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The choice between fixed and random effects

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The choice between fixed and random effects In prior chapters, group-specific effects were assumed to be drawn from a distribution, typically Gaussian. In applied research in the social and behavioral sciences, economics, public health, public policy, and many other fields, alternatives to this choice are often made, with the most common being the “fixed effects” approach. In its most basic formulation, group-specific intercepts are modeled using indicator variables, effectively making them open parameters in the model and not assigning them a distributional form. The choice between random and fixed effects has implications for the interpretation of parameter estimates and for estimation efficiency. 1 Specifically, the effect estimates for each group will be different depending on the modeling choice; the β estimates for the predictors in the model may be different depending on the modeling choice; the reliability with which one can make predictions for new groups differs; and there is ...

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... Observed covariates divide naturally into two types: X ij , which vary only within groups (e.g., the student's sex), and W j , which vary only between groups (e.g., average teacher salary in the school). Note that any observed covariate can be split into these two parts via group mean centering (Enders and Tofighi 2007;Neuhaus and Kalbfleisch 1998;Raudenbush 2009;Townsend et al. 2013), and we do this automatically in software. 2 ...
... •τ W (within estimator): This can be implemented in many ways, one of which is to group-mean center all variables (including Z) and then regress the outcome on these transformed predictors, reporting the coefficient on centered Z. The resulting estimateτ W is equivalent to the so-called fixed effects estimate,τ F E , obtained by OLS estimation on a model with indicators for each group (see Townsend et al. 2013, for further details.). Translating the exogeneity assumption to the notation of our DGP, this estimator assumes E(Z ij (ζ y U ij + y ij )) = 0, which will only hold when either ζ y = 0 or ζ z = 0. ...
... It is a weighted-average of the within-and between-estimators, but the weights introduce an additional parameter that we are unable to unbiasedly estimate, and as such, we cannot easily "learn" from this model. See Townsend et al. (2013) and Appendix B for some discussion. Causal researchers may note that these estimators are sometimes use to target different estimands. ...
... and group means ( x j ) in the model, which will lead to within estimates for the household level indicators that are similar to those resulting from a model including country fixed effects (e.g., Allison, 2009;Schunk & Perales, 2017;Townsend, Buckley, Harada, & Scott, 2013). 8 For the application of this approach to nonlinear models, additional assumptions should be tested (Schunk & Perales, 2017). ...
... We repeat the main models with interactions by income groups separately for different scenarios. To assess the impact of the choice of governance indicator, we estimate the main models using three alternative measures: the rule of law from the Worldwide Governance Indicators (Kaufmann et al., 2010, p. 4); the mean of three political risk components, namely corruption, law and order, and bureaucracy quality, of the International Country Risk Guide (ICRG) rating (Teorell et al., 2016;The PRS Group, 2016); and the corruption perception index by Transparency International (2015). While we do not find a significant positive effect for either low-or middle-income countries in case of the corruption indicator, we find the same substantive results for the rule of law and ICRG indicators. ...
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... One question for controlling omitted household heterogeneity and contrasting children within the same household is whether household fixed or household random effect models should be adopted. The sample has a large number of households with a small number of children within each one, which means fixed effect may not be the most appropriate choice (Goldstein, 2003;Snijders and Bosker, 1999;Townsend et al., 2013). Furthermore, households that only have children of the same gender (no within-group variation) will be ignored when using fixed effect models (Bartels, 2008;Townsend et al., 2013). ...
... The sample has a large number of households with a small number of children within each one, which means fixed effect may not be the most appropriate choice (Goldstein, 2003;Snijders and Bosker, 1999;Townsend et al., 2013). Furthermore, households that only have children of the same gender (no within-group variation) will be ignored when using fixed effect models (Bartels, 2008;Townsend et al., 2013). More importantly, fixed effect logistic model also requires within-group variation in dependent variables. ...
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This research studies whether children’s gender influences household adults’ perceptions of their illnesses and the pattern of seeking medical treatments for them, the aim of which is to understand to what extent minor girls (under 15) are discriminated against in Chinese rural households’ allocation of curative healthcare. Using the 2014 wave of China Family Panel Studies (CFPS), we find that households in rural China do allocate more medical resources to boys than girls. Gender differences mainly exist in children’s hospitalization rates and medical expenditures. Girls are especially disadvantaged as they grow older. They also face an added problem of sibling rivalry that leads to sick girls being less likely to be taken to a hospital when they have siblings of the same gender. These results suggest that sick girls in rural China may not be able to receive sufficient curative healthcare due to son preference. This is not only a threat to girls’ well-being, but also a potential cause of the imbalanced sex-ratio of the Chinese population.
... These typical assumptions 16, 44 are generally not too restrictive. 34,47 The associated standard deviation τ > 0 is estimated along with the other parameters and deserves some attention: It explicitly quantifies the heterogeneity between providers beyond the heterogeneity that can be explained by variations in volume and other providerspecific as well as patient-specific effects incorporated in the model. This enables direct comparisons between the two components of the provider-specific effect b i (see (3)), i.e. the volume effect f vol and the volume-independent effect represented by the u i . ...
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Despite the ongoing strong interest in associations between quality of care and the volume of health care providers, a unified statistical framework for analyzing them is missing, and many studies suffer from poor statistical modelling choices. We propose a flexible, additive mixed model for studying volume-outcome associations in health care that takes into account individual patient characteristics as well as provider-specific effects through a multi-level approach. More specifically, we treat volume as a continuous variable, and its effect on the considered outcome is modelled as a smooth function. We take account of different case-mixes by including patient-specific risk factors and of clustering on the provider level through random intercepts. This strategy enables us to extract a smooth volume effect as well as volume-independent provider effects. These two quantities can be compared directly in terms of their magnitude, which gives insight into the sources of variability of quality of care. Based on a causal DAG, we derive conditions under which the volume-effect can be interpreted as a causal effect. The paper provides confidence sets for each of the estimated quantities relying on joint estimation of all effects and parameters. Our approach is illustrated through simulation studies and an application to German health care data. Keywords: health care quality measurement, volume-outcome analysis, minimum provider volume, additive regression models, random intercept
... Although useful for making inferences about each specific level-2 unit (see Cushing et al. 2014), an unfortunate limitation is that other cluster-level-2 predictors then cannot be examined. Recent work has shown how endogeneity tests indicating the superiority of a fixed effects model over a random effects model actually indicate that fixed slopes of constant-centered level-1 predictors have been smushed (Bell et al. 2019, Hamaker & Muthén 2020, McNeish & Kelly 2019, Townsend et al. 2013. The term correlated random effects model describes the addition of a contextual-level-2 fixed slope to remove the correlation of a constant-centered level-1 predictor with a level-2 random intercept, thus preventing predictor endogeneity bias (Antonakis et al. 2021). ...
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This review focuses on the use of multilevel models in psychology and other social sciences. We target readers who are catching up on current best practices and sources of controversy in the specification of multilevel models. We first describe common use cases for clustered, longitudinal, and cross-classified designs, as well as their combinations. Using examples from both clustered and longitudinal designs, we then address issues of centering for observed predictor variables: its use in creating interpretable fixed and random effects of predictors, its relationship to endogeneity problems (correlations between predictors and model error terms), and its translation into multivariate multilevel models (using latent-centering within multilevel structural equation models). Finally, we describe novel extensions—mixed-effects location–scale models—designed for predicting differential amounts of variability.
... Using a regularization term with positive tuning parameter, the parameters are defined and estimable, compare also Friedman et al. (2010), where this procedure has been used in overparameterized multinomial regression models. The choice between fixed and random effects was discussed extensively in Townsend et al. (2013). One advantage of fixed effects models is that they do not have to assume that random ...
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A general paired comparison model for the evaluation of sport competitions is proposed. It efficiently uses the available information by allowing for ordered response categories and team-specific home advantage effects. Penalized estimation techniques are used to identify clusters of teams that share the same ability. The model is extended to include team-specific explanatory variables. It is shown that regularization techniques allow to identify the contribution of explanatory variables to the success of teams. The usefulness of the methods is demonstrated by investigating the performance and its dependence on the budget for football teams of the German Bundesliga.
... They can also reduce heterogeneity bias (Hsiao, 1986). And also, since we expect substantial differences to be present within the groups, fixed-effects models can capture these within-group variations (Townsend, Buckley, Harada, & Scott, 2013). The approach is also expected to absorb the impacts of demographic variables not explicitly accounted. ...
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Frozen seafood marketing in grocery stores in the United States (U.S.) has undergone substantial transformation as a result of the introduction of value-added and convenience products into the category. However, it is not yet clear whether consumers perceive these value-added products to be substitutes for the traditional unbreaded products. We model the demand for frozen seafood in the United States using the linear approximate almost ideal demand system (LA-AIDS) employing market-level monthly retail scanner panel data. Our emphasis is on the demand relationships between the three aggregate frozen seafood categories, namely, breaded products, entrées and unbreaded products, and on the demand relationships for these categories when disaggregated as finfish and shellfish. We use fixed effects on the spatial and temporal variation in demand and incorporated demographic shifter variables. Our results show that unbreaded products, as compared to value-added categories of breaded seafood and seafood entrées, would gain market share if expenditure on frozen seafood increases. We also find that unbreaded products are stronger substitutes for value-added products than vice versa. We explore similar relationships between frozen shellfish and finfish products. Unbreaded shellfish can be expected to gain market share if expenditure on frozen seafood were to increase.
... would violate the assumption that fixed and random effects are orthogonal (Wooldridge, 2010). We thus used fixed effects models to more accurately measure relationships between traits, site characteristics and a species' presence (Townsend, Buckley, Harada, & Scott, 2013). ...
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Anthropogenic land use change is an important driver of impacts to biological communities and the ecosystem services they provide. Pollination is one ecosystem service that may be threatened by community disassembly. Relatively little is known about changes in bee community composition in the tropics, where pollination limitation is most severe and land use change is rapid. Understanding how anthropogenic changes alter community composition and functioning has been hampered by high variability in responses of individual species. Trait-based approaches, however, are emerging as a potential method for understanding responses of ecologically-similar species to global change. 2. We studied how communities of tropical, eusocial stingless bees (Apidae: Meliponini) disassemble when forest is lost. These bees are vital tropical pollinators that exhibit high trait diversity, but are under considerable threat from human activities. 3. We compared functional traits of stingless bee species found in pastures surrounded by differing amounts forest in an extensively deforested landscape in southern Costa Rica. 4. Our results suggest that foraging traits modulate competitive interactions that underlie community disassembly patterns. In contrast to both theoretical predictions and temperate bee communities, we found that stingless bee species with the widest diet breadths were less likely to persist in sites with less forest. These wide-diet-breadth species also tend to be solitary foragers, and are competitively subordinate to group-foraging stingless bee species. Thus, displacement by dominant, group-foraging species may make subordinate species more dependent on the larger or more diversified resource pool that natural habitats offer. We also found that traits that may reduce reliance on trees – nesting in the ground or inside nests of other species – correlated with persistence in highly deforested landscapes. 5. The functional trait perspective we employed enabled capturing community processes in analyses and suggests that land use change may disassemble bee communities via different mechanisms in temperate and tropical areas. Our results further suggest that community processes, such as competition, can be important regulators of community disassembly under land use change. A better understanding of community disassembly processes is critical for conserving and restoring pollinator communities and the ecosystem services and functions they provide. This article is protected by copyright. All rights reserved.
... Finally, I investigate the causality between DnSp and economic growth through instrumental regressionthat is, whether the broadband speed increase can be shown to cause economic changes in towns (section 4.5.1.c). Comcast (also known by its trade name Xfinity), the largest internet provider in Based on a Hausman Specification Test (Townsend et al, 2013) result that rejected (p < .01) the null hypothesis that a random effects model was preferred, I used a fixed effects model formulation throughout. Table 4.5.1 presents the estimation results for overall and pre vs post analyses. ...
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Information and Communication Technologies (ICT) form the backbone of today’s technological and economic growth. Advances in ICT enabled technologies can be beneficial to the economic development of different sectors of a society. However, to take full advantage of ICT, availability of high-speed broadband is critical. Infrastructure development to provide high speed connection has huge upfront cost, and therefore posit a significant challenge for local governments, who are the primary decision makers with respect to broadband expansion plans. Scant research is available to measure the impact of broadband speed on local economies. In a related vein, the explosion in broadband speed enables rapid data sharing among multiple entities. Despite significant benefits of such data sharing, concerns over proper data protection are mounting. Legislative frameworks around such data access and protection are difficult to pass, averaging around 6% success rate over the last 20 years. This dissertation combines data from state, federal, and commercial sources to develop predictive models to quantify the impact of bandwidth expansion at a local level. The predictive analysis demonstrates differing impacts related to town size and local geographies, indicating the necessity to calculate benefits before implementing bandwidth expansion decisions. A causal analysis using an instrumental variable approach finds that broadband speed significantly improves a town’s household income and real estate development. Turning to the legislative action, analysis of U.S. Congressional bills shows that legislation addressing data protection and associated ICT issues may take several attempts and more than two years to pass. The study also finds that the textual information in bills augments traditional metadata in determining the key influential factors of such legislative success. Overall, to the best of my knowledge, this study is the first to provide significant insights to decision makers for ICT investment at local level, and legislative success factors for ICT related bills at the national level. These issues greatly affect technology availability and associated regulatory framework in different sectors of the society, and impact both personal lives and markets.
... We do not use multilevel or hierarchical models because they assume the error terms are uncorrelated with the independent variables. If this assumption is violated, the estimates are biased(Townsend et al. 2013). As explained in this section, due to individual heterogeneity, the error terms are likely to be correlated with homeworking decisions. ...
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With the expansion of high-speed internet during the recent decades, a growing number of people are working from home. Yet there is no consensus on how working from home affects workers’ well-being in the literature. Using data from the 2010, 2012, and 2013 American Time Use Survey Well-Being Modules, this paper examines how subjective well-being varies among wage/salary workers between working at home and working in the workplace using individual fixed-effects models. We find that compared to working in the workplace, bringing work home on weekdays is associated with less happiness, and telework on weekdays or weekends/holidays is associated with more stress. The effect of working at home on subjective well-being also varies by parental status and gender. Parents, especially fathers, report a lower level of subjective well-being when working at home on weekdays but a higher level of subjective well-being when working at home on weekends/holidays. Non-parents’ subjective well-being does not vary much by where they work on weekdays, but on weekends/holidays childless males feel less painful whereas childless females feel more stressed when teleworking instead of working in the workplace. This paper provides new evidence on the impact of working at home and sheds lights for policy makers and employers to re-evaluate the benefits of telework.
... This result can be framed with respect to the bias-variance tradeoff. For predictions of elements in groups, such as the predicted HTCB in a plot, mixed-effects models will compensate between a "global" part of the model, common to all groups, and the "local" observations within the group via the predicted random effect, which can lead to bias over all predictions (Townsend et al., 2013;Clark and Linzer, 2015). This is structurally different from the OLS correction factor that regresses purely on the local residuals and does not incorporate the relative share of random effect variance as a weight, reducing potential bias for plot-level predictions, and driving up the variance, resulting in larger RMSEs relative to the NMEM. ...
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Height-to-crown-base (HTCB) measurements are frequently used as inputs for growth and yield models. They are essential for reliable projections of stand structure over time required for sustainable forest management. Often, HTCB is measured alongside total height (HT) for only a subsample of trees and must be imputed for unobserved measurements in the sample. Typically, HTCB models require HT as a covariate, requiring a “double” imputation for missing HTCB measurements, leading to potentially substantial error propagation issues. We compared the effects of subsample size, imputation method, and use of imputed rather than measured HT on the accuracy of HTCB predictions. HT and HTCB were imputed using a nonlinear fixed-effects model (NFEM), a NFEM with a correction factor estimated using an ordinary least squares (OLS) regression on the subsampled measurements, and a nonlinear mixed-effects model (NMEM) with stand- and plot-level random parameters. Using cross-validated bias and root mean squared error (RMSE), our results indicated that NMEM obtained the smallest RMSE at subsample sizes greater than one, with RMSEs ranging between 2.19 and 2.51 m. However, while NMEM provided generally smaller RMSEs, biases ranging between −0.92 and −0.29 m existed at subsample sizes less than three trees per plot, and −2.86 m when no subsample was available. We observed negative bias when using imputed rather than measured HT for both the correction factor and mixed-effects model that is mitigated to a range of −0.51 m to −0.08 m with subsample sizes of at least two trees. Following this, we recommend using NMEMs for HTCB imputation with at least four trees per plot.
... A priori, modeling a grouping variable as fixed or random effect are equally well suited for 74 multilevel analysis (Townsend et al. 2013) and strict rules don't exist because the best 75 strategy generally depends on the goal of the analysis (Gelman & Hill 2007, see Box 2). 76 ...
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Biological data are often intrinsically hierarchical. Due to their ability to account for such dependencies, mixed-effect models have become a common analysis technique in ecology and evolution. While many questions around their theoretical foundations and practical applications are solved, one fundamental question is still highly debated: When having a low number of levels should we model a grouping variable as a random or fixed effect? In such situation, the variance of the random effect is presumably underestimated, but whether this affects the statistical properties of the fixed effects is unclear. Here, we analyze the consequences of including a grouping variable as fixed or random effect and possible other modeling options (over and underspecified models) for data with small number of levels in the grouping variable (2 - 8). For all models, we calculated type I error rates, power and coverage. Moreover, we show the influence of possible study designs on these statistical properties. We found that mixed-effect models already for two groups correctly estimate variance for two groups. Moreover, model choice does not influence the statistical properties when there is no random slope in the data-generating process. However, if an ecological effect differs among groups, using a random slope and intercept model, and switching to a fixed-effect model only in case of a singular fit avoids overconfidence in the results. Additionally, power and type I error are strongly influenced by the number of and the difference between groups. We conclude that inferring the correct random effect structure is of high importance to get correct statistical properties. When in doubt, we recommend starting with the simpler model and using model diagnostics to identify missing components. When having identified the correct structure, we encourage to start with a mixed-effects model independent of the number of groups and only in case of a singular fit switch to a fixed-effect model. With these recommendations, we allow for more informative choices about study design and data analysis and thus make ecological inference with mixed-effects models more robust for low number of groups.
... One further aspect is that it is assumed that the random effects and the covariates observed per second level unit are independent; a criticism that has a long tradition, in particular in the econometric literature. For an overview on the choice between fixed and random effects models, see, Townsend et al (2013). As an alternative we consider the fixed effect or subject-specific model ...
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