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Multilevel models are an increasingly popular method to analyze data that originate from a clustered or hierarchical structure. To effectively utilize multilevel models, one must have an adequately large number of clusters; otherwise, some model parameters will be estimated with bias. The goals for this paper are to (1) raise awareness of the problems associated with a small number of clusters, (2) review previous studies on multilevel models with a small number of clusters, (3) to provide an illustrative simulation to demonstrate how a simple model becomes adversely affected by small numbers of clusters, (4) to provide researchers with remedies if they encounter clustered data with a small number of clusters, and (5) to outline methodological topics that have yet to be addressed in the literature.

Background: The article focuses on the relationship between students’ expectations and persistence in the context of higher education. It explores the role that high expectations play in increasing the probability of adult students’ persistence, controlling for individual sociodemographic attributes, skills preparation, values, and commitments. Method: A multilevel logistic model was applied to data on 2,697 first-year students who were enrolled in 54 programmes at a Portuguese public university during 2015-2016. Results: The findings suggest that high academic expectations are relevant to older students, since such expectations increase their likelihood of persistence. Being admitted to their first-choice programmes and differences in their study habits also contribute to increasing the probability of persistence. In the presence of such motivational and behavioural attributes, we did not find statistically significant differences according to students’ socioeconomic background or gender. Our results also suggest that the relationship between prior academic achievement and persistence varies randomly across programmes. Conclusions: This institutional research study gives evidence towards the relevance of taking into account the level of programmes/courses in order to support interventions that effectively meet the students´ expectations and, thus, could increase the probability of persistence for all students entering HE.

Digital literacy refers to a set of competencies related to the skilled use of computers and information technology. Low digital skills can be a barrier for older adults’ full participation in a digital society, and COVID-19 has increased this risk of social exclusion. Older adults’ digital inclusion is a complex process that consists of the interplay of structural and individual factors. The ACCESS project unwrapped the complexity of the process and developed an innovative, multilevel model that illustrates how societal, institutional, material and pedagogical aspects shape adults’ appropriation of digital literacy. A holistic model describes factors contributing to older adults’ digital literacy, acknowledging sociocultural contexts, environments, learning settings and instruction practices for learning digital literacy. Instead of seeing older adults’ reasons for learning digital skills purely as individual choice, this model recognizes the interpersonal, institutional and societal aspects that implicitly or explicitly influence older adults’ acquisition of digital literacy. The results offer a tool for stakeholders, the research community, companies, designers and other relevant stakeholders to consider digital skills and the given support. It demands diverse communication between different stakeholders about the things that should be discussed when organizing digital support in digitalized societies.

ABSTRACT Aversive conditioning changes visuocortical responses to conditioned cues, and the generalization of these changes to perceptually similar cues may provide mechanistic insights into anxiety and fear disorders. Yet, as in many areas of cognitive neuroscience, testing hypotheses about trial‐by‐trial dynamics in conditioning paradigms is challenged by poor single‐trial signal‐to‐noise ratios (SNR), missing trials, and inter‐individual differences. The present technical report demonstrates how a state‐of‐the‐art Bayesian workflow can overcome these issues, using a preliminary sample of simultaneously recorded EEG‐fMRI data. A preliminary group of observers (N = 24) viewed circular gratings varying in orientation, with only one orientation paired with an aversive outcome (noxious electric pulse). Gratings were flickered at 15 Hz to evoke steady‐state visual evoked potentials (ssVEPs), recorded with 31 channels of EEG in an MRI scanner. First, the benefits of a Bayesian multilevel structure are demonstrated on the fMRI data by improving a standard fMRI first‐level multiple regression. Next, the Bayesian modeling approach is demonstrated by applying a theory‐driven learning model to the EEG data. The multilevel structure of the Bayesian learning model informs and constrains estimates per participant, providing an interpretable generative model. In the example analysis provided in this report, it showed superior cross‐validation accuracy and provided insights into participant‐level learning dynamics. It also isolated the generalization effects of conditioning, providing improved statistical certainty. Lastly, missing trials were interpolated and weighted appropriately using the full model's structure. This is a critical aspect for single‐trial analyses of simultaneously recorded physiological measures because each added measure will typically increase the number of trials missing a complete set of observations. The present report aims to illustrate the utility of this analytical framework. It shows how models may be iteratively built and compared in a modern Bayesian workflow. Future models may use different conceptualizations of learning, allow integration of clinically relevant factors, and enable the fusion of different simultaneous recordings such as EEG, autonomic, behavioral, and hemodynamic data. A Bayesian model had superior cross‐validation accuracy of ssVEPs while also being more informative. Left, Model 2 holds the effect of cue constant across participants. Right, the learning model (3) better isolates the aversive conditioning and generalization by allowing learning rates to differ per participant.

Analysis of variance (ANOVA) is an extremely important method in exploratory and confirmatory data analysis. Unfortunately, in complex problems (e.g., split-plot designs), it is not always easy to set up an appropriate ANOVA. We propose a hierarchical analysis that automatically gives the correct ANOVA comparisons even in complex scenarios. The inferences for all means and variances are performed under a model with a separate batch of effects for each row of the ANOVA table. We connect to classical ANOVA by working with finite-sample variance components: fixed and random effects models are characterized by inferences about existing levels of a factor and new levels, respectively. We also introduce a new graphical display showing inferences about the standard deviations of each batch of effects. We illustrate with two examples from our applied data analysis, first illustrating the usefulness of our hierarchical computations and displays, and second showing how the ideas of ANOVA are helpful in understanding a previously fit hierarchical model.

Improving the estimation accuracy for the energy consumption of electric vehicles (EVs) would greatly contribute to alleviating the range anxiety of drivers and serve as a critical basis for the planning, operation, and management of charging infrastructures. To address the challenges in energy consumption estimation encountered due to sparse Global Positioning System (GPS) observations, an estimation model is proposed that considers both the kinetic characteristics from sparse GPS observations and the unique attributes of EVs: (1) work opposing the rolling resistance; (2) aerodynamic friction losses; (3) energy consumption/generation depending on the grade of the route; (4) auxiliary load consumption; and (5) additional energy losses arising from the unstable power output of the electric motor. Two quantities, the average energy consumption per kilometer and the energy consumption for an entire trip, were focused on and compared for model fitness, parameter, and effectiveness, and the latter showed a higher fitness. Based on sparse GPS observations of 68 EVs in Aichi Prefecture, Japan, the traditional linear regression approach and a multilevel mixed-effects linear regression approach were used for model calibration. The proposed model showed a high accuracy and demonstrated a great potential for application in using sparse GPS observations to predict the energy consumption of EVs.

Multilevel modelling is sometimes used for data from complex surveys involving multistage sampling, unequal sampling probabilities and stratification. We consider generalized linear mixed models and particularly the case of dichotomous responses. A pseudolikelihood approach for accommodating inverse probability weights in multilevel models with an arbitrary number of levels is implemented by using adaptive quadrature. A sandwich estimator is used to obtain standard errors that account for stratification and clustering. When level 1 weights are used that vary between elementary units in clusters, the scaling of the weights becomes important. We point out that not only variance components but also regression coefficients can be severely biased when the response is dichotomous. The pseudolikelihood methodology is applied to complex survey data on reading proficiency from the American sample of the 'Program for international student assessment' 2000 study, using the Stata program gllamm which can estimate a wide range of multilevel and latent variable models. Performance of pseudo-maximum-likelihood with different methods for handling level 1 weights is investigated in a Monte Carlo experiment. Pseudo-maximum-likelihood estimators of (conditional) regression coefficients perform well for large cluster sizes but are biased for small cluster sizes. In contrast, estimators of marginal effects perform well in both situations. We conclude that caution must be exercised in pseudo-maximum-likelihood estimation for small cluster sizes when level 1 weights are used.

While linear mixed modeling methods are foundational concepts introduced in any statistical education, adequate general methods for interval estimation involving models with more than a few variance components are lacking, especially in the unbalanced setting. Generalized fiducial inference provides a possible framework that accommodates this absence of methodology. Under the fabric of generalized fiducial inference along with sequential Monte Carlo methods, we present an approach for interval estimation for both balanced and unbalanced Gaussian linear mixed models. We compare the proposed method to classical and Bayesian results in the literature in a simulation study of two-fold nested models and two-factor crossed designs with an interaction term. The proposed method is found to be competitive or better when evaluated based on frequentist criteria of empirical coverage and average length of confidence intervals for small sample sizes. A MATLAB implementation of the proposed algorithm is available from the authors.

BACKGROUND When analysing effects of maternal age on child outcomes, many researchers estimate sibling models to control for unobserved factors shared between siblings. Some have included birth year in these models, as it is linked to maternal age and may also have independent effects. However, this creates a linear-dependence problem. OBJECTIVE One aim is to illustrate how misleading the results may actually be when attempts are made to separate effects of maternal age and birth year in a sibling analysis. Another goal is to present and discuss the multilevel-multiprocess model as an alternative. METHODS Infant mortality was chosen as the outcome. Births and infant deaths were simulated from a multilevel-multiprocess model that included two equations for fertility, with a joint random effect, and one equation for infant mortality, with another random effect. The two random effects were set to be correlated. The effects of the independent variables were taken from simpler models estimated from register data. Various sibling models and multilevel-multiprocess models were estimated from these simulated births and deaths. In some simulations, two standard assumptions about the random effects were intentionally violated. CONTRIBUTIONS The paper illustrates how problematic it is to include both maternal age and birth year in a sibling model. Also, if only maternal age is included, but along with other reproductive variables, small categories should be used. It is argued that a multilevel-multiprocess model may be used instead to separate effects of maternal age and birth year, but this approach also has limitations, which are discussed.