Risk‐Group‐Specific Dose Finding Based on an Average Toxicity Score

Summary We propose a Bayesian dose‐finding design that accounts for two important factors, the severity of toxicity and heterogeneity in patients' susceptibility to toxicity. We consider toxicity outcomes with various levels of severity and define appropriate scores for these severity levels. We then use a multinomial‐likelihood function and a Dirichlet prior to model the probabilities of these toxicity scores at each dose, and characterize the overall toxicity using an average toxicity score (ATS) parameter. To address the issue of heterogeneity in patients' susceptibility to toxicity, we categorize patients into different risk groups based on their susceptibility. A Bayesian isotonic transformation is applied to induce an order‐restricted posterior inference on the ATS. We demonstrate the performance of the proposed dose‐finding design using simulations based on a clinical trial in multiple myeloma.     

A New Class of Completely Self‐Designing Clinical Trials

A flexible method is proposed for group sequentially performed clinical trials which allows for an adaptive, data‐driven sample size reassessment at each stage. By also adaptively assigning different weights to the several stages the total number of study parts can be steered to an intended early or late end of the trial in dependence on all information available prior to a stage. Although at each stage the null hypothesis is tested on rejection, the full level‐α‐test is preserved at the end of the study. The proposed method is not restricted to normally distributed responses. The discussed adaptive designing is a useful tool provided that a priori information about parameters involved in the trial are not available or subject to uncertainty. The presented learning algorithm enables the complete self‐designing of a study.     

Blinded Sample Size Recalculation for Normally Distributed Outcomes Using Long‐ and Short‐term Data

Clinical trials are often planned with high uncertainty about the variance of the primary outcome variable. A poor estimate of the variance, however, may lead to an over‐ or underpowered study. In the internal pilot study design, the sample variance is calculated at an interim step and the sample size can be adjusted if necessary. The available recalculation procedures use the data of those patients for sample size recalculation that have already completed the study. In this article, we consider a variance estimator that takes into account both the data at the endpoint and at an intermediate point of the treatment phase. We derive asymptotic properties of this estimator and the relating sample size recalculation procedure. In a simulation study, the performance of the proposed approach is evaluated and compared with the procedure that uses only long‐term data. Simulation results demonstrate that the sample size resulting from the proposed procedure shows in general a smaller variability. At the same time, the Type I error rate is not inflated and the achieved power is close to the desired value.     

Time‐to‐event continual reassessment method incorporating treatment cycle information with application to an oncology phase I trial

Delayed dose limiting toxicities (i.e. beyond first cycle of treatment) is a challenge for phase I trials. The time‐to‐event continual reassessment method (TITE‐CRM) is a Bayesian dose‐finding design to address the issue of long observation time and early patient drop‐out. It uses a weighted binomial likelihood with weights assigned to observations by the unknown time‐to‐toxicity distribution, and is open to accrual continually. To avoid dosing at overly toxic levels while retaining accuracy and efficiency for DLT evaluation that involves multiple cycles, we propose an adaptive weight function by incorporating cyclical data of the experimental treatment with parameters updated continually. This provides a reasonable estimate for the time‐to‐toxicity distribution by accounting for inter‐cycle variability and maintains the statistical properties of consistency and coherence. A case study of a First‐in‐Human trial in cancer for an experimental biologic is presented using the proposed design. Design calibrations for the clinical and statistical parameters are conducted to ensure good operating characteristics. Simulation results show that the proposed TITE‐CRM design with adaptive weight function yields significantly shorter trial duration, does not expose patients to additional risk, is competitive against the existing weighting methods, and possesses some desirable properties.     

Equilibrium and non‐equilibrium phases in the radiation of Hakea and the drivers of diversity in Mediterranean‐type ecosystems

Mediterranean‐type ecosystems (MTEs) contain exceptional plant diversity. Explanations for this diversity are usually classed as either “equilibrium,” with elevated MTE diversity resulting from greater ecological carrying capacities, or “non‐equilibrium,” with MTEs having a greater accumulation of diversity over time than other types of ecosystems. These models have typically been considered as mutually exclusive. Here, we present a trait‐based explanatory framework that incorporates both equilibrium and non‐equilibrium dynamics. Using a large continental Australian plant radiation (Hakea) as a case study, we identify traits associated with niche partitioning in coexisting species (α‐traits) and with environmental filtering (β‐traits), and reconstruct the mode and relative timing of diversification of these traits. Our results point to a radiation with an early non‐equilibrium phase marked by divergence of β‐traits as Hakea diversified exponentially and expanded from the southwest Australian MTE into biomes across the Australian continent. This was followed from seven million years ago by an equilibrium phase, marked by diversification of α‐traits and a slowdown in lineage diversification as MTE‐niches became saturated. These results suggest that processes consistent with both equilibrium and non‐equilibrium models have been important during different stages of the radiation of Hakea, and together they provide a richer explanation of present‐day diversity patterns.     

A Comparison of Procedures for Adaptive Choice of Location Tests in Flexible Two‐Stage Designs

Although linear rank statistics for the two‐sample problem are distribution free tests, their power depends on the distribution of the data. In the planning phase of an experiment, researchers are often uncertain about the shape of this distribution and so the choice of test statistic for the analysis and the determination of the required sample size are based on vague information. Adaptive designs with interim analysis can potentially overcome both problems. And in particular, adaptive tests based on a selector statistic are a solution to the first. We investigate whether adaptive tests can be usefully implemented in flexible two‐stage designs to gain power. In a simulation study, we compare several methods for choosing a test statistic for the second stage of an adaptive design based on interim data with the procedure that applies adaptive tests in both stages. We find that the latter is a sensible approach that leads to the best results in most situations considered here. The different methods are illustrated using a clinical trial example.     

SIZE‐DEPENDENT MORTALITY AND COMPETITION INTERACTIVELY SHAPE COMMUNITY DIVERSITY

Body size is recognized as a major factor in evolutionary processes mediating sympatric diversification and community structuring. Life‐history types with distinct body sizes can result from two fundamental mechanisms, size‐dependent competition and size‐dependent mortality. While previous theoretical studies investigated these two processes in separation, the model analyzed here allows both selective forces to affect body‐size evolution interactively. Here we show for the first time that in the presence of size‐dependent competition, size‐dependent mortality can give rise to multiple, coexisting size morphs representing the final outcomes of evolution. Moreover, our results demonstrate that interactions between size‐dependent competition and mortality can create characteristic abrupt changes in size structure and nonmonotonic patterns of biological diversity along continuous and monotonic environmental gradients. We find that the two selective forces differentially affect the body‐size ratios of coexisting morphs: size‐dependent competition results in small and relatively constant ratios, whereas size‐dependent mortality can open niches for morphs that greatly differ in body size. We show that these differential effects result in characteristic distributions of size ratios across communities, which we suggest can help detect the concurrent action and relative influence of size‐dependent competition and mortality in nature.     

An Adaptive Location‐Scale Test

Increasing locations are often accompanied by an increase in variability. In this case apparent heteroscedasticity can indicate that there are treatment effects and it is appropriate to consider an alternative involving differences in location as well as in scale. As a location‐scale test the sum of a location and a scale test statistic can be used. However, the power can be raised through weighting the sum. In order to select values for this weighting an adaptive design with an interim analysis is proposed: The data of the first stage are used to calculate the weights and with the second stage's data a weighted location‐scale test is carried out. The p‐values of the two stages are combined through Fisher's combination test. With a Lepage‐type location‐scale test it is illustrated that the resultant adaptive test can be more powerful than the ‘optimum’ test with no interim analysis. The principle to calculate weights, which cannot be reasonably chosen a priori, with the data of the first stage may be useful for other tests which utilize weighted statistics, too. Furthermore, the proposed test is illustrated with an example from experimental ecology.     

A PARAMETRIC METHOD FOR ASSESSING DIVERSIFICATION‐RATE VARIATION IN PHYLOGENETIC TREES

Phylogenetic hypotheses are frequently used to examine variation in rates of diversification across the history of a group. Patterns of diversification‐rate variation can be used to infer underlying ecological and evolutionary processes responsible for patterns of cladogenesis. Most existing methods examine rate variation through time. Methods for examining differences in diversification among groups are more limited. Here, we present a new method, parametric rate comparison (PRC), that explicitly compares diversification rates among lineages in a tree using a variety of standard statistical distributions. PRC can identify subclades of the tree where diversification rates are at variance with the remainder of the tree. A randomization test can be used to evaluate how often such variance would appear by chance alone. The method also allows for comparison of diversification rate among a priori defined groups. Further, the application of the PRC method is not restricted to monophyletic groups. We examined the performance of PRC using simulated data, which showed that PRC has acceptable false‐positive rates and statistical power to detect rate variation. We apply the PRC method to the well‐studied radiation of North American Plethodon salamanders, and support the inference that the large‐bodied Plethodon glutinosus clade has a higher historical rate of diversification compared to other Plethodon salamanders.     

On the Use of the Shapiro‐Wilk Test in Two‐Stage Adaptive Inference for Paired Data from Moderate to Very Heavy Tailed Distributions

Paired data arises in a wide variety of applications where often the underlying distribution of the paired differences is unknown. When the differences are normally distributed, the t‐test is optimum. On the other hand, if the differences are not normal, the t‐test can have substantially less power than the appropriate optimum test, which depends on the unknown distribution. In textbooks, when the normality of the differences is questionable, typically the non‐parametric Wilcoxon signed rank test is suggested. An adaptive procedure that uses the Shapiro‐Wilk test of normality to decide whether to use the t‐test or the Wilcoxon signed rank test has been employed in several studies. Faced with data from heavy tails, the U.S. Environmental Protection Agency (EPA) introduced another approach: it applies both the sign and t‐tests to the paired differences, the alternative hypothesis is accepted if either test is significant. This paper investigates the statistical properties of a currently used adaptive test, the EPA's method and suggests an alternative technique. The new procedure is easy to use and generally has higher empirical power, especially when the differences are heavy‐tailed, than currently used methods.     

Predator‐induced spine length and exocuticle thickness in Leucorrhinia dubia (Insecta: Odonata): a simple physiological trade‐off?

1. Morphological defence structures evolve against predators but are costly to the individual, and are induced only when required. A well‐studied example is the development of longer abdominal spines in dragonfly larvae in the presence of fish. Numerous attempts to discover trade‐offs between spine size and behaviour, development time or body size have, however, produced little evidence. 2. We considered a physiological trade‐off. Spines consist of cuticle and using material to build longer structures may result in less material remaining elsewhere. We therefore measured exocuticle thickness at nine locations on Leucorrhinia dubia larvae from habitats with and without fish. 3. Our results show a significant effect of the interaction between fish presence and spine length on head and fore leg exocuticle thickness. Relative thickness increased with relative length of lateral spine 9 in the absence of fish, whereas no such relationship existed with fish. Hence, synthesis and secretion of cuticle material occur as a trade‐off when larvae react to fish presence. 4. We assume the mechanism to be a selective synthesis of material with different responses in different parts of the larval body. These findings offer a new angle to the fish/spine trade off debate.     

Stem tilting in the inter‐tropical cactus Echinocactus platyacanthus: an adaptive solution to the trade‐off between radiation acquisition and temperature control

While plants require radiation for photosynthesis, radiation in warm deserts can have detrimental effects from high temperatures. This dilemma may be solved through plant morphological attributes. In cold deserts, stem tilting keeps reproductive organs warm by increasing radiation interception at the cost of decreased annual light interception. Conversely, little is known about stem tilting in warm deserts. We hypothesised that stem tilting in Echinocactus platyacanthus prevents high temperatures near the apex, where reproduction occurs. The study was conducted in the warm, inter‐tropical portion of the Chihuahuan Desert, Mexico. We found that cacti preferentially tilted towards the south, which reduced temperatures of reproductive organs during the hot season, but increased total annual near‐apex PAR interception. Tilting also maximised reproduction, a likely consequence of temperature control but perhaps also of the difficulty in translocating photosynthates in cacti; therefore, annual energy acquisition near floral meristems may be largely allocated to reproduction. Unlike plants of higher latitudes, in inter‐tropical deserts sunlight at noon comes either from the north or the south, depending on the season, and thus stem tilting may more strongly affect total annual radiation received in different portions of the stem. Inter‐tropical cacti can synchronise reproduction with irradiance peaks if flowering occurs in a specific (north or south) portion of the stem; also, they effectively solve the conflict between maximising annual PAR interception and minimising temperature at the hottest time of day. Notably, the two inter‐tropical cacti in which stem tilting has been studied successfully solve this conflict.     

Evolution of cranial shape in a continental‐scale evolutionary radiation of Australian lizards

A defining character of adaptive radiations is the evolution of a diversity of morphological forms that are associated with the use of different habitats, following the invasion of vacant niches. Island adaptive radiations have been thoroughly investigated but continental scale radiations are more poorly understood. Here, we use 52 species of Australian agamid lizards and their Asian relatives as a model group, and employ three‐dimensional geometric morphometrics to characterize cranial morphology and investigate whether variation in cranial shape reflects patterns expected from the ecological process of adaptive radiation. Phylogenetic affinity, evolutionary allometry, and ecological life habit all play major roles in the evolution of cranial shape in the sampled lizards. We find a significant association between cranial shapes and life habit. Our results are in line with the expectations of an adaptive radiation, and this is the first time detailed geometric morphometric analyses have been used to understand the selective forces that drove an adaptive radiation at a continental scale.     

Frequency‐dependent two‐sex models: a new approach to sex ratio evolution with multiple maternal conditions

Mothers that experience different individual or environmental conditions may produce different proportions of male to female offspring. The Trivers‐Willard hypothesis, for instance, suggests that mothers with different qualities (size, health, etc.) will use different sex ratios if maternal quality differentially affects sex‐specific reproductive success. Condition‐dependent, or facultative, sex ratio strategies like these allow multiple sex ratios to coexist within a population. They also create complex population structure due to the presence of multiple maternal conditions. As a result, modeling facultative sex ratio evolution requires not only sex ratio strategies with multiple components, but also two‐sex population models with explicit stage structure. To this end, we combine nonlinear, frequency‐dependent matrix models and multidimensional adaptive dynamics to create a new framework for studying sex ratio evolution. We illustrate the applications of this framework with two case studies where the sex ratios depend one of two possible maternal conditions (age or quality). In these cases, we identify evolutionarily singular sex ratio strategies, find instances where one maternal condition produces exclusively male or female offspring, and show that sex ratio biases depend on the relative reproductive value ratios for each sex.