Foraging juncos: interaction of reward mean and variability

Recent stochastic theories of feeding strategy propose that a risk-averse forager will trade off increases in reward variability against increases in average reward. We show that two models of risk-aversion (variance discounting and the z-score model) imply different forms for this interaction of mean and variability. Variance discounting suggests constant risk-aversion, since the effect of reward variance on expected fitness is assumed to be independent of the mean. The z-score model suggests decreasing risk-aversion, since the effect of a given level of reward variance on expected fitness is assumed to decrease as the mean increases. We report two series of experiments with dark-eyed juncos (Junco hyemalis). Each series investigated the mean-variability trade-off under physiological conditions promoting risk-aversion; reward sizes were generally greater in the second series. The results clearly indicate a trade-off, but within a series the data do not discriminate between the two models' predictions. Comparing the two series suggests that the juncos' level of aversion to reward variability decreases after an overall increment in environmental profitability.     

Sensitivity of mean annual primary production to precipitation

In many terrestrial ecosystems, variation in aboveground net primary production (ANPP) is positively correlated with variation in interannual precipitation. Global climate change will alter both the mean and the variance of annual precipitation, but the relative impact of these changes in precipitation on mean ANPP remains uncertain. At any given site, the slope of the precipitation‐ANPP relationship determines the sensitivity of mean ANPP to changes in mean precipitation, whereas the curvature of the precipitation‐ANPP relationship determines the sensitivity of ANPP to changes in precipitation variability. We used 58 existing long‐term data sets to characterize precipitation‐ANPP relationships in terrestrial ecosystems and to quantify the sensitivity of mean ANPP to the mean and variance of annual precipitation. We found that most study sites have a nonlinear, saturating relationship between precipitation and ANPP, but these nonlinearities were not strong. As a result of these weak nonlinearities, ANPP was nearly 40 times more sensitive to precipitation mean than variance. A 1% increase in mean precipitation caused a ?0.2% to 1.8% change in mean ANPP, with a 0.64% increase on average. Sensitivities to precipitation mean peaked at sites with a mean annual precipitation near 500 mm. Changes in species composition and increased intra‐annual precipitation variability could lead to larger ANPP responses to altered precipitation regimes than predicted by our analysis.     

Enhanced Cumulative Sum Charts for Monitoring Process Dispersion

The cumulative sum (CUSUM) control chart is widely used in industry for the detection of small and moderate shifts in process location and dispersion. For efficient monitoring of process variability, we present several CUSUM control charts for monitoring changes in standard deviation of a normal process. The newly developed control charts based on well-structured sampling techniques - extreme ranked set sampling, extreme double ranked set sampling and double extreme ranked set sampling, have significantly enhanced CUSUM chart ability to detect a wide range of shifts in process variability. The relative performances of the proposed CUSUM scale charts are evaluated in terms of the average run length (ARL) and standard deviation of run length, for point shift in variability. Moreover, for overall performance, we implore the use of the average ratio ARL and average extra quadratic loss. A comparison of the proposed CUSUM control charts with the classical CUSUM R chart, the classical CUSUM S chart, the fast initial response (FIR) CUSUM R chart, the FIR CUSUM S chart, the ranked set sampling (RSS) based CUSUM R chart and the RSS based CUSUM S chart, among others, are presented. An illustrative example using real dataset is given to demonstrate the practicability of the application of the proposed schemes.     

Selection for uniformity in livestock by exploiting genetic heterogeneity of residual variance

In some situations, it is worthwhile to change not only the mean, but also the variability of traits by selection. Genetic variation in residual variance may be utilised to improve uniformity in livestock populations by selection. The objective was to investigate the effects of genetic parameters, breeding goal, number of progeny per sire and breeding scheme on selection responses in mean and variance when applying index selection. Genetic parameters were obtained from the literature. Economic values for the mean and variance were derived for some standard non-linear profit equations, e.g. for traits with an intermediate optimum. The economic value of variance was in most situations negative, indicating that selection for reduced variance increases profit. Predicted responses in residual variance after one generation of selection were large, in some cases when the number of progeny per sire was at least 50, by more than 10% of the current residual variance. Progeny testing schemes were more efficient than sib-testing schemes in decreasing residual variance. With optimum traits, selection pressure shifts gradually from the mean to the variance when approaching the optimum. Genetic improvement of uniformity is particularly interesting for traits where the current population mean is near an intermediate optimum.     

The influence of mean climate trends and climate variance on beaver survival and recruitment dynamics

Ecologists are increasingly aware of the importance of environmental variability in natural systems. Climate change is affecting both the mean and the variability in weather and, in particular, the effect of changes in variability is poorly understood. Organisms are subject to selection imposed by both the mean and the range of environmental variation experienced by their ancestors. Changes in the variability in a critical environmental factor may therefore have consequences for vital rates and population dynamics. Here, we examine ≥90‐year trends in different components of climate (precipitation mean and coefficient of variation (CV); temperature mean, seasonal amplitude and residual variance) and consider the effects of these components on survival and recruitment in a population of Eurasian beavers (n = 242) over 13 recent years. Within climatic data, no trends in precipitation were detected, but trends in all components of temperature were observed, with mean and residual variance increasing and seasonal amplitude decreasing over time. A higher survival rate was linked (in order of influence based on Akaike weights) to lower precipitation CV (kits, juveniles and dominant adults), lower residual variance of temperature (dominant adults) and lower mean precipitation (kits and juveniles). No significant effects were found on the survival of nondominant adults, although the sample size for this category was low. Greater recruitment was linked (in order of influence) to higher seasonal amplitude of temperature, lower mean precipitation, lower residual variance in temperature and higher precipitation CV. Both climate means and variance, thus proved significant to population dynamics; although, overall, components describing variance were more influential than those describing mean values. That environmental variation proves significant to a generalist, wide‐ranging species, at the slow end of the slow‐fast continuum of life histories, has broad implications for population regulation and the evolution of life histories.     

Guanine Nucleotide Exchange Factor OSG-1 Confers Functional Aging via Dysregulated Rho Signaling in Caenorhabditis elegans Neurons

Genetic studies usually focus on quantifying and understanding the existence of genetic control on expected phenotypic outcomes. However, there is compelling evidence suggesting the existence of genetic control at the level of environmental variability, with some genotypes exhibiting more stable and others more volatile performance. Understanding the mechanisms responsible for environmental variability not only informs medical questions but is relevant in evolution and in agricultural science. In this work fully sequenced inbred lines of Drosophila melanogaster were analyzed to study the nature of genetic control of environmental variance for two quantitative traits: starvation resistance (SR) and startle response (SL). The evidence for genetic control of environmental variance is compelling for both traits. Sequence information is incorporated in random regression models to study the underlying genetic signals, which are shown to be different in the two traits. Genomic variance in sexual dimorphism was found for SR but not for SL. Indeed, the proportion of variance captured by sequence information and the contribution to this variance from four chromosome segments differ between sexes in SR but not in SL. The number of studies of environmental variation, particularly in humans, is limited. The availability of full sequence information and modern computationally intensive statistical methods provides opportunities for rigorous analyses of environmental variability.     

Frog population viability under present and future climate conditions: a Bayesian state-space approach

1.?World-wide extinctions of amphibians are at the forefront of the biodiversity crisis, with climate change figuring prominently as a potential driver of continued amphibian decline. As in other taxa, changes in both the mean and variability of climate conditions may affect amphibian populations in complex, unpredictable ways. In western North America, climate models predict a reduced duration and extent of mountain snowpack and increased variability in precipitation, which may have consequences for amphibians inhabiting montane ecosystems. 2.?We used Bayesian capture-recapture methods to estimate survival and transition probabilities in a high-elevation population of the Columbia spotted frog (Rana luteiventris) over 10?years and related these rates to interannual variation in peak snowpack. Then, we forecasted frog population growth and viability under a range of scenarios with varying levels of change in mean and variance in snowpack. 3.?Over a range of future scenarios, changes in mean snowpack had a greater effect on viability than changes in the variance of snowpack, with forecasts largely predicting an increase in population viability. Population models based on snowpack during our study period predicted a declining population. 4.?Although mean conditions were more important for viability than variance, for a given mean snowpack depth, increases in variability could change a population from increasing to decreasing. Therefore, the influence of changing climate variability on populations should be accounted for in predictive models. The Bayesian modelling framework allows for the explicit characterization of uncertainty in parameter estimates and ecological forecasts, and thus provides a natural approach for examining relative contributions of mean and variability in climatic variables to population dynamics. 5.?Longevity and heterogeneous habitat may contribute to the potential for this amphibian species to be resilient to increased climatic variation, and shorter-lived species inhabiting homogeneous ecosystems may be more susceptible to increased variability in climate conditions.     

ANOVA Method for Variance Component Decomposition and Diagnosis in Batch Manufacturing Processes

In batch manufacturing processes, the total process variation is generally decomposed into batch-by-batch variation and within-batch variation. Since different variation components may be caused by different sources, separation, testing, and estimation of each variance component are essential to the process improvement. Most of the previous SPC research emphasized reducing variations due to assignable causes by implementing control charts for process monitoring. Different from this focus, this article aims to analyze and reduce inherent natural process variations by applying the ANOVA method. The key issue of using the ANOVA method is how to develop appropriate statistical models for all variation components of interest. The article provides a generic framework for decomposition of three typical variation components in batch manufacturing processes. For the purpose of variation root causes diagnosis, the corresponding linear contrasts are defined to represent the possible site variation patterns and the statistical nested effect models are developed accordingly. The article shows that the use of a full factor decomposition model can expedite the determination of the number of nested effect models and the model structure. Finally, an example is given for the variation reduction in the screening conductive gridline printing process for solar battery fabrication.     

Variability in Abundance of Temperate Reef Fishes Estimated by Visual Census

Identifying sources of sampling variation and quantifying their magnitude is critical to the interpretation of ecological field data. Yet, most monitoring programs of reef fish populations based on underwater visual censuses (UVC) consider only a few of the factors that may influence fish counts, such as the diver or census methodology. Recent studies, however, have drawn attention to a broader range of processes that introduce variability at different temporal scales. This study analyzes the magnitude of different sources of variation in UVCs of temperate reef fishes off Patagonia (Argentina). The variability associated with time-of-day, tidal state, and time elapsed between censuses (minutes, days, weeks and months) was quantified for censuses conducted on the five most conspicuous and common species: Pinguipes brasilianus, Pseudopercis semifasciata, Sebastes oculatus, Acanthistius patachonicus and Nemadactylus bergi. Variance components corresponding to spatial heterogeneity and to the different temporal scales were estimated using nested random models. The levels of variability estimated for the different species were related to their life history attributes and behavior. Neither time-of-day nor tidal state had a significant effect on counts, except for the influence of tide on P. brasilianus. Spatial heterogeneity was the dominant source of variance in all but one species. Among the temporal scales, the intra-annual variation was the highest component for most species due to marked seasonal fluctuations in abundance, followed by the weekly and the instantaneous variation; the daily component was not significant. The variability between censuses conducted at different tidal levels and time-of-day was similar in magnitude to the instantaneous variation, reinforcing the conclusion that stochastic variation at very short time scales is non-negligible and should be taken into account in the design of monitoring programs and experiments. The present study provides baseline information to design and interpret results from visual census programs in temperate reefs.     

A Control Chart Based on Cluster-Regression Adjustment for Retrospective Monitoring of Individual Characteristics

The tendency for experimental and industrial variables to include a certain proportion of outliers has become a rule rather than an exception. These clusters of outliers, if left undetected, have the capability to distort the mean and the covariance matrix of the Hotelling’s T ² multivariate control charts constructed to monitor individual quality characteristics. The effect of this distortion is that the control chart constructed from it becomes unreliable as it exhibits masking and swamping, a phenomenon in which an out-of-control process is erroneously declared as an in-control process or an in-control process is erroneously declared as out-of-control process. To handle these problems, this article proposes a control chart that is based on cluster-regression adjustment for retrospective monitoring of individual quality characteristics in a multivariate setting. The performance of the proposed method is investigated through Monte Carlo simulation experiments and historical datasets. Results obtained indicate that the proposed method is an improvement over the state-of-art methods in terms of outlier detection as well as keeping masking and swamping rate under control.     

A penalized spline approach to functional mixed effects model analysis

In this article, we propose penalized spline (P-spline)-based methods for functional mixed effects models with varying coefficients. We decompose longitudinal outcomes as a sum of several terms: a population mean function, covariates with time-varying coefficients, functional subject-specific random effects, and residual measurement error processes. Using P-splines, we propose nonparametric estimation of the population mean function, varying coefficient, random subject-specific curves, and the associated covariance function that represents between-subject variation and the variance function of the residual measurement errors which represents within-subject variation. Proposed methods offer flexible estimation of both the population- and subject-level curves. In addition, decomposing variability of the outcomes as a between- and within-subject source is useful in identifying the dominant variance component therefore optimally model a covariance function. We use a likelihood-based method to select multiple smoothing parameters. Furthermore, we study the asymptotics of the baseline P-spline estimator with longitudinal data. We conduct simulation studies to investigate performance of the proposed methods. The benefit of the between- and within-subject covariance decomposition is illustrated through an analysis of Berkeley growth data, where we identified clearly distinct patterns of the between- and within-subject covariance functions of children's heights. We also apply the proposed methods to estimate the effect of antihypertensive treatment from the Framingham Heart Study data.     

Physical factors influencing microbial interactions and biochemical changes during the Baceman stage of Indonesian Kecap (soy sauce) production

The composition of samples taken from the baceman stage of traditional Indonesian kecap (soy sauce) production can vary greatly from batch to batch and from manufacturer to manufacturer. This variability could be caused by physical factors such as salt concentration, temperature, dissolved oxygen tension and pH. The effect of these factors on the changes in microflora and biochemical composition during the baceman stage are described in this report. Salt concentration was found to have a large influence on the spoilage of the baceman. At low salt concentrations pellicle-forming yeasts were able to grow. The pellicle on the liquid surface formed by these yeasts provided an acrobic environment in which coryneform bacteria could grow. These bacteria consumed amino acids and fermentation products like acetate and lactate, which resulted in a rise in pH and subsequent spoilage of the baceman. An aerobic baceman showed the same development as those with a low salt concentration. Salt concentration as well as temperature had a large influence on the rate of growth of bacteria. Growth of bacteria and associated lactate and acetate production were also stimulated by a high pH, while growth of yeasts and ethanol and glycerol formation was better at low pH. Also, the production of formol nitrogen by enzymes resulting from the previous bungkil stage was greatly influenced by temperature and pH, with higher temperature and pH giving rise to higher formol nitrogen contents. The consequences of these physical factors on the production of traditional Indonesian kecap are discussed.     

Long Recording Sequences: How to Track the Intra-Individual Variability of Acoustic Signals

Recently developed acoustic technologies - like automatic recording units - allow the recording of long sequences in natural environments. These devices are used for biodiversity survey but they could also help researchers to estimate global signal variability at various (individual, population, species) scales. While sexually-selected signals are expected to show a low intra-individual variability at relatively short time scale, this variability has never been estimated so far. Yet, measuring signal variability in controlled conditions should prove useful to understand sexual selection processes and should help design acoustic sampling schedules and to analyse long call recordings. We here use the overall call production of 36 male treefrogs (Hyla arborea) during one night to evaluate within-individual variability in call dominant frequency and to test the efficiency of different sampling methods at capturing such variability. Our results confirm that using low number of calls underestimates call dominant frequency variation of about 35% in the tree frog and suggest that the assessment of this variability is better by using 2 or 3 short and well-distributed records than by using samples made of consecutive calls. Hence, 3 well-distributed 2-minutes records (beginning, middle and end of the calling period) are sufficient to capture on average all the nightly variability, whereas a sample of 10 000 consecutive calls captures only 86% of it. From a biological point of view, the call dominant frequency variability observed in H. arborea (116Hz on average but up to 470 Hz of variability during the course of the night for one male) challenge about its reliability in mate quality assessment. Automatic acoustic recording units will provide long call sequences in the near future and it will be then possible to confirm such results on large samples recorded in more complex field conditions.     

The effect of past changes in inter‐annual temperature variability on tree distribution limits

Aim The northern limits of temperate broadleaved species in Fennoscanndia are controlled by their requirements for summer warmth for successful regeneration and growth as well as by the detrimental effects of winter cold on plant tissue. However, occurrences of meteorological conditions with detrimental effects on individual species are rare events rather than a reflection of average conditions. We explore the effect of changes in inter‐annual temperature variability on the abundances of the tree species Tilia cordata, Quercus robur and Ulmus glabra near their distribution limits using a process‐based model of ecosystem dynamics. Location A site in central Sweden and a site in southern Finland were used as examples for the ecotone between boreal and temperate forests in Fennoscandia. The Finnish site was selected because of the availability of varve‐thickness data. Methods The dynamic vegetation model LPJ‐GUESS was run with four scenarios of inter‐annual temperature forcing for the last 10,000 years. In one scenario the variability in the thickness of summer and winter varves from the annually laminated lake in Finland was used as a proxy for past inter‐annual temperature variability. Two scenarios were devised to explore systematically the effect of stepwise changes in the variance and shape parameter of a probability distribution. All variability scenarios were run both with and without the long‐term trend in Holocene temperature change predicted by an atmospheric general circulation model. Results Directional changes in inter‐annual temperature variability have significant effects on simulated tree distribution limits through time. Variations in inter‐annual temperature variability alone are shown to alter vegetation composition by magnitudes similar to the magnitude of changes driven by variation in mean temperatures. Main conclusions The varve data indicate that inter‐annual climate variability has changed in the past. The model results show that past changes in species abundance can be explained by changes in the inter‐annual variability of climate parameters as well as by mean climate. Because inter‐annual climatic variability is predicted to change in the future, this component of climate change should be taken into account both when making projections of future plant distributions and when interpreting vegetation history.     

The Cohesive Population Genetics of Molecular Drive

The long-term population genetics of multigene families is influenced by several biased and unbiased mechanisms of nonreciprocal exchanges (gene conversion, unequal exchanges, transposition) between member genes, often distributed on several chromosomes. These mechanisms cause fluctuations in the copy number of variant genes in an individual and lead to a gradual replacement of an original family of n genes (A) in N number of individuals by a variant gene (a). The process for spreading a variant gene through a family and through a population is called molecular drive. Consideration of the known slow rates of nonreciprocal exchanges predicts that the population variance in the copy number of gene a per individual is small at any given generation during molecular drive. Genotypes at a given generation are expected only to range over a small section of all possible genotypes from one extreme (n number of A) to the other (n number of a). A theory is developed for estimating the size of the population variance by using the concept of identity coefficients. In particular, the variance in the course of spreading of a single mutant gene of a multigene family was investigated in detail, and the theory of identity coefficients at the state of steady decay of genetic variability proved to be useful. Monte Carlo simulations and numerical analysis based on realistic rates of exchange in families of known size reveal the correctness of the theoretical prediction and also assess the effect of bias in turnover. The population dynamics of molecular drive in gradually increasing the mean copy number of a variant gene without the generation of a large variance (population cohesion) is of significance regarding potential interactions between natural selection and molecular drive.     

Microevolution of Aedes aegypti

Scientific research into the epidemiology of dengue frequently focuses on the microevolution and dispersion of the mosquito Aedes aegypti. One of the world’s largest urban agglomerations infested by Ae. aegypti is the Brazilian megalopolis of Sao Paulo, where >26,900 cases of dengue were reported until June 2015. Unfortunately, the dynamics of the genetic variability of Ae. aegypti in the Sao Paulo area have not been well studied. To reduce this knowledge gap, we assessed the morphogenetic variability of a population of Ae. aegypti from a densely urbanised neighbourhood of Sao Paulo. We tested if allelic patterns could vary over a short term and if wing shape could be a predictor of the genetic variation. Over a period of 14 months, we examined the variation of genetic (microsatellites loci) and morphological (wing geometry) markers in Ae. aegypti. Polymorphisms were detected, as revealed by the variability of 20 microsatellite loci (115 alleles combined; overall F_st = 0.0358) and 18 wing landmarks (quantitative estimator Q_st = 0.4732). These levels of polymorphism are higher than typically expected to an exotic species. Allelic frequencies of the loci changed over time and temporal variation in the wing shape was even more pronounced, permitting high reclassification levels of chronological samples. In spite of the fact that both markers underwent temporal variation, no correlation was detected between their dynamics. We concluded that microevolution was detected despite the short observational period, but the intensities of change of the markers were discrepant. Wing shape failed from predicting allelic temporal variation. Possibly, natural selection (Q_st>F_st) or variance of expressivity of wing phenotype are involved in this discrepancy. Other possibly influential factors on microevolution of Ae. aegypti are worth searching. Additionally, the implications of the rapid evolution and high polymorphism of this mosquito vector on the efficacy of control methods have yet to be investigated.     

Optimal Designs of the Median Run Length Based Double Sampling X? Chart for Minimizing the Average Sample Size

Designs of the double sampling (DS) chart are traditionally based on the average run length (ARL) criterion. However, the shape of the run length distribution changes with the process mean shifts, ranging from highly skewed when the process is in-control to almost symmetric when the mean shift is large. Therefore, we show that the ARL is a complicated performance measure and that the median run length (MRL) is a more meaningful measure to depend on. This is because the MRL provides an intuitive and a fair representation of the central tendency, especially for the rightly skewed run length distribution. Since the DS chart can effectively reduce the sample size without reducing the statistical efficiency, this paper proposes two optimal designs of the MRL-based DS chart, for minimizing (i) the in-control average sample size (ASS) and (ii) both the in-control and out-of-control ASSs. Comparisons with the optimal MRL-based EWMA and Shewhart charts demonstrate the superiority of the proposed optimal MRL-based DS chart, as the latter requires a smaller sample size on the average while maintaining the same detection speed as the two former charts. An example involving the added potassium sorbate in a yoghurt manufacturing process is used to illustrate the effectiveness of the proposed MRL-based DS chart in reducing the sample size needed.     

Genetic basis of between‐individual and within‐individual variance of docility

Between‐individual variation in phenotypes within a population is the basis of evolution. However, evolutionary and behavioural ecologists have mainly focused on estimating between‐individual variance in mean trait and neglected variation in within‐individual variance, or predictability of a trait. In fact, an important assumption of mixed‐effects models used to estimate between‐individual variance in mean traits is that within‐individual residual variance (predictability) is identical across individuals. Individual heterogeneity in the predictability of behaviours is a potentially important effect but rarely estimated and accounted for. We used 11 389 measures of docility behaviour from 1576 yellow‐bellied marmots (Marmota flaviventris) to estimate between‐individual variation in both mean docility and its predictability. We then implemented a double hierarchical animal model to decompose the variances of both mean trait and predictability into their environmental and genetic components. We found that individuals differed both in their docility and in their predictability of docility with a negative phenotypic covariance. We also found significant genetic variance for both mean docility and its predictability but no genetic covariance between the two. This analysis is one of the first to estimate the genetic basis of both mean trait and within‐individual variance in a wild population. Our results indicate that equal within‐individual variance should not be assumed. We demonstrate the evolutionary importance of the variation in the predictability of docility and illustrate potential bias in models ignoring variation in predictability. We conclude that the variability in the predictability of a trait should not be ignored, and present a coherent approach for its quantification.     

Impact of Spatial and Temporal Variation in Calf Survival on the Growth of Elk Populations

Abstract: The realized impact of a vital rate on population growth (λ) is determined by both the relative influence of the vital rate on λ (elasticity) and its magnitude of variability. We estimated mean survival and reproductive rates in elk (Cervus elaphus) and spatial and temporal variation in these rates from 37 sources located primarily across the Rocky Mountain region and northwestern United States. We removed sampling variance from estimates of process variance both within and across vital-rate data sets using the variance discounting method developed by White (2000). Deterministic elasticities calculated from a population matrix model parameterized with these mean vital rates ranked adult female survival (e_Scow = 0.869) much higher than calf survival (e_Scalf = 0.131). However, process variance in calf survival was >11 times greater than process variance in female survival across data sets and 10 times greater on average within studies. We conducted Life-Stage Simulation Analysis to incorporate both vital-rate elasticity patterns and empirical estimates of variability to identify those vital rates most influential in elk population dynamics. The overwhelming magnitude of variation in calf survival explained 75% of the variation in the population growth rates generated from 1,000 matrix replicates, compared to just 16% of the variation in λ explained by variation in female survival. Variation in calf survival greatly impacts elk population growth and calls into question the utility of classical elasticity analysis alone for guiding elk management. These results also suggest that the majority of interannual variability that wildlife managers document in late-winter and spring elk surveys is attributable to variation in calf survival over the previous year and less influenced by variation in the harvest of females during the preceding autumn. To meet elk population size objectives, managers should consider the inherent variation in calf survival, and its apparent sensitivity to management, in addition to female harvest.     

Zooplankton sampling strategies for environmental studies

This study characterizes sources of variation in total zooplankton abundance estimates at seven stations within the 5–10 m depth contour of southeastern Lake Michigan which were sampled monthly, April through October, for the 1975 to 1979 period. Month, year, and station were statistically significant factors affecting abundance estimates as were all interactions. Month was the largest source of variance either as a main effect or interaction. Smallest coefficients of variation were associated with subsampling (mean 6.1%) and replicate sampling (mean 15.1%). The between-station coefficient of variation averaged 39.0% and tended to be highest during the summer. For a given station and month, the between-year coefficient of variation averaged 73.4% while the between-month coefficient of variation for a single station in a given year averaged 95.1%. A table shows the estimated number of replications necessary to detect a true difference in two population means as a function of coefficient of variation. Environmental studies designed to detect spatial alterations should conduct such analyses on a cruise-by-cruise basis. Cruises should consist of a large number of stations and be conducted at least once during each season. Studies designed to detect temporal alterations require more frequent sampling because of the greater variability associated with temporal data sets. Because spatial variability adds little to the overall variability of such data sets, only a few representative stations need be sampled during each cruise.