The block numerical range of analytic operator functions

We introduce the block numerical range Wn(L) of an operator function L with respect to a decomposition H = H1⊕. . .⊕Hn of the underlying Hilbert space. Our main results include the spectral inclusion property and estimates of the norm of the resolvent for analytic L . They generalise, and improve, the corresponding results for the numerical range (which is the case n = 1) since the block numerical range is contained in, and may be much smaller than, the usual numerical range. We show that refinements of the decomposition entail inclusions between the corresponding block numerical ranges and that the block numerical range of the operator matrix function L contains those of its principal subminors. For the special case of operator polynomials, we investigate the boundedness of Wn(L) and we prove a Perron-Frobenius type result for the block numerical radius of monic operator polynomials with coefficients that are positive in Hilbert lattice sense.

Elucidating the Functional Relationship Between Working Memory Capacity and Psychometric Intelligence: A Fixed-Links Modeling Approach for Experimental Repeated-Measures Designs

Numerous studies reported a strong link between working memory capacity (WMC) and fluid intelligence (Gf), although views differ in respect to how close these two constructs are related to each other. In the present study, we used a WMC task with five levels of task demands to assess the relationship between WMC and Gf by means of a new methodological approach referred to as fixed-links modeling. Fixed-links models belong to the family of confirmatory factor analysis (CFA) and are of particular interest for experimental, repeated-measures designs. With this technique, processes systematically varying across task conditions can be disentangled from processes unaffected by the experimental manipulation. Proceeding from the assumption that experimental manipulation in a WMC task leads to increasing demands on WMC, the processes systematically varying across task conditions can be assumed to be WMC-specific. Processes not varying across task conditions, on the other hand, are probably independent of WMC. Fixed-links models allow for representing these two kinds of processes by two independent latent variables. In contrast to traditional CFA where a common latent variable is derived from the different task conditions, fixed-links models facilitate a more precise or purified representation of the WMC-related processes of interest. By using fixed-links modeling to analyze data of 200 participants, we identified a non-experimental latent variable, representing processes that remained constant irrespective of the WMC task conditions, and an experimental latent variable which reflected processes that varied as a function of experimental manipulation. This latter variable represents the increasing demands on WMC and, hence, was considered a purified measure of WMC controlled for the constant processes. Fixed-links modeling showed that both the purified measure of WMC (β = .48) as well as the constant processes involved in the task (β = .45) were related to Gf. Taken together, these two latent variables explained the same portion of variance of Gf as a single latent variable obtained by traditional CFA (β = .65) indicating that traditional CFA causes an overestimation of the effective relationship between WMC and Gf. Thus, fixed-links modeling provides a feasible method for a more valid investigation of the functional relationship between specific constructs.

Bargaining for power and information : an experimental and theoretical analysis of the interaction between institutions, externalities and adverse selection

Bargaining is the building block of many economic interactions, ranging from bilateral to multilateral encounters and from situations in which the actors are individuals to negotiations between firms or countries. In all these settings, economists have been intrigued for a long time by the fact that some projects, trades or agreements are not realized even though they are mutually beneficial. On the one hand, this has been explained by incomplete information. A firm may not be willing to offer a wage that is acceptable to a qualified worker, because it knows that there are also unqualified workers and cannot distinguish between the two types. This phenomenon is known as adverse selection. On the other hand, it has been argued that even with complete information, the presence of externalities may impede efficient outcomes. To see this, consider the example of climate change. If a subset of countries agrees to curb emissions, non-participant regions benefit from the signatories’ efforts without incurring costs. These free riding opportunities give rise to incentives to strategically improve ones bargaining power that work against the formation of a global agreement. This thesis is concerned with extending our understanding of both factors, adverse selection and externalities. The findings are based on empirical evidence from original laboratory experiments as well as game theoretic modeling. On a very general note, it is demonstrated that the institutions through which agents interact matter to a large extent. Insights are provided about which institutions we should expect to perform better than others, at least in terms of aggregate welfare. Chapters 1 and 2 focus on the problem of adverse selection. Effective operation of markets and other institutions often depends on good information transmission properties. In terms of the example introduced above, a firm is only willing to offer high wages if it receives enough positive signals about the worker’s quality during the application and wage bargaining process. In Chapter 1, it will be shown that repeated interaction coupled with time costs facilitates information transmission. By making the wage bargaining process costly for the worker, the firm is able to obtain more accurate information about the worker’s type. The cost could be pure time cost from delaying agreement or cost of effort arising from a multi-step interviewing process. In Chapter 2, I abstract from time cost and show that communication can play a similar role. The simple fact that a worker states to be of high quality may be informative. In Chapter 3, the focus is on a different source of inefficiency. Agents strive for bargaining power and thus may be motivated by incentives that are at odds with the socially efficient outcome. I have already mentioned the example of climate change. Other examples are coalitions within committees that are formed to secure voting power to block outcomes or groups that commit to different technological standards although a single standard would be optimal (e.g. the format war between HD and BlueRay). It will be shown that such inefficiencies are directly linked to the presence of externalities and a certain degree of irreversibility in actions. I now discuss the three articles in more detail. In Chapter 1, Olivier Bochet and I study a simple bilateral bargaining institution that eliminates trade failures arising from incomplete information. In this setting, a buyer makes offers to a seller in order to acquire a good. Whenever an offer is rejected by the seller, the buyer may submit a further offer. Bargaining is costly, because both parties suffer a (small) time cost after any rejection. The difficulties arise, because the good can be of low or high quality and the quality of the good is only known to the seller. Indeed, without the possibility to make repeated offers, it is too risky for the buyer to offer prices that allow for trade of high quality goods. When allowing for repeated offers, however, at equilibrium both types of goods trade with probability one. We provide an experimental test of these predictions. Buyers gather information about sellers using specific price offers and rates of trade are high, much as the model’s qualitative predictions. We also observe a persistent over-delay before trade occurs, and this mitigates efficiency substantially. Possible channels for over-delay are identified in the form of two behavioral assumptions missing from the standard model, loss aversion (buyers) and haggling (sellers), which reconcile the data with the theoretical predictions. Chapter 2 also studies adverse selection, but interaction between buyers and sellers now takes place within a market rather than isolated pairs. Remarkably, in a market it suffices to let agents communicate in a very simple manner to mitigate trade failures. The key insight is that better informed agents (sellers) are willing to truthfully reveal their private information, because by doing so they are able to reduce search frictions and attract more buyers. Behavior observed in the experimental sessions closely follows the theoretical predictions. As a consequence, costless and non-binding communication (cheap talk) significantly raises rates of trade and welfare. Previous experiments have documented that cheap talk alleviates inefficiencies due to asymmetric information. These findings are explained by pro-social preferences and lie aversion. I use appropriate control treatments to show that such consideration play only a minor role in our market. Instead, the experiment highlights the ability to organize markets as a new channel through which communication can facilitate trade in the presence of private information. In Chapter 3, I theoretically explore coalition formation via multilateral bargaining under complete information. The environment studied is extremely rich in the sense that the model allows for all kinds of externalities. This is achieved by using so-called partition functions, which pin down a coalitional worth for each possible coalition in each possible coalition structure. It is found that although binding agreements can be written, efficiency is not guaranteed, because the negotiation process is inherently non-cooperative. The prospects of cooperation are shown to crucially depend on i) the degree to which players can renegotiate and gradually build up agreements and ii) the absence of a certain type of externalities that can loosely be described as incentives to free ride. Moreover, the willingness to concede bargaining power is identified as a novel reason for gradualism. Another key contribution of the study is that it identifies a strong connection between the Core, one of the most important concepts in cooperative game theory, and the set of environments for which efficiency is attained even without renegotiation.

Nonparametric location estimators in the randomized complete block design

Several tests for the comparison of different groups in the randomized complete block design exist. However, there is a lack of robust estimators for the location difference between one group and all the others on the original scale. The relative marginal effects are commonly used in this situation, but they are more difficult to interpret and use by less experienced people because of the different scale. In this paper two nonparametric estimators for the comparison of one group against the others in the randomized complete block design will be presented. Theoretical results such as asymptotic normality, consistency, translation invariance, scale preservation, unbiasedness, and median unbiasedness are derived. The finite sample behavior of these estimators is derived by simulations of different scenarios. In addition, possible confidence intervals with these estimators are discussed and their behavior derived also by simulations.

Asset Pricing with Heterogeneous Agents, Incomplete Markets and Trading Constraints

The consumption capital asset pricing model is the standard economic model used to capture stock market behavior. However, empirical tests have pointed out to its inability to account quantitatively for the high average rate of return and volatility of stocks over time for plausible parameter values. Recent research has suggested that the consumption of stockholders is more strongly correlated with the performance of the stock market than the consumption of non-stockholders. We model two types of agents, non-stockholders with standard preferences and stock holders with preferences that incorporate elements of the prospect theory developed by Kahneman and Tversky (1979). In addition to consumption, stockholders consider fluctuations in their financial wealth explicitly when making decisions. Data from the Panel Study of Income Dynamics are used to calibrate the labor income processes of the two types of agents. Each agent faces idiosyncratic shocks to his labor income as well as aggregate shocks to the per-share dividend but markets are incomplete and agents cannot hedge consumption risks completely. In addition, consumers face both borrowing and short-sale constraints. Our results show that in equilibrium, agents hold different portfolios. Our model is able to generate a time-varying risk premium of about 5.5% while maintaining a low risk free rate, thus suggesting a plausible explanation for the equity premium puzzle reported by Mehra and Prescott (1985).

Bayesian doubly optimal group sequential designs for randomized clinical trials

When conducting a randomized comparative clinical trial, ethical, scientific or economic considerations often motivate the use of interim decision rules after successive groups of patients have been treated. These decisions may pertain to the comparative efficacy or safety of the treatments under study, cost considerations, the desire to accelerate the drug evaluation process, or the likelihood of therapeutic benefit for future patients. At the time of each interim decision, an important question is whether patient enrollment should continue or be terminated; either due to a high probability that one treatment is superior to the other, or a low probability that the experimental treatment will ultimately prove to be superior. The use of frequentist group sequential decision rules has become routine in the conduct of phase III clinical trials. In this dissertation, we will present a new Bayesian decision-theoretic approach to the problem of designing a randomized group sequential clinical trial, focusing on two-arm trials with time-to-failure outcomes. Forward simulation is used to obtain optimal decision boundaries for each of a set of possible models. At each interim analysis, we use Bayesian model selection to adaptively choose the model having the largest posterior probability of being correct, and we then make the interim decision based on the boundaries that are optimal under the chosen model. We provide a simulation study to compare this method, which we call Bayesian Doubly Optimal Group Sequential (BDOGS), to corresponding frequentist designs using either O'Brien-Fleming (OF) or Pocock boundaries, as obtained from EaSt 2000. Our simulation results show that, over a wide variety of different cases, BDOGS either performs at least as well as both OF and Pocock, or on average provides a much smaller trial. ^

Bayesian dual threshold design with Dirichlet distribution: An alternative way to frequentist multi-stage phase II designs in oncology

Many phase II clinical studies in oncology use two-stage frequentist design such as Simon's optimal design. However, they have a common logistical problem regarding the patient accrual at the interim. Strictly speaking, patient accrual at the end of the first stage may have to be suspended until all patients have events, success or failure. For example, when the study endpoint is six-month progression free survival, patient accrual has to be stopped until all outcomes from stage I is observed. However, study investigators may have concern when accrual is suspended after the first stage due to the loss of accrual momentum during this hiatus. We propose a two-stage phase II design that resolves the patient accrual problem due to an interim analysis, and it can be used as an alternative way to frequentist two-stage phase II studies in oncology. ^

Bayesian adaptive designs for drug combinations in early phase clinical trials

Treating patients with combined agents is a growing trend in cancer clinical trials. Evaluating the synergism of multiple drugs is often the primary motivation for such drug-combination studies. Focusing on the drug combination study in the early phase clinical trials, our research is composed of three parts: (1) We conduct a comprehensive comparison of four dose-finding designs in the two-dimensional toxicity probability space and propose using the Bayesian model averaging method to overcome the arbitrariness of the model specification and enhance the robustness of the design; (2) Motivated by a recent drug-combination trial at MD Anderson Cancer Center with a continuous-dose standard of care agent and a discrete-dose investigational agent, we propose a two-stage Bayesian adaptive dose-finding design based on an extended continual reassessment method; (3) By combining phase I and phase II clinical trials, we propose an extension of a single agent dose-finding design. We model the time-to-event toxicity and efficacy to direct dose finding in two-dimensional drug-combination studies. We conduct extensive simulation studies to examine the operating characteristics of the aforementioned designs and demonstrate the designs' good performances in various practical scenarios.^

Bayesian Adaptive Designs for Early Phase Clinical Trials

My dissertation focuses mainly on Bayesian adaptive designs for phase I and phase II clinical trials. It includes three specific topics: (1) proposing a novel two-dimensional dose-finding algorithm for biological agents, (2) developing Bayesian adaptive screening designs to provide more efficient and ethical clinical trials, and (3) incorporating missing late-onset responses to make an early stopping decision. Treating patients with novel biological agents is becoming a leading trend in oncology. Unlike cytotoxic agents, for which toxicity and efficacy monotonically increase with dose, biological agents may exhibit non-monotonic patterns in their dose-response relationships. Using a trial with two biological agents as an example, we propose a phase I/II trial design to identify the biologically optimal dose combination (BODC), which is defined as the dose combination of the two agents with the highest efficacy and tolerable toxicity. A change-point model is used to reflect the fact that the dose-toxicity surface of the combinational agents may plateau at higher dose levels, and a flexible logistic model is proposed to accommodate the possible non-monotonic pattern for the dose-efficacy relationship. During the trial, we continuously update the posterior estimates of toxicity and efficacy and assign patients to the most appropriate dose combination. We propose a novel dose-finding algorithm to encourage sufficient exploration of untried dose combinations in the two-dimensional space. Extensive simulation studies show that the proposed design has desirable operating characteristics in identifying the BODC under various patterns of dose-toxicity and dose-efficacy relationships. Trials of combination therapies for the treatment of cancer are playing an increasingly important role in the battle against this disease. To more efficiently handle the large number of combination therapies that must be tested, we propose a novel Bayesian phase II adaptive screening design to simultaneously select among possible treatment combinations involving multiple agents. Our design is based on formulating the selection procedure as a Bayesian hypothesis testing problem in which the superiority of each treatment combination is equated to a single hypothesis. During the trial conduct, we use the current values of the posterior probabilities of all hypotheses to adaptively allocate patients to treatment combinations. Simulation studies show that the proposed design substantially outperforms the conventional multi-arm balanced factorial trial design. The proposed design yields a significantly higher probability for selecting the best treatment while at the same time allocating substantially more patients to efficacious treatments. The proposed design is most appropriate for the trials combining multiple agents and screening out the efficacious combination to be further investigated. The proposed Bayesian adaptive phase II screening design substantially outperformed the conventional complete factorial design. Our design allocates more patients to better treatments while at the same time providing higher power to identify the best treatment at the end of the trial. Phase II trial studies usually are single-arm trials which are conducted to test the efficacy of experimental agents and decide whether agents are promising to be sent to phase III trials. Interim monitoring is employed to stop the trial early for futility to avoid assigning unacceptable number of patients to inferior treatments. We propose a Bayesian single-arm phase II design with continuous monitoring for estimating the response rate of the experimental drug. To address the issue of late-onset responses, we use a piece-wise exponential model to estimate the hazard function of time to response data and handle the missing responses using the multiple imputation approach. We evaluate the operating characteristics of the proposed method through extensive simulation studies. We show that the proposed method reduces the total length of the trial duration and yields desirable operating characteristics for different physician-specified lower bounds of response rate with different true response rates.

Bayesian adaptive designs in early phase clinical trials

There are two practical challenges in the phase I clinical trial conduct: lack of transparency to physicians, and the late onset toxicity. In my dissertation, Bayesian approaches are used to address these two problems in clinical trial designs. The proposed simple optimal designs cast the dose finding problem as a decision making process for dose escalation and deescalation. The proposed designs minimize the incorrect decision error rate to find the maximum tolerated dose (MTD). For the late onset toxicity problem, a Bayesian adaptive dose-finding design for drug combination is proposed. The dose-toxicity relationship is modeled using the Finney model. The unobserved delayed toxicity outcomes are treated as missing data and Bayesian data augment is employed to handle the resulting missing data. Extensive simulation studies have been conducted to examine the operating characteristics of the proposed designs and demonstrated the designs' good performances in various practical scenarios.^

On Bayesian seamless phase I-II designs

Early phase clinical trial designs have long been the focus of interest for clinicians and statisticians working in oncology field. There are several standard phse I and phase II designs that have been widely-implemented in medical practice. For phase I design, the most commonly used methods are 3+3 and CRM. A newly-developed Bayesian model-based mTPI design has now been used by an increasing number of hospitals and pharmaceutical companies. The advantages and disadvantages of these three top phase I designs have been discussed in my work here and their performances were compared using simulated data. It was shown that mTPI design exhibited superior performance in most scenarios in comparison with 3+3 and CRM designs. ^ The next major part of my work is proposing an innovative seamless phase I/II design that allows clinicians to conduct phase I and phase II clinical trials simultaneously. Bayesian framework was implemented throughout the whole design. The phase I portion of the design adopts mTPI method, with the addition of futility rule which monitors the efficacy performance of the tested drugs. Dose graduation rules were proposed in this design to allow doses move forward from phase I portion of the study to phase II portion without interrupting the ongoing phase I dose-finding schema. Once a dose graduated to phase II, adaptive randomization was used to randomly allocated patients into different treatment arms, with the intention of more patients being assigned to receive more promising dose(s). Again simulations were performed to compare the performance of this innovative phase I/II design with a recently published phase I/II design, together with the conventional phase I and phase II designs. The simulation results indicated that the seamless phase I/II design outperform the other two competing methods in most scenarios, with superior trial power and the fact that it requires smaller sample size. It also significantly reduces the overall study time. ^ Similar to other early phase clinical trial designs, the proposed seamless phase I/II design requires that the efficacy and safety outcomes being able to be observed in a short time frame. This limitation can be overcome by using validated surrogate marker for the efficacy and safety endpoints.^

Journeying Beyond the Balcony to the Block Party and Up to 50,000 Feet: Disparities in Children’s Health and Healthcare

Invited Commentary on “Observations from the Balcony: Directions for Pediatric Health Disparities Research and Policy".