Mixed-effects Location Scale Model Research Articles

A mixed-effects location scale model for time-to-event data: A smoking behavior application

In general, mixed-effects location scale models (MELS) allow assessment of within-person and between-person variability with time-to-event data for outcomes that follow a normal or ordinal distribution. In this article, we extend the mixed-effects location scale model to time-to-event data in relation to smoking data. Better understanding of the time-graded within-person variability of factors involved in nicotine dependence can be helpful to researchers in their efforts to fine-tune smoking cessation programs. We illustrate the MELS model with data on time to first cigarette measured every day for 7 days in smokers randomized to two groups: a) those asked to keep smoking, or b) those asked to stop. Our results show that some individuals remain very stable in their time to first cigarette over the week, while others show variable patterns. The stable individuals smoked every day, did not smoke immediately upon waking, and were all in the group asked to keep smoking. Conversely, the variable individuals had at least one day during which they did not smoke, other days during which they smoked within the first 5 min of waking, and they were almost all in the group asked to quit smoking. These findings suggested that MELS have the potential to provide insights on how people try to stop smoking. More importantly, this model can be applied to other clinically important outcomes such as time to relapse in a range of cessation programs.

Modeling Intraindividual Variability in Three-Level Multilevel Models

Abstract. The mixed-effects location scale model is an extension of a multilevel model for longitudinal data. It allows covariates to affect both the within-subject variance and the between-subject variance (i.e., the intercept variance) beyond their influence on the means. Typically, the model is applied to two-level data (e.g., the repeated measurements of persons), although researchers are often faced with three-level data (e.g., the repeated measurements of persons within specific situations). Here, we describe an extension of the two-level mixed-effects location scale model to such three-level data. Furthermore, we show how the suggested model can be estimated with Bayesian software, and we present the results of a small simulation study that was conducted to investigate the statistical properties of the suggested approach. Finally, we illustrate the approach by presenting an example from a psychological study that employed ecological momentary assessment.

The Power to Detect and Predict Individual Differences in Intra-Individual Variability Using the Mixed-Effects Location-Scale Model

ABSTRACTOur goal is to provide empirical scientists with practical tools and advice with which to test hypotheses related to individual differences in intra-individual variability using the mixed-effects location-scale model. To that end, we evaluate Type I error rates and power to detect and predict individual differences in intra-individual variability using this model and provide empirically-based guidelines for building scale models that include random and/or systematically-varying fixed effects. We also provide two power simulation programs that allow researchers to conduct a priori empirical power analyses. Our results aligned with statistical power theory, in that, greater power was observed for designs with more individuals, more repeated occasions, greater proportions of variance available to be explained, and larger effect sizes. In addition, our results indicated that Type I error rates were acceptable in situations when individual differences in intra-individual variability were not initially detectable as well as when the scale-model individual-level predictor explained all initially detectable individual differences in intra-individual variability. We conclude our paper by providing study design and model building advice for those interested in using the mixed-effects location-scale model in practice.

A 3-level Bayesian mixed effects location scale model with an application to ecological momentary assessment data.

Ecological momentary assessment studies usually produce intensively measured longitudinal data with large numbers of observations per unit, and research interest is often centered around understanding the changes in variation of people's thoughts, emotions and behaviors. Hedeker et al developed a 2-level mixed effects location scale model that allows observed covariates as well as unobserved variables to influence both the mean and the within-subjects variance, for a 2-level data structure where observations are nested within subjects. In some ecological momentary assessment studies, subjects are measured at multiple waves, and within each wave, subjects are measured over time. Li and Hedeker extended the original 2-level model to a 3-level data structure where observations are nested within days and days are then nested within subjects, by including a random location and scale intercept at the intermediate wave level. However, the 3-level random intercept model assumes constant response change rate for both the mean and variance. To account for changes in variance across waves, as well as clustering attributable to waves, we propose a more comprehensive location scale model that allows subject heterogeneity at baseline as well as across different waves, for a 3-level data structure where observations are nested within waves and waves are then further nested within subjects. The model parameters are estimated using Markov chain Monte Carlo methods. We provide details on the Bayesian estimation approach and demonstrate how the Stan statistical software can be used to sample from the desired distributions and achieve consistent estimates. The proposed model is validated via a series of simulation studies. Data from an adolescent smoking study are analyzed to demonstrate this approach. The analyses clearly favor the proposed model and show significant subject heterogeneity at baseline as well as change over time, for both mood mean and variance. The proposed 3-level location scale model can be widely applied to areas of research where the interest lies in the consistency in addition to the mean level of the responses.

Do fluctuations in positive affective and physical feeling states predict physical activity and sedentary time?

ObjectivesHigher levels of positive affect and feelings of energy are associated with greater physical activity (PA) and lower sedentary time (ST). However, whether fluctuations in these feelings contribute to the regulation of these behaviors is unclear. This study examined the extent to which subject-level variability (i.e., degree of intraindividual variability) in positive affect and feeling energetic predicted participants’ overall levels of PA and ST. DesignThis analysis combined data from four ecological momentary assessment (EMA) studies (agerange: 8–73 years) with ambulatory monitoring via waist-worn accelerometry (N = 661). MethodsPositive affect and energy were assessed through EMA several times per day across 4–7 days. Accelerometer data was used to create the following behavioral outcomes: (1) meeting MVPA guidelines (children: 60 minutes/day, adults: 30 minutes/day) and (2) minutes of ST per hour of accelerometer wear. A two-stage analytic approach was used to test the study aim. In the first stage, mixed-effects location scale modeling decomposed mean levels and variability in positive affect and energy. In the second stage, a linear or logistic regression (depending on whether the outcome was continuous or dichotomous, respectively) was tested to investigate associations between subject-level mean and variability in EMA ratings and the behavioral outcome. ResultsGreater subject-level variability but not subject-level mean of feeling energetic was associated with lower odds of meeting MVPA guidelines (β = −0.43, p < 0.05). ConclusionsFluctuations in physical feeling states may deplete self-regulatory resources involved in planning and implementing PA behavior. Alternatively, being more physically active may stabilize one's perceived energy levels.

Puffed-up but shaky selves: State self-esteem level and variability in narcissists.

Different theoretical conceptualizations characterize grandiose narcissists by high, yet fragile self-esteem. Empirical evidence, however, has been inconsistent, particularly regarding the relationship between narcissism and self-esteem fragility (i.e., self-esteem variability). Here, we aim at unraveling this inconsistency by disentangling the effects of two theoretically distinct facets of narcissism (i.e., admiration and rivalry) on the two aspects of state self-esteem (i.e., level and variability). We report on data from a laboratory-based and two field-based studies (total N = 596) in realistic social contexts, capturing momentary, daily, and weekly fluctuations of state self-esteem. To estimate unbiased effects of narcissism on the level and variability of self-esteem within one model, we applied mixed-effects location scale models. Results of the three studies and their meta-analytical integration indicated that narcissism is positively linked to self-esteem level and variability. When distinguishing between admiration and rivalry, however, an important dissociation was identified: Admiration was related to high (and rather stable) levels of state self-esteem, whereas rivalry was related to (rather low and) fragile self-esteem. Analyses on underlying processes suggest that effects of rivalry on self-esteem variability are based on stronger decreases in self-esteem from one assessment to the next, particularly after a perceived lack of social inclusion. The revealed differentiated effects of admiration and rivalry explain why the analysis of narcissism as a unitary concept has led to the inconsistent past findings and provide deeper insights into the intrapersonal dynamics of grandiose narcissism governing state self-esteem. (PsycINFO Database Record

Bayesian inference on mixed-effects location scale models with skew-t distribution and mismeasured covariates for longitudinal data.

In AIDS studies, heterogeneous between and within subject variations are often observed on longitudinal endpoints. To accommodate heteroscedasticity in the longitudinal data, statistical methods have been developed to model the mean and variance jointly. Most of these methods assume (conditional) normal distributions for random errors, which is not realistic in practice. In this article, we propose a Bayesian mixed-effects location scale model with skew-t distribution and mismeasured covariates for heterogeneous longitudinal data with skewness. The proposed model captures the between-subject and within-subject (WS) heterogeneity by modeling the between-subject and WS variations with covariates as well as a random effect at subject level in the WS variance. Further, the proposed model also takes into account the covariate measurement errors, and commonly assumed normal distributions for model errors are substituted by skew-t distribution to account for skewness. Parameter estimation is carried out in a Bayesian framework. The proposed method is illustrated with a Multicenter AIDS Cohort Study. Simulation studies are performed to assess the performance of the proposed method. Copyright © 2017 John Wiley & Sons, Ltd.

Detecting and Understanding Interviewer Effects on Survey data by using a Cross-classified Mixed Effects Location–scale Model

Summary We propose a cross-classified mixed effects location–scale model for the analysis of interviewer effects in survey data. The model extends the standard two-way cross-classified random-intercept model (respondents nested in interviewers crossed with areas) by specifying the residual variance to be a function of covariates and an additional interviewer random effect. This extension provides a way to study interviewers’ effects on not just the ‘location’ (mean) of respondents’ responses, but additionally on their ‘scale’ (variability). It therefore allows researchers to address new questions such as ‘Do interviewers influence the variability of their respondents’ responses in addition to their average, and if so why?’. In doing so, the model facilitates a more complete and flexible assessment of the factors that are associated with interviewer error. We illustrate this model by using data from wave 3 of the UK Household Longitudinal Survey, which we link to a range of interviewer characteristics measured in an independent survey of interviewers. By identifying both interviewer characteristics in general, but also specific interviewers who are associated with unusually high or low or homogeneous or heterogeneous responses, the model provides a way to inform improvements to survey quality.

Intra-Individual Variability of Physical Activity in Older Adults With and Without Mild Alzheimer's Disease.

Physical activity shows promise for protection against cognitive decline in older adults with and without Alzheimer’s disease (AD). To better understand barriers to adoption of physical activity in this population, a clear understanding of daily and weekly activity patterns is needed. Most accelerometry studies report average physical activity over an entire wear period without considering the potential importance of the variability of physical activity. This study evaluated individual differences in the amount and intra-individual variability of physical activity and determined whether these differences could be predicted by AD status, day of wear, age, gender, education, and cardiorespiratory capacity. Physical activity was measured via accelerometry (Actigraph GT3X+) over one week in 86 older adults with and without AD (n = 33 and n = 53, respectively). Mixed-effects location-scale models were estimated to evaluate and predict individual differences in the amount and intra-individual variability of physical activity. Results indicated that compared to controls, participants with AD averaged 21% less activity, but averaged non-significantly greater intra-individual variability. Women and men averaged similar amounts of physical activity, but women were significantly less variable. The amount of physical activity differed significantly across days of wear. Increased cardiorespiratory capacity was associated with greater average amounts of physical activity. Investigation of individual differences in the amount and intra-individual variability of physical activity provided insight into differences by AD status, days of monitor wear, gender, and cardiovascular capacity. All individuals regardless of AD status were equally consistent in their physical activity, which may have been due to a highly sedentary sample and/or the early disease stage of those participants with AD. These results highlight the value of considering individual differences in both the amount and intra-individual variability of physical activity.

The Power to Detect and Predict Individual Differences in Within-Person Variability Using the Mixed-Effects Location-Scale Model

Our goal is to provide empirical scientists with practical tools and advice with which to test hypotheses related to individual differences in intra-individual variability using the mixed-effects location-scale model. To that end, we evaluate Type I error rates and power to detect and predict individual differences in intra-individual variability using this model and provide empirically-based guidelines for building scale models that include random and/or systematically-varying fixed effects. We also provide two power simulation programs that allow researchers to conduct a priori empirical power analyses. Our results aligned with statistical power theory, in that, greater power was observed for designs with more individuals, more repeated occasions, greater proportions of variance available to be explained, and larger effect sizes. In addition, our results indicated that Type I error rates were acceptable in situations when individual differences in intra-individual variability were not initially detectable as well as when the scale-model individual-level predictor explained all initially detectable individual differences in intra-individual variability. We conclude our paper by providing study design and model building advice for those interested in using the mixed-effects location-scale model in practice.

Physical Activity and Variation in Momentary Behavioral Cognitions: An Ecological Momentary Assessment Study.

Decisions to perform moderate-to-vigorous physical activity (MVPA) involve behavioral cognitive processes that may differ within individuals depending on the situation. Ecological momentary assessment (EMA) was used to examine the relationships of momentary behavioral cognitions (ie, self-efficacy, outcome expectancy, intentions) with MVPA (measured by accelerometer). A sample of 116 adults (mean age, 40.3 years; 72.4% female) provided real-time EMA responses via mobile phones across 4 days. Multilevel models were used to test whether momentary behavioral cognitions differed across contexts and were associated with subsequent MVPA. Mixed-effects location scale models were used to examine whether subject-level means and within-subjects variances in behavioral cognitions were associated with average daily MVPA. Momentary behavioral cognitions differed across contexts for self-efficacy (P = .007) but not for outcome expectancy (P = .53) or intentions (P = .16). Momentary self-efficacy, intentions, and their interaction predicted MVPA within the subsequent 2 hours (Ps < .01). Average daily MVPA was positively associated with within-subjects variance in momentary self-efficacy and intentions for physical activity (Ps < .05). Although momentary behavioral cognitions are related to subsequent MVPA, adults with higher average MVPA have more variation in physical activity self-efficacy and intentions. Performing MVPA may depend more on how much behavioral cognitions vary across the day than whether they are generally high or low.

Why Are Children Different in Their Daily Sedentariness? An Approach Based on the Mixed-Effects Location Scale Model.

This study aimed to investigate the between- and within-individual variability in sedentary time over seven days, using a mixed-effects location scale model. The sample comprised 686 Portuguese children (381 girls) aged 9–11 years, from 23 schools. Sedentary time was estimated by the Actigraph GT3X+ accelerometer, which was used 24 hours/day for 7 consecutive days; height, sitting height, and weight were measured, BMI was computed (WHO cut-points were used to classify subjects as normal weight or overweight/obese), and maturity offset was estimated. Information regarding the home environment was obtained by questionnaire. Results revealed that (i) children were more sedentary on Friday, but less so on Saturday and Sunday (compared to Monday), with significant variation between- and within-subjects (between-subject variance=0.800, within-subject variance=1.793, intra-subject correlation=0.308); (ii) there is a sex effect on sedentariness, with boys being less sedentary than girls (p<0.001), and the between-subject variance was 1.48 times larger for boys than girls; (iii) in terms of the within-subject variance, or erraticism, Tuesday, Wednesday and Friday have similar erraticism levels as Monday (Thursday has less, while Saturday and Sunday have more); in addition, girls (variance ratio=0.632, p<0.001), overweight/obese children (variance ratio=0.861, p=0.019), and those later mature (variance ratio=0.849, p=0.013) have less erraticism than their counterparts; (iv) the within-subject variance varied significantly across subjects (scale std dev=0.342±0.037, p<0.001); and (v) in the fixed part of the model, only biological maturation was positively related to sedentariness. This study demonstrated that there is significant between- and within-subject variability in sedentariness across a whole week. This implies that a focus on intra-individual variability, instead of only on mean values, would provide relevant information towards a more complete map of children’s sedentary behaviour, which can be helpful when developing more efficient strategies to reduce sedentariness.

Association of usual self-reported dietary intake with ecological momentary measures of affective and physical feeling states in children

Background: Little is known about the relationship between dietary intake and affective and physical feeling states in children. Purpose: The current study used Ecological Momentary Assessment (EMA) to examine how usual dietary intake is cross-sectionally associated with both average affective and physical feeling state ratings and rating variability in children. Methods: Children (N = 110, mean age = 11.0 ± 1.2 years, 52.5% male, 30.1% Hispanic/Latino) completed EMA measures of affective and physical feeling states 3–7 times per day for a full or partial day (weekday evenings and weekend days and evenings) over a 4-day period. Usual intake of pre-selected dietary components was measured prior to the EMA measurement period using the Block Kids Food Screener. Statistical analyses included mixed models and mixed-effects location scale models. Results: Greater usual fiber intake was cross-sectionally associated with higher average positive affect (PA) ratings, lower variability of NA ratings, and higher variability of physical fatigue ratings. Lower usual glycemic load of diet was cross-sectionally associated with lower variability of NA ratings. Lower usual added sugar intake was cross-sectionally associated with higher average physical energy ratings and lower variability of NA ratings. Conclusions: Although temporal precedence was not established by these findings, they indicate that characteristics of children's usual dietary intake are cross-sectionally associated with both the average and variability of affective and physical feeling states. EMA offers a promising avenue through which to explore the associations between affective states and diet and has the potential to provide insight into nuances of this relationship.

A Bivariate Mixed-Effects Location-Scale Model with application to Ecological Momentary Assessment (EMA) data.

A bivariate mixed-effects location-scale model is proposed for estimation of means, variances, and covariances of two continuous outcomes measured concurrently in time and repeatedly over subjects. Modeling the two outcomes jointly allows examination of BS and WS association between the outcomes and whether the associations are related to covariates. The variance-covariance matrices of the BS and WS effects are modeled in terms of covariates, explaining BS and WS heterogeneity. The proposed model relaxes assumptions on the homogeneity of the within-subject (WS) and between-subject (BS) variances. Furthermore, the WS variance models are extended by including random scale effects. Data from a natural history study on adolescent smoking are used for illustration. 461 students, from 9th and 10th grades, reported on their mood at random prompts during seven consecutive days. This resulted in 14,105 prompts with an average of 30 responses per student. The two outcomes considered were a subject's positive affect and a measure of how tired and bored they were feeling. Results showed that the WS association of the outcomes was negative and significantly associated with several covariates. The BS and WS variances were heterogeneous for both outcomes, and the variance of the random scale effects were significantly different from zero.

Runmixregls: A Program to Run theMIXREGLSMixed-Effects Location Scale Software from withinStata

Hedeker and Nordgren (2013) present the stand-alone MIXREGLS program for fitting the mixed-effects location scale model to continuous longitudinal and other clustered data. This model can be used when interest lies in joint modeling the mean and dispersion of subjects' responses over time. The model extends the standard two-level randomintercept mixed model by allowing both the within- and between-subject variances to be influenced by the covariates and for the within-subject variance to additionally depend on a subject random-scale effect. In this article we present the runmixregls command to run MIXREGLS seamlessly from within Stata. We illustrate the notable advantages of using runmixregls by replicating and extending the two example analyses presented in Hedeker and Nordgren (2013). We then use runmixregls to demonstrate a new and important research finding. Namely, that ignoring the random-scale effect in the withinsubject variance function will lead to the regression coefficients in this function to be estimated with spurious precision, especially the regression coefficients of subject-level covariates.

Applying Mixed-Effects Location Scale Modeling to Examine Within-Person Variability in Physical Activity Self-Efficacy

Background: Physical activity self-efficacy is conceptualized as a construct that is changeable and responsive to contextual factors. The current study applied mixed-effects location scale modeling to examine within-person variability in physical activity self-efficacy among middle-aged and older adults (N = 14 adults, mean age = 59.4 years) who were attempting behavior change. Methods: An electronic diary was used to record self-reported self-efficacy and physical activity via Ecological Momentary Assessment (EMA) twice a day (2:00 pm and 9:00 pm). Data from weeks 1-6 were analyzed using a Mixed-Effects Location Scale Model in SAS PROC NLMIXED. Results: Participants differed from each other in the degree to which physical activity self-efficacy varied from day to day (p = .03). Within-person variation in self-efficacy was negatively related to levels of brisk walking each week (p = .002), and decreased over time (p = .03). Conclusions: Preliminary results suggest that fluctuations in self-efficacy may be as important for predicting short-term behavior as the overall or mean level of self-efficacy.

MIXREGLS: A Program for Mixed-Effects Location Scale Analysis

MIXREGLS is a program which provides estimates for a mixed-effects location scale model assuming a (conditionally) normally-distributed dependent variable. This model can be used for analysis of data in which subjects may be measured at many observations and interest is in modeling the mean and variance structure. In terms of the variance structure, covariates can by specified to have effects on both the between-subject and within-subject variances. Another use is for clustered data in which subjects are nested within clusters (e.g., clinics, hospitals, schools, etc.) and interest is in modeling the between-cluster and within-cluster variances in terms of covariates. MIXREGLS was written in Fortran and uses maximum likelihood estimation, utilizing both the EM algorithm and a Newton-Raphson solution. Estimation of the random effects is accomplished using empirical Bayes methods. Examples illustrating stand-alone usage and features of MIXREGLS are provided, as well as use via the SAS and R software packages.

A three‐level mixed‐effects location scale model with an application to ecological momentary assessment data

In studies using ecological momentary assessment (EMA), or other intensive longitudinal data collection methods, interest frequently centers on changes in the variances, both within-subjects and between-subjects. For this, Hedeker et al. (Biometrics 2008; 64: 627-634) developed an extended two-level mixed-effects model that treats observations as being nested within subjects and allows covariates to influence both the within-subjects and between-subjects variance, beyond their influence on means. However, in EMA studies, subjects often provide many responses within and across days. To account for the possible systematic day-to-day variation, we developed a more flexible three-level mixed-effects location scale model that treats observations within days within subjects, and allows covariates to influence the variance at the subject, day, and observation level (over and above their usual effects on means) using a log-linear representation throughout. We provide details of a maximum likelihood solution and demonstrate how SAS PROC NLMIXED can be used to achieve maximum likelihood estimates in an alternative parameterization of our proposed three-level model. The accuracy of this approach using NLMIXED was verified by a series of simulation studies. Data from an adolescent mood study using EMA were analyzed to demonstrate this approach. The analyses clearly show the benefit of the proposed three-level model over the existing two-level approach. The proposed model has useful applications in many studies with three-level structures where interest centers on the joint modeling of the mean and variance structure.

Modeling Individual Differences in Within-Person Variation of Negative and Positive Affect in a Mixed Effects Location Scale Model Using BUGS/JAGS

A mixed effects location scale model was used to model and explain individual differences in within-person variability of negative and positive affect across 7 days (N=178) within a measurement burst design. The data come from undergraduate university students and are pooled from a study that was repeated at two consecutive years. Individual differences in level and change in mood was modeled with a random intercept and random slope where the residual within-person variability was allowed to vary across participants. Additionally changes in within-person variability were explained by the inclusion of a time-varying predictor indicating the severity of daily stressors. This model accounted for 2 location and 2 scale effects and provided evidence that individuals who reported higher severity in daily stressors also exhibited greater variability in affect—but only for participants who showed low overall affect variability and who reported low average negative affect. Those who were more variable in their affect reports overall were less reactive to daily stressors in the sense that their high levels of affect variability remained high. We describe the utility of this model for further research on individual variation and change.

Modeling between‐subject and within‐subject variances in ecological momentary assessment data using mixed‐effects location scale models

Ecological momentary assessment and/or experience sampling methods are increasingly used in health studies to study subjective experiences within changing environmental contexts. In these studies, up to 30 or 40 observations are often obtained for each subject. Because there are so many measurements per subject, one can characterize a subject's mean and variance and can specify models for both. In this article, we focus on an adolescent smoking study using ecological momentary assessment where interest is on characterizing changes in mood variation. We describe how covariates can influence the mood variances and also extend the statistical model by adding a subject-level random effect to the within-subject variance specification. This permits subjects to have influence on the mean, or location, and variability, or (square of the) scale, of their mood responses. These mixed-effects location scale models have useful applications in many research areas where interest centers on the joint modeling of the mean and variance structure.