Ordinal Approach Research Articles

An ordination approach to explore similarities among communities

Analysis of similarities among communities can help to decipher the biogeographical, evolutionary, and ecological factors that drive local diversity. Recent indices of similarity among communities incorporate not only information on species presence and abundance but also information on how similar species are in their traits and how closely related they are in terms of taxonomy or phylogeny. Towards this aim, trait-based, taxonomic or phylogenetic similarities among species have been defined and bounded between 0 (species are maximally distinct) and 1 (species are similar). A required property for an index of similarity between two communities is that it must provide minimum similarity (0) where communities have maximally distinct species, as well as maximum similarity (1) where communities are equivalent in their trait, taxonomic or phylogenetic compositions. Here, I developed a new ordination methodology that conforms to the requirement: double similarity principal component analysis (DSPCA). DSPCA summarizes multidimensional trait-based, taxonomic or phylogenetic similarities among communities into orthogonal axes. The species that drive each similarity pattern can be identified together with their traits or with their taxonomic or phylogenetic positions. I applied this methodology to theoretical examples and to empirical data sets on bird and bat communities to illustrate key properties of DSPCA. I compared the results obtained with DSPCA with those provided by related approaches. Theoretical and empirical case studies highlight the following additional properties of DSPCA: (i) axes are orthogonal and identify independent (dis)similarity patterns between communities; (ii) the more functionally, taxonomically or phylogenetically similar communities are, the closer they are on an axis; (iii) the coordinate of a species on an axis expresses how representative the species is of the pattern identified by the axis; and (iv) a species is representative of x communities if the functional, taxonomic or phylogenetic characteristics of this species are very common within each of these x communities. DSPCA is an efficient approach to visualize functional, taxonomic and phylogenetic similarities between communities. It is also a useful alternative to recent methods dedicated to phylogenetic diversity patterns. It will be an asset for all studies that aim to compare functional, taxonomic, genetic and phylogenetic diversity.

Ordinal vs dichotomous analyses of modified Rankin Scale, 5-year outcome, and cost of stroke.

ObjectiveTo compare how 3 common representations (ordinal vs dichotomized as 0–1/2–6 or 0–2/3–6) of the modified Rankin Scale (mRS)—a commonly used trial outcome measure—relate to long-term outcomes, and quantify trial ineligibility rates based on premorbid mRS.MethodsIn consecutive patients with ischemic stroke in a population-based, prospective, cohort study (Oxford Vascular Study; 2002–2014), we related 3-month mRS to 1-year and 5-year disability and death (logistic regressions), and health/social care costs (generalized linear model), adjusted for age/sex, and compared goodness-of-fit values (C statistic, mean absolute error). We also calculated the proportion of patients in whom premorbid mRS score >1 or >2 would result in exclusion from trials using dichotomous analysis.ResultsAmong 1,607 patients, the ordinal mRS was more strongly related to 5-year mortality than both the 0–1/2–6 and 0–2/3–6 dichotomies (all p < 0.0001). Results were similar for 5-year disability, and 5-year care costs were also best captured by the ordinal model (change in mean absolute error vs age/sex: −$3,059 for ordinal, −$2,805 for 0–2/3–6, −$1,647 for 0–1/2–6). Two hundred forty-four (17.1%) 3-month survivors had premorbid mRS score >2 and 434 (30.5%) had mRS score >1; both proportions increased with female sex, socioeconomic deprivation, and age (all p < 0.0001).ConclusionThe ordinal form of the 3-month mRS relates better to long-term outcomes and costs in survivors of ischemic stroke than either dichotomy. This finding favors using ordinal approaches in trials analyzing the mRS. Exclusion of patients with higher premorbid disability by use of dichotomous primary outcomes will also result in unrepresentative samples.

Evaluation of redundancy analysis to identify signatures of local adaptation.

Ordination is a common tool in ecology that aims at representing complex biological information in a reduced space. In landscape genetics, ordination methods such as principal component analysis (PCA) have been used to detect adaptive variation based on genomic data. Taking advantage of environmental data in addition to genotype data, redundancy analysis (RDA) is another ordination approach that is useful to detect adaptive variation. This study aims at proposing a test statistic based on RDA to search for loci under selection. We compare redundancy analysis to pcadapt, which is a nonconstrained ordination method, and to a latent factor mixed model (LFMM), which is a univariate genotype-environment association method. Individual-based simulations identify evolutionary scenarios where RDA genome scans have a greater statistical power than genome scans based on PCA. By constraining the analysis with environmental variables, RDA performs better than PCA in identifying adaptive variation when selection gradients are weakly correlated with population structure. In addition, we show that if RDA and LFMM have a similar power to identify genetic markers associated with environmental variables, the RDA-based procedure has the advantage to identify the main selective gradients as a combination of environmental variables. To give a concrete illustration of RDA in population genomics, we apply this method to the detection of outliers and selective gradients on an SNP data set of Populus trichocarpa (Geraldes etal., ). The RDA-based approach identifies the main selective gradient contrasting southern and coastal populations to northern and continental populations in the north-western American coast.

Re-thinking ‘normal’ development in the early learning of number

In this article we suggest that, notwithstanding noted differences, one unmarked similarity across psychology and mathematics education is the continued dominance of the view that there is a ‘normal’ path of development. We focus particularly on the case of the early learning of number and point to evidence that puts into question the dominant narrative of how number sense develops through the concrete and the cardinal. Recent neuroscience findings have raised the potential significance of ordinal approaches to learning number, which in privileging the symbolic—and hence the abstract—reverse one aspect of the ‘normal’ development order. We draw on empirical evidence to suggest that what children can do, and in what order, is sensitive to, among other things, the curriculum approach—and also the tools they have at their disposition. We draw out implications from our work for curriculum organisation in the early years of schooling, to disrupt taken-for-granted paths.

Pengaruh Sikap dan Keyakinan Konsumen dalam Keputusan Pembelian Tepung Terigu Merek Gunung Bromo Produk PT. Bogasari Flour Mils Surabaya

This research aim to know, the influence of customer believe n and customer attitude toward customer buying decision during choice raw material of wheat flour for making noodle and bread. Object of this research is noodle manufacture in Surabaya which is PT Bogasari Flour Mills. In the same way, this research using dependence and independence variables. Dependence variable is buying decision and independence variabels are customer believe ( x1) and customer attitude (x2) . Model of analysis aproach is linear multiple regresion with 30 respondance which is a customer of PT Bogasari Flour Mills Surabaya. Sample using random sampling with samples take without any selection between respondance. Answer alternatif using ordinal approach with likert scale method. Validation test using correction item total and shown that all item are valid because value of corrected item total correlation biger than valu of r tabel. It’s mean quetion items can be uses to show something .In the other hand for reliable test using Alpha Cronbach’s trough cronbach alpha item test and the result is reliable because result of all items bigger than 0.7 ( standart correlation ) it’s mean if items of question will be use continously will give respond or result consistatly or never change .Furher more, multiple regresion result shown that customer believe ( x1) and customer attitude (x2) has influence to customer buying decision (y) with partial hypotesa test and simultaneously has significan result . Between both variable independence, customerbelieve (x1) has dominant influence trough customer buying decision (y). Value corelation between idependent and dependent variable with partial and simultanous indicator together has strong value above 0.7. Further more, according this research can show that PT Bogasari Flour Mills Surabaya should increase and growing positive perception of customer during build attitude image and believe image trough customer buying decison

Predicting multi-level drug response with gene expression profile in multiple myeloma using hierarchical ordinal regression

BackgroundMultiple myeloma (MM), like other cancers, is caused by the accumulation of genetic abnormalities. Heterogeneity exists in the patients’ response to treatments, for example, bortezomib. This urges efforts to identify biomarkers from numerous molecular features and build predictive models for identifying patients that can benefit from a certain treatment scheme. However, previous studies treated the multi-level ordinal drug response as a binary response where only responsive and non-responsive groups are considered.MethodsIt is desirable to directly analyze the multi-level drug response, rather than combining the response to two groups. In this study, we present a novel method to identify significantly associated biomarkers and then develop ordinal genomic classifier using the hierarchical ordinal logistic model. The proposed hierarchical ordinal logistic model employs the heavy-tailed Cauchy prior on the coefficients and is fitted by an efficient quasi-Newton algorithm.ResultsWe apply our hierarchical ordinal regression approach to analyze two publicly available datasets for MM with five-level drug response and numerous gene expression measures. Our results show that our method is able to identify genes associated with the multi-level drug response and to generate powerful predictive models for predicting the multi-level response.ConclusionsThe proposed method allows us to jointly fit numerous correlated predictors and thus build efficient models for predicting the multi-level drug response. The predictive model for the multi-level drug response can be more informative than the previous approaches. Thus, the proposed approach provides a powerful tool for predicting multi-level drug response and has important impact on cancer studies.

Discrete ordinates simulation of radiative participating nanofluid natural convection in an enclosure

Natural convection was studied by discrete ordinates approach within a radiation enclosure in which nanoparticles absorption and scattering were taken into consideration. Nanofluid was prepared in deionized water as the base fluid and Al2O3 nanoparticles. In order to reach a semitransparent medium suspension, nanofluid was prepared by adding small amount of Al2O3 nanoparticles volume concentration. In addition, the Navier–Stokes equations were solved based on SIMPLE algorithm of finite volume method to simulate the natural convection and then coupling to radiation by discrete ordinates approach. Two-dimensional inclined rectangular enclosure was assumed, while the inner emissive walls were found to be diffuse-gray. The various values of Ra number and volume concentration of nanoparticles at different wavelengths were studied. The results declared that the wavelength had positive impact on the radiation heat flux. More accurate results were obtained in comparison with the previous researches used Roseland approximation, due to using the discrete ordinates method to consider the adsorption and scattering coefficients. With increasing nanoparticles volume concentration, the scattering coefficient increased; hence, the heat transfer of radiation declined. On the other hand, it enhanced the natural convective heat transfer. By coupling these two conclusions, it was observed that more volume concentration corresponded to less total heat transfer which implied the main outcome of present work and made it different from the present works used nanofluid.

Optimality of the uniform rule under single-peaked preferences

Consider the problem of distributing a fixed amount of a divisible resource among agents whose preferences are single-peaked. The uniform rule has been widely characterized under an ordinal utility approach. Instead, in a cardinal utility approach, we show that the uniform rule is the only consistent rule that maximizes the worst-case surplus among strategy-proof and ordinally efficient mechanisms.

Utility-Weighted Modified Rankin Scale as Primary Outcome in Stroke Trials: A Simulation Study.

The utility-weighted modified Rankin Scale (UW-mRS) has been proposed as a new patient-centered primary outcome in stroke trials. We aimed to describe utility weights for the mRS health states and to evaluate the statistical efficiency of the UW-mRS to detect treatment effects in stroke intervention trials. We used data of the 500 patients enrolled in the MR CLEAN (Multicenter Randomized Clinical Trial of Endovascular Treatment for Acute Ischemic Stroke in the Netherlands). Utility values were elicited from the EuroQol Group 5-Dimension Self-Report Questionnaire assessed at 90 days after inclusion, simultaneously with the mRS. Utility weights were determined by averaging the utilities of all patients within each mRS category. We performed simulations to evaluate statistical efficiency. The simulated treatment effect was an odds ratio of 1.65 in favor of the treatment arm, similar for all mRS cutoffs. This treatment effect was analyzed using 3 approaches: linear regression with the UW-mRS as outcome, binary logistic regression with a dichotomized mRS (0-1/2-6, 0-2/3-6, and 0-4/5-6), and proportional odds logistic regression with the ordinal mRS. The statistical power of the 3 approaches was expressed as the proportion of 10 000 simulations that resulted in a statistically significant treatment effect (P≤0.05). The mean utility values (SD) for mRS categories 0 to 6 were: 0.95 (0.08), 0.93 (0.13), 0.83 (0.21), 0.62 (0.27), 0.42 (0.28), 0.11 (0.28), and 0 (0), respectively, but varied substantially between individual patients within each category. The UW-mRS approach was more efficient than the dichotomous approach (power 85% versus 71%) but less efficient than the ordinal approach (power 85% versus 87%). The UW-mRS as primary outcome does not capture individual variation in utility values and may reduce the statistical power of a randomized trial.

Factors contributing to the decision to perform a cesarean section in Labrador retrievers

BackgroundIn the past 10 years, the frequency of unplanned cesarean sections in the Labrador Retriever breeding colony at Guiding Eyes for the Blind stayed around 10% (range 5% to 28%). To reduce the number of cesarean sections, factors influencing the occurrence of a cesarean section need to be known. The goal of this study was to identify factors that contribute to the decision to perform a cesarean section.ResultsOf the 688 Labrador Retriever litters whelped between 2003 and 2016, 667 litters had sufficient data and remained in the analysis. The target trait was ordinal with the three levels “normal whelping”, “assisted whelping” and “cesarean section”. A general ordinal logistic regression approach was used to analyze the data. Model selection with possible predictors resulted in a final model including weight of the dam, the weight of the heaviest puppy of a litter, the number of fetuses malpositioned and the quality of uterine contractions. Weight and size of a litter, parity, maternal inbreeding coefficient, whelping season, dam and sire were dropped from the model because they were not significant. The risk of a cesarean section was influenced by the combination of the weight of the dam and the weight of the heaviest puppy in the litter, as well as by the number of malpositioned fetuses and the quality of the contractions. Larger puppies increased the risk of cesarean section especially when the dam had a lighter weight. For dams weighing 23.6 kg and 32.8 kg the predicted probability of a cesarean section was low, with 0.06 and 0.02, respectively, when the heaviest puppy in a litter was light (0.42 kg), contractions were normal and no fetus was malpositioned. However, the probability of a cesarean section was much higher, ranging from 0.24 to 0.08, when the heaviest puppy in a litter was heavy (0.66 kg).ConclusionsMeans to reduce the cesarean section frequency in this Labrador Retriever breeding colony should include genetic selection for ideal puppy weight. In addition, dams with an adult body weight substantially below average should not be selected as breeders in this colony.

Comparing assembly processes for multimetric indices of biotic integrity

Anthropogenic alterations to global ecosystems necessitate management action to conserve or restore biodiversity and ecosystem services. A major advancement in ecosystem management was the development of multimetric indices of biotic integrity (MMIBI) used to guide development of, and measure progress towards, restoration goals. Despite considerable refinement of MMIBI applications over the past three decades, a central challenge remains concerning the method of selecting ecological indicators for inclusion in MMIBI. We quantitatively compared MMIBI metric assembly processes across four sub-regions for fish assemblages in western Tennessee, USA to assess relative performance of three metric selection approaches. Metric selection methods we assessed included “filter gradient” using a multi-step approach to filter candidate metrics down to only the most reproducible and responsive, “indirect gradient” using a correlative unconstrained ordination approach, and “direct gradient” involving an automated constrained ordination approach. For each method, we calculated MMIBI using the selected metrics and compared their precision (i.e., stability across multiple samples), responsiveness (i.e., discrimination between most- and least-altered sites), and sensitivity (i.e., ability to detect landscape alterations). We found metric selection using the filter gradient approach produced MMIBI that were most responsive across all four sub-regions, while the indirect gradient approach produced the most sensitive and precise MMIBI for three of four sub-regions. The direct gradient metric selection approach produced the most sensitive MMIBI only for a single sub-region with a relatively short gradient in landscape alterations. These results reveal a tradeoff between filter and indirect gradient selection methods in which filter gradient metric selection provides high MMIBI responsiveness, but at the cost of increased number of steps and reduced precision and sensitivity. The “middle of the road”, indirect gradient metric selection approach produced precise and sensitive MMIBI, but at the cost of reduced responsiveness. These findings highlight the necessity to pair well-developed ecosystem management goals with MMIBI application, and provide a road map for the most appropriate assembly process for managers developing MMIBI. For example, identification of least- and most-altered sites might best be accomplished with MMIBI developed using the filter gradient approach, but assessing the factors contributing to alteration and precisely measuring progress towards restoration endpoints might best be accomplished with MMIBI developed using the indirect gradient approach. Restoration and management actions guided by MMIBI will become increasingly prevalent with increased future alteration to global ecosystems, and this work provides important insight into how technological and quantitative advances will improve application of ecological indicators.

A General Symbolic Approach to Kolmogorov-Sinai Entropy

It is popular to study a time-dependent nonlinear system by encoding outcomes of measurements into sequences of symbols following certain symbolization schemes. Mostly, symbolizations by threshold crossings or variants of it are applied, but also, the relatively new symbolic approach, which goes back to innovative works of Bandt and Pompe—ordinal symbolic dynamics—plays an increasing role. In this paper, we discuss both approaches novelly in one breath with respect to the theoretical determination of the Kolmogorov-Sinai entropy (KS entropy). For this purpose, we propose and investigate a unifying approach to formalize symbolizations. By doing so, we can emphasize the main advantage of the ordinal approach if no symbolization scheme can be found that characterizes KS entropy directly: the ordinal approach, as well as generalizations of it provide, under very natural conditions, a direct route to KS entropy by default.

Multiple performance measures are needed to evaluate triage systems in the emergency department

ObjectivesEmergency department triage systems can be considered prediction rules with an ordinal outcome, where different directions of misclassification have different clinical consequences. We evaluated strategies to compare the performance of triage systems and aimed to propose a set of performance measures that should be used in future studies. Study Design and SettingWe identified performance measures based on literature review and expert knowledge. Their properties are illustrated in a case study evaluating two triage modifications in a cohort of 14,485 pediatric emergency department visits. Strengths and weaknesses of the performance measures were systematically appraised. ResultsCommonly reported performance measures are measures of statistical association (34/60 studies) and diagnostic accuracy (17/60 studies). The case study illustrates that none of the performance measures fulfills all criteria for triage evaluation. Decision curves are the performance measures with the most attractive features but require dichotomization. In addition, paired diagnostic accuracy measures can be recommended for dichotomized analysis, and the triage-weighted kappa and Nagelkerke's R2 for ordinal analyses. Other performance measures provide limited additional information. ConclusionWhen comparing modifications of triage systems, decision curves and diagnostic accuracy measures should be used in a dichotomized analysis, and the triage-weighted kappa and Nagelkerke's R2 in an ordinal approach.

Multivariate ordination identifies vegetation types associated with spider conservation in brassica crops.

Conservation biological control emphasizes natural and other non-crop vegetation as a source of natural enemies to focal crops. There is an unmet need for better methods to identify the types of vegetation that are optimal to support specific natural enemies that may colonize the crops. Here we explore the commonality of the spider assemblage—considering abundance and diversity (H)—in brassica crops with that of adjacent non-crop and non-brassica crop vegetation. We employ spatial-based multivariate ordination approaches, hierarchical clustering and spatial eigenvector analysis. The small-scale mixed cropping and high disturbance frequency of southern Chinese vegetation farming offered a setting to test the role of alternate vegetation for spider conservation. Our findings indicate that spider families differ markedly in occurrence with respect to vegetation type. Grassy field margins, non-crop vegetation, taro and sweetpotato harbour spider morphospecies and functional groups that are also present in brassica crops. In contrast, pumpkin and litchi contain spiders not found in brassicas, and so may have little benefit for conservation biological control services for brassicas. Our findings also illustrate the utility of advanced statistical approaches for identifying spatial relationships between natural enemies and the land uses most likely to offer alternative habitats for conservation biological control efforts that generates testable hypotheses for future studies.

Using light-dependent scleractinia to define the upper boundary of mesophotic coral ecosystems on the reefs of Utila, Honduras.

Shallow water zooxanthellate coral reefs grade into ecologically distinct mesophotic coral ecosystems (MCEs) deeper in the euphotic zone. MCEs are widely considered to start at an absolute depth limit of 30m deep, possibly failing to recognise that these are distinct ecological communities that may shift shallower or deeper depending on local environmental conditions. This study aimed to explore whether MCEs represent distinct biological communities, the upper boundary of which can be defined and whether the depth at which they occur may vary above or below 30m. Mixed-gas diving and closed-circuit rebreathers were used to quantitatively survey benthic communities across shallow to mesophotic reef gradients around the island of Utila, Honduras. Depths of up to 85m were sampled, covering the vertical range of the zooxanthellate corals around Utila. We investigate vertical reef zonation using a variety of ecological metrics to identify community shifts with depth, and the appropriateness of different metrics to define the upper MCE boundary. Patterns observed in scleractinian community composition varied between ordination analyses and approaches utilising biodiversity indices. Indices and richness approaches revealed vertical community transition was a gradation. Ordination approaches suggest the possibility of recognising two scleractinian assemblages. We could detect a mesophotic and shallow community while illustrating that belief in a static depth limit is biologically unjustified. The switch between these two communities occurred across bathymetric gradients as small as 10m and as large as 50m in depth. The difference between communities appears to be a loss of shallow specialists and increase in depth-generalist taxa. Therefore, it may be possible to define MCEs by a loss of shallow specialist species. To support a biological definition of mesophotic reefs, we advocate this analytical framework should be applied around the Caribbean and extended into other ocean basins where MCEs are present.

Radiative heat transfer in processing of glass‐forming melts

AbstractThe paper summarizes the findings of a study on mathematical modeling of heat transfer in glass melts using the commonly employed diffusion or Rosseland approximation (R) for optically thick media and the more general, spectral discrete ordinates (DO) approach. The systems considered were the melting of an ultra‐clear glass in a furnace (optical thickness ~ 10) and heat transfer in 2 glass melts with different levels of iron oxide flowing through a delivery channel (optical thicknesses of ~17 and ~80). The results showed that the use of the R approach for modeling heat transfer in the ultra‐clear glass will lead to serious errors in the prediction of important practical parameters such as the refractory temperature, energy efficiency, and the quality of glass melt. The results for the channels showed that differences between the 2 thermal radiation models began to narrow as the optical thickness value approached 17. The paper also presents brief reviews of high‐temperature absorption spectra of commercial glass melts and the theoretical background on thermal radiation in participating media. In addition, a relatively simple system involving 1‐dimensional radiation heat transfer in a participating medium is used to illustrate the implications of using the diffusion approximation.

The Automation of Regression Modeling in Osteometric Sorting: An Ordination Approach.

This study compares the original osteometric sorting association method with an ordination approach across all combinations of the humerus, ulna, radius, femur, tibia, and fibula. This includes both the original prediction interval and t-statistic approaches. Standard measurements are utilized in the models with full measurements combined and without length measurements. The sample is the osteometric sorting reference from the Defense POW/MIA Accounting Agency. A full set of performance statistics is provided. Results indicate the ordination approach outperforms the original in the majority of bone combinations. Models with length measurements have more exclusion power than those without. It is recommended for the ordination approach to supersede the original when applied to large commingled assemblages.

Economic Analysis of Factors Responsible for the Attainment of Food Security Levels among Beneficiaries of Selected Agricultural Cooperatives in the Limpopo Province of South Africa

ABSTRACTThis study determined factors responsible for the determination of four levels of food security: never, seldom, sometimes and often or always among beneficiaries of two farmer cooperatives in the Limpopo Province of South Africa using the ordinal regression approach. A sample size of 178 respondents was used for the analysis. Respondents were purposively selected from beneficiaries of two cooperatives which had vegetable production as their main enterprise. Semi-structured questionnaires were used to collect information from respondents during the 2014/5 farming season. Data were analyzed using the SPSS version 22. Predictors which included, educational level, perceived high impact of cooperatives on food security, farming experience and ownership of equipment were identified as being positively associated with the odds of being food secure. It was recommended that for policy development and appropriate interventions in food security, factors identified to be positively associated with the attainm...

Demand Price Setting

This study describes the specifics of demand price setting. At the end of the chapter, you will be able: - to define the main characteristics of the cardinal approach to utility analysis, to determine the main analytical factors and to calculate the optimal structure of individual purchases for a fixed revenue structure and real market prices - to describe the main tool for analysing customer preferences of the ordinal theory – the indifference curve; to analyse the main characteristics of the ordinal approach and customer reactions to changes of the ratio between their income and the supply prices - based on observations of customer behaviour to determine the specific demand characteristics and the degree of utility maximisation. The study includes the following sections: 1. Quantitative (cardinal) approach to utility and demand analysis 2. Defining the utility function based on ordinary utility hypotheses. 3. Customer preferences depending on market price levels.

Breakdown of equipartition in diffuse fields caused by energy leakage

Equipartition is a central concept in the analysis of random wavefields which stipulates that in an infinite scattering medium all modes and propagation directions become equally probable at long lapse time in the coda. The objective of this work is to examine quantitatively how this conclusion is affected in an open waveguide geometry, with a particular emphasis on seismological applications. To carry our this task, the problem is recast as a spectral analysis of the radiative transfer equation. Using a discrete ordinate approach, the smallest eigenvalue and associated eigenfunction of the transfer equation, which control the asymptotic intensity distribution in the waveguide, are determined numerically with the aid of a shooting algorithm. The inverse of this eigenvalue may be interpreted as the leakage time of the diffuse waves out of the waveguide. The associated eigenfunction provides the depth and angular distribution of the specific intensity. The effect of boundary conditions and scattering anisotropy is investigated in a series of numerical experiments. Two propagation regimes are identified, depending on the ratio H∗ between the thickness of the waveguide and the transport mean path in the layer. The thick layer regime H∗ > 1 has been thoroughly studied in the literature in the framework of diffusion theory and is briefly considered. In the thin layer regime H∗ < 1, we find that both boundary conditions and scattering anisotropy leave a strong imprint on the leakage effect. A parametric study reveals that in the presence of a flat free surface, the leakage time is essentially controlled by the mean free time of the waves in the layer in the limit H∗ → 0. By contrast, when the free surface is rough, the travel time of ballistic waves propagating through the crust becomes the limiting factor. For fixed H∗, the efficacy of leakage, as quantified by the inverse coda quality factor, increases with scattering anisotropy. For sufficiently thin layers H∗≈ 1/5, the energy flux is predominantly directed parallel to the surface and equipartition breaks down. Qualitatively, the anisotropy of the intensity field is found to increase with the inverse non-dimensional leakage time, with the scattering mean free time as time scale. Because it enhances leakage, a rough free surface may result in stronger anisotropy of the intensity field than a flat surface, for the same bulk scattering properties. Our work identifies leakage as a potential explanation for the large deviation from isotropy observed in the coda of body waves.