs. Comparing these two methods for model selection would tell us which method for model selection is more accurate and thus preferred.
The dataset used in this study comprised of students from the University of Leuven (Pe & Kuppens, 2012). No comparison is made between healthy participants and non-healthy participants in this study. It is quite possible that analyses run in this study might yield other results when performing these analyses with data from non-healthy participants. Kuppens, Allen, and Sheeber (2010) showed that depressed participants displayed higher emotional inertia dan non-depressed participants. It is plausible then that the correlation between momentary and day-to-day inertia –a correlation that was not found in this study– does exist in depressed participants. Future research should redo these analyses with a sample of non-healthy participants and, if possible, compare these results with a matching sample of healthy participants.
Despite these limitations, the current study provides a new insight into the dynam-ics of emotion and emotion regulation. Our results imply that people who have high momentary inertia do not necessarily also have to have high day-to-day inertia or vice versa. However, people with a high PA tend to have decreased momentary inertia. Peo-ple with a high NA on the other hand, tend to have increased day-to-day inertia. Our results highlight the need for more research into estimation techniques for momentary and day-to-day inertia. Such research would help to accurately estimate momentary and day-to-day inertia and thereby accurately estimate the correlation between these two processes.
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Appendix I: Prior selection
In this study we perform a Bayesian Multilevel analysis to test whether day-to-day emotional inertia and momentary emotional inertia are related. Because we work within a Bayesian framework, we need to specify priors for all parameters at level 3 of the models as well as all residual error terms. We want the priors to minimally influence the data as possible and express minimal information, and therefore we will use uninformative priors (Lesaffre & Lawson, 2012). The prior for the residual error terms is a uniform
distribution that ranges between zero and fifty. The prior of the level 3 parameter γ000 is
a normal distribution with a set mean and a small precision (the inverse of the variance). By reducing the precision, the distribution becomes flat and uninformative.
The prior of the random parameters in the AR(1)-model (µi, βi and φi) is a
mul-tivariate normal distribution with set means and a precision matrix that will make the prior uninformative. The precision matrix has a prior as well and is an Inverse-Wishart (IW) distribution. An often used specification for the IW is an identity matrix (IM) in which diagonal elements are one and off-diagonal elements zero. The problem with this specification arises when the variances of the random parameters are small, as is the case in this study. When these variances are small, the IW will become very informative, as is shown by Schuurman, Grasman, and Hamaker (2014). To ensure that our prior stays uninformative, we will estimate the input of the IW by running our model twice. First, we fit the model without any covariations to estimate the variances of the random parameters. Second, we will use these estimations as input for the IW and then run our model again for the main data analysis (Schuurman et al., 2014).
The fixed parameters γ000, γ010 and γ100also have a prior; a normal distribution with
a set mean and a small precision. Note that the priors of the residual error terms and the fixed parameters are equal for the empty model and the AR(1)-model.
