Simple, defensible sample sizes based on cost efficiency

Summary .   The conventional approach of choosing sample size to provide 80% or greater power ignores the cost implications of different sample size choices. Costs, however, are often impossible for investigators and funders to ignore in actual practice. Here, we propose and justify a new approach for choosing sample size based on cost efficiency, the ratio of a study's projected scientific and/or practical value to its total cost. By showing that a study's projected value exhibits diminishing marginal returns as a function of increasing sample size for a wide variety of definitions of study value, we are able to develop two simple choices that can be defended as more cost efficient than any larger sample size. The first is to choose the sample size that minimizes the average cost per subject. The second is to choose sample size to minimize total cost divided by the square root of sample size. This latter method is theoretically more justifiable for innovative studies, but also performs reasonably well and has some justification in other cases. For example, if projected study value is assumed to be proportional to power at a specific alternative and total cost is a linear function of sample size, then this approach is guaranteed either to produce more than 90% power or to be more cost efficient than any sample size that does. These methods are easy to implement, based on reliable inputs, and well justified, so they should be regarded as acceptable alternatives to current conventional approaches.     

SPCalc: A web-based calculator for sample size and power calculations in micro-array studies

Calculation of the appropriate sample size in planning microarray studies is important because sample collection can be expensive and time-consuming. Sample-size calculation is also a challenging issue for microarray studies because the number of genes is far larger than the number of samples so that traditional methods of sample-size calculation cannot be directly applied. To help investigators answer the question of how many samples are needed in their microarray studies, we developed a user-friendly web-based calculator, SPCalc, for calculating sample size and power for a variety of commonly used experimental designs, including completely randomized treatmentcontrol design, matched-pairs design, multiple-treatment design having an isolated treatment effect, and randomized block design. AVAILABILITY: The web-based calculator SPCalc is publicly available at http://www.biostat.harvard.edu /people/faculty/mltlee/webfront-r.html.     

Heterogeneous treatment effects in stratified clinical trials with time‐to‐event endpoints

When analyzing clinical trials with a stratified population, homogeneity of treatment effects is a common assumption in survival analysis. However, in the context of recent developments in clinical trial design, which aim to test multiple targeted therapies in corresponding subpopulations simultaneously, the assumption that there is no treatment‐by‐stratum interaction seems inappropriate. It becomes an issue if the expected sample size of the strata makes it unfeasible to analyze the trial arms individually. Alternatively, one might choose as primary aim to prove efficacy of the overall (targeted) treatment strategy. When testing for the overall treatment effect, a violation of the no‐interaction assumption renders it necessary to deviate from standard methods that rely on this assumption. We investigate the performance of different methods for sample size calculation and data analysis under heterogeneous treatment effects. The commonly used sample size formula by Schoenfeld is compared to another formula by Lachin and Foulkes, and to an extension of Schoenfeld's formula allowing for stratification. Beyond the widely used (stratified) Cox model, we explore the lognormal shared frailty model, and a two‐step analysis approach as potential alternatives that attempt to adjust for interstrata heterogeneity. We carry out a simulation study for a trial with three strata and violations of the no‐interaction assumption. The extension of Schoenfeld's formula to heterogeneous strata effects provides the most reliable sample size with respect to desired versus actual power. The two‐step analysis and frailty model prove to be more robust against loss of power caused by heterogeneous treatment effects than the stratified Cox model and should be preferred in such situations.     

Size‐induced phenotypic reaction norms in a parasitoid wasp: an examination of life‐history and behavioural traits

The amount of resources available during development often affects body size, causing phenotypic variation in life‐history traits and reproductive behaviours. However, past studies have seldom examined the reaction norms of both life‐history and behavioural traits versus body size. We measured the phenotypic plasticity of several life‐history (age‐specific egg load, egg size, longevity) and behavioural (oviposition rate, host marking rate, walking speed) traits of the egg parasitoid Telenomus podisi Ashmead (Hymenoptera: Scelionidae) in response to body size variation. We predicted that life‐history traits would show more evidence of size compensation than behavioural traits, resulting in fewer positively‐sloped size versus trait reaction norms among the former. As predicted by life‐history models, smaller wasps appear to shift resource allocation towards early‐life reproduction, having a similar egg load to large individuals 9 days after emergence. Surprisingly, longevity was unaffected by body size. However, egg size, the number of offspring produced during oviposition bouts, and the rate of subsequent egg synthesis were greater for larger individuals. In addition, as predicted, the reaction norms of behavioural traits versus body size were all positively sloped. Thus, despite possible adaptive compensatory plasticity of life‐history traits by small individuals, behavioural constraints directly related to body size would contribute to maintaining a positive size–fitness relationship.     

Assessment method for a power analysis to identify differentially expressed pathways

Gene expression data can provide a very rich source of information for elucidating the biological function on the pathway level if the experimental design considers the needs of the statistical analysis methods. The purpose of this paper is to provide a comparative analysis of statistical methods for detecting the differentially expression of pathways (DEP). In contrast to many other studies conducted so far, we use three novel simulation types, producing a more realistic correlation structure than previous simulation methods. This includes also the generation of surrogate data from two large-scale microarray experiments from prostate cancer and ALL. As a result from our comprehensive analysis of 41,004 parameter configurations, we find that each method should only be applied if certain conditions of the data from a pathway are met. Further, we provide method-specific estimates for the optimal sample size for microarray experiments aiming to identify DEP in order to avoid an underpowered design. Our study highlights the sensitivity of the studied methods on the parameters of the system.     

SNP allele frequency estimation in DNA pools and variance components analysis

The estimation of single nucleotide polymorphism (SNP) allele frequency in pooled DNA samples has been proposed as a cost-effective approach to whole genome association studies. However, the key issue is the allele frequency window in which a genotyping method operates and provides a statistically reliable answer. We assessed the homogeneous mass extend assay and estimated the variance associated with each experimental stage. We report that a relationship between estimated allele frequency and variance might exist, suggesting that high statistical power can be retained at low, as well as high, allele frequencies. Assuming this relationship, the formation of subpools consisting of 100 samples retains an effective sample size greater than 70% of the true sample size, with a savings of 11-fold the cost of an individual genotyping study, regardless of allele frequency.     

The yellow stingray, Urobatis jamaicensis, as a model for studying cerebellar function in vertebrates

Fishes, in general, have several advantages as vertebrate models for basic brain function. They are phylogenetically closer than mammals to the basic vertebrate blueprint and thus allow behavioural and neurological studies of fundamental brain systems without the interaction of more recently evolved functions. Further, the absence of a highly developed telencephalon allows ready access to many structures without cerebral interference. A disadvantage of working with most fishes is, however, the relatively small size of the brain that often hinders or precludes the use of many standard neurological techniques. In contrast, a group of chondrichthians, the stingrays, Dasyatoidea, has a brain size rivaling mammalian rodent models. Of particular interest to our research, stingrays, like mammals, have a large, complex, three‐lobed cerebellum. However, in the yellow stingray these lobes are completely separated. Thus, the lobes can be individually manipulated to examine behavioural correlates of specific lobes. For example, ablation of the centre lobe (also known as anterior caudal lobule) causes a fixed‐pattern hyperactivity. Yellow stingrays are abundant in many areas, they are hardy, and tolerate anaesthesia and the surgical procedures well. A more complete elucidation of cerebellar function awaits further physical and pharmacological ablative studies but the potential for these animals as vertebrate models of cerebellar‐controlled behaviour is clear.     

Fatty acid–genotype interactions and cardiovascular risk

Cardiovascular disease (CVD) risk and rate of progression is determined by genetic, environmental and behavioural factors. Majority of genotype–diet–CVD phenotype research till date has focussed on the interactive impact of single nucleotide polymorphisms (SNP) and dietary fat composition, on blood lipids levels, with strong evidence of the existence of hypo- and hyper-responders. However, a recognised concern in the field of nutrigenetics is a lack of consistency between findings of different studies. This apparent lack of consistency is likely to be attributable to the impact of factors such as ethnicity and gender on the ‘size’ of nutrigenetic interactions, a clear understanding of which needs to be gained. Although not yet ready for widespread use, in the future a greater use of genetic profiling is likely to enhance current strategies of CVD prediction, and improve the design of more personalised approaches to minimise risk in the individual.     

How Reliable are the Methods for Estimating Repertoire Size?

Quantifying signal repertoire size is a critical first step towards understanding the evolution of signal complexity. However, counting signal types can be so complicated and time consuming when repertoire size is large, that this trait is often estimated rather than measured directly. We studied how three common methods for repertoire size quantification (i.e., simple enumeration, curve‐fitting and capture‐recapture analysis) are affected by sample size and presentation style using simulated repertoires of known sizes. As expected, estimation error decreased with increasing sample size and varied among presentation styles. More surprisingly, for all but one of the presentation styles studied, curve‐fitting and capture–recapture analysis yielded errors of similar or greater magnitude than the errors researchers would make by simply assuming that the number of types in an incomplete sample is the true repertoire size. Our results also indicate that studies based on incomplete samples are likely to yield incorrect ranking of individuals and spurious correlations with other parameters regardless of the technique of choice. Finally, we argue that biological receivers face similar difficulties in quantifying repertoire size than human observers and we explore some of the biological implications of this hypothesis.     

Sample size and sampling error in geometric morphometric studies of size and shape

Geometric morphometric studies are increasingly becoming common in systematics and palaeontology. The samples in such studies are often small, due to the paucity of material available for analysis. However, very few studies have tried to assess the impact of sampling error on analytical results. Here, this issue is addressed empirically using repeated randomized selection experiments to build progressively smaller samples from an original dataset of ∼400 vervet monkey (Cercopithecus aethiops) skulls. Size and shape parameters (including mean size and shape, size and shape variances, angles of allometric trajectories) that are commonly used in geometric morphometric studies, are estimated first in the original sample and then in the random subsamples. Estimates are then compared to give an indication of what is the minimum desirable sample size for each parameter. Mean size, standard deviation of size and variance of shape are found to be fairly accurate even in relatively small samples. In contrast, mean shapes and angles between static allometric trajectories are strongly affected by sampling error. If confirmed in other groups, our findings may have substantial implications for studies of morphological variation in present and fossil species. By performing rarefaction analyses like those presented in our study, morphometricians can be easily provided with important clues on how a simple but crucial factor like sample size can alter results of their studies.     

Taking a Comparative Approach: Analysing Personality as a Multivariate Behavioural Response across Species

Animal personality, repeatable behaviour through time and across contexts, is ecologically and evolutionarily important as it can account for the exhibition of sub-optimal behaviours. Interspecific comparisons have been suggested as important for understanding the evolution of animal personality; however, these are seldom accomplished due, in part, to the lack of statistical tools for quantifying differences and similarities in behaviour between groups of individuals. We used nine species of closely-related coral reef fishes to investigate the usefulness of ecological community analyses for the analysis of between-species behavioural differences and behavioural heterogeneity. We first documented behavioural carryover across species by observing the fishes' behaviour and measuring their response to a threatening stimulus to quantify boldness. Bold fish spent more time away from the reef and fed more than shy fish. We then used ecological community analysis tools (canonical variate analysis, multi-response permutation procedure, and permutational analysis of multivariate dispersion) and identified four 'clusters' of behaviourally similar fishes, and found that the species differ in the behavioural variation expressed; some species are more behaviourally heterogeneous than others. We found that ecological community analysis tools are easily and fruitfully applied to comparative studies of personality and encourage their use by future studies.     

Translocation for conservation: Neonates are less suitable than adults

Animal behaviour can affect the outcome of conservation translocations. It is important to understand the behaviour of the species being considered for translocation and how its behaviour varies over life stages. There may be uncertainty about what life stages are best as founders for release back into wild populations. A technique called head‐starting whereby juvenile life stages are raised in captivity and then released is one potential pre‐release strategy. However, juveniles of many species have a dispersive role in the life cycle, potentially raising difficulties for establishing new populations due to dispersal from the intended habitat following release. For this study, we compared aspects of the behaviour of captive adult and neonate pygmy bluetongue lizards (Tiliqua adelaidensis) – an endangered species for which translocation is likely to be an important management strategy – to determine if neonate behavioural characteristics are appropriate for their translocation. We filmed adult and neonate pygmy bluetongue lizards and compared their behaviour. We also filmed adults over an activity season to compare seasonal behaviour. Behavioural parameters measured included basking time, burrow exits, burrows occupied and walking the perimeter wall. Neonates basked significantly more than adults in summer and autumn. Neonates are likely to be basking more than adults because they are in a stage of rapid growth and need to gain body mass before the winter inactivity period. Neonates exited burrows more often than adults and used a greater number of burrows. These results indicate neonate lizards are actively exploring their habitat. Neonates are unlikely to be as suitable for translocation as they are actively moving about and more likely to be predated upon or disperse from the translocation site. Our finding can be applied to other species that have active juvenile life stages and are at particular risk of predation due to their small size.     

Impact of an ecological factor on the costs of resource acquisition: fighting and metabolic physiology of crabs

1. Current game theory models and recent experimental evidence suggests that the strategy an animal adopts in agonistic encounters is determined by individual state. Therefore manipulation of an individual's state should elicit different behavioural responses. In this paper, mechanisms are examined that underlie state-dependent strategies using Shore Crabs, Carcinus maenas , and how, by altering the environment, behaviour and physiology are affected.
2. Fights were staged between pairs of male crabs under normoxic and severely hypoxic (<15 torr) conditions to determine if the metabolic costs of fighting and resource acquisition are affected by water P _O2. After fighting, blood and tissue samples from each crab were taken and analysed for metabolites associated with anaerobiosis ( L -lactate, glucose and glycogen).
3. The spectrum of behavioural acts performed during contests was unaffected by hypoxic conditions. However, fight duration was significantly shorter in the hypoxic treatment.
4. The phenomenon of being of a larger relative size and winning had a greater influence in the contests staged under hypoxia with 93% of the victors being of a larger size compared to 78% in normoxic conditions. Fight duration and intensity had no relationship with relative size in either treatments.
5. The accumulation of L -lactate was significantly greater in the blood and tissues of crabs after fighting under hypoxia than in normoxic conditions. In addition, there was greater glycolytic activity in the tissues of these crabs, shown by elevated concentrations of glucose in the blood and increased breakdown of glycogen.
6. This study demonstrates that the internal state of the crabs altered the length of time they were willing to engage in fighting and that fighting was energetically more expensive under hypoxic conditions.     

Body size,but not age-at-maturation or context,affects the expression of predator-induced behavioural plasticity in female green swordtails (Xiphophorus hellerii)

Behavioural plasticity is a form of reversible phenotypic plasticity in which a genotype can express different behavioural phenotypes under different environmental conditions. Though an interest in among-individual differences in behavioural plasticity has flourished in recent decades, few studies have considered the effects of intrinsic factors, such as life-history or morphological traits, in tandem with extrinsic factors, such as presence of conspecifics in different social contexts, on predator-induced behavioural plasticity. Here, we present a study conducted with female green swordtail fishes, Xiphophorus hellerii, designed to assess the effects of age-at-maturation and body size on the expression of predator-induced behavioural plasticity in two social contexts: (a) female-only (two females) and (b) female-and-male (two females and a male). We further examined the extent to which individual expression of behavioural plasticity is consistent across these two social contexts. We found that in the presence of a predator, focal females were more timid in response to the stimulus and more tolerant of the non-focal female, and small females expressed this change from bold/less tolerant to timid/more tolerant to a greater degree than large females, regardless of age-at-maturation. However, individuals were not consistent in the degree or direction of plasticity expressed in the behaviours of interest between the female-only and the female-and-male context. Here, we show that within- and among-individual differences in behavioural expression are common but inconsistent. How intrinsic and extrinsic factors independently or together drive expression of plasticity in antipredator and agonistic behaviours is varied and warrants further study.     

Accounting for the capacity of common and rare species to contribute to diversity spatial patterns: Is it a sampling issue or a biological effect?

The contribution of common species to overall species richness in many cases is greater than that of rare species. However, the explanation of this phenomenon remains vague. One hypothesis is that this is a sampling issue and not a biological one. Therefore standardization methods like the information index and empirical variance have been proposed. But, these standardizations do not explicitly compare the significance of the dataset size of the common and rare sub-assemblage. Here, we investigate the role of dataset size in accounting for the capacity of common and rare species to contribute to diversity spatial patterns. We used a dataset of 5148 vascular plant species recorded in 16,439 sample plots in the Greek Natura 2000 network. Species were ranked according to the number of sample plots they occupied in ascending (rare to common), descending (common to rare) and random order. We analyzed the correlation between the richness of each sub-assemblage and total species richness. When comparing among sub-assemblages with equal number of species, common species are clearly the better predictors of total species richness. But, when comparing among sub-assemblages with equal number of occurrence records, the patterns changed. Common and rare species contribution to the overall richness pattern was comparable, with rare species contributing slightly less than widespread species in some cases and the opposite in other cases. However in all cases, sub-assemblages of random species remarkably outperformed the equal sized sub-assemblages of common or rare species. Our results suggest that common and rare species are biased samples of the community and that equal sized random samples are more representative; thus the greater contribution of common species than rare species to biodiversity patterns might be more a sampling issue than a biological effect of commonness or rarity.     

Statistical tests for taxonomic distinctiveness from observations of monophyly

The observation of monophyly for a specified set of genealogical lineages is often used to place the lineages into a distinctive taxonomic entity. However, it is sometimes possible that monophyly of the lineages can occur by chance as an outcome of the random branching of lineages within a single taxon. Thus, especially for small samples, an observation of monophyly for a set of lineages--even if strongly supported statistically--does not necessarily indicate that the lineages are from a distinctive group. Here I develop a test of the null hypothesis that monophyly is a chance outcome of random branching. I also compute the sample size required so that the probability of chance occurrence of monophyly of a specified set of lineages lies below a prescribed tolerance. Under the null model of random branching, the probability that monophyly of the lineages in an index group occurs by chance is substantial if the sample is highly asymmetric, that is, if only a few of the sampled lineages are from the index group, or if only a few lineages are external to the group. If sample sizes are similar inside and outside the group of interest, however, chance occurrence of monophyly can be rejected at stringent significance levels (P < 10(-5)) even for quite small samples (approximately 20 total lineages). For a fixed total sample size, rejection of the null hypothesis of random branching in a single taxon occurs at the most stringent level if samples of nearly equal size inside and outside the index group--with a slightly greater size within the index group--are used. Similar results apply, with smaller sample sizes needed, when reciprocal monophyly of two groups, rather than monophyly of a single group, is of interest. The results suggest minimal sample sizes required for inferences to be made about taxonomic distinctiveness from observations of monophyly.     

Egg size and offspring quality: a meta‐analysis in birds

Parents affect offspring fitness by propagule size and quality, selection of oviposition site, quality of incubation, feeding of dependent young, and their defence against predators and parasites. Despite many case studies on each of these topics, this knowledge has not been rigorously integrated into individual parental care traits for any taxon. Consequently, we lack a comprehensive, quantitative assessment of how parental care modifies offspring phenotypes. This meta‐analysis of 283 studies with 1805 correlations between egg size and offspring quality in birds is intended to fill this gap. The large sample size enabled testing of how the magnitude of the relationship between egg size and offspring quality depends on a number of variables. Egg size was positively related to nearly all studied offspring traits across all stages of the offspring life cycle. Not surprisingly, the relationship was strongest at hatching but persisted until the post‐fledging stage. Morphological traits were the most closely related to egg size but significant relationships were also found with hatching success, chick survival, and growth rate. Non‐significant effect sizes were found for egg fertility, chick immunity, behaviour, and life‐history or sexual traits. Effect size did not depend on whether chicks were raised by their natural parents or were cross‐fostered to other territories. Effect size did not depend on species‐specific traits such as developmental mode, clutch size, and relative size of the egg, but was larger if tested in captive compared to wild populations and between rather than within broods. In sum, published studies support the view that egg size affects juvenile survival. There are very few studies that tested the relationship between egg size and the fecundity component of offspring fitness, and no studies on offspring survival as adults or on global fitness. More data are also needed for the relationships between egg size and offspring behavioural and physiological traits. It remains to be established whether the relationship between egg size and offspring performance depends on the quality of the offspring environment. Positive effect sizes found in this study are likely to be driven by a causal effect of egg size on offspring quality. However, more studies that control for potential confounding effects of parental post‐hatching care, genes, and egg composition are needed to establish firmly this causal link.     

New Ranked Set Sampling for One‐Parameter Family of Distributions

Bhoj (1997c) proposed a new ranked set sampling (NRSS) procedure for a specific two‐parameter family of distributions when the sample size is even. This NRSS procedure can be applied to one‐parameter family of distributions when the sample size is even. However, this procedure cannot be used if the sample size is odd. Therefore, in this paper, we propose a modified version of the NRSS procedure which can be used for one‐parameter distributions when the sample size is odd. Simple estimator for the parameter based on proposed NRSS is derived. The relative precisions of this estimator are higher than those of other estimators which are based on other ranked set sampling procedures and the best linear unbiased estimator using all order statistics.     

How does habitat complexity affect ant foraging success? A test using functional measures on three continents

Habitat complexity can mediate key processes that structure local assemblages through effects on factors such as competition, predation and foraging behaviour. While most studies address assemblage responses to habitat complexity within one locality, a more global approach allows conclusions with greater independence from the phylogenetic constraints of the target assemblages, thus allowing greater generality. We tested the effects of natural and manipulated habitat complexities on ant assemblages from South Africa, Australia and Sweden, in order to determine if there were globally consistent responses in how functional measures of foraging success are regulated by habitat complexity. Specifically, we considered how habitat complexity affected ant foraging rates including the speed of discovery and rate of monopolisation. We also tested if habitat complexity affected the body size index, a size-related morphological trait, of ants discovering resources and occupying and monopolising the resources after 180 min. Ants were significantly slower to discover baits in the more complex treatments, consistent with predictions that they would move more slowly through more complex environments. The monopolisation index was also lower in the more complex treatments, suggesting that resources were more difficult to defend. Our index of ant body size showed trends in the predicted direction for complexity treatments. In addition, ants discovering, occupying and monopolising resources were smaller in simple than in complex natural habitats. Responses of discovering ants to resources in natural habitats were clear in only one of three regions. Consistent with our predictions, habitat complexity thus affected functional measures of the foraging success of ants in terms of measures of discovery and monopolisation rates and body size traits of successful ants. However, patterns were not always equally clear in manipulative and mensurative components of the study.