Stochastic finite element analysis of biological systems: comparison of a simple intervertebral disc model with experimental results

Statistical methods allow the effects of uncertainty to be incorporated into finite element models. This has potential benefits for the analysis of biological systems where natural variability can give rise to substantial uncertainty in both material and geometrical properties. In this study, a simple model of the intervertebral disc under compression was created and analysed as both a deterministic and a stochastic system. Factorial analysis was used to determine the important parameters to be included in the stochastic analysis. The predictions from the model were compared to experimental results from 21 sheep discs. The size and shape of the distribution of the axial deformations predicted by the model was consistent with the experimental results given that the number of model solutions far exceeded the number of experimental results. Stochastic models could be valuable in determining the range and most likely value of stress in a tissue or implant.     

Stochastic Finite Element Analysis of Biological Systems: Comparison of a Simple Intervertebral Disc Model with Experimental Results

Statistical methods allow the effects of uncertainty to be incorporated into finite element models. This has potential benefits for the analysis of biological systems where natural variability can give rise to substantial uncertainty in both material and geometrical properties. In this study, a simple model of the intervertebral disc under compression was created and analysed as both a deterministic and a stochastic system. Factorial analysis was used to determine the important parameters to be included in the stochastic analysis. The predictions from the model were compared to experimental results from 21 sheep discs. The size and shape of the distribution of the axial deformations predicted by the model was consistent with the experimental results given that the number of model solutions far exceeded the number of experimental results. Stochastic models could be valuable in determining the range and most likely value of stress in a tissue or implant.     

单木生物量模型估计区域尺度生物量的不确定性

采用系统抽样体系江西省固定样地杉木连续观测数据和生物量数据,通过Monte Carlo法反复模拟由单木生物量模型推算区域尺度地上生物量的过程,估计了江西省杉木地上总生物量。基于不同水平建模样本量n及不同决定系数R~2的设计,分别研究了单木生物量模型参数变异性及模型残差变异性对区域尺度生物量估计不确定性的影响。研究结果表明:2009年江西省杉木地上生物量估计值为(19.84±1.27)t/hm~2,不确定性占生物量估计值约6.41%。生物量估计值和不确定性值达到平稳状态所需的运算时间随建模样本量及决定系数R~2的增大而减小;相对于模型参数变异性,残差变异性对不确定性的影响更小。     

Implications of Uncertainty and Variability in the Life Cycle Assessment of Pig Production Systems(7 pp)

Goal, Scope and Background Calculating LCA outcomes implies the use of parameters, models, choices and scenarios which introduce uncertainty, as they imperfectly account for the variability of both human and environmental systems. The analysis of the uncertainty of LCA results, and its reduction by an improved estimation of key parameters and through the improvement of the models used to convert emissions into regional impacts, such as eutrophication, are major issues for LCA. Methods In a case study of pig production systems, we propose a simple quantification of the uncertainty of LCA results (intra-system variability) and we explore the inter-system variability to produce more robust LCA outcomes. The quantification of the intra-system uncertainty takes into account the variability of the technical performance (crop yield, feed efficiency) and of emission factors (for NH3, N2O and NO3) and the influence of the functional unit (FU) (kg of pig versus hectare used). For farming systems, the inter-system variability is investigated through differentiating the production mode (conventional, quality label, organic (OA)), and the farmer practices (Good Agricultural Practice (GAP) versus Over Fertilised (OF)), while for natural systems, variability due to physical and climatic characteristics of catchments expected to modify nitrate fate is explored. Results and Conclusion For the eutrophication and climate change impact categories, the uncertainty associated with field emissions contributes more to the overall uncertainty than the uncertainty associated with emissions from livestock buildings, with crop yield and with feed efficiency. For acidification, the uncertainty of emissions from livestock buildings is the single most important contributor to the overall uncertainty. The influence of the FU on eutrophication results is very important when comparing systems with different degrees of intensification such as GAP and OA. Concerning the inter-system variability, differences in farmer practices have a larger effect on eutrophication than differences between production modes. Finally, the physical characteristics of the catchment and the climate strongly affect the results for eutrophication. In conclusion, in this case study, the main sources of uncertainty are in the estimation of emission factors, due both to the variability of environmental conditions and to lack of knowledge (emissions of N2O at the field level), but also in the model used for assessing regional impacts such as eutrophication. Recommendation and Perspective Suitable deterministic simulation models integrating the main controlling variables (environmental conditions, farmer practices, technology used) should be used to predict the emissions of a given system as well as their probabilistic distribution allowing the use of stochastic modelling. Finally, our simulations on eutrophication illustrate the necessity of integrating the fate of pollutants in models of impact assessment and highlight the important margin of improvement existing for the eutrophication impact assessment model.     

Dependence of mechanical behavior of the murine tail disc on regional material properties: a parametric finite element study

In vivo rodent tail models are becoming more widely used for exploring the role of mechanical loading on the initiation and progression of intervertebral disc degeneration. Historically, finite element models (FEMs) have been useful for predicting disc mechanics in humans. However, differences in geometry and tissue properties may limit the predictive utility of these models for rodent discs. Clearly, models that are specific for rodent tail discs and accurately simulate the disc's transient mechanical behavior would serve as important tools for clarifying disc mechanics in these animal models. An FEM was developed based on the structure, geometry, and scale of the mouse tail disc. Importantly, two sources of time-dependent mechanical behavior were incorporated: viscoelasticity of the matrix, and fluid permeation. In addition, a novel strain-dependent swelling pressure was implemented through the introduction of a dilatational stress in nuclear elements. The model was then validated against data from quasi-static tension-compression and compressive creep experiments performed previously using mouse tail discs. Finally, sensitivity analyses were performed in which material parameters of each disc subregion were individually varied. During disc compression, matrix consolidation was observed to occur preferentially at the periphery of the nucleus pulposus. Sensitivity analyses revealed that disc mechanics was greatly influenced by changes in nucleus pulposus material properties, but rather insensitive to variations in any of the endplate properties. Moreover, three key features of the model-nuclear swelling pressure, lamellar collagen viscoelasticity, and interstitial fluid permeation-were found to be critical for accurate simulation of disc mechanics. In particular, collagen viscoelasticity dominated the transient behavior of the disc during the initial 2200 s of creep loading, while fluid permeation governed disc deformation thereafter. The FEM developed in this study exhibited excellent agreement with transient creep behavior of intact mouse tail motion segments. Notably, the model was able to produce spatial variations in nucleus pulposus matrix consolidation that are consistent with previous observations in nuclear cell morphology made in mouse discs using confocal microscopy. Results of this study emphasize the need for including nucleus swelling pressure, collagen viscoelasticity, and fluid permeation when simulating transient changes in matrix and fluid stress/strain. Sensitivity analyses suggest that further characterization of nucleus pulposus material properties should be pursued, due to its significance in steady-state and transient disc mechanical response.     

Effects of sample size on the performance of species distribution models

A wide range of modelling algorithms is used by ecologists, conservation practitioners, and others to predict species ranges from point locality data. Unfortunately, the amount of data available is limited for many taxa and regions, making it essential to quantify the sensitivity of these algorithms to sample size. This is the first study to address this need by rigorously evaluating a broad suite of algorithms with independent presence–absence data from multiple species and regions. We evaluated predictions from 12 algorithms for 46 species (from six different regions of the world) at three sample sizes (100, 30, and 10 records). We used data from natural history collections to run the models, and evaluated the quality of model predictions with area under the receiver operating characteristic curve (AUC). With decreasing sample size, model accuracy decreased and variability increased across species and between models. Novel modelling methods that incorporate both interactions between predictor variables and complex response shapes (i.e. GBM, MARS-INT, BRUTO) performed better than most methods at large sample sizes but not at the smallest sample sizes. Other algorithms were much less sensitive to sample size, including an algorithm based on maximum entropy (MAXENT) that had among the best predictive power across all sample sizes. Relative to other algorithms, a distance metric algorithm (DOMAIN) and a genetic algorithm (OM-GARP) had intermediate performance at the largest sample size and among the best performance at the lowest sample size. No algorithm predicted consistently well with small sample size ( n  < 30) and this should encourage highly conservative use of predictions based on small sample size and restrict their use to exploratory modelling.     

Uncertainty analysis of material properties and morphology parameters in numerical models regarding the motion of lumbar vertebral segments

Abstract

The kinematics of a spinal motion segment is determined by the material properties of the soft-tissue and the morphology. The material properties can vary within subjects and between vertebral levels, leading to a wide possible range of motion of a spinal segment independently on its morphology. The goal of this numerical study was to identify the most influential material parameters concerning the kinematics of a spinal motion segment and their plausible ranges. Then, a method was tested to deduce the material properties automatically, based on a given ROM and morphology. A fully parametric finite element model of the morphology and material properties of a lumbar spinal motion segment was developed. The impact of uncertainty of twelve spinal material parameters, as well as the size of the gap between the articular surfaces of the facet joints was examined. The simulation results were compared to our own in vitro data. The flexibility of a lumbar segment was especially influenced by the properties of the anterior annulus region, the facet gap size and the interspinous ligament. The high degree of uncertainty in the material properties and facet gap size published in the literature can lead to a wide scatter in the motion of a spinal segment, with a range of 6°-17° in the intact condition in flexion/extension, from 5°-22° in lateral bending and from 3°-14° in axial rotation. Statistical analysis of the variability might help to estimate the sensitivity and total uncertainty propagated through biomechanical simulations, affecting the reliability of the predictions.     

Application of reaction-diffusion models to cell patterning in Xenopus retina. Initiation of patterns and their biological stability

We have examined the behavior of two reaction-diffusion models, originally proposed by Gierer & Meinhardt (1972) and by Kauffman, Shymko & Trabert (1978), for biological pattern formation. Calculations are presented for pattern formation on a disc (approximating the geometry of a number of embryonic anlagen including the frog eye rudiment), emphasizing the sensitivity of patterns to changes in initial conditions and to perturbations in the geometry of the morphogen-producing space. Analysis of the linearized equations from the models enabled us to select appropriate parameters and disc size for pattern growth. A computer-implemented finite element method was used to solve the non-linear model equations reiteratively. For the Gierer-Meinhardt model, initial activation (varying in size over two orders of magnitude) of one point on the disc's edge was sufficient to generate the primary gradient. Various parts of the disc were removed (remaining only as diffusible space) from the morphogen-producing cycle to investigate the effects of cells dropping out of the cycle due to cell death or malfunction (single point removed) or differentiation (center removed), as occur in the Xenopus eye rudiment. The resulting patterns had the same general shape and amplitude as normal gradients. Nor did a two-fold increase in disc size affect the pattern-generating ability of the model. Disc fragments bearing their primary gradient patterns were fused (with gradients in opposite directions, but each parallel to the fusion line). The resulting patterns generated by the model showed many similarities to results of "compound eye" experiments in Xenopus. Similar patterns were obtained with the model of Kauffman's group (1978), but we found less stability of the pattern subject to simulations of central differentiation. However, removal of a single point from the morphogen cycle (cell death) did not result in any change. The sensitivity of the Kauffman et al. model to shape perturbations is not surprising since the model was originally designed to use shape and increasing size during growth to generate a sequence of transient patterns. However, the Gierer-Meinhardt model is remarkably stable even when subjected to a wide range of perturbations in the diffusible space, thus allowing it to cope with normal biological variability, and offering an exciting range of possibilities for reaction-diffusion models as mechanisms underlying the spatial patterns of tissue structures.     

Characterization and Simulation of Uncertain Frequency Distributions: Effects of Distribution Choice,Variability, Uncertainty,and Parameter Dependence

We use bootstrap simulation to characterize uncertainty in parametric distributions, including Normal, Lognormal, Gamma, Weibull, and Beta, commonly used to represent variability in probabilistic assessments. Bootstrap simulation enables one to estimate sampling distributions for sample statistics, such as distribution parameters, even when analytical solutions are not available. Using a two-dimensional framework for both uncertainty and variability, uncertainties in cumulative distribution functions were simulated. The mathematical properties of uncertain frequency distributions were evaluated in a series of case studies during which the parameters of each type of distribution were varied for sample sizes of 5, 10, and 20. For positively skewed distributions such as Lognormal, Weibull, and Gamma, the range of uncertainty is widest at the upper tail of the distribution. For symmetric unbounded distributions, such as Normal, the uncertainties are widest at both tails of the distribution. For bounded distributions, such as Beta, the uncertainties are typically widest in the central portions of the distribution. Bootstrap simulation enables complex dependencies between sampling distributions to be captured. The effects of uncertainty, variability, and parameter dependencies were studied for several generic functional forms of models, including models in which two-dimensional random variables are added, multiplied, and divided, to show the sensitivity of model results to different assumptions regarding model input distributions, ranges of variability, and ranges of uncertainty and to show the types of errors that may be obtained from mis-specification of parameter dependence. A total of 1,098 case studies were simulated. In some cases, counter-intuitive results were obtained. For example, the point value of the 95th percentile of uncertainty for the 95th percentile of variability of the product of four Gamma or Weibull distributions decreases as the coefficient of variation of each model input increases and, therefore, may not provide a conservative estimate. Failure to properly characterize parameter uncertainties and their dependencies can lead to orders-of-magnitude mis-estimates of both variability and uncertainty. In many cases, the numerical stability of two-dimensional simulation results was found to decrease as the coefficient of variation of the inputs increases. We discuss the strengths and limitations of bootstrap simulation as a method for quantifying uncertainty due to random sampling error.     

Spinal facet joint biomechanics and mechanotransduction in normal, injury and degenerative conditions

The facet joint is a crucial anatomic region of the spine owing to its biomechanical role in facilitating articulation of the vertebrae of the spinal column. It is a diarthrodial joint with opposing articular cartilage surfaces that provide a low friction environment and a ligamentous capsule that encloses the joint space. Together with the disc, the bilateral facet joints transfer loads and guide and constrain motions in the spine due to their geometry and mechanical function. Although a great deal of research has focused on defining the biomechanics of the spine and the form and function of the disc, the facet joint has only recently become the focus of experimental, computational and clinical studies. This mechanical behavior ensures the normal health and function of the spine during physiologic loading but can also lead to its dysfunction when the tissues of the facet joint are altered either by injury, degeneration or as a result of surgical modification of the spine. The anatomical, biomechanical and physiological characteristics of the facet joints in the cervical and lumbar spines have become the focus of increased attention recently with the advent of surgical procedures of the spine, such as disc repair and replacement, which may impact facet responses. Accordingly, this review summarizes the relevant anatomy and biomechanics of the facet joint and the individual tissues that comprise it. In order to better understand the physiological implications of tissue loading in all conditions, a review of mechanotransduction pathways in the cartilage, ligament and bone is also presented ranging from the tissue-level scale to cellular modifications. With this context, experimental studies are summarized as they relate to the most common modifications that alter the biomechanics and health of the spine-injury and degeneration. In addition, many computational and finite element models have been developed that enable more-detailed and specific investigations of the facet joint and its tissues than are provided by experimental approaches and also that expand their utility for the field of biomechanics. These are also reviewed to provide a more complete summary of the current knowledge of facet joint mechanics. Overall, the goal of this review is to present a comprehensive review of the breadth and depth of knowledge regarding the mechanical and adaptive responses of the facet joint and its tissues across a variety of relevant size scales.     

Age and size outperform topographic effects on growth-climate responses of trees in two Central European coniferous forest types

Trees do not respond to climatic conditions uniformly, but instead show individualistic growth responses. The extent of and causes behind this within-stand variability represents significant uncertainty in predictions of how forests will respond to future climate change. We analyzed patterns of individualistic tree growth within two types of conifer stands of Central Europe – high-elevation Picea abies and low-elevation Pinus sylvestris forests. We quantified the relative effect of age, size, and topographic conditions on variability in growth patterns and climate-growth responses using principal component analysis and linear models, considering both year-to-year and decadal growth variability. Our results show that Picea abies stands with dominant temperature limitation exhibit greater growth coherency than Pinus sylvestris stands characterized by drought-limited growth. Growth variability and individual climate-growth responses in both forest types were mainly driven by tree size and age, while the effect of topographic conditions was less important. The effect of size and age variables was dominant considering decadal growth trends, whereas intermediate importance of topographic variables was observed for high-frequency growth variability and climate-growth responses. Our results highlight that between-tree variability in climatic signal and growth trends also reflects the specific distribution of the age/size and topographic parameters within the stand. We suggest careful selection of datasets used for large-scale assessments of growth trends and climate-growth responses which should consider the age and size representation of sampled trees.     

Accounting for variability in sample size estimation with applications to nonadherence and estimation of variance and effect size

We consider sample size calculations for testing differences in means between two samples and allowing for different variances in the two groups. Typically, the power functions depend on the sample size and a set of parameters assumed known, and the sample size needed to obtain a prespecified power is calculated. Here, we account for two sources of variability: we allow the sample size in the power function to be a stochastic variable, and we consider estimating the parameters from preliminary data. An example of the first source of variability is nonadherence (noncompliance). We assume that the proportion of subjects who will adhere to their treatment regimen is not known before the study, but that the proportion is a stochastic variable with a known distribution. Under this assumption, we develop simple closed form sample size calculations based on asymptotic normality. The second source of variability is in parameter estimates that are estimated from prior data. For example, we account for variability in estimating the variance of the normal response from existing data which are assumed to have the same variance as the study for which we are calculating the sample size. We show that we can account for the variability of the variance estimate by simply using a slightly larger nominal power in the usual sample size calculation, which we call the calibrated power. We show that the calculation of the calibrated power depends only on the sample size of the existing data, and we give a table of calibrated power by sample size. Further, we consider the calculation of the sample size in the rarer situation where we account for the variability in estimating the standardized effect size from some existing data. This latter situation, as well as several of the previous ones, is motivated by sample size calculations for a Phase II trial of a malaria vaccine candidate.     

How does the geometry affect the internal biomechanics of a lumbar spine bi-segment finite element model? Consequences on the validation process

Numerical modelling can provide a thorough understanding of the mechanical influence on the spinal tissues and may offer explanations to mechanically linked pathologies. Such objective might be achieved only if the models are carefully validated. Sensitivity study must be performed in order to evaluate the influence of the approximations inherent to modelling. In this study, a new geometrically accurate L3-L5 lumbar spine bi-segmental finite-element model was acquired by modifying a previously existing model. The effect of changes in bone geometry, ligament fibres distribution, nucleus position and disc height was investigated in flexion and extension by comparison of the results obtained from the model before and after the geometrical update. Additional calculations were performed in axial rotation and lateral bending in order to compare the computed ranges of motion (ROM) with experimental results. It was found that the geometrical parameters affected the stress distribution and strain energy in the zygapophysial joints, the ligaments, and the intervertebral disc, changing qualitatively and quantitatively their relative role in resisting the imposed loads. The predicted ROM were generally in good agreement with the experimental results, independently of the geometrical changes. Hence, although the model update affected its internal biomechanics, no conclusions could be drawn from the experimental data about the validation of a particular geometry. Hence the validation of the lumbar spine model should be based on the relative role of its structural components and not only on its global mobility.     

How People Use Social Information to Find out What to Want in the Paradigmatic Case of Inter-temporal Preferences

The weight with which a specific outcome feature contributes to preference quantifies a person’s ‘taste’ for that feature. However, far from being fixed personality characteristics, tastes are plastic. They tend to align, for example, with those of others even if such conformity is not rewarded. We hypothesised that people can be uncertain about their tastes. Personal tastes are therefore uncertain beliefs. People can thus learn about them by considering evidence, such as the preferences of relevant others, and then performing Bayesian updating. If a person’s choice variability reflects uncertainty, as in random-preference models, then a signature of Bayesian updating is that the degree of taste change should correlate with that person’s choice variability. Temporal discounting coefficients are an important example of taste–for patience. These coefficients quantify impulsivity, have good psychometric properties and can change upon observing others’ choices. We examined discounting preferences in a novel, large community study of 14–24 year olds. We assessed discounting behaviour, including decision variability, before and after participants observed another person’s choices. We found good evidence for taste uncertainty and for Bayesian taste updating. First, participants displayed decision variability which was better accounted for by a random-taste than by a response-noise model. Second, apparent taste shifts were well described by a Bayesian model taking into account taste uncertainty and the relevance of social information. Our findings have important neuroscientific, clinical and developmental significance.     

Temporal variation of juvenile survival in a long-lived species: the role of parasites and body condition

Studies of population dynamics of long-lived species have generally focused on adult survival because population growth should be most sensitive to this parameter. However, actual variations in population size can often be driven by other demographic parameters, such as juvenile survival, when they show high temporal variability. We used capture–recapture data from a long-term study of a hunted, migratory species, the greater snow goose (Chen caerulescens atlantica), to assess temporal variability in first-year survival and the relative importance of natural and hunting mortality. We also conducted a parasite-removal experiment to determine the effect of internal parasites and body condition on temporal variation in juvenile survival. We found that juvenile survival showed a higher temporal variability than adult survival and that natural mortality was more important than hunting mortality, unlike in adults. Parasite removal increased first-year survival and reduced its annual variability in females only. Body condition at fledging was also positively correlated with first-year survival in treated females. With reduced parasite load, females, which are thought to invest more in their immune system than males according to Bateman’s principle, could probably reallocate more energy to growth than males, leading to a higher survival. Treated birds also had a higher survival than control ones during their second year, suggesting a developmental effect that manifested later in life. Our study shows that natural factors such as internal parasites may be a major source of variation in juvenile survival of a long-lived, migratory bird, which has implications for its population dynamics.     

Astragalar morphology of Afradapis, a large adapiform primate from the earliest late Eocene of Egypt

The ～37 million-year-old Birket Qarun Locality 2 (BQ-2), in the Birket Qarun Formation of Egypt's Fayum Depression, yields evidence for a diverse primate fauna, including the earliest known lorisiforms, parapithecoid anthropoids, and Afradapis longicristatus, a large folivorous adapiform. Phylogenetic analysis has placed Afradapis as a stem strepsirrhine within a clade of caenopithecine adapiforms, contradicting the recently popularized alternative hypothesis aligning adapiforms with haplorhines or anthropoids. We describe an astragalus from BQ-2 (DPC 21445C), attributable to Afradapis on the basis of size and relative abundance. The astragalus is remarkably similar to those of extant lorises, having a low body, no posterior shelf, a broad head and neck. It is like extant strepsirrhines more generally, in having a fibular facet that slopes gently away from the lateral tibial facet, and in having a groove for the tendon of flexor fibularis that is lateral to the tibial facet. Comparisons to a sample of euarchontan astragali show the new fossil to be most similar to those of adapines and lorisids. The astragali of other adapiforms are most similar to those of lemurs, but distinctly different from those of all anthropoids. Our measurements show that in extant strepsirrhines and adapiforms the fibular facet slopes away from the lateral tibial facet at a gradual angle (112-126°), in contrast to the anthropoid fibular facet, which forms a sharper angle (87-101°). Phylogenetic analyses incorporating new information from the astragalus continue to support strepsirrhine affinities for adapiforms under varying models of character evolution.     

Exploring Approximate Bayesian Computation for inferring recent demographic history with genomic markers in nonmodel species

Approximate Bayesian computation (ABC) is widely used to infer demographic history of populations and species using DNA markers. Genomic markers can now be developed for nonmodel species using reduced representation library (RRL) sequencing methods that select a fraction of the genome using targeted sequence capture or restriction enzymes (genotyping‐by‐sequencing, GBS). We explored the influence of marker number and length, knowledge of gametic phase, and tradeoffs between sample size and sequencing depth on the quality of demographic inferences performed with ABC. We focused on two‐population models of recent spatial expansion with varying numbers of unknown parameters. Performing ABC on simulated data sets with known parameter values, we found that the timing of a recent spatial expansion event could be precisely estimated in a three‐parameter model. Taking into account uncertainty in parameters such as initial population size and migration rate collectively decreased the precision of inferences dramatically. Phasing haplotypes did not improve results, regardless of sequence length. Numerous short sequences were as valuable as fewer, longer sequences, and performed best when a large sample size was sequenced at low individual depth, even when sequencing errors were added. ABC results were similar to results obtained with an alternative method based on the site frequency spectrum (SFS) when performed with unphased GBS‐type markers. We conclude that unphased GBS‐type data sets can be sufficient to precisely infer simple demographic models, and discuss possible improvements for the use of ABC with genomic data.     

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.