Testing for environmentally induced bias in phenotypic estimates of natural selection: theory and practice

Measuring natural selection has been a fundamental goal of evolutionary biology for more than a century, and techniques developed in the last 20 yr have provided relatively simple means for biologists to do so. Many of these techniques, however, share a common limitation: when applied to phenotypic data, environmentally induced covariances between traits and fitness can lead to biased estimates of selection and misleading predictions about evolutionary change. Utilizing estimates of breeding values instead of phenotypic data with these methods can eliminate environmentally induced bias, although this approach is more difficult to implement. Despite this potential limitation to phenotypic methods and the availability of a potential solution, little empirical evidence exists on the extent of environmentally induced bias in phenotypic estimates of selection. In this article, we present a method for detecting bias in phenotypic estimates of selection and demonstrate its use with three independent data sets. Nearly 25% of the phenotypic selection gradients estimated from our data are biased by environmental covariances. We find that bias caused by environmental covariances appears mainly to affect quantitative estimates of the strength of selection based on phenotypic data and that the magnitude of these biases is large. As our estimates of selection are based on data from spatially replicated field experiments, we suggest that our findings on the prevalence of bias caused by environmental covariances are likely to be conservative.     

Precision and accuracy of divergence time estimates from STR and SNPSTR variation

Inference of intraspecific population divergence patterns typically requires genetic data for molecular markers with relatively high mutation rates. Microsatellites, or short tandem repeat (STR) polymorphisms, have proven informative in many such investigations. These markers are characterized, however, by high levels of homoplasy and varying mutational properties, often leading to inaccurate inference of population divergence. A SNPSTR is a genetic system that consists of an STR polymorphism closely linked (typically < 500 bp) to one or more single-nucleotide polymorphisms (SNPs). SNPSTR systems are characterized by lower levels of homoplasy than are STR loci. Divergence time estimates based on STR variation (on the derived SNP allele background) should, therefore, be more accurate and precise. We use coalescent-based simulations in the context of several models of demographic history to compare divergence time estimates based on SNPSTR haplotype frequencies and STR allele frequencies. We demonstrate that estimates of divergence time based on STR variation on the background of a derived SNP allele are more accurate (3% to 7% bias for SNPSTR versus 11% to 20% bias for STR) and more precise than STR-based estimates, conditional on a recent SNP mutation. These results hold even for models involving complex demographic scenarios with gene flow, population expansion, and population bottlenecks. Varying the timing of the mutation event generating the SNP revealed that estimates of divergence time are sensitive to SNP age, with more recent SNPs giving more accurate and precise estimates of divergence time. However, varying both mutational properties of STR loci and SNP age demonstrated that multiple independent SNPSTR systems provide less biased estimates of divergence time. Furthermore, the combination of estimates based separately on STR and SNPSTR variation provides insight into the age of the derived SNP alleles. In light of our simulations, we interpret estimates from data for human populations.     

Reducing the cost of benthic sample processing by using sieve retention probability models

Estimation of abundance or biomass of benthic invertebrates requires considerable effort to process samples. Consequently, it has been suggested to process only organisms retained by a relatively coarse meshed sieve and apply size-specific correction factors based on the probability that a sieve retains individual organisms. Benthic samples were collected from 10 sites in 2 regions and processed to validate an existing empirical model predicting sieve retention probabilities, to test whether periphyton biomass affects probability of retention, and to determine the optimal strategy that minimizes both cost and variability of estimates. The existing model predicting sieve retention probabilities corrected for organisms lost through sieves and mostly corrected for underestimation of biomass, but this model lead to overestimates of the frequency of the smallest organisms. Inclusion of algal biomass improved slightly the proportion of correct predictions (whether an organism is retained or not by a sieve) by 0.6% relative to the existing model (from 90.8% to 91.4%), and removed the bias. Density and biomass estimates obtained by only processing organisms retained by 1- or 2-mm sieves and applying correction factors derived from the predicted retention probabilities were accurate and only marginally less precise than estimates obtained by processing all organisms. The reduced precision of estimates from subsets of organisms could be compensated by increasing sample size and still lead to a reduction of 40–60% of the number of organisms processed. Even though the use of subsets introduces additional analytical variability, this variability is relatively small compared to the natural spatial variability among replicates. Handling editor: K. Martens     

Finding confidence limits on population growth rates: Bootstrap and analytic methods

When predicting population dynamics, the value of the prediction is not enough and should be accompanied by a confidence interval that integrates the whole chain of errors, from observations to predictions via the estimates of the parameters of the model. Matrix models are often used to predict the dynamics of age- or size-structured populations. Their parameters are vital rates. This study aims (1) at assessing the impact of the variability of observations on vital rates, and then on model’s predictions, and (2) at comparing three methods for computing confidence intervals for values predicted from the models. The first method is the bootstrap. The second method is analytic and approximates the standard error of predictions by their asymptotic variance as the sample size tends to infinity. The third method combines use of the bootstrap to estimate the standard errors of vital rates with the analytical method to then estimate the errors of predictions from the model. Computations are done for an Usher matrix models that predicts the asymptotic (as time goes to infinity) stock recovery rate for three timber species in French Guiana. Little difference is found between the hybrid and the analytic method. Their estimates of bias and standard error converge towards the bootstrap estimates when the error on vital rates becomes small enough, which corresponds in the present case to a number of observations greater than 5000 trees.     

Harnessing stratigraphic bias at the section scale: conodont diversity in the Homerian (Silurian) of the Midland Platform,England

Fossil abundance and diversity in geological successions are subject to bias arising from shifting depositional and diagenetic environments, resulting in variable rates of fossil accumulation and preservation. In simulations, this bias can be constrained based on sequence‐stratigraphic architecture. Nonetheless, a practical quantitative method of incorporating the contribution of sequence‐stratigraphic architecture in community palaeoecology and diversity analyses derived from individual successions is missing. As a model of faunal turnover affected by the stratigraphic bias, we use the ‘Mulde event’, a postulated mid‐Silurian interval of elevated conodont turnover, which coincides with global eustatic sea‐level changes and which has been based on regionally constrained observations. We test whether conodont turnover is highest at the boundary corresponding to the ‘event’ and post‐‘event’ interval against the alternative that conodont turnover reflects habitat tracking and peaks at facies shifts. Based on the previously documented, parasequence‐level stratigraphic framework of sections in the northern and central part of the Midland Platform, the relative controls of sequence‐stratigraphic architecture, time and depositional environment over conodont distribution are evaluated using permutational multivariate analysis of variance. The depositional environment controls the largest part of variability in conodont assemblage composition, whereas the postulated ‘Mulde event’, or genuine temporal change in conodont diversity, cannot be detected. Depending on the binning of the stratigraphic succession, contrasting diversity and turnover patterns can be produced. The simple approach proposed here, emulating partitioning of β diversity into spatial and temporal components, may help to constrain the stratigraphic bias, even at the scale of an individual section.     

Estimating receptive fields from responses to natural stimuli with asymmetric intensity distributions

The reasons for using natural stimuli to study sensory function are quickly mounting, as recent studies have revealed important differences in neural responses to natural and artificial stimuli. However, natural stimuli typically contain strong correlations and are spherically asymmetric (i.e. stimulus intensities are not symmetrically distributed around the mean), and these statistical complexities can bias receptive field (RF) estimates when standard techniques such as spike-triggered averaging or reverse correlation are used. While a number of approaches have been developed to explicitly correct the bias due to stimulus correlations, there is no complementary technique to correct the bias due to stimulus asymmetries. Here, we develop a method for RF estimation that corrects reverse correlation RF estimates for the spherical asymmetries present in natural stimuli. Using simulated neural responses, we demonstrate how stimulus asymmetries can bias reverse-correlation RF estimates (even for uncorrelated stimuli) and illustrate how this bias can be removed by explicit correction. We demonstrate the utility of the asymmetry correction method under experimental conditions by estimating RFs from the responses of retinal ganglion cells to natural stimuli and using these RFs to predict responses to novel stimuli.     

How Long Depends on How Fast—Perceived Flicker Dilates Subjective Duration

How do humans perceive the passage of time and the duration of events without a dedicated sensory system for timing? Previous studies have demonstrated that when a stimulus changes over time, its duration is subjectively dilated, indicating that duration judgments are based on the number of changes within an interval. In this study, we tested predictions derived from three different accounts describing the relation between a changing stimulus and its subjective duration as either based on (1) the objective rate of changes of the stimulus, (2) the perceived saliency of the changes, or (3) the neural energy expended in processing the stimulus. We used visual stimuli flickering at different frequencies (4–166 Hz) to study how the number of changes affects subjective duration. To this end, we assessed the subjective duration of these stimuli and measured participants'' behavioral flicker fusion threshold (the highest frequency perceived as flicker), as well as their threshold for a frequency-specific neural response to the flicker using EEG. We found that only consciously perceived flicker dilated perceived duration, such that a 2 s long stimulus flickering at 4 Hz was perceived as lasting as long as a 2.7 s steady stimulus. This effect was most pronounced at the slowest flicker frequencies, at which participants reported the most consistent flicker perception. Flicker frequencies higher than the flicker fusion threshold did not affect perceived duration at all, even if they evoked a significant frequency-specific neural response. In sum, our findings indicate that time perception in the peri-second range is driven by the subjective saliency of the stimulus'' temporal features rather than the objective rate of stimulus changes or the neural response to the changes.     

Sea ice phenology and timing of primary production pulses in the Arctic Ocean

Arctic organisms are adapted to the strong seasonality of environmental forcing. A small timing mismatch between biological processes and the environment could potentially have significant consequences for the entire food web. Climate warming causes shrinking ice coverage and earlier ice retreat in the Arctic, which is likely to change the timing of primary production. In this study, we test predictions on the interactions among sea ice phenology and production timing of ice algae and pelagic phytoplankton. We do so using the following (1) a synthesis of available satellite observation data; and (2) the application of a coupled ice‐ocean ecosystem model. The data and model results suggest that, over a large portion of the Arctic marginal seas, the timing variability in ice retreat at a specific location has a strong impact on the timing variability in pelagic phytoplankton peaks, but weak or no impact on the timing of ice‐algae peaks in those regions. The model predicts latitudinal and regional differences in the timing of ice algae biomass peak (varying from April to May) and the time lags between ice algae and pelagic phytoplankton peaks (varying from 45 to 90 days). The correlation between the time lag and ice retreat is significant in areas where ice retreat has no significant impact on ice‐algae peak timing, suggesting that changes in pelagic phytoplankton peak timing control the variability in time lags. Phenological variability in primary production is likely to have consequences for higher trophic levels, particularly for the zooplankton grazers, whose main food source is composed of the dually pulsed algae production of the Arctic.     

Climate change and heat-related mortality in six cities Part 2: climate model evaluation and projected impacts from changes in the mean and variability of temperature with climate change

Previous assessments of the impacts of climate change on heat-related mortality use the “delta method” to create temperature projection time series that are applied to temperature–mortality models to estimate future mortality impacts. The delta method means that climate model bias in the modelled present does not influence the temperature projection time series and impacts. However, the delta method assumes that climate change will result only in a change in the mean temperature but there is evidence that there will also be changes in the variability of temperature with climate change. The aim of this paper is to demonstrate the importance of considering changes in temperature variability with climate change in impacts assessments of future heat-related mortality. We investigate future heat-related mortality impacts in six cities (Boston, Budapest, Dallas, Lisbon, London and Sydney) by applying temperature projections from the UK Meteorological Office HadCM3 climate model to the temperature–mortality models constructed and validated in Part 1. We investigate the impacts for four cases based on various combinations of mean and variability changes in temperature with climate change. The results demonstrate that higher mortality is attributed to increases in the mean and variability of temperature with climate change rather than with the change in mean temperature alone. This has implications for interpreting existing impacts estimates that have used the delta method. We present a novel method for the creation of temperature projection time series that includes changes in the mean and variability of temperature with climate change and is not influenced by climate model bias in the modelled present. The method should be useful for future impacts assessments. Few studies consider the implications that the limitations of the climate model may have on the heat-related mortality impacts. Here, we demonstrate the importance of considering this by conducting an evaluation of the daily and extreme temperatures from HadCM3, which demonstrates that the estimates of future heat-related mortality for Dallas and Lisbon may be overestimated due to positive climate model bias. Likewise, estimates for Boston and London may be underestimated due to negative climate model bias. Finally, we briefly consider uncertainties in the impacts associated with greenhouse gas emissions and acclimatisation. The uncertainties in the mortality impacts due to different emissions scenarios of greenhouse gases in the future varied considerably by location. Allowing for acclimatisation to an extra 2°C in mean temperatures reduced future heat-related mortality by approximately half that of no acclimatisation in each city.     

Variability and coding efficiency of noisy neural spike encoders

Encoding synaptic inputs as a train of action potentials is a fundamental function of nerve cells. Although spike trains recorded in vivo have been shown to be highly variable, it is unclear whether variability in spike timing represents faithful encoding of temporally varying synaptic inputs or noise inherent in the spike encoding mechanism. It has been reported that spike timing variability is more pronounced for constant, unvarying inputs than for inputs with rich temporal structure. This could have significant implications for the nature of neural coding, particularly if precise timing of spikes and temporal synchrony between neurons is used to represent information in the nervous system. To study the potential functional role of spike timing variability, we estimate the fraction of spike timing variability which conveys information about the input for two types of noisy spike encoders--an integrate and fire model with randomly chosen thresholds and a model of a patch of neuronal membrane containing stochastic Na(+) and K(+) channels obeying Hodgkin-Huxley kinetics. The quality of signal encoding is assessed by reconstructing the input stimuli from the output spike trains using optimal linear mean square estimation. A comparison of the estimation performance of noisy neuronal models of spike generation enables us to assess the impact of neuronal noise on the efficacy of neural coding. The results for both models suggest that spike timing variability reduces the ability of spike trains to encode rapid time-varying stimuli. Moreover, contrary to expectations based on earlier studies, we find that the noisy spike encoding models encode slowly varying stimuli more effectively than rapidly varying ones.     

Prospective timing under dual-task paradigms: attentional and contextual-change mechanisms

The effect of a concurrent memory task on prospective time estimates by human participants was investigated in two experiments. The objective was to isolate task effects from those of participant timing strategy (self-paced counting) and number of contextual changes during the temporal stimulus. Accordingly, self-paced counting was suppressed by requiring participants to perform a word-reading task during the temporal stimuli, while number of stimulus changes presented during temporal stimuli was controlled. Presence versus absence of the concurrent memory task was manipulated in Experiment 1, and instruction to focus on timing or to focus on memory was manipulated in Experiment 2. There was no significant effect of presence versus absence of the concurrent memory task on time estimates; however, time estimates were shorter when participants were instructed to focus on memory versus timing. In both experiments, time estimates were positively correlated with participants' estimates of the number of words presented during the interval, even though number of words presented was invariant. These findings were generally consistent with resource-allocation attentional accounts of concurrent task effects; however, support for a contextual-change model of timing was also obtained.     

Tracking a changing environment: optimal sampling, adaptive memory and overnight effects

Foraging in a variable environment presents a classic problem of decision making with incomplete information. Animals must track the changing environment, remember the best options and make choices accordingly. While several experimental studies have explored the idea that sampling behavior reflects the amount of environmental change, we take the next logical step in asking how change influences memory. We explore the hypothesis that memory length should be tied to the ecological relevance and the value of the information learned, and that environmental change is a key determinant of the value of memory. We use a dynamic programming model to confirm our predictions and then test memory length in a factorial experiment. In our experimental situation we manipulate rates of change in a simple foraging task for blue jays over a 36 h period. After jays experienced an experimentally determined change regime, we tested them at a range of retention intervals, from 1 to 72 h. Manipulated rates of change influenced learning and sampling rates: subjects sampled more and learned more quickly in the high change condition. Tests of retention revealed significant interactions between retention interval and the experienced rate of change. We observed a striking and surprising difference between the high and low change treatments at the 24 h retention interval. In agreement with earlier work we find that a circadian retention interval is special, but we find that the extent of this ‘specialness’ depends on the subject's prior experience of environmental change. Specifically, experienced rates of change seem to influence how subjects balance recent information against past experience in a way that interacts with the passage of time.     

Single nucleotide polymorphisms derived from ancestral populations show no evidence for biased diversity estimates in Drosophila melanogaster

Single nucleotide polymorphisms (SNPs) are about to become one of the most popular genetic markers for genetic model organisms. To test the usefulness of SNPs for estimating genetic diversity, we surveyed three genomic regions in two Drosophila melanogaster populations, one from Africa and one European, collected in Austria. Diversity estimates based on the full SNP set indicated higher levels of variability in the African than in the European flies. When the analysis was based on the European SNP set, European and African flies had similar levels of variability. Interestingly, this bias was not observed for diversity estimates using SNPs derived from the ancestral African population. This result suggests that diversity estimates based on SNPs from ancestral populations could provide a general strategy to avoid biased SNP diversity estimates. Finally, the potential of SNPs for nonmodel organisms is discussed.     

How does temporal variability affect predictions of weed population numbers?

1. Population models that are used to predict weed population dynamics or the impact of control measures on weed abundance typically ignore temporal variability in life-history parameters and control measures, and utilize mean arithmetic population growth rates to predict population abundance.
2. We demonstrate that the persistence of weeds in a stochastically varying environment depends on the geometric mean population growth rate being greater than zero, rather than the arithmetic mean population growth rate being greater than zero.
3. In a stochastically varying environment we show that temporal variability in fecundity, germination and survivorship will tend to decrease population size, relative to predictions based on arithmetic means. Conversely, variability in competitive effects and weed control will tend to increase population size, relative to predictions based on arithmetic mean values. The distinction between these two sets of parameters is that increases in the former will increase population growth rate, whereas increases in the latter will decrease it.
4. We argue that population models based on arithmetic mean population growth rates will tend to over-estimate population size. Numerical simulations indicate that this bias may be considerable.
5. Since short-term studies cannot, in general, estimate the geometric mean growth rate of a population we suggest several approaches for estimating the degree of bias in the predictions of models owing to the effects of variability. Accounting for such variability is necessary since current models for the dynamics of weed populations are based on arithmetic mean measures of population growth and hence likely to be biased.     

Testing the efficiency of sensory coding with optimal stimulus ensembles

According to Barlow's seminal "efficient coding hypothesis," the coding strategy of sensory neurons should be matched to the statistics of stimuli that occur in an animal's natural habitat. Using an automatic search technique, we here test this hypothesis and identify stimulus ensembles that sensory neurons are optimized for. Focusing on grasshopper auditory receptor neurons, we find that their optimal stimulus ensembles differ from the natural environment, but largely overlap with a behaviorally important sub-ensemble of the natural sounds. This indicates that the receptors are optimized for peak rather than average performance. More generally, our results suggest that the coding strategies of sensory neurons are heavily influenced by differences in behavioral relevance among natural stimuli.     

Task-induced neural covariability as a signature of approximate Bayesian learning and inference

Perception is often characterized computationally as an inference process in which uncertain or ambiguous sensory inputs are combined with prior expectations. Although behavioral studies have shown that observers can change their prior expectations in the context of a task, robust neural signatures of task-specific priors have been elusive. Here, we analytically derive such signatures under the general assumption that the responses of sensory neurons encode posterior beliefs that combine sensory inputs with task-specific expectations. Specifically, we derive predictions for the task-dependence of correlated neural variability and decision-related signals in sensory neurons. The qualitative aspects of our results are parameter-free and specific to the statistics of each task. The predictions for correlated variability also differ from predictions of classic feedforward models of sensory processing and are therefore a strong test of theories of hierarchical Bayesian inference in the brain. Importantly, we find that Bayesian learning predicts an increase in so-called “differential correlations” as the observer’s internal model learns the stimulus distribution, and the observer’s behavioral performance improves. This stands in contrast to classic feedforward encoding/decoding models of sensory processing, since such correlations are fundamentally information-limiting. We find support for our predictions in data from existing neurophysiological studies across a variety of tasks and brain areas. Finally, we show in simulation how measurements of sensory neural responses can reveal information about a subject’s internal beliefs about the task. Taken together, our results reinterpret task-dependent sources of neural covariability as signatures of Bayesian inference and provide new insights into their cause and their function.     

Possible biases in heritability estimates from intraclass correlation

Summary There is an inherent bias in intraclass correlations since the expectation of a ratio does not equal the ratio of expectations. A simple accurate approximation for this bias is derived, and it is found that the inherent bias is usually negligible. Selection of sires is known to bias half-sib heritability estimates, and appropriate formulae are given and discussed.     

Effective number of breeders and maintenance of genetic diversity in the captive bearded vulture population

We combined pedigree data with data derived from 14 microsatellite loci to investigate genetic diversity and its maintenance in the captive source population for the reintroduction of the bearded vulture into the Alps. We found the captive population to be genetically more variable than the largest natural population in Europe, both in terms of mean number of alleles per locus and mean observed and expected heterozygosity. Allelic diversity of the captive population was higher than, and mean heterozygosity measurements were comparable with the ones found in two large, extinct populations from Sardinia and the Alps represented by museum specimens. The amount of genetic variability recruited with the founders was still present in the captive population of the year 2000, mainly because the carriers of rare alleles were still alive. However, the decline in expected heterozygosity and the loss of alleles over generations in captivity was significant. Point estimates of effective population size, N(e), based on pedigree data and estimates of effective number of breeders, N(b), based on allele frequency changes, ranged from 20 to 30 and were significantly smaller than the census size. The results demonstrate that the amount of genetic variability in the captive bearded vulture population is comparable or even larger than the amount present in natural populations. However, the population is in danger to lose genetic variability over time because of genetic drift. Management strategies should therefore aim at preserving genetic variability by minimising kinship, and at increasing N(e) by recruiting additional founders and enhancing gene flow between the released, the captive and natural populations.     

Modeling habituation in rat EEG-evoked responses via a neural mass model with feedback

Habituation is a generic property of the neural response to repeated stimuli. Its strength often increases as inter-stimuli relaxation periods decrease. We propose a simple, broadly applicable control structure that enables a neural mass model of the evoked EEG response to exhibit habituated behavior. A key motivation for this investigation is the ongoing effort to develop model-based reconstruction of multi-modal functional neuroimaging data. The control structure proposed here is illustrated and validated in the context of a biophysical neural mass model, developed by Riera et?al. (Hum Brain Mapp 27(11):896-914, 2006; 28(4):335-354, 2007), and of simplifications thereof, using data from rat EEG response to medial nerve stimuli presented at frequencies from 1 to 8?Hz. Performance was tested by predictions of both the response to the next stimulus based on the current one, and also of continued stimuli trains over 4-s time intervals based on the first stimulus in the interval, with similar success statistics. These tests demonstrate the ability of simple generative models to capture key features of the evoked response, including habituation.     

Estimating infection prevalence: Best practices and their theoretical underpinnings

Accurately estimating infection prevalence is fundamental to the study of population health, disease dynamics, and infection risk factors. Prevalence is estimated as the proportion of infected individuals (“individual‐based estimation”), but is also estimated as the proportion of samples in which evidence of infection is detected (“anonymous estimation”). The latter method is often used when researchers lack information on individual host identity, which can occur during noninvasive sampling of wild populations or when the individual that produced a fecal sample is unknown. The goal of this study was to investigate biases in individual‐based versus anonymous prevalence estimation theoretically and to test whether mathematically derived predictions are evident in a comparative dataset of gastrointestinal helminth infections in nonhuman primates. Using a mathematical model, we predict that anonymous estimates of prevalence will be lower than individual‐based estimates when (a) samples from infected individuals do not always contain evidence of infection and/or (b) when false negatives occur. The mathematical model further predicts that no difference in bias should exist between anonymous estimation and individual‐based estimation when one sample is collected from each individual. Using data on helminth parasites of primates, we find that anonymous estimates of prevalence are significantly and substantially (12.17%) lower than individual‐based estimates of prevalence. We also observed that individual‐based estimates of prevalence from studies employing single sampling are on average 6.4% higher than anonymous estimates, suggesting a bias toward sampling infected individuals. We recommend that researchers use individual‐based study designs with repeated sampling of individuals to obtain the most accurate estimate of infection prevalence. Moreover, to ensure accurate interpretation of their results and to allow for prevalence estimates to be compared among studies, it is essential that authors explicitly describe their sampling designs and prevalence calculations in publications.