Different Sources Of Heterogeneity Research Articles

Machine Learning Models for Blood Glucose Level Prediction in Patients With Diabetes Mellitus: Systematic Review and Network Meta-Analysis.

Machine learning (ML) models provide more choices to patients with diabetes mellitus (DM) to more properly manage blood glucose (BG) levels. However, because of numerous types of ML algorithms, choosing an appropriate model is vitally important. In a systematic review and network meta-analysis, this study aimed to comprehensively assess the performance of ML models in predicting BG levels. In addition, we assessed ML models used to detect and predict adverse BG (hypoglycemia) events by calculating pooled estimates of sensitivity and specificity. PubMed, Embase, Web of Science, and Institute of Electrical and Electronics Engineers Explore databases were systematically searched for studies on predicting BG levels and predicting or detecting adverse BG events using ML models, from inception to November 2022. Studies that assessed the performance of different ML models in predicting or detecting BG levels or adverse BG events of patients with DM were included. Studies with no derivation or performance metrics of ML models were excluded. The Quality Assessment of Diagnostic Accuracy Studies tool was applied to assess the quality of included studies. Primary outcomes were the relative ranking of ML models for predicting BG levels in different prediction horizons (PHs) and pooled estimates of the sensitivity and specificity of ML models in detecting or predicting adverse BG events. In total, 46 eligible studies were included for meta-analysis. Regarding ML models for predicting BG levels, the means of the absolute root mean square error (RMSE) in a PH of 15, 30, 45, and 60 minutes were 18.88 (SD 19.71), 21.40 (SD 12.56), 21.27 (SD 5.17), and 30.01 (SD 7.23) mg/dL, respectively. The neural network model (NNM) showed the highest relative performance in different PHs. Furthermore, the pooled estimates of the positive likelihood ratio and the negative likelihood ratio of ML models were 8.3 (95% CI 5.7-12.0) and 0.31 (95% CI 0.22-0.44), respectively, for predicting hypoglycemia and 2.4 (95% CI 1.6-3.7) and 0.37 (95% CI 0.29-0.46), respectively, for detecting hypoglycemia. Statistically significant high heterogeneity was detected in all subgroups, with different sources of heterogeneity. For predicting precise BG levels, the RMSE increases with a rise in the PH, and the NNM shows the highest relative performance among all the ML models. Meanwhile, current ML models have sufficient ability to predict adverse BG events, while their ability to detect adverse BG events needs to be enhanced. PROSPERO CRD42022375250; https://www.crd.york.ac.uk/prospero/display_record.php?RecordID=375250.

Cumulative information on quality and willingness to pay: a study on wine evaluation

ABSTRACT Availability of information about a good with uncertain quality can influence the way consumers perceive its quality, hence, their willingness to pay (WTP) for it. We present a study to investigate whether and to what extent WTP is impacted by the degree of information available to consumers who are exposed first to extrinsic and then intrinsic information regarding a variety of Italian wines. We implement linear mixed models in a Bayesian framework, which provides a flexible tool to account for different sources of heterogeneity, e.g. correlation within groups of observations and spatial correlation between participants sitting nearby. Based on data collected in Italy, results show that the availability of extrinsic and intrinsic information yields relevant changes in WTP, but this effect also depends on age, gender, drinking habits, wine quality, and connoisseurship of the agents. According to the findings, the analyzed wines cannot be considered search goods, although this might not hold for more experienced consumers.

A spatial semiparametric M-quantile regression for hedonic price modelling

This paper proposes an M-quantile regression approach to address the heterogeneity of the housing market in a modern European city. We show how M-quantile modelling is a rich and flexible tool for empirical market price data analysis, allowing us to obtain a robust estimation of the hedonic price function whilst accounting for different sources of heterogeneity in market prices. The suggested methodology can generally be used to analyse nonlinear interactions between prices and predictors. In particular, we develop a spatial semiparametric M-quantile model to capture both the potential nonlinear effects of the cultural environment on pricing and spatial trends. In both cases, nonlinearity is introduced into the model using appropriate bases functions. We show how the implicit price associated with the variable that measures cultural amenities can be determined in this semiparametric framework. Our findings show that the effect of several housing attributes and urban amenities differs significantly across the response distribution, suggesting that buyers of lower-priced properties behave differently than buyers of higher-priced properties.

Evaluation of mixed traffic performance on Croatian railways

In many countries, including Croatia, the fundamental daily activities on the railway network are predetermined by a conflict-free schedule that combines passenger and freight routes of various railway undertakings of a particular railway market. When scheduling railway traffic on a given network, the aim is to offer a suitable quality of service to the growing demand volume in a capacity-optimized manner. However, capacity-relevant operating conditions in a market with multiple undertaking competition are not the same for all conventional railway systems mostly due to different sources of heterogeneity. Heterogeneity in railways is hard to unambiguously define due to its complex multidimensional properties. Mostly it is assessed in the context of traffic conditions, which is not completely accurate. In this article, we demarcate different layers of traffic-related heterogeneity, which are present in the literature and attach the most important related research to each. There is an evident lack of practical measures for decreasing the negative impacts of heterogeneity on timetable performance and capacity utilization. Therefore, we performed a detailed analysis of mixed rail traffic on selected main corridor routes in Croatia to depict more clearly the importance of this network-specific operational problem and influencing factors and get insights into the dynamic effect of real interactions between mixed trains on a typical conventional railway line in the deregulated railway market.

Efficient Estimation of the Marginal Mean of Recurrent Events

AbstractRecurrent events are often encountered in clinical and epidemiological studies where a terminal event is also observed. With recurrent events data it is of great interest to estimate the marginal mean of the cumulative number of recurrent events experienced prior to the terminal event. The standard nonparametric estimator was suggested in Cook and Lawless and further developed in Ghosh and Lin. We here investigate the efficiency of this estimator that, surprisingly, has not been studied before. We rewrite the standard estimator as an inverse probability of censoring weighted estimator. From this representation we derive an efficient augmented estimator using efficient estimation theory for right-censored data. We show that the standard estimator is efficient in settings with no heterogeneity. In other settings with different sources of heterogeneity, we show theoretically and by simulations that the efficiency can be greatly improved when an efficient augmented estimator based on dynamic predictions is employed, at no extra cost to robustness. The estimators are applied and compared to study the mean number of catheter-related bloodstream infections in heterogeneous patients with chronic intestinal failure who can possibly die, and the efficiency gain is highlighted in the resulting point-wise confidence intervals.

Greedy optimization for growing spatially embedded oscillatory networks.

The coupling of some types of oscillators requires the mediation of a physical link between them, rendering the distance between oscillators a critical factor to achieve synchronization. In this paper, we propose and explore a greedy algorithm to grow spatially embedded oscillator networks. The algorithm is constructed in such a way that nodes are sequentially added seeking to minimize the cost of the added links' length and optimize the linear stability of the growing network. We show that, for appropriate parameters, the stability of the resulting network, measured in terms of the dynamics of small perturbations and the correlation length of the disturbances, can be significantly improved with a minimal added length cost. In addition, we analyze numerically the topological properties of the resulting networks, and we find that, while being more stable, their degree distribution is approximately exponential and independent of the algorithm parameters. Moreover, we find that other topological parameters related with network resilience and efficiency are also affected by the proposed algorithm. Finally, we extend our findings to more general classes of networks with different sources of heterogeneity. Our results are a step in the development of algorithms for the directed growth of oscillatory networks with desirable stability, dynamical and topological properties.

Modelling approaches for histology-independent cancer drugs to inform NICE appraisals: a systematic review and decision-framework.

The first histology-independent marketing authorisation in Europe was granted in 2019. This was the first time that a cancer treatment was approved based on a common biomarker rather than the location in the body at which the tumour originated. This research aims to explore the implications for National Institute for Health and Care Excellence appraisals. Targeted reviews were undertaken to determine the type of evidence that is likely to be available at the point of marketing authorisation and the analyses required to support National Institute for Health and Care Excellence appraisals. Several challenges were identified concerning the design and conduct of trials for histology-independent products, the greater levels of heterogeneity within the licensed population and the use of surrogate end points. We identified approaches to address these challenges by reviewing key statistical literature that focuses on the design and analysis of histology-independent trials and by undertaking a systematic review to evaluate the use of response end points as surrogate outcomes for survival end points. We developed a decision framework to help to inform approval and research policies for histology-independent products. The framework explored the uncertainties and risks associated with different approval policies, including the role of further data collection, pricing schemes and stratified decision-making. We found that the potential for heterogeneity in treatment effects, across tumour types or other characteristics, is likely to be a central issue for National Institute for Health and Care Excellence appraisals. Bayesian hierarchical methods may serve as a useful vehicle to assess the level of heterogeneity across tumours and to estimate the pooled treatment effects for each tumour, which can inform whether or not the assumption of homogeneity is reasonable. Our review suggests that response end points may not be reliable surrogates for survival end points. However, a surrogate-based modelling approach, which captures all relevant uncertainty, may be preferable to the use of immature survival data. Several additional sources of heterogeneity were identified as presenting potential challenges to National Institute for Health and Care Excellence appraisal, including the cost of testing, baseline risk, quality of life and routine management costs. We concluded that a range of alternative approaches will be required to address different sources of heterogeneity to support National Institute for Health and Care Excellence appraisals. An exemplar case study was developed to illustrate the nature of the assessments that may be required. Adequately designed and analysed basket studies that assess the homogeneity of outcomes and allow borrowing of information across baskets, where appropriate, are recommended. Where there is evidence of heterogeneity in treatment effects and estimates of cost-effectiveness, consideration should be given to optimised recommendations. Routine presentation of the scale of the consequences of heterogeneity and decision uncertainty may provide an important additional approach to the assessments specified in the current National Institute for Health and Care Excellence methods guide. Further exploration of Bayesian hierarchical methods could help to inform decision-makers on whether or not there is sufficient evidence of homogeneity to support pooled analyses. Further research is also required to determine the appropriate basis for apportioning genomic testing costs where there are multiple targets and to address the challenges of uncontrolled Phase II studies, including the role and use of surrogate end points. This project was funded by the National Institute for Health Research (NIHR) Evidence Synthesis programme and will be published in full in Health Technology Assessment; Vol. 25, No. 76. See the NIHR Journals Library website for further project information.

Hybrid multigroup partial least squares structural equation modelling: an application to bank employee satisfaction and loyalty

We describe a practical approach to tackling observed heterogeneity using partial least squares structural equation modelling (PLS-SEM) when the number of categorical variables is high and the context of the research is exploratory. The approach is based on combining classical multigroup PLS-SEM approach and pathmox analysis. We provide practical guidance on using our hybrid multigroup PLS-SEM and illustrate its application using real data for bank employees. In investigating work climate, specifically the relationship between satisfaction and loyalty considering specific drivers (empowerment, company reputation, leadership, pay, and work conditions) and different sources of heterogeneity (gender, age, marital status, education, job level, and antiquity), the hybrid multigroup PLS-SEM identified three partitions defined by juniors, seniors, and managers, and identified significant differences between those groups, specifically in indicating that leadership and pay were more important for juniors, empowerment for seniors, and company reputation and work conditions for managers.

The Zika Virus Individual Participant Data Consortium: A Global Initiative to Estimate the Effects of Exposure to Zika Virus during Pregnancy on Adverse Fetal, Infant, and Child Health Outcomes.

This commentary describes the creation of the Zika Virus Individual Participant Data Consortium, a global collaboration to address outstanding questions in Zika virus (ZIKV) epidemiology through conducting an individual participant data meta-analysis (IPD-MA). The aims of the IPD-MA are to (1) estimate the absolute and relative risks of miscarriage, fetal loss, and short- and long-term sequelae of fetal exposure; (2) identify and quantify the relative importance of different sources of heterogeneity (e.g., immune profiles, concurrent flavivirus infection) for the risk of adverse fetal, infant, and child outcomes among infants exposed to ZIKV in utero; and (3) develop and validate a prognostic model for the early identification of high-risk pregnancies and inform communication between health care providers and their patients and public health interventions (e.g., vector control strategies, antenatal care, and family planning programs). By leveraging data from a diversity of populations across the world, the IPD-MA will provide a more precise estimate of the risk of adverse ZIKV-related outcomes within clinically relevant subgroups and a quantitative assessment of the generalizability of these estimates across populations and settings. The ZIKV IPD Consortium effort is indicative of the growing recognition that data sharing is a central component of global health security and outbreak response.

University choice and the attractiveness of the study area: Insights on the differences amongst degree programmes in Italy based on generalised mixed-effect models

We investigate the determinants of students' university choices (where to study and which field) by linking micro-data provided by the Italian National Student Archive (ANS) on a cohort of students enrolled for the first time at the university in a.y. 2017/18 with socioeconomic indicators related to the territorial areas and other information concerning universities, fields of study and degree programmes provided by national surveys on graduated students. The aim of the analysis is to propose a set of indicators of attractiveness of tertiary education institutions that are suitable for making comparisons among fields of study and degree programmes between and within universities. To this end, students' choices to attend bachelor's degree studies outside their region of residence have been modelled within the generalised mixed-effect models framework. We show how the setting of alternative model parametrisations and the adjustment for different sources of heterogeneity in the data enable us to build up measures that are suitable to investigate the characteristics of institutions attractiveness. The results of the analysis enable us to assess and distinguish the roles played by the university, the field of study and other factors that are endogenous to tertiary education institutions in determining the power to attract students from other regions.

What Can Surface-Slip Distributions Tell Us about Fault Connectivity at Depth?

ABSTRACT Fault systems with stepovers and gaps along strike are ubiquitous in nature, and many modern earthquakes (e.g., 1992 Landers, 1999 Hector Mine, 2016 Kaikōura, and 2019 Ridgecrest) have shown that ruptures can readily propagate across some disconnections, while being halted by others. It is quite possible, however, that many faults that appear discontinuous at the surface are in fact connected at depth, facilitating throughgoing rupture, and potentially increasing earthquake size. The present work explores whether the mapped surface slip in an earthquake is indicative of the connectivity of the fault system at depth. If there is a signal of subsurface connectivity in the surface-slip pattern, then the connectivity of the system could potentially be inferred. Through 3D dynamic rupture modeling of faults with along-strike gaps of various depths, I explore whether the amplitude or the spatial distribution of slip after an earthquake could be used to diagnose fault connectivity at depth. I find that, in general, fault segments that are connected up to shallow depths of 1–2 km and are relatively long along strike compared to the seismogenic depth tend to have higher slip gradients at their edges than faults that are connected at greater depth, or that are disconnected to the bottom of the seismogenic zone. Systematic slip gradient differences at fault segment edges have been recorded in past earthquakes, giving hope that the modeled effect can be detected in many cases, even though mapped surface slip is affected by a number of different sources of heterogeneity. The results provide an alternative explanation for observations that stepovers that allow throughgoing rupture tend to have higher slip gradients than those at which rupture terminates: perhaps many such stepovers are connected at depth, which could persistently favor throughgoing rupture. There may be implications for interpretation of apparent fault discontinuities worldwide.

Mind the Gaps: How Organization Design Shapes the Sourcing of Inventions

In their quest to sustain their innovativeness, firms pursue multiple inventions, with only a small proportion of them achieving fruition. In addition to the challenge of developing their inventions, firms also face the challenge of replenishing and maintaining the flow of inventions within their pipelines. This replenishment could be done via internally or externally sourced inventions through licensing, alliance or acquisition modes. Existing research has considered this decision to take place at the firm-level and the logic for decision-making at the transaction level. We integrate the incentive- and knowledge-based views of the firm to offer a new theory to explain this decision. Within our theory, we consider that firms may decentralize such decisions within specific R&D units and that the decision-making might operate at the pipeline-level rather than at the individual transaction level. This allows us to consider different sources of heterogeneity within firms’ decision-making processes, and show how organization design can have significant implications for firms’ invention sourcing. We explore our arguments using a novel dataset of firms’ invention sourcing decisions in the pharmaceutical industry between 1995 and 2015. We find that decentralized designs with multiple R&D units are associated with a higher proportion of externally sourced inventions. This difference is primarily driven by differences in the propensity to license, and for inventions of moderate novelty. Our findings highlight an important linkage between firms’ internal organization design and their sourcing of inventions, and in doing so, show how such decision-making is impacted by both managerial incentives and intra-organizational knowledge flows.

Leveraging heterogeneity for neural computation with fading memory in layer 2/3 cortical microcircuits.

Complexity and heterogeneity are intrinsic to neurobiological systems, manifest in every process, at every scale, and are inextricably linked to the systems’ emergent collective behaviours and function. However, the majority of studies addressing the dynamics and computational properties of biologically inspired cortical microcircuits tend to assume (often for the sake of analytical tractability) a great degree of homogeneity in both neuronal and synaptic/connectivity parameters. While simplification and reductionism are necessary to understand the brain’s functional principles, disregarding the existence of the multiple heterogeneities in the cortical composition, which may be at the core of its computational proficiency, will inevitably fail to account for important phenomena and limit the scope and generalizability of cortical models. We address these issues by studying the individual and composite functional roles of heterogeneities in neuronal, synaptic and structural properties in a biophysically plausible layer 2/3 microcircuit model, built and constrained by multiple sources of empirical data. This approach was made possible by the emergence of large-scale, well curated databases, as well as the substantial improvements in experimental methodologies achieved over the last few years. Our results show that variability in single neuron parameters is the dominant source of functional specialization, leading to highly proficient microcircuits with much higher computational power than their homogeneous counterparts. We further show that fully heterogeneous circuits, which are closest to the biophysical reality, owe their response properties to the differential contribution of different sources of heterogeneity.

The impact of different sources of heterogeneity on loss of accuracy from genomic prediction models.

Cross-study validation (CSV) of prediction models is an alternative to traditional cross-validation (CV) in domains where multiple comparable datasets are available. Although many studies have noted potential sources of heterogeneity in genomic studies, to our knowledge none have systematically investigated their intertwined impacts on prediction accuracy across studies. We employ a hybrid parametric/non-parametric bootstrap method to realistically simulate publicly available compendia of microarray, RNA-seq, and whole metagenome shotgun microbiome studies of health outcomes. Three types of heterogeneity between studies are manipulated and studied: (i) imbalances in the prevalence of clinical and pathological covariates, (ii) differences in gene covariance that could be caused by batch, platform, or tumor purity effects, and (iii) differences in the "true" model that associates gene expression and clinical factors to outcome. We assess model accuracy, while altering these factors. Lower accuracy is seen in CSV than in CV. Surprisingly, heterogeneity in known clinical covariates and differences in gene covariance structure have very limited contributions in the loss of accuracy when validating in new studies. However, forcing identical generative models greatly reduces the within/across study difference. These results, observed consistently for multiple disease outcomes and omics platforms, suggest that the most easily identifiable sources of study heterogeneity are not necessarily the primary ones that undermine the ability to accurately replicate the accuracy of omics prediction models in new studies. Unidentified heterogeneity, such as could arise from unmeasured confounding, may be more important.

Identification of influencers through the wisdom of crowds.

Identifying individuals who are influential in diffusing information, ideas or products in a population remains a challenging problem. Most extant work can be abstracted by a process in which researchers first decide which features describe an influencer and then identify them as the individuals with the highest values of these features. This makes the identification dependent on the relevance of the selected features and it still remains uncertain if triggering the identified influencers leads to a behavioral change in others. Furthermore, most work was developed for cross-sectional or time-aggregated datasets, where the time-evolution of influence processes cannot be observed. We show that mapping the influencer identification to a wisdom of crowds problem overcomes these limitations. We present a framework in which the individuals in a social group repeatedly evaluate the contribution of other members according to what they perceive as valuable and not according to predefined features. We propose a method to aggregate the behavioral reactions of the members of the social group into a collective judgment that considers the temporal variation of influence processes. Using data from three large news providers, we show that the members of the group surprisingly agree on who are the influential individuals. The aggregation method addresses different sources of heterogeneity encountered in social systems and leads to results that are easily interpretable and comparable within and across systems. The approach we propose is computationally scalable and can be applied to any social systems where behavioral reactions are observable.

The stability and slow dynamics of spot patterns in the 2D Brusselator model: The effect of open systems and heterogeneities

Spot patterns, whereby the activator field becomes spatially localized near certain dynamically-evolving discrete spatial locations in a bounded multi-dimensional domain, is a common occurrence for two-component reaction–diffusion (RD) systems in the singular limit of a large diffusivity ratio. In previous studies of 2-D localized spot patterns for various specific well-known RD systems, the domain boundary was assumed to be impermeable to both the activator and inhibitor, and the reaction-kinetics were assumed to be spatially uniform. As an extension of this previous theory, we use formal asymptotic methods to study the existence, stability, and slow dynamics of localized spot patterns for the singularly perturbed 2-D Brusselator RD model when the domain boundary is only partially impermeable, as modeled by an inhomogeneous Robin boundary condition, or when there is an influx of inhibitor across the domain boundary. In our analysis, we will also allow for the effect of a spatially variable bulk feed term in the reaction kinetics. By applying our extended theory to the special case of one-spot patterns and ring patterns of spots inside the unit disk, we provide a detailed analysis of the effect on spot patterns of these three different sources of heterogeneity. In particular, when there is an influx of inhibitor across the boundary of the unit disk, a ring pattern of spots can become pinned to a ring-radius closer to the domain boundary. Under a Robin condition, a quasi-equilibrium ring pattern of spots is shown to exhibit a novel saddle–node bifurcation behavior in terms of either the inhibitor diffusivity, the Robin constant, or the ambient background concentration. A spatially variable bulk feed term, with a concentrated source of “fuel” inside the domain, is shown to yield a saddle–node bifurcation structure of spot equilibria, which leads to qualitatively new spot-pinning behavior. Results from our asymptotic theory are validated from full numerical simulations of the Brusselator model.

Military Expenditure, Endogeneity and Economic Growth

While not always a concern for the general economic growth literature, the debate over the effects of military spending on growth continues to develop, with no consensus, but a deepening understanding of the limitations of previous work. One important issue that has not been adequately dealt with is the endogeneity of military spending in the growth equation, mainly because of the difficulty of finding any variables that would make adequate instruments. This paper considers this issue, using an endogenous growth model estimated on a large sample of 109 non-high-income countries for the period 1998–2012. The empirical analysis is framed within an instrumental variable setting that exploits the increase in military spending that occurs when unrest in a country escalates to turmoil. The estimation results show that endogeneity arising from reverse causality is a crucial issue, with the instrumental variable estimates providing a larger significant negative effect of military spending on growth than OLS would. This result is found to be robust to different sources of heterogeneity and different time periods.

Cue versus independent food attributes: the effect of adding attributes in choice experiments

We examine the effects of adding an independent food attribute on consumers’ willingness to pay (WTP) estimates for both cue and independent food attributes. In three separate choice experiments, a cue attribute present along the entire sequence of choices had independent food attributes enucleated and made explicit from the cue at later stages. Logit models were estimated using (i) a complete panel approach; (ii) error components and (iii) utility in WTP-space. Results suggest that the way a subject processes food attributes depends not only on the design dimensions but also on food attributes’ functional roles. When complexity of designs increases, models that account for different sources of heterogeneity have better fit to the data.

Explaining heterogeneity in the predictive value of Type D personality for cardiac events and mortality

BackgroundType D personality has been associated with adverse outcomes in patients with coronary artery disease (CAD). However, large heterogeneity exists between Type D studies, including some studies reporting null-findings. ObjectivesThe aim of this study was to examine i) choice of endpoint and ii) age as two study characteristics that may partly explain this large heterogeneity in the Type D associated prognostic effect. MethodsWe used four existing data cohorts of 1503 CAD patients (89% male, mean age=57.2±9.1) with baseline measures of Type D and endpoints >5years follow-up: major adverse cardiac events (MACE), cardiac death/MI, and non-cardiac death. Patients were classified in 4 age categories: <50y, 50–59y, 60–69y and ≥70y. Multiple logistic regression models included age, sex, and clinical covariates. ResultsAt follow-up, there were 295 events, including 116 cardiac death/MI, and 37 non-cardiac deaths. Both continuous and categorical measures of Type D predicted adverse events. Type D was independently associated with MACE (OR=1.82; 95%CI 1.33–2.50) and cardiac death/MI (OR=2.49; 95%CI 1.55–3.99). However, Type D was not associated with non-cardiac death (OR=1.23; 95%CI 0.57–2.69). Regarding age, Type D consistently predicted MACE in the lower age groups (all ORs≥2.20, all ps≤.004), but not in patients aged ≥70y (OR=1.43, p=.57). ConclusionsChoice of endpoint and age modulated the risk conferred by Type D personality. Type D was associated with an increased risk of cardiac events, but not with non-cardiac death, or with events in patients aged ≥70y. Research on psychosocial risk in CAD should account for different sources of heterogeneity in study characteristics.

Optimal equity auctions with heterogeneous bidders

I analyze the effects of heterogeneity in terms of bidders' valuation distributions and standalone values in equity auctions—auctions in which bidders offer equities rather than cash. Heterogeneity misaligns equity bids' face values, monetary values, and bidder types, making the seller's revenues sensitive to the auction design. Given heterogeneity, I identify the mechanism that maximizes the expected revenue among all incentive-compatible mechanisms of equity auctions. I show how different sources of heterogeneity alter the optimal design, revenues, and allocational efficiency. Unlike optimal cash auctions (Myerson, 1981), where bidders' standalone values are irrelevant, the allocation of optimal equity auctions favors bidders with lower standalone values, because the seller can extract a larger proportion of their rents. By contrast, when bidders differ only in valuation distributions, optimal equity auctions feature more efficient allocations than optimal cash auctions.