How to statistically test the predictability of evolution?

How to statistically test the predictability of evolution?

We are searching data for your request:

Forums and discussions:
Manuals and reference books:
Data from registers:
Wait the end of the search in all databases.
Upon completion, a link will appear to access the found materials.

Can anyone recommend me an experimental study which tries to test the predictability of evolution? The closest works I found are the studies of fluctuation tests (f.e. classical study of Luria & Delbrück, 1943) which demonstrates that in bacteria, genetic mutations arise in the absence of selection, rather than being a response to selection. Is there any other point of view from which the predictability of evolution can be studied?

Luria & Delbrück (1943) Mutations of bacteria from virus sensitivity to virus resistance. Genetics 28: 491-511.

I think Lenski's evolutionary long-term experiment should be a good example, without being familiar with Luria & Delbrück (1943). There they use replicated populations of E-coli to study (among other things) the "repeatability" (historical contingency) of evolutionary changes. Nice overviews of the project are given in Blount et al. (2008) and Philippe et al (2008), where they study to what extent certain mutations are repeated in replicate populations, in relation to citrate metabolism and size/shape.


  • Blount, Borland & Lenski. 2008. Historical contingency and the evolution of a key innovation in an experimental population of Escherichia coli. PNAS vol. 105 no. 23
  • Philippe et al (2008). Evolution of Penicillin-Binding Protein 2 Concentration and Cell Shape during a Long-Term Experiment with Escherichia coli. Journal of Bacteriology 191(3):909-21

Several methods have been proposed to test for introgression across genomes. One method tests for a genome-wide excess of shared derived alleles between taxa using Patterson’s D statistic, but does not establish which loci show such an excess or whether the excess is due to introgression or ancestral population structure. Several recent studies have extended the use of D by applying the statistic to small genomic regions, rather than genome-wide. Here, we use simulations and whole-genome data from Heliconius butterflies to investigate the behavior of D in small genomic regions. We find that D is unreliable in this situation as it gives inflated values when effective population size is low, causing D outliers to cluster in genomic regions of reduced diversity. As an alternative, we propose a related statistic f ^ d ⁠ , a modified version of a statistic originally developed to estimate the genome-wide fraction of admixture. f ^ d is not subject to the same biases as D, and is better at identifying introgressed loci. Finally, we show that both D and f ^ d outliers tend to cluster in regions of low absolute divergence (dXY), which can confound a recently proposed test for differentiating introgression from shared ancestral variation at individual loci.

Hybridization and gene flow between taxa play a major role in evolution, acting as a force against divergence, and as a potential source of adaptive novelty ( Abbott et al. 2013). Although identifying gene flow between species has been a long-standing problem in population genetics, the issue has received considerable recent attention with the analysis of shared ancestry between humans and Neanderthals (e.g., Yang et al. 2012 Wall et al. 2013 Sankararaman et al. 2014). With genomic data sets becoming available in a wide variety of other taxonomic groups, there is a need for reliable, computationally tractable methods that identify, quantify, and date gene flow between species in large data sets.

A sensitive and widely used approach to test for gene flow is to fit coalescent models using maximum-likelihood or Bayesian methods ( Pinho and Hey 2010). However, simulation and model fitting are computationally intensive tasks and are not easily applied on a genomic scale. A simpler and more computationally efficient approach that is gaining in popularity is to test for an excess of shared derived variants using a four-taxon test ( Kulathinal et al. 2009 Green et al. 2010 Durand et al. 2011). The test considers ancestral (“A”) and derived (“B”) alleles and is based on the prediction that two particular single nucleotide polymorphism (SNP) patterns, termed “ABBA” and “BABA” (see Materials and Methods), should be equally frequent under a scenario of incomplete lineage sorting without gene flow. An excess of ABBA or BABA patterns is indicative of gene flow between two of the taxa and can be detected using Patterson’s D statistic ( Green et al. 2010 Durand et al. 2011 see Materials and Methods for details). However, an excess of shared derived variants can arise from factors other than recent introgression, in particular nonrandom mating in the ancestral population due to population structure ( Eriksson and Manica 2012). It is therefore important to make use of additional means to distinguish between these alternative hypotheses, for example, by examining the size of introgressed tracts ( Wall et al. 2013), or the level of absolute divergence in introgressed regions ( Smith and Kronforst 2013).

The D statistic was originally designed to be applied on a genome-wide or chromosome-wide scale, with block-jackknifing used to overcome the problem of nonindependence between loci ( Green et al. 2010). However, many researchers are interested in identifying particular genomic regions subject to gene flow, rather than simply estimating a genome-wide parameter. Theory predicts that the rate of gene flow should vary across the genome, both in the case of secondary contact after isolation ( Barton and Gale 1993) as well as continuous gene flow during speciation ( Wu 2001). Indeed, a maximum-likelihood test for speciation with gene flow devised by Yang (2010) is based on detecting this underlying heterogeneity. Moreover, adaptive introgression might lead to highly localized signals of introgression, limited to the particular loci under selection.

Many methods for characterizing heterogeneity in patterns of introgression across the genome have been proposed. Several genomic studies have used FST to characterize heterogeneity in divergence across the genome, often interpreting the variation in FST as indicative of variation in rates of gene flow (e.g., Ellegren et al. 2012). However, it is well established that, as a relative measure of divergence, FST is dependent on within-population genetic diversity ( Charlesworth 1998), and is therefore an unreliable indicator of how migration rates vary across the genome. In particular, heterogeneity in purifying selection and recombination rate could confound FST-based studies ( Noor and Bennett 2009 Hahn et al. 2012 Roesti et al. 2012 Cruickshank and Hahn 2014). Various studies of admixture among human populations, or between humans and Neanderthals, have used probabilistic methods to assign ancestry to haplotypes, and infer how this ancestry changes across a chromosome ( Sankararaman et al. 2008 Price et al. 2009 Henn et al. 2012 Lawson et al. 2012 Omberg et al. 2012 Churchhouse and Marchini 2013 Maples et al. 2013 Sankararaman et al. 2014). Other methods have modeled speciation with the allowance for variable introgression rates among loci ( Garrigan et al. 2012 Roux et al. 2013), allowing the detection of more ancient gene flow.

There have also been recent attempts to characterize heterogeneity in patterns of introgression across the genome using the D statistic, calculated either in small windows ( Kronforst et al. 2013 Smith and Kronforst 2013) or for individual SNPs ( Rheindt et al. 2014). The robustness of the D statistic for detecting a genome-wide excess of shared derived alleles has been thoroughly explored ( Green et al. 2010 Durand et al. 2011 Yang et al. 2012 Eaton and Ree 2013 Martin et al. 2013). However, it has not been established whether D provides a robust and unbiased means to identify individual loci with an excess of shared derived alleles, or to demonstrate that these loci have been subject to introgression. Any inherent biases of the D statistic when applied to specific loci have implications for methods that assume its robustness.

For example, Smith and Kronforst (2013) made use of the D statistic in a proposed test to distinguish between the hypotheses of introgression and shared ancestral variation at wing-patterning loci of Heliconius butterflies. Two wing-patterning loci are known to show an excess of shared derived alleles between comimetic populations of Heliconius melpomene and H. timareta ( Heliconius Genome Consortium 2012). At one of these loci, phylogenetic evidence and patterns of linkage disequilibrium are consistent with recent gene flow ( Pardo-Diaz et al. 2012). Nevertheless, Smith and Kronforst (2013) argue that this shared variation might represent an ancestral polymorphism that was maintained through the speciation event by balancing selection. Conceptually, this is not unlike the population structure argument of Eriksson and Manica (2012), except that here structure is limited to one or a few individual loci.

Smith and Kronforst proposed that the alternative explanations of introgression or ancestral polymorphism could be distinguished by considering absolute divergence within and outside of the loci of interest. Both hypotheses predict an excess of shared derived alleles at affected loci, but introgression should lead to reduced absolute divergence due to more recent coalescence at these loci, whereas the locus-specific population structure hypothesis predicts no reduction in absolute divergence at these loci compared with other loci in the genome. Loci with an excess of shared derived alleles, and therefore showing evidence of shared ancestry, were located by calculating the D statistic in nonoverlapping 5-kb windows across genomic regions of interest, and identifying outliers using an arbitrary cutoff (the 10% of windows with the highest D values). The mean absolute genetic divergence (dXY) was then compared between the outliers and nonoutliers, and found to be significantly lower in outlier windows, consistent with recent introgression ( Smith and Kronforst 2013). This method makes two assumptions. First, that the D statistic can accurately identify regions that carry a significant excess of shared variation, and second, that D outliers do not have inherent biases leading to their cooccurrence with regions of low absolute divergence. These assumptions, which extend the use of D beyond its original definition, may be made by other researchers for similar purposes, but they remain to be tested.

Here, we first assess the reliability of the D statistic as a means to quantify introgression at individual loci. Using simulations of small sequence windows, we compare D to a related statistic that was developed by Green et al. (2010) specifically for estimating f, the proportion of the genome that has been shared, and we propose improvements to this statistic. We then use whole-genome data from several Heliconius species to investigate how these statistics perform on empirical data, and specifically how they are influenced by underlying heterogeneity in diversity across the genome. Lastly, we use a large range of simulated data sets to test the proposal that recent gene flow can be distinguished from shared ancestral variation based on absolute divergence in D outlier regions.

Knowledge and Skills Required

Questions on the Biology exam require test takers to demonstrate one or more of the following abilities.

  • Knowledge of facts, principles, and processes of biology
  • Understanding the means by which information is collected, how it is interpreted, how one hypothesizes from available information, and how one draws conclusions and makes further predictions
  • Understanding that science is a human endeavor with social consequences

The subject matter of the Biology exam is drawn from the following topics. The percentages next to the main topics indicate the approximate percentage of exam questions on that topic.

Molecular and Cellular Biology (33%)

Chemical composition of organisms

  • Simple chemical reactions and bonds
  • Properties of water
  • Chemical structure of carbohydrates, lipids, proteins, nucleic acids
  • Origin of life
  • Structure and function of cell organelles
  • Properties of cell membranes
  • Comparison of prokaryotic and eukaryotic cells
  • Enzyme-substrate complex
  • Roles of coenzymes
  • Inorganic cofactors
  • Inhibition and regulation

Chemical nature of the gene

  • Watson-Crick model of nucleic acids
  • DNA replication
  • Mutations
  • Control of protein synthesis: transcription, translation, posttranscriptional processing
  • Structural and regulatory genes
  • Transformation
  • Viruses

Organismal Biology (34%)

Structure and function in plants with emphasis on angiosperms

  • Root, stem, leaf, flower, seed, fruit
  • Water and mineral absorption and transport
  • Food translocation and storage

Plant reproduction and development

  • Alternation of generations in ferns, conifers, and flowering plants
  • Gamete formation and fertilization
  • Growth and development: hormonal control
  • Tropisms and photoperiodicity

Structure and function in animals with emphasis on vertebrates

  • Major systems (e.g., digestive, gas exchange, skeletal, nervous, circulatory, excretory, immune)
  • Homeostatic mechanisms
  • Hormonal control in homeostasis and reproduction

Animal reproduction and development

  • Gamete formation, fertilization
  • Cleavage, gastrulation, germ layer formation, differentiation of organ systems
  • Experimental analysis of vertebrate development
  • Extraembryonic membranes of vertebrates
  • Formation and function of the mammalian placenta
  • Blood circulation in the human embryo
  • Mendelian inheritance (dominance, segregation, independent assortment)
  • Chromosomal basis of inheritance
  • Linkage, including sex-linked
  • Polygenic inheritance (height, skin color)
  • Multiple alleles (human blood groups)

Population Biology (33%)

  • Energy flow and productivity in ecosystems
  • Biogeochemical cycles
  • Population growth and regulation (natality, mortality, competition, migration, density, r- and K-selection)
  • Community structure, growth, regulation (major biomes and succession)
  • Habitat (biotic and abiotic factors)
  • Concept of niche
  • Island biogeography
  • Evolutionary ecology (life history strategies, altruism, kin selection)
  • History of evolutionary concepts
  • Concepts of natural selection (differential reproduction, mutation, Hardy-Weinberg equilibrium, speciation, punctuated equilibrium)
  • Adaptive radiation
  • Major features of plant and animal evolution
  • Concepts of homology and analogy
  • Convergence, extinction, balanced polymorphism, genetic drift
  • Classification of living organisms
  • Evolutionary history of humans
  • Human population growth (age composition, birth and fertility rates, theory of demographic transition)
  • Human intervention in the natural world (management of resources, environmental pollution)
  • Biomedical progress (control of human reproduction, genetic engineering)

Scientists engineer fruit flies with ancient genes to test causes of evolution

Scientists at the University of Chicago have created the first genetically modified animals containing reconstructed ancient genes, which they used to test the evolutionary effects of past genetic changes on the animals’ biology and fitness.

The research, published online in Nature Ecology & Evolution on Jan. 13, is a major step forward for efforts to study the genetic basis of adaptation and evolution. The specific findings, involving the fruit fly’s ability to break down alcohol in rotting fruit, overturn a widely held hypothesis about the molecular causes of one of evolutionary biology’s classic cases of adaptation.

“One of the major goals of modern evolutionary biology is to identify the genes that caused species to adapt to new environments, but it’s been hard to do that directly, because we’ve had no way to test the effects of ancient genes on animal biology,” said Mo Siddiq, a graduate student in ecology and evolution at the University of Chicago, one of the study’s lead scientists.

“We realized we could overcome this problem by combining two recently developed methods—statistical reconstruction of ancient gene sequences and engineering of transgenic animals,” he said.

Until recently, most studies of molecular adaptation have analyzed gene sequences to identify “signatures of selection”—patterns suggesting that a gene changed so quickly during its evolution that selection is likely to have been the cause. The evidence from this approach is only circumstantial, however, because genes can evolve quickly for many reasons, such as chance, fluctuations in population size or selection for functions unrelated to the environmental conditions to which the organism is thought to have adapted.

Siddiq and his adviser, Joe Thornton, professor of ecology and evolution and human genetics, wanted to directly test the effects of a gene’s evolution on adaptation. Thornton has pioneered methods for reconstructing ancestral genes—statistically determining their sequences from large databases of present-day sequences, then synthesizing them and experimentally studying their molecular properties in the laboratory. This strategy has yielded major insights into the mechanisms by which biochemical functions evolve.

Thornton and Siddiq reasoned that by combining ancestral gene reconstruction with techniques for engineering transgenic animals, they could study how genetic changes that occurred in the deep past affected whole organisms—their development, physiology and even their fitness.

“This strategy of engineering ‘ancestralized animals’ could be applied to many evolutionary questions,” Thornton said. “For the first test case, we chose a classic example of adaptation—how fruit flies evolved the ability to survive the high alcohol concentrations found in rotting fruit. We found that the accepted wisdom about the molecular causes of the flies’ evolution is simply wrong.”

Challenging earlier thinking

The fruit fly Drosophila melanogaster is one of the most studied organisms in genetics and evolution. In the wild, D. melanogaster lives in alcohol-rich rotting fruit, tolerating far higher alcohol concentrations than its closest relatives, which live on other food sources. Twenty-five years ago at the University of Chicago, biologists Martin Kreitman and John McDonald invented a new statistical method for finding signatures of selection, which remains to this day one of the most widely used methods in molecular evolution. They demonstrated it on the alcohol dehydrogenase (Adh) gene—the gene for the enzyme that breaks down alcohol inside cells—from this group of flies. Adh had a strong signature of selection, and it was already known that D. melanogaster flies break down alcohol faster than their relatives. So, the idea that the Adh enzyme was the cause of the fruit fly’s adaptation to ethanol became the first accepted case of a specific gene that mediated adaptive evolution of a species.

Siddiq and Thornton realized that this hypothesis could be tested directly using the new technologies. Siddiq first inferred the sequences of ancient Adh genes from just before and just after D. melanogaster evolved its ethanol tolerance, some two to four million years ago. He synthesized these genes biochemically, expressed them and used biochemical methods to measure their ability to break down alcohol in a test tube. The results were surprising: The genetic changes that occurred during the evolution of D. melanogaster had no detectable effect on the protein’s function.

Working with collaborators David Loehlin at the University of Wisconsin and Kristi Montooth at the University of Nebraska, Siddiq then created and characterized transgenic flies containing the reconstructed ancestral forms of Adh. They bred thousands of these “ancestralized” flies, tested how quickly they could break down alcohol, and how well the larvae and adult flies survived when raised on food with high alcohol content. Surprisingly, the transgenic flies carrying the more recent Adh were no better at metabolizing alcohol than flies carrying the more ancient form of Adh. Even more strikingly, they were no better able to grow or survive on increasing alcohol concentrations. Thus, none of the predictions of the classic version of the story were fulfilled. There is no doubt that D. melanogaster did adapt to high-alcohol food sources during its evolution, but not because of changes in the Adh enzyme.

“The Adh story was accepted because the ecology, physiology and the statistical signature of selection all pointed in the same direction. But three lines of circumstantial evidence don’t make an airtight case,” Thornton said. “That’s why we wanted to test the hypothesis directly, now that we finally have the means to do so.”

Siddiq and Thornton hope that the strategy of making ancestralized transgenics will become the gold standard in the field to decisively determine the historical changes in genes to their changes on organisms’ biology and fitness.

For his part, Kreitman, who is still a professor of ecology and evolution at UChicago, has been supportive of the new research, helping advise Siddiq on the project and sharing his vast knowledge about molecular evolution and Drosophila genetics.

“From the beginning, Marty was excited about our experiments, and he was just as supportive when our results overturned well-known conclusions based on his past work,” Siddiq said. “I think that’s extremely inspiring.”

The study, “Experimental test and refutation of a classic case of molecular adaptation in Drosophila melanogaster,” was supported by the National Science Foundation, the National Institutes of Health, the Howard Hughes Medical Institute, and the Life Sciences Research Foundation.


Statistical results of EFA and CFA

Exploratory factor analysis (EFA)

Exploratory factor analysis rendered three factors from our religious measure with eigenvalues above 1 that explained over 5% of the variance seen in the data. The resulting religiosity factors were defined as religious practice, religious influence, and religious hope. The remaining measures, creationist views, scientific reasoning and evolution acceptance, each resulted in only one factor with an eigenvalue higher than 1 that explained more than 5% of the variance. From this, we concluded that the survey instrument measured six factors: religious practice, religious influence, religious hope, creationist views, scientific reasoning, and evolution acceptance.

Confirmatory factor analysis (CFA)

We used CFA to confirm that each instrument measured the identified factor. The following items were excluded from this analysis due to lack of fit or redundancy: one item regarding frequency of prayer (i.e., How often do you pray?) was removed from the religious practice factor due to poor fit, one item was removed from the religious hope factor due to poor fit (i.e., Do you believe it is possible for all humans to live in harmony together?), and two items were removed from the creationist views factor due to redundancy (lack of uniqueness i.e., All creatures on earth were created in the last 10,000 years and All present day humans are direct descendants of Adam and Eve). The resulting fit statistics show that each instrument model fit the data well (see Table 1). Figure 1 displays the measurement model with correlation coefficients displayed between each factor and factor loadings shown between each item and the factor of which it is a part. The factor loadings for each item across all instruments were high (above .5) with few exceptions. One cross loading was added between items 5 and 6 for creationist views indicating an overlap in what they are measuring. From our CFA, we found that our instruments are valid in measuring distinct and identifiable factors.

Culminating measurement model from our SEM calculation. Bidirectional arrows indicate correlation coefficients while directional arrows indicate factor loading values for each item on their respective factor. Significance is noted for the correlations: *p < .001. All factor loadings were significant at p < .001

Results of structural equation modeling

Using structural equation modeling, we found that scientific reasoning ability does not correlate with religiosity among religious individuals (Fig. 2). This means that religiosity does not predict scientific reasoning ability and scientific reasoning ability does not predict religiosity. Each is an independent measure uncorrelated with the other. In addition, we found that scientific reasoning ability was a non-significant predictor of acceptance of creationist views or acceptance of evolutionary theory among religious individuals (p > .05). In other words, a religious individual’s scientific reasoning ability plays no predictive role in their decision to accept or reject either creationist views or evolutionary theory.

Illustration of structural equation model that characterizes relationships among factors. Bidirectional lines (dotted line) indicate correlation coefficients, while directional lines (solid lines) indicate predictive relationships. Significance is noted: *p < 0.001

The model shows that all three factors of religiosity (religious practice, religious influence, and religious hope) are strongly correlated with one another (p < .001). It also shows that two components of religiosity (religious hope and religious influence) are significant positive predictors of creationist views (p < .001) and that acceptance of creationist views is a significant negative predictor of evolution acceptance (i.e., higher religiosity and acceptance of creationist views predicts lower acceptance of evolution p < .001). The indirect negative effect of religious hope and religious influence on acceptance of evolution is also significant (p < .001). The structural model demonstrated a robust fit for the data as indicated by fit statistics and probability scores (TLI = .988 CFI = .973 RMSEA = .067 χ 2 = 544.647, p < .001). Further, 92.3% of parameter estimates were statistically significant.

Quantifying thermal extremes and biological variation to predict evolutionary responses to changing climate

Central ideas from thermal biology, including thermal performance curves and tolerances, have been widely used to evaluate how changes in environmental means and variances generate changes in fitness, selection and microevolution in response to climate change. We summarize the opportunities and challenges for extending this approach to understanding the consequences of extreme climatic events. Using statistical tools from extreme value theory, we show how distributions of thermal extremes vary with latitude, time scale and climate change. Second, we review how performance curves and tolerances have been used to predict the fitness and evolutionary responses to climate change and climate gradients. Performance curves and tolerances change with prior thermal history and with time scale, complicating their use for predicting responses to thermal extremes. Third, we describe several recent case studies showing how infrequent extreme events can have outsized effects on the evolution of performance curves and heat tolerance. A key issue is whether thermal extremes affect reproduction or survival, and how these combine to determine overall fitness. We argue that a greater focus on tails—in the distribution of environmental extremes, and in the upper ends of performance curves—is needed to understand the consequences of extreme events.

This article is part of the themed issue ‘Behavioural, ecological and evolutionary responses to extreme climatic events’.

1. Introduction

In the US, debate has raged since the intense heat waves in the summer of 1988 over whether a ‘signal’ of global warming has finally been detected against the background ‘noise’ of natural climatic variation… (Stephen Schneider, 1990 [1, p. 9])

Models for adaptive evolution require information about selection—how phenotypic (or genotypic) variation causes variation in fitness [8,9]. In this framework, changes in environmental conditions (including climate) alter the relationships between phenotypic traits and fitness, and thereby change the form, direction and magnitude of selection. Environmental change may also alter phenotypic and genetic variation, which can alter both the strength and evolutionary responses to selection [10,11]. This framework has facilitated a wealth of empirical studies quantifying phenotypic selection (and to a lesser extent, genetic variation) in different environmental conditions [12,13]. But a major limitation to applying this framework to climate change and climate extremes is that the causal connections among climate conditions, phenotypes and fitness are rarely known. What is lacking is a quantitative theory of phenotypic selection that would allow predictions of how changes in environment generate changes in selection.

The field of thermal biology has provided a useful test case for developing a quantitative framework for selection in the context of climate change [14,15]. These studies focus on two main types of phenotypic traits: thermal performance curves (TPCs), which relate performance or fitness as a function of body temperature and thermal tolerances, which represent body temperature thresholds at which survival (or performance) changes precipitously. By combining data on changes in weather or climate with information on TPCs and tolerances, we can predict the fitness and selective consequences of environmental change. This approach has been used to quantify the empirical relationships between temperature changes and changes in mean fitness, phenotypic selection and evolutionary responses and to predict how recent and future climate changes will alter mean fitness, selection and evolution [14–19]. Some of these studies highlight the potential importance of climatic extremes for selection and evolutionary responses to climate change [16,18,19]. However, we suggest that there are some important challenges in using TPCs and tolerances to model responses to extreme conditions.

In this perspective we highlight the challenges of connecting climate extremes and thermal biology to understand selection and evolutionary responses of ectotherms to climate change. First, we discuss climate ‘extremes’ in the context of variation in weather and climate. In this paper we define extremes in terms of the upper end (or tail) of the distribution of climatic variables, focusing on temporal variation in temperature and how it changes geographically [20]. We summarize some key concepts and tools from the statistics of extreme values, and apply these to environmental temperature data along two climatic (latitudinal) gradients. One message is that temporal distributions of temperatures are frequently skewed and have ‘fat’ or ‘thin’ tails, and that these properties vary with geographical region and with time scale. This has important consequences for the nature of climate extremes and their biological consequences. Second, we briefly summarize the use of TPCs and thermal thresholds for quantifying the effects of climate variation and extremes on mean fitness, selection and evolutionary responses. An important challenge is that TPCs and thermal thresholds can vary with prior thermal history and with the time scale at which they are measured, making it difficult to integrate the effects of climate variation and extremes across the life cycle to quantify fitness and selection. Third, we review several recent field and modelling studies that document or predict evolutionary responses in performance curves. We use extreme value analyses to quantify how extreme thermal events contribute to the evolution of thermal tolerance and performance curves in these studies. The analyses illustrate how environmental extremes and unpredictability can impact evolutionary responses to climate change, but their predictions depend strongly on key assumptions about fitness consequences of higher temperatures. We highlight several key areas that limit current progress in understanding the role of climate extremes in rapid adaptive evolution.

2. Variation in weather and climate

There is a well-developed statistical framework for analysing variation in extreme values [21]. Denny and colleagues provide an excellent introduction to this framework for biologists [22,23]. Here we use environmental data on daily maximum air temperatures at sites along latitudinal gradients to determine the distributions of extreme temperatures at each site, and illustrate how tools from extreme value theory can characterize extreme thermal events. Our presentation focuses on how latitude and time scale alter the distribution and frequency of extreme thermal events.

(a) Weather and climate extremes are not normal

Daily maximum temperatures are relevant to short-term thermal stress in many ectotherms [24,25]. We quantify the distribution of daily maximum temperatures using weather stations in the Global Historical Climatology Network (GHCN). We accessed the data using the R package rnoaa [26]. We restricted our analysis to weather stations below 500 m in elevation, with data more recent than 2010, and with at least 10 (and up to 60) years of nearly (more than 85%) complete data. Using data only since 1980 yielded very similar results, so we report analyses of the full data here. Because we are primarily interested in high temperatures that may cause heat stress, we restricted our analyses to summer months (June, July and August: all sites we consider are in the northern hemisphere). We examine weather stations along latitudinal transects in the centres of North America (−100 °E) and Asia (77.5 °E) to explore continental rather than coastal climate conditions.

The breadth, skewness and shape of the daily maximum temperature distribution varies with latitude (figure 1). Lower latitude distributions are relatively narrow and shift little as latitude increases. The location of the 99th percentile also tends to aggregate at lower latitudes. At higher latitudes, distributions broaden and shift steadily to lower mean temperatures with increasing latitude. Many distributions depart from normality, increasingly so with climate change [27].

Figure 1. The distribution (and 99% quantiles: dashed vertical lines) of maximum daily temperature (°C) broadens and shifts toward lower temperatures as the latitudes of weather stations increase along latitudinal gradients (at 5° intervals) in the centres of (a) North America (−100 °E) and (b) Asia (77.5 °E).

Generalized extreme value (GEV) distributions can describe temperature distributions that depart from normality and have thick or bounded tails (see below), and have been used to assess the incidence of extreme climatic events [21,27]. GEV analyses are increasingly applied to daily maximum or minimum temperature data to quantify thermal extremes in studies of climate change [28,29]. GEV distributions are described by three parameters: location indicates the position, scale indicates the breadth (figure 2a), and shape indicates the heaviness of the tail. Shape parameter values near zero correspond to a Gumbel (type I) distribution characterized by a light tail shape parameter values greater than zero correspond to a Frechet (type II) distribution characterized by a heavy tail and shape parameter values less than zero correspond to a Weibull (type III) distribution characterized by a bounded tail (figure 2d).

Figure 2. The parameters of the generalized extreme value (GEV) distribution describing maximum daily temperatures (°C) vary across latitude for transects in North America and Asia. GEVs are characterized by three parameters: (a) location, which indicates position scale, which indicates breadth and (d) shape, which indicates the thickness of the tail. The (b) 99% quantiles and (c) GEV locations decline steadily with latitude, whereas the (e) scale parameter increases. (f) The GEV shape parameter is variable across intermediate latitudes. Low and high latitude stations, particularly in Asia, tend to have heavier tails.

We use GEV distributions to characterize distributions of maximum daily temperature across the latitudinal gradient. We fit GEV distributions using maximum likelihood and the function in the ismev R package [26]. We fit stationary distributions, but note that non-stationary fits can be used to account for shifts in the distribution due to climate change [27]. Both the 99th distribution percentiles and GEV location are relatively constant across latitude up to approximately 40°N, before declining steadily toward the poles (figure 2b,c). The breadth (scale parameter) increases steadily toward the poles (figure 2e). The GEV shape parameter varies, but shows little pattern, across intermediate latitudes (figure 2f). The temperature distributions, particularly in Asia, tend to have a heavier tail at both low and high latitudes. These results about daily maximum temperatures suggest that average maximum temperatures are similar across a wide latitudinal band (up to 30–40° latitude), and variation in maximum temperature increases consistently with latitude. By contrast, thermal extremes (tails) are strongly bounded over a wide range of intermediate latitudes (approx. 20–60°), with fatter tails at some low and high latitude sites (figure 2f).

(b) Environmental variability depends on time scale

The time scale of temperature data influences the distribution and the incidence of climatic extremes [30,31]. Because different organismal processes respond to environmental variation at different time scales [32,33] (see below), this has important consequences for the biological consequences of climate extremes. For example, the stressful impacts of heat waves are often determined by repeated exposures to high daily maximum temperatures rather than to overall mean temperatures. In addition, the biological effects of single versus repeated exposures to extreme temperatures can be qualitatively different [33–38]. To explore this issue, we average daily maximum temperatures for two North American sites across weeks (moving average), months and years. As the temperature data are aggregated at longer time scales, distributions necessarily narrow, but the thinning of the tails is more pronounced than the narrowing breadth (figure 3a). In addition, the effect of time scale is more pronounced in the thermally variable higher latitude site (45°N) relative to the lower latitude site (24°N).

Figure 3. (a) The maximum temperature (°C) distribution narrows and tails thin as data are averaged across weeks, months and years. The effect of time scale is more pronounced in the thermally variable higher latitude site (45°N) relative to the lower latitude site (24°N). (b) The maximum temperature experienced increases with the duration of the return period (years) and is greater when data are less aggregated. (c) The distribution of days between heat events shifts to longer intervals as the magnitude of the extremes increases (from the 90% to 98% percentiles).

Appropriately characterizing the tails of the temperature distribution is central to understanding how often organisms will experience extreme events. We use the generalized Pareto distribution to characterize the tails of the distribution. We fit the distribution using maximum likelihood with the fpot function from the R package evd [26]. We examine the maximum temperature expected to be experienced over a given duration of time (return period). Averaging over time decreases the magnitude of maximum temperatures experienced, particularly for the lower elevation, less thermally variable site (figure 3a). For both daily and weekly data, the magnitude of temperature extremes increases with the duration of the return period, with the slope shallowing.

Extremes are rare on average but can occur repeatedly. Repeat thermal stress events can prevent recovery in between the events and otherwise amplify thermal stress [39]. The interval between heat events is described well as a Poisson distribution [27]. As the magnitude of the extremes increases (higher quantile of the temperature distribution), the peak of the distribution shifts to longer intervals and the thickness of the tail (longer intervals between extremes) increases (figure 3c). The flat distribution of rare heat events makes it difficult to anticipate biological responses.

3. Responses of ectotherms to variable weather and climate

(a) Performance, tolerance and thermal thresholds

Most aspects of organismal performance—e.g. rates of locomotion, feeding, growth, reproduction and survival—depend on the organism's body temperature this relationship is called a thermal performance curve [14,40]. Performance curves frequently have a characteristic shape in which performance initially increases with increasing temperature, reaches maximal performance at some intermediate (optimal) temperature, then declines rapidly with further increases in temperature (figure 4). The basic shape reflects responses to both average and stressful temperatures: the effects of temperature on enzymatic rate process, and on enzyme activation and stability at high temperatures [42]. Comparative and experimental studies in a variety of systems demonstrate adaptive variation in both optimal temperature (Topt) and in thermal breadth (Tbr): optimal temperatures are greater in systems where mean environmental temperatures are higher (and less variable) and thermal breadths are wider in systems where environmental variation is greater [14,43]. The upper thermal limit (Tu) for performance can be defined as the temperature at which performance reaches (or approaches) zero, and is sometimes used as a measure of thermal limits (see below). Most empirical studies of performance curves focus on quantifying Topt, Tbr and lower thermal limits, rather than upper limits estimates of thermal limits frequently involve extrapolation beyond the data [19], resulting in large statistical uncertainties in our estimates of Tu. As we discuss below, this has important consequences for our understanding of responses to climate extremes.

Figure 4. Mean (±1 s.e.) growth rate of Manduca sexta larvae as a function of temperature. (a) Short-term (24 h) growth rate (corrected for initial mass) at the start of the 5th instar, for larvae reared from hatching at constant (25°C: solid line, circles) or fluctuating (25°C ± 10°C: dotted line, squares) rearing temperatures. (b) Long-term (hatching to wandering) larval growth rate at constant temperatures. From [41].

Tolerance curves can be considered a special case of performance curves in which the measure of performance is survival [44]. Tolerance curves (at least on a linear scale) are typically less skewed (i.e. more symmetric) and platykurtotic (i.e. flat-topped) when compared to other performance curves: survival is high and relatively constant over a range of temperatures, but declines rapidly at lower and higher temperatures. The high temperature at which survival reaches or approaches zero, Tu, is an important measure of heat tolerance. A complementary approach to characterizing heat tolerance is to measure the critical thermal maximal temperature (CTmax): the threshold temperature at which an organism ‘fails’ some relevant assay of performance (e.g. body posture or righting response, locomotory activity, neuromuscular control, survival). Both static (constant) and dynamic (ramping) temperature experiments can be used to estimate CTmax, resulting in an extensive literature on the topic [25,45–47]. Recent comparative studies indicate that, unlike metrics of lower thermal limits, mean CTmax does not decrease with increasing latitude in most ectotherms [24,25]. High CTmax may reflect the need to tolerate rare heat events [48], but historical patterns of colonization, selection favouring ‘hotter is better’ and warming associated with solar radiation likely also maintain high CTmax [25].

TPCs and threshold temperatures provide a useful framework for quantifying and predicting the effects of body temperature and thermal variation for ectotherms, but have several important limitations. First, the effects of (current) temperature on performance or tolerance may depend on previous thermal history, as a result of stress and acclimation responses. Many studies have demonstrated that higher developmental temperatures or acute heat shocks can alter CTmax and other metrics of heat tolerance [38,49] and exposure to increased maximum temperatures during development can also change optimal temperatures, upper thermal limits and maximum temperatures in some organisms [14,41,50]. Second, temperature may interact with other environmental factors to alter performance curves. For example, food availability and nutritional quality change optimal temperatures, upper thermal limits and maximal performance in fish and insects [51,52].

A third, less appreciated limitation is that performance curves and thresholds often reflect particular time scales. Some aspects of performance, such as rates of locomotion, feeding, growth, metabolism, oviposition and survival can be measured over short time scales (minutes to hours), whereas rates of growth, development, survival or fitness over a lifestage or the lifespan of individuals involve longer time scales (days to months or even years) [32,33,53]. As an example, the TPCs for larval growth rates in Manduca sexta measured over short (24 h) or long (duration of larval growth period, 15–50 days) time scales differ in optimal temperature, thermal breadth and upper thermal limits: temperatures that maximize growth at short time scales are deleterious or lethal at longer time scales (figure 4) [41]. Similarly, thermal thresholds of larval M. sexta are much higher at shorter than at longer time scales: the mean upper thermal limit for survival (through the larval period) is 35–36°C, whereas mean CTmax and upper lethal limits are 44–46°C [34].

Measurements of CTmax are also confounded by the temporal and thermal conditions in which they are measured. Many recent studies use a ramping protocol in which individuals are acclimated to a starting temperature, then the temperature is increased (ramped) at some linear rate CTmax is then defined as the temperature at which failure is observed. Studies with Drosophila show that changes in starting temperature and ramping rate can systematically alter mean estimates of CTmax by more than 5°C [46]. The CTmax that an organism can tolerate declines with the duration of thermal stress [54]. Both statistical and biological reasons underlie these methodological effects, highlighting the need to develop ‘ecologically relevant’ thermal tolerance metrics [45,47].

The effects of time scale become particularly important when using estimates of performance curves and thresholds to quantify mean and variation in performance in fluctuating thermal environments. In principle, information about the performance curve, P(T), and changes in temperature over time t, T(t), can be used to predict mean performance in fluctuating environments over some time period of interest. This simple model has been widely applied in thermal biology, including for predictions about responses to climate change (see below). But recent tests of this model question whether performance curves are constructed in a manner appropriate for assessing responses to diurnally fluctuating temperatures. For example, TPCs based on experiments using constant temperatures throughout development yielded poor predictions about mean development rates during diurnal fluctuating conditions in marsh frogs [55]. Similarly in M. sexta, neither short-term (24 h) nor long-term (larval duration) TPCs for growth rates based on constant temperatures gave accurate predictions for mean growth in diurnally fluctuating temperature conditions [41]. Predictions were particularly inaccurate for higher mean temperatures with large diurnal fluctuations—precisely the situation in which thermal extremes may be relevant. These predictions fail because this simple model ignores time-dependent effects: the effects of prior thermal history on current performance that result from stress, acclimation and similar processes [32,53]. These results call into question the common practice of using TPCs measured at constant temperatures to predict responses of ectotherms to diurnal fluctuations and climate change.

(b) Predicting the fitness consequences of climate change and climate extremes

The past decade has seen a burst of modelling studies that use TPCs (primarily for insect fitness) to predict responses of ectotherms to recent and future climate change [16–19]. These studies reveal the need to filter climate change responses through the lens of organismal physiology. Even small temperature increases may cause declines in the fitness of tropical ectotherms, which have evolved narrow thermal breadth and optimal temperature that are already near mean environmental temperatures in relatively constant environments [16]. Ectotherms at mid- and higher latitudes, with broad thermal breadth and optimal temperatures well above mean environmental temperatures, will be positively (or at least less negatively) impacted by future climate warming. Responses to environmental variation and extremes may cause deviations from these predictions.

The TPCs may inadequately capture responses to environmental variation. First, the TPCs for fitness (e.g. intrinsic rate of increase, r) used in these studies were estimated from data at constant temperatures over the entire lifespan. Because such long-term curves have lower optimal temperatures and upper thermal limits than shorter-term curves (figure 4) and omit acclimation, applying these curves to short-term (diurnally fluctuating) thermal variation will overestimate the negative consequences of high daily maximal temperatures [41]. Second, depending on the functional form chosen to represent the TPC, fitness at high temperatures declines to zero but is never negative [16]. Models that allow fitness to decline below zero predict that environmental variation may drive mid-latitude rather than tropical insects to suffer the greatest negative fitness consequences of climate warming [19].

A third, related issue is that different fitness components contribute in different, nonlinear ways to total fitness, so that computing mean fitness is not straightforward when there is environmental variation at time scales shorter than a generation. For example, within a generation, the arithmetic mean of reproductive rates (given survival) is appropriate for estimating overall reproduction, whereas the geometric mean of survival rates is more appropriate for estimating overall survival. As a result, the arithmetic mean of r may be a poor indicator of fitness responses to short-term thermal variation and modelling the separate effects of temperature on each fitness component may be needed [19]. As we discuss below, whether thermal stress causes reductions in reproduction or increases in mortality has important impacts on the evolutionary responses to thermal extremes.

These limitations are particularly important when considering responses to climatic extremes. Cumulative thermal effects and threshold temperature effects in response to thermal extremes decrease the accuracy of predictions of climate change responses based on mean temperatures [56]. Using TPCs to accurately predict responses to thermal extremes will require better characterizing performance above thermal optima and limits quantifying the effects of time scale and time-dependent effects and assessing how extremes will reduce performance beyond levels predicted by arithmetic means.

(c) Microevolutionary responses to climate extremes: data and models

As summarized by Grant et al. [6], both laboratory (and mesocosm) evolution studies and artificial selection experiments have been widely used to document evolutionary responses to increased mean temperature and high temperatures. These studies demonstrate evolution responses in mean fitness, optimal temperature and heat tolerance, but the results are of limited relevance to evolutionary responses of natural climatic extremes [6]. For example, artificial selection experiments typically maintain a constant selection intensity (e.g. upper 5% of the distribution) on heat tolerance each generation, resulting in a linearly increasing cumulative selection differential over time [57]. Laboratory and mesocosm evolution studies typically use a step change to a new, constant mean temperature over time. But as described above (figures 1 –3), natural climatic extremes occur infrequently and unpredictably and theoretical models show that stochastic variation in selection reduces the evolutionary responses of populations to sustained, directional environmental change [10,11,58,59]. More realistic experimental designs will be needed to evaluate the evolutionary responses to extreme climatic events, and to identify their genetic bases [6]. In addition, extreme and low quality environmental conditions can sometimes reduce genetic variation and evolutionary potential of ecologically important traits [48,60,61].

Historical and long-term studies can provide invaluable information about phenotypic and evolutionary responses to recent climate change. Such studies have documented shifts in body size, coloration, phenology, life history and other traits [6]. A recent historical study of TPCs illustrates the potential importance of changes in extreme temperatures [62]. Common-garden experiments with populations of Colias butterflies (C. eriphyle from Colorado and C. eurytheme from California) were used to determine mean TPCs for short-term larval feeding at two time points: 1972 [63] and 2012 [62]. The upper thermal limits of the performance of each species increased by 3–6°C during this 40 year period. Data from GHCN weather stations (USC00055722 in Montrose, CO and USW00023271 in Sacramento, CA) were used to quantify air temperature distributions at each site in the decade prior to each time point. Mean environmental temperatures during the active growing season did not change substantially (less than 1°C) over the time period at either site however, the frequency of high temperatures (more than 28°C) more than doubled at each site during this period. In contrast, there was little change in the frequency of low temperatures.

To estimate changes in GEV distributions during this 40-year period, we used daily maximum temperatures across summer months (June through September) from each site. The GEVs shifted to higher temperatures and narrowed in both Colorado (means ± s.e. of maximum-likelihood fits 1961–1971: location = 26.92 ± 0.14 and scale = 4.73 ± 0.10 2001–2011: location = 28.14 ± 0.14 and scale = 4.71 ± 0.10) and California (1961–1971: location = 30.64 ± 0.14 and scale = 4.87 ± 0.10 2001–2011: location = 31.58 ± 0.13 and scale = 4.60 ± 0.09). Small increases in the thickness of the tail suggest increases in the incidence of thermal extremes in both Colorado (1961–1971: shape = −0.51 ± 0.01 2001–2011: shape = −0.47 ± 0.01) and California (1961–1971: shape = −0.36 ± 0.01 2001–2011: shape = −0.35 ± 0.01). Only the increase in location in Colorado and the decrease in breadth in California are significant. More notable is the increase in the proportion of heat events. The percentage of years reaching maximum temperatures exceeding the 1961–1971 95th percentile increased from 2.9% to 9.7% in Colorado and from 4.8% to 6.5% in California (exceedance rate from generalized Pareto distribution). This highlights the utility of GEVs in characterizing shifts in the incidence of extreme events relevant to selection on thermal tolerance.

These findings suggest that the increasing frequency of high temperatures during the past 40 years has led to increased upper thermal limits in these populations. Interestingly, the evolutionary shifts in the performance curves were quite different in the two populations: in C. eriphyle from Colorado, the optimal temperature but not thermal breadth increased whereas in C. eurytheme from California, thermal breadth but not optimal temperature increased. These different responses may stem from the growth season remaining restricted to summer in Colorado but expanding in recent decades in California.

As discussed above (and see [14]), the evolutionary consequences of thermal extremes depend on whether thermal stress causes variation in survival (e.g. viability selection) or in reproduction (e.g. mating success or fecundity selection). In varying thermal environments, viability selection favours the evolution of thermal generalists [44], whereas fecundity selection favours the evolution of thermal specialists [64]. Several recent studies have combined these two effects and integrated performance curves, thermal tolerances and simple evolutionary models to explore how climate variation and extremes affect selection and evolutionary responses for ectotherms [65–67]. We will briefly describe two of these models to illustrate how analyses of extreme events can inform the results of these models.

Denny and Dowd [67] developed a model for the evolution of thermal tolerance (lethal temperature Tlethal, the body temperature as which an individual dies), assuming a polygenic (10 additive loci) basis for genetic variation in Tlethal. They assume a simple trade-off in which higher Tlethal is associated with a cost to reproduction at lower (non-lethal) temperatures. They implement this model for a large intertidal limpet (Lottia gigantea) that can sometimes experience deleteriously high body temperatures during low, midday tides. They combined a biophysical model that predicts the body temperature of limpets with fine-grained (10 min) data on thermal conditions in the intertidal zone at one site in central California. Using a bootstrapping approach to generate stochastic simulations of long-term environmental data [23], their evolutionary models predict that infrequent, extreme thermal events have important impacts on the evolution of thermal tolerance. For example, random, stressful events with mean return times of 2–8 years contributed strongly to the evolution of increased heat tolerance because of the larger impacts of short-term mortality on total fitness and selection.

Following Kingsolver and Buckley [65], Buckley and Huey [66] combined additive (TPC for reproduction) and multiplicative (thermal threshold for survival) components of fitness in a discrete-generation, quantitative genetic model. Using environmental temperature data along latitudinal clines in Australia, they apply the model to the evolution of thermal performance and tolerance in Drosophila [66]. Even rare thermal extremes substantially influenced the evolution of TPCs, particularly when the extremes caused mortality or persistent physiological injury, or when organisms were unable to use behaviour to buffer exposure to extremes. The latitudinal gradient in thermal extremes is much shallower than that of mean temperatures in Australia the model correctly predicted the evolution of a shallow cline in thermal tolerance in Drosophila. Their analyses illustrate how the evolution of tolerance, and of the upper limits of TPCs, is driven more by infrequent extremes than by environmental means or variances [66]. Extending the model to include beneficial acclimatization and cumulative damage revealed that substantial mortality or other reductions in fitness differences among individuals lessen the evolutionary impacts of thermal extremes [33].

To further characterize latitudinal gradients in thermal extremes in this system, we use GEV distributions of environmental temperatures along continental and coastal sites in Australia (see [66] for a description of the environmental data used). With movement toward the equator within Australia, GEV location shifts to warmer temperatures (figure 5). The GEV tails steadily thin in continental sites, but show less of a gradient and are more variable in coastal sites. The annual rate of exceedance increases steadily for continental sites, but remains relatively flat for coastal sites. The shallow gradients in the thickness of the tails of the distribution and exceedance rate revealed by GEVs are consistent with extremes influencing the evolution of thermal tolerance in Australia.

Figure 5. The parameters of the generalized extreme value (GEV) distribution describing maximum daily temperatures (°C) show variable patterns across coastal and continental latitudinal gradients in Australia. GEVs are characterized by three parameters: location, which indicates position scale, which indicates breadth and shape, which indicates the thickness of the tail. We additionally depict the annual rate of exceeding a threshold of 40°C.

These selected historical and modelling studies illustrate how infrequent, extreme thermal events can drive the evolution of both thermal tolerance and TPCs in ectotherms. Our analyses demonstrate how statistical analyses of extreme events using the GEV framework can aid our understanding of such events and their biological consequences.

4. Suggestions for future directions

Thermal biology, including performance curves and tolerances, has provided a productive, trait-based framework for quantifying the effects of climate variation and climate change on fitness and evolution for ectotherms. In this perspective we have highlighted several important challenges in extending this framework to understand climate extremes, suggesting several avenues for future research.

First, a greater focus on the tails is needed. Extremes involve the upper end of the distribution of environmental conditions, and characterizing these tails requires different statistical tools from those used to quantify means and variances. We have illustrated how the GEV analyses can quantify the frequency and temporal patterns of extreme events and inform their biological consequences, and urge that these tools be applied more widely by biologists interested in climate change [18,22]. Similarly, the shapes of performance curves above optimal temperatures are poorly characterized, and biologists often make convenient but arbitrary assumptions about curve shape near and above the upper limits. As a result, our inferences and predictions about responses to extreme temperatures may be weak or misleading. Characterizing the upper tails of performance curves will require changes in the design of experiments used to measure these curves.

Because extreme events are temporally structured (figure 3), the time-dependence of biological responses is also important. The effects of time scale on both TPCs and tolerances are rarely considered, but they can have major impacts on mean performance and fitness in variable environments that include extreme conditions. In addition, because TPCs and tolerances are often measured at different time scales, integrating information from upper performance limits to lethal temperatures is problematic [33]. More explicit description of both performance and tolerances as rates at specific time scales is needed in both empirical and modelling studies. Similarly, the effects of prior thermal history on performance and tolerance are widely documented, but rarely incorporated into models of climate change response.

Third, extreme thermal events may impact different components of fitness, and thus have major consequences for evolutionary responses. Extreme events may have both additive and multiplicative effects on overall fitness, so that quantifying the separate effects of performance and tolerance on survival, mating success and reproduction may be needed, instead of aggregate fitness metrics such as r. Integrating the effects of variation in generation time on overall fitness will also be important [19].

Finally, most models for evolutionary responses to climate change, including those summarized here, assume constant population size and constant phenotypic and genetic variation of performance and tolerance. Both of these factors are important, but assuming constant population sizes is particularly unrealistic in the current context: because most extreme events are stressful, they may generate large declines in mean absolute fitness and in population size, and strongly limit adaptive evolutionary responses [10]. Integrating ecological and evolutionary responses into models for population extinction and the evolution of thermal performance and tolerance will be a major challenge for thermal biologists and evolutionary ecologists alike.

Welcome to Statistics How To!

Looking for elementary statistics help? You’ve come to the right place. Statistics How To has more than 1,000 articles and hundreds of videos for elementary statistics, probability, AP and advanced statistics topics. Looking for a specific topic? Type it into the search box at the top of the page.

Watch the welcome video:

What is Probability and Statistics?

Probability and Statistics usually refers to an introductory course in probability and statistics.

The “probability” part of the class includes calculating probabilities for events happening. While it’s usual for the class to include basic scenarios like playing cards and dice rolling at first, these basic tools are used later in the class to find more complex probabilities, like the probability of contracting a certain disease.

The “statistics” part of probability and statistics includes a wide variety of methods to find actual statistics, which are numbers you can use to generalize about a population.

Statistics How To example: you could calculate the height of all your male classmates and find the mean height to be 5𔄃″ — this is a statistic. But then you could take that statistic and say “I think the average height of an American male is 5𔄃″ “. How accurate your guess is depends on many factors, including how many men you measured and how many men are in the entire population. Statistics are useful because we often don’t have the resources to measure, survey or poll every member of a population, so instead we take a sample (a small amount).

Need help with a specific homework or test question? Check out our Statistics How To tutoring page.

Feel like “cheating” at Statistics? Check out our Practically Cheating Statistics Handbook, which gives you hundreds of easy-to-follow answers in a PDF format. Recommended reading at top universities!

Need help with a homework or test question? With Facebook page.

The Biomathematics Program at NCSU

The Biomathematics Graduate Program at NCSU offers majors, co-majors, and minors at the Masters and PhD levels. Coursework includes theory and methods of biological modeling, mathematical and statistical methods, and applications to substantive problems in biology. The degree programs are flexible, to accomodate students with backgrounds in the biological, mathematical, or physical sciences and with diverse career interests.

The Biomathematics Program is jointly administered by the Departments of Statistics and Mathematics, but it has a long history of cooperation with other departments at NCSU, including Zoology, Entomology, Crop Science, Genetics, and the Veterinary School, and with research laboratories in the Research Triangle Park. Graduates of the program have found professional employment in university mathematics, biology, and statistics departments, in government, and in industries such as pharmaceuticals and environmental consulting.

Fossils and Converging Evidence

That the fossil record, in general, suggests evolution is certainly an important piece of evidence, but it becomes even more telling when it is combined with other evidence for evolution. For example, the fossil record is consistent in terms of biogeography — and if evolution is true, we would expect that the fossil record would be in harmony with current biogeography, the phylogenetic tree, and the knowledge of ancient geography suggested by plate tectonics. In fact, some finds, such as fossil remains of marsupials in Antarctica are strongly supportive of evolution, given that Antarctica, South America, and Australia were once part of the same continent.

If evolution did happen, then you would expect not just that the fossil record would show a succession of organisms as described above, but that the succession seen in the record would be compatible with that derived by looking at currently living creatures. For example, when examining the anatomy and biochemistry of living species, it appears that the general order of development for the major types of vertebrate animals was fish -> amphibians -> reptiles -> mammals. If current species developed as a result of common descent then the fossil record should show the same order of development.

In fact, the fossil record does show the same order of development. In general, the fossil record is consistent with the developmental order suggested by looking at the characteristics of living species. As such it represents another independent piece of evidence for common descent and a very significant one since the fossil record is a window to the past.

Using statistics to predict rogue waves

Scientists have developed a mathematical model to derive the probability of extreme waves. This model uses multi-point statistics, the joint statistics of multiple points in time or space, to predict how likely extreme waves are.

The results, published today, Friday 11 March, in the New Journal of Physics, demonstrate that evolution of these probabilities obey a well-known function, greatly reducing the complexity of the results."It's common in science and engineering to consider noise and fluctuations as something we need to avoid or eliminate in order to gain the best results" explains Matthias Wächter, an author on the paper. "For us, understanding noise and fluctuations is helpful for understanding complex systems."

'Rogue' waves are large and spontaneous waves which occur in the open water, and can be extremely dangerous, even to large ships and ocean liners. They are typically defined by oceanographers as waves whose height is twice the 'significant wave height' - itself defined as the average of the largest third of the waves in the current sea state.

"Multi-point statistics allows us to capture a high level of complexity, such as wave heights or turbulent air flows" continues Wächter. "A key point of our work is that we were able to reduce the complexity of these so that they obey the well-known Fokker-Planck equation."

Sadly, it is unlikely that this approach could be applied to Tsunami-type events. "Typically, a Tsunami is the consequence of an isolated earthquake event" explains Wächter. "It is likely that their statistics differ significantly from common ocean waves, so this approach cannot capture them."

Further work remains for the researchers to extend the range of these predictions to a scale of minutes or hours. They are also working on expanding their model to encompass atmospheric wind data.

"This has tremendous practical relevance in wind energy applications, where knowing about an impending large gust of wind will help wind turbines adjust their operation accordingly" concludes Wächter. "But there is still a lot of research to do!"

The researchers would like to acknowledge the support of the Volkswagen Foundation, and their fruitful collaboration with Norbert Hoffman and partners in the project "Extreme Ocean Gravity Waves".

Watch the video: Πώς Να Εμφανίσετε Τα Στατιστικά Ανάλυσης Ενός Εγγράφου Στο Microsoft Word; (August 2022).