Equal Sample Sizes Research Articles

Do Immune-Mediated TTP Patients Presenting with No Detectable ADAMTS13 Antibody Represent a Unique Clinical Phenotype?

Background:Immune-mediated thrombotic thrombocytopenic purpura (iTTP) is characterised by antibody mediated activity against ADAMTS13. Prompt diagnosis and treatment is critical in this rare condition which has an untreated mortality of around 90%. An ADAMTS13 activity level of <10% is consistent with a diagnosis of TTP and for a significant proportion of patients, ADAMTS13 antibodies can also be demonstrated, confirming iTTP. Recent literature suggests that the sub-group of patients with low ADAMTS13 antigen levels and high ADAMTS13 antibody have the highest mortality (Ferras et al., Blood 2017.); however, the sub-group of patients who have never had a detectable inhibitor remain poorly defined. With therapies becoming increasingly available as adjuvants to plasma exchange, in addition to elective agents for suspected relapsing disease, there is a need to better risk stratify this heterogeneous patient group and identify which patients may benefit most from additional therapies. In most literature to date, patients with low ADAMTS13 inhibitor levels have been excluded from further data analysis and therefore remain an under-represented population. Aim: To define the characteristics of the iTTP patient sub-group where an ADAMTS13 inhibitor has never been demonstrated throughout their clinical course Methods: This was retrospective study of data from a single UK TTP tertiary referral centre from 2011 to 2018. Inclusion criteria for patients with iTTP included ADAMTS13 activity ≤10% at acute presentation. Patients with congenital TTP were excluded from data analysis. A positive ADAMTS13 antibody was defined as >12 U/ml using the Technoclone ADAMTS13 inhibitor ELISA assay. Data collected included patient demographics, presenting symptoms, laboratory markers at first presentation, ADAMTS13 activity and antibody and number of relapses. Clinical relapse was defined as reappearance of clinical manifestations or laboratory parameters consistent with acute TTP and serological relapse was defined as ADAMTS13 activity of <15%. Patients with no detectable antibody were compared with a like group of TTP patients with a detectable ADAMTS13 antibody. Results: A total of 61 patients and 72 acute patient episodes were analysed, of which 7 patients had never demonstrated an ADAMTS13 inhibitor in 10 acute episodes. This sub-group was subsequently compared to an equal sample size of patients with a detectable ADAMTS13 antibody and similar demographics. The average antibody level for the antibody negative group was 5.8 U/ml (range 3 - 10 U/ml) compared to 98.7 U/ml (range 37 - 147U/ml) in the antibody positive group. The respective average ADAMTS13 activity was 4.5% (0-10%), compared with 0.3% (0-1%). 100% of the patients in the antibody positive group had neurological symptoms at presentation compared with 43% of the patients with no detectable antibody. The average presenting platelet count and Troponin T for the antibody negative cohort was 27 x 109/L and 21 ng/L respectively; compared with 10 x 109/L and 206 ng/L in the antibody positive group. 43% of patients in the antibody negative group had multiple acute clinical relapses (1-3) compared with 0% patients presenting with a detectable antibody. 14% of the antibody positive group had a serological relapse treated with elective Rituximab compared with 43% of the patients in the antibody negative group. Conclusion: This data specifically reports the apparent characteristics of patients with ADAMTS13 antibody negative iTTP. Up to 10% of patients may never demonstrate an inhibitor against ADAMTS13 and appear to have a less severe clinical presentation with fewer neurological and cardiac manifestations. This study further suggests that this sub-group may represent a more ‘grumbling‘/ relapsing phenotype with a tendency for more frequent clinical or serological relapse of TTP. Although TTP is rare, further study of this sub-population of iTTP patients is important in risk stratification, and to support treatment decisions, in particular towards preventing future relapse. DisclosuresNo relevant conflicts of interest to declare.

Bayesian evaluation of informative hypotheses for multiple populations.

The software package Bain can be used for the evaluation of informative hypotheses with respect to the parameters of a wide range of statistical models. For pairs of hypotheses the support in the data is quantified using the approximate adjusted fractional Bayes factor (BF). Currently, the data have to come from one population or have to consist of samples of equal size obtained from multiple populations. If samples of unequal size are obtained from multiple populations, the BF can be shown to be inconsistent. This paper examines how the approach implemented in Bain can be generalized such that multiple-population data can properly be processed. The resulting multiple-population approximate adjusted fractional Bayes factor is implemented in the R package Bain.

The effect of foot type on the Achilles tendon moment arm and biomechanics

BackgroundThe aim was to calculate the Achilles tendon moment arm in different degrees of plantarflexion for pes planus, pes cavus and normal arched feet. Methods99 patients (99 radiographs; 40 males, 59 females; mean age 49 years, SD 15) with a healthy ankle joint and a preoperative weightbearing lateral radiograph of the foot were included. Three groups (pes planus, pes cavus and normal-arched feet) with equal sample sizes (n=33) were formed. On radiographs, the angle formed between a horizontal line and the line connecting the insertion of the Achilles tendon with the center of rotation of the ankle, was measured. The interrater reliabilities (ICC) of the angle alpha were compared on radiographs and on MRIs. Using the angle alpha, the Achilles tendon moment arm was calculated in different plantarflexion positions. ResultsThe ICC of alpha was higher on radiographs (0.84, [0.73–0.91]) than on MRIs (0.61, [0.27–0.81]). The average alpha was statistically significantly different (normal arched foot 31 degrees (°), pes planus 24°, pes cavus 36°, p=0.021), resulting in a significant shorter Achilles tendon moment arm for pes cavus than for pes planus (p<0.0001) and normal arched feet (p=0.006) in neutral position. ConclusionThe data suggests that it is feasible to use radiographs to measure the Achilles tendon moment arm. The maximum Achilles tendon moment arm is reached at different angles of ankle flexion for pes cavus, pes planus and normal-arched feet. This has to be taken into consideration when planning surgeries.

An optimized two-stage spatial sampling scheme for winter wheat acreage estimation using remotely sensed imagery

ABSTRACTTimely and reliable information on crop acreage is essential for formulating grain production policies and ensuring national food security. The combination of available satellite-based remotely sensed images and traditional sampling methods offers the possibility of improved crop acreage estimation at a regional scale. Due to the administrative convenience, reduced survey cost and workload, two-stage sampling has widely been used for crop acreage survey at the large-scale regions. However, compared with single-stage sampling, the two-stage sampling can introduce larger estimation errors, since it has multiple sampling stages. This study’s aim is to optimize the two-stage sampling scheme using satellite-based remotely sensed imagery to improve the accuracy of crop acreage estimation. Taking Mengcheng County, Anhui Province, China, as the study area, this study explored the influence of stratum boundary and sample selection method on the sampling efficiency at the first sampling stage, analysed the impact of sample size on population extrapolation accuracy and then optimized the sample size of the second sampling stage using crop thematic map retrieved by ALOS (Advanced Land Observing Satellite) AVNIR (Advanced Visible light and Near Infrared Radiometer)-2 images in 2009. The results showed that the relative error (RE), coefficient of variation (CV), standard error (SE) of population extrapolation, and sampling fraction (f) using the cumulative square root of frequency (CSRF) method is the minimum among three methods for the stratum boundary determination at the first sampling stage, followed by the equal interval (EI) and equal sample size (ESS) method. Moreover, the RE, CV, and SE of population extrapolation using the ST sampling method is the minimum, compared with simple random (SI) and systematic (SY) sampling method. Therefore, the sampling scheme of the first stage can be optimized by CSRF method for stratum boundary determination and stratified sampling (ST) sampling method for samples selection. At the second sampling stage, RE and CV values of population extrapolation decrease as the sample size increases. Comprehensively considering the accuracy, stability of population extrapolation and sampling cost, the most cost-effective sample size for estimating the winter wheat acreage of the study area is 4. From the perspective of the reasonable selection of sample selection methods, sample size and determination of stratum boundaries, this study provides an important basis for formulating a cost-effective two-stage spatial sampling scheme for crop acreage estimation.

How should we estimate diversity in the fossil record? Testing richness estimators using sampling‐standardised discovery curves

AbstractTo infer genuine patterns of biodiversity change in the fossil record, we must be able to accurately estimate relative differences in numbers of taxa (richness) despite considerable variation in sampling between time intervals. Popular subsampling (=interpolation) methods aim to standardise diversity samples by rarefying the data to equal sample size or equal sample completeness (=coverage). Standardising by sample size is misleading because it compresses richness ratios, thereby flattening diversity curves. However, standardising by coverage reconstructs relative richness ratios with high accuracy. Asymptotic richness extrapolators are widely used in ecology, but rarely applied to fossil data. However, a recently developed parametric extrapolation method,TRiPS(True Richness estimation using Poisson Sampling), specifically aims to estimate the true richness of fossil assemblages.Here, we examine the suitability of a range of richness estimators (both interpolators and extrapolators) for fossil datasets, using simulations and a novel method for comparing the performance of richness estimators with empirical data. We constructed sampling‐standardised discovery curves (SSDCs) for two datasets, each spanning 150 years of palaeontological research: Mesozoic dinosaurs at global scale, and Mesozoic–early Cenozoic tetrapods from North America. These approaches reveal how each richness estimator responds to both simulated best‐case and empirical real‐world accumulation of fossil occurrences.We find that extrapolators can only truly standardise diversity data once sampling is sufficient for richness estimates to have asymptoted. Below this point, directly comparing extrapolated estimates derived from samples of different sizes may not accurately reconstruct relative richness ratios. When abundance distributions are not perfectly flat and sampling is moderate to good, but not perfect,TRiPSdoes not extrapolate, because it overestimates binomial sampling probabilities. Coverage‐based interpolators, by contrast, generally yield more stable subsampled diversity estimates, even in the face of dramatic increases in face‐value counts of species richness. Richness estimators that standardise by coverage are among the best currently available methods for reconstructing deep‐time biodiversity patterns. However, we recommend the use of sampling‐standardised discovery curves to understand how biased reporting of fossil occurrences may affect sampling‐standardised diversity estimates.

Tavern proximity, tavern density and socio-economic status as predictors of assault occurrence within New Zealand: a temporal comparison

ABSTRACTThis paper investigates how the associations between tavern proximity, tavern density and area unit socio-economic status with assault occurrence vary in a temporal sense. Using New Zealand Police data specifying the day, time and location of assaults in 2016 and Ministry of Justice data specifying the location of on-licenced taverns, we construct logistic regression models to determine how well tavern proximity, tavern density and socio-economic status predict the occurrence of assaults at peak (Fri 22:00–Sat 03:00 and Sat 22:00–Sun 03:00) and off-peak times. An equal-sized sample of traffic generators (public venues whose primary function is not the sale of alcohol) is constructed and similar procedures applied. We find that tavern proximity and tavern density are stronger predictors of assault occurrence at peak, compared to off-peak, times. Conversely, socio-economic status is a better predictor of assault occurrence at off-peak times. We also find that whilst tavern proximity and density are stronger predictors of assault occurrence relative to traffic generator proximity and density at peak times, the opposite is true at off-peak times. These results suggest that in order to minimise alcohol-related harm, there is a need for policy-makers to take into account the temporal nature of these relationships.

Joint estimation of multiple Gaussian graphical models across unbalanced classes

The problem of jointly estimating unbalanced multi-class Gaussian graphical models is considered. Most existing methods require equal or similar sample sizes among classes. However, many real applications do not have similar sample sizes. Hence, the joint adaptive graphical lasso, a weighted l1 penalized approach is proposed for unbalanced multi-class problems. The joint adaptive graphical lasso approach combines information across classes so that their common characteristics can be shared during the estimation process. Regularization is also introduced into the adaptive term. Simulation studies show that the new approach performs better than existing methods in terms of false positive rate, accuracy, Mathews correlation coefficient, and false discovery rate. The advantages of the new approach are also demonstrated using a liver cancer data set.

Non-normal data: Is ANOVA still a valid option?

The robustness of F-test to non-normality has been studied from the 1930s through to the present day. However, this extensive body of research has yielded contradictory results, there being evidence both for and against its robustness. This study provides a systematic examination of F-test robustness to violations of normality in terms of Type I error, considering a wide variety of distributions commonly found in the health and social sciences. We conducted a Monte Carlo simulation study involving a design with three groups and several known and unknown distributions. The manipulated variables were: Equal and unequal group sample sizes; group sample size and total sample size; coefficient of sample size variation; shape of the distribution and equal or unequal shapes of the group distributions; and pairing of group size with the degree of contamination in the distribution. The results showed that in terms of Type I error the F-test was robust in 100% of the cases studied, independently of the manipulated conditions.

Evaluating and improving recruitment and retention in an mHealth clinical trial: an example of iterating methods during a trial.

Recruitment and retention strategy investigations in mHealth clinical trials are rare. Technology presents an opportunity to intensely and remotely evaluate recruitment, use of mobile apps, and retention, leading to new insights for continuous improvement of mHealth trials. The objective of this paper is to present a case study in which a trial evaluated and changed strategies during a clinical trial to improve recruitment, adherence to study protocols, and retention in the mHealth trial. In Fall 2015, the NUYou trial enrolled 150 college freshmen in an mHealth protocol. Three months after study initiation, NUYou struggled to meet recruitment goals and maintain anticipated usage levels of the study smartphone application. Two sets of data were collected to improve recruitment and retention: a survey about recruitment was sent to the target population and surveys regarding usability of the app was sent to the study sample. Survey results informed improvements in recruitment strategies, the study retention protocol, and the smartphone application. Survey results revealed several insights including misunderstanding components of the trial by potential participants, low perceived usefulness of the app, and little recall or impact of the incentive structure. After implementation of user-centered improvements, the second cohort of NUYou recruitment in the fall of 2016 produced an equal sample size in 4 weeks less time. Winter quarter of 2016 compared to 2017 demonstrated an improvement in retention via app use and completion of weekly in-app surveys. Recruitment and retention in clinical trials continues to be a critical challenge and mHealth trials may present both unique challenges and opportunities. To our knowledge, this is the first study to describe a systematic evaluation followed by changes and further evaluation to recruitment, use of the mHealth application, adherence to study protocol, and retention during an mHealth clinical trial. Future work should adopt and explicitly study these processes to optimize both enrollment and retention in these types of trials to preserve validity and reliability of research results.

Comparing the performance of normality tests with ROC analysis and confidence intervals

ABSTRACTThere are several statistical hypothesis tests available for assessing normality assumptions, which is an a priori requirement for most parametric statistical procedures. The usual method for comparing the performances of normality tests is to use Monte Carlo simulations to obtain point estimates for the corresponding powers. The aim of this work is to improve the assessment of 9 normality hypothesis tests. For that purpose, random samples were drawn from several symmetric and asymmetric nonnormal distributions and Monte Carlo simulations were carried out to compute confidence intervals for the power achieved, for each distribution, by two of the most usual normality tests, Kolmogorov–Smirnov with Lilliefors correction and Shapiro–Wilk. In addition, the specificity was computed for each test, again resorting to Monte Carlo simulations, taking samples from standard normal distributions. The analysis was then additionally extended to the Anderson–Darling, Cramer-Von Mises, Pearson chi-square Shapiro–Francia, Jarque–Bera, D'Agostino and uncorrected Kolmogorov–Smirnov tests by determining confidence intervals for the areas under the receiver operating characteristic curves. Simulations were performed to this end, wherein for each sample from a nonnormal distribution an equal-sized sample was taken from a normal distribution. The Shapiro–Wilk test was seen to have the best global performance overall, though in some circumstances the Shapiro–Francia or the D'Agostino tests offered better results. The differences between the tests were not as clear for smaller sample sizes. Also to be noted, the SW and KS tests performed generally quite poorly in distinguishing between samples drawn from normal distributions and t Student distributions.

THE LONG TERM EFFECT OF DIFFERENT MEDICAMENTS ON THE BOND STRENGTH OF FIBER POST-ROOT CANAL DENTIN

Purpose: The aim of this study was to investigate the long-term effects of different medicaments on the bond strength of fiber post-root canal dentin interface. Materials and Methods: Eighty single-rooted mandibular premolars were randomly allocated into four groups of equal sample size, which were described as Group 1: no medicament (control group), group 2: Metapex, group 3: calcium hydroxide (CH), and group 4: Ledermix. The specimens were stored for 21 days in 100% humidity at 37°C. Teeth were then filled with either one of the above mentioned materials and fiber posts were inserted. Two subgroups (n=10) were created from the original samples depending on the waiting time, which were described as the immediate (24 hours) and long-term test (6 months) groups. All specimens were subjected to push-out tests by using a universal testing machine. Results: Significant differences were observed between Ledermix and other groups in the tests performed after 24 hours (immediate test) (p<0.05). In long-term test groups, Metapex and control groups showed significantly higher bond strength values than those found in Ledermix and CH groups (p<0.05). Conclusion: Root canals with no medicament and those with Metapex showed higher bond strength values in the long-term tests when compared to immediate tests, in contrast to the canals filled with Ledermix and CH.

Semisupervised JITL Framework for Nonlinear Industrial Soft Sensing Based on Locally Semisupervised Weighted PCR

Just-in-time learning (JITL) is a commonly used technique for industrial soft sensing of nonlinear processes. However, traditional JITL approaches mainly focus on equal sample sizes between process (input) variables and quality (output) variables, which may not be practical in industrial processes since quality variables are usually much harder to obtain than other process variables. In order to handle unequal length dataset with only a few labeled data, a novel semisupervised JITL framework is proposed for soft sensor modeling for nonlinear processes, which is based on semisupervised weighted probabilistic principal component regression (SWPPCR). In the new semisupervised JITL framework, traditional Mahalanobis distance and a new proposed scaled Mahalanobis distance are used for similarity measurement and weight assignment. By selecting the most relevant labeled and unlabeled samples and assigning them with the corresponding weights, a local SWPPCR can be built to estimate the output variables of the query sample. Case studies are carried out to evaluate the prediction performance of the proposed semisupervised JITL framework on a numerical example and an industrial process. The effectiveness and flexibility of the proposed method are demonstrated by the prediction results.

English

Experimental studies that aimed to determine the effects of crude aqueous resin extracts of the Commiphora swynnertonii on plasma cholesterol levels and weight changes were carried out in rats (Rattus rattus). A total of 24 experimental rats divided into four groups with equal sample size (n=6) were used. Group one (G1) served as negative control that received 0.5ml of distilled water (0 mg/kg) orally. Groups 2 (G2), 3 (G3) and 4 (G4) received 50, 100, and 200 mg/kg body weight orally on daily basis for 21 days respectively. Results revealed a significant decrease (P &lt; 0.05) in the body weight and on cholestrol levels between the treated and the control groups in a dose dependent manner (R2 = 0.89). Commiphora swynnertonii resin lowered cholesterol level by 54, 76 and 79% and weight changes by 18, 31 and 23% for the exposed rats at concentrations of 50, 100 and 200 mg/kg BW respectively. The rats were able to tolerate resin at concentrations lower than 100 mg/kg BW. At higher (&gt;100 mg/kg) doses, few rats showed signs of illness including diarrhoea and finally death. Based on these results, C. swynnertonii has a potential to serve as an anti-cholesterol agent with body weight lowering properties. &nbsp; Key words: Oltemwai, Cardiovascular diseases, rats, Tanzania, blood chemistry.

The multivariate multisample nonparametric rank statistics for the location alternatives

Multisample testing problems are among the most important topics in nonparametric statistics. Various nonparametric tests have been proposed for multisample testing problems involving location parameters, and the analysis of multivariate data is important in many scientific fields. One type of multivariate multisample testing problem based on Jureckova-Kalina-type rank of distance is discussed in this paper. A multivariate Kruskal-Wallis-type statistic is proposed for testing the location parameter with both equal and unequal sample sizes. Simulations are used to compare the power of proposed nonparametric statistics with the Wilks' lambda, the Pillai's trace and the Lawley-Hotelling trace for various population distributions.

Farklı Dağılımlar İçin Varyans Homojenlik Testlerinin Karşılaştırılması

Objective: The reliability of the hypothesis tests to be performed using variance analysis depends on the holding of the assumptions. In the analysis of variance, the sensitivity of the results obtained for the test statistic is most influenced by the heterogeneity of the variances. In this study, it is aimed to compare the variance homogeneity tests widely used in different literature for different distributions in terms of type-I error level and power. Material and Methods: 1000 repetitive simulations were performed for the Levene median, Levene mean, Levene trimmed mean, bootstrap Levene median, Bartlett and Cochran tests. Four cases where the sample sizes are equal were taken as n₁=n₂=n₃=15; n₁=n₂=n₃=30; n₁=n₂=n₃=45; n₁=n₂=n₃=100 and the two cases where the sample sizes are not equal were taken as n₁=15, n₂=30, n₃=45; n₁=15, n₂=30, n₃=100. For different distributions, the data were generated by taking into account the skewness and kurtosis coefficients. For power, the variance ratios of 1:1:2, 1:2:4, 1:2:2 and 1:4:4 were taken and different variances were corresponded to different sample sizes. Results: In terms of type-I error rates, the bootstrap Levene median test gave the best result for all distributions. For power; if the sample sizes were equal, for medium and large sample sizes bootstrap Levene median and Levene median tests gave the best results. For large sample sizes Bartlett test gave similar results to these two tests. Conclusion: It was seen that, while the performances of the tests were affected from the distribution, the best result against the deviations from normality was obtained for the bootstrap Levene median test and the performance of the tests was affected from the variance-sample size combination.

Case study of residual stresses distribution in steel welded parts using ultrasound

Residual stresses occur in every welded manufactured structure. Different studies aimed to classify the methods of investigating the residual stresses and to highlight the advantages and disadvantages/limits of each of them. The conclusion is that ultrasonic investigation permits the evaluation of bulk stress state of welded components with acceptable accuracy. Our past research has addressed various aspects of ultrasonic investigation of residual stresses in welded steel parts: different cases of the analysis of residual stresses, edge effect on analyses of residual stresses. The purpose of this paper is to continue the previous research by introducing a new case: comparative study of equal size samples obtained by splitting an initial sample. The probe was conceived as a frame with a detachable side. After welding, the assembly is processed on a grinding machine to ensure parallel surfaces. The sample, as obtained, is ultrasonic investigated as part. After that, the detachable side is removed and the resulted part is split in six equal samples. Every so obtained sample was investigated and results are presented. New direction for future research is proposed.

Baseline EDSS proportions in MS clinical trials affect the overall outcome and power: A cautionary note

Background: In randomized clinical trials, when treatments do not work equally effectively across stratifications of participants, observed event rates may differ from those hypothesized leading to deviations in estimated power. Objectives: To investigate the effect of distributions of baseline Expanded Disability Status Scale (EDSS) proportions in relapsing-remitting multiple sclerosis (RRMS) on the trial outcome, confirmed disability progression rate (CDPR), and power. Methods: We reported CDPRs in the CombiRx trial by baseline EDSS and by groups (1st (0, 1), 2nd (1.5, 2), 3rd (2.5, 3), and 4th (⩾3.5)) and investigated the effect of different combinations of baseline EDSS proportions on the trial outcome and power. Results: There were 244 (25.4%) participants in the 1st group, 368 (38.4%) in the 2nd group, 223 (23.3%) in the 3rd group, and 124 (12.9%) in the 4th group with CDPRs of 40.1%, 13.9%, 11.2%, and 16.9%, respectively. Both CDPR and power increased when the proportion of the 1st group increased in hypothetical trials with equal sample sizes in each arm, and a 10% increase in the 1st group led to a 5% increase in power. Conclusion: Various baseline EDSS proportions yielded different CDPRs and power, suggesting caution in interpretation of treatment effects across trials that enrolled participants with different proportions of baseline EDSS.

A Comparison of the False Discovery Rate with Dunnett’s Test for a Large Number of Treatments

Probabilities of two types of errors are estimated for the False Discovery Rate Procedure (FDR), Dunnett’s test, and conducting individual t-tests when treatments are being compared against a control. Probabilities for two types of powers are also estimated. The number of treatments considered is 50, 100, 150, and 300 with the first treatment being the control. Differing numbers of treatments equal to the control are considered with equal sample sizes of 3 and 5.

Spatial configuration matters: a test of the habitat amount hypothesis for plants in calcareous grasslands

A recent hypothesis, the habitat amount hypothesis, predicts that the total amount of habitat in the landscape can replace habitat patch size and isolation in studies of species richness in fragmented landscapes. To test the habitat amount hypothesis by first evaluating at which spatial scale the relationship between species richness in equal-sized sample quadrats and habitat amount was the strongest, and then test the importance of spatial configuration of habitat—measured as local patch size and isolation—when habitat amount was taken into account. A quasi-experimental setup with 20 habitat patches of dry calcareous grasslands varying in patch size, patch isolation and habitat amount at the landscape scale was established in the inner Oslo fjord, Southern Norway. We recorded species richness of habitat specialists of vascular plants in equal-sized sample quadrats and analysed the relationship between species richness, habitat amount in the landscape and patch size and isolation. Although the total amount of habitat in a 3 km-radius around the local patch was positively related to species richness in the sample quadrats, local patch size had an additional positive effect, and the effect of patch size was higher when the amount of habitat within the 3 km-radius was high than when it was low. In our study system of specialist vascular plants in dry calcareous grasslands, we do not find support for the habitat amount hypothesis.

An exact Kolmogorov–Smirnov test for whether two finite populations are the same

We develop an algorithm for finding exact critical values for the two-sample Kolmogorov–Smirnov test in the finite-population case. We then compare these exact critical values to the asymptotic values that are available in the literature. The asymptotic critical values work well for equal sample sizes, but can be excessively conservative when the sample sizes differ.