Agricultural Statistics & Data Science · Statistical genetics and biometrics · also Statistical Ecology, Environmental Statistics
Counting what cannot be counted directly: capture-recapture and distance sampling, quadrat and line-transect methods, diversity indices and species-abundance models, fish stock assessment, forest inventory sampling and environmental monitoring designs.
In the exams: AICE-JRF Ag Stat Unit 5 'livestock and fishery statistics' (present); ICAR MSc/PhD elective; light in national exams
Much of what agriculture, fisheries, forestry and conservation need to know cannot be counted directly. Nobody can count every fish in a lake, every tree in a forest region, every pest in a field or every bird on a farm; nobody can wait for a hundred years of rainfall records to design a culvert. Statistical ecology and environmental statistics supply the methods: sampling designs that estimate abundance from a fraction of the area, capture–recapture and distance methods that correct for animals that are missed, indices that summarise the diversity of communities, models of population growth and harvest, fish stock assessment, forest inventory, monitoring designs that can detect environmental change, and extreme value theory for floods and droughts.
This manual covers spatial patterns and dispersion indices; quadrat, line-transect and distance sampling; capture–recapture with the Lincoln–Petersen estimator and open-population models; diversity indices and species richness estimators; species-abundance and species–area models; population dynamics with the logistic and Leslie matrix models; fish stock assessment with growth, mortality and yield-per-recruit; forest inventory with plot sampling, point sampling and volume estimation; environmental monitoring designs and trend detection; and extreme value analysis of rainfall and floods. Each method is presented with its assumptions, its estimator and a worked example, because in ecology the assumptions, such as equal catchability or certain detection on the transect line, decide whether a number is useful.
Doctoral examinations in agricultural statistics include livestock and fishery statistics and related sampling problems; postgraduate programmes in agricultural statistics offer this material as an elective; and professional work in fisheries, forestry, wildlife and environmental agencies uses it daily. Short numericals are typical: dispersion indices, Lincoln–Petersen estimates, density from transects, Shannon and Simpson indices, Chao1, maximum sustainable yield, von Bertalanffy lengths, survival from instantaneous mortality, basal area from point sampling, sample sizes and return periods.
The arrangement of organisms in space carries information about their biology and determines how they should be sampled. In a random pattern, each individual's position is independent of the others, and counts in equal-sized quadrats follow a Poisson distribution, whose variance equals its mean. In a clumped (aggregated, contagious) pattern, individuals occur in groups, as with insect colonies, seedlings around parent trees, weeds spreading from patches or schooling fish; quadrat counts are more variable than Poisson, with variance greater than the mean, and are often described by the negative binomial distribution. In a regular (uniform, even) pattern, individuals are more evenly spaced than random, as with territorial animals, competing desert shrubs or planted crops; variance is less than the mean, and the positive binomial may fit. Pattern depends on scale: plants that are regular at the scale of their crowns may be clumped at the scale of soil patches, so quadrat size should be chosen with the question in mind.
The simplest measure is the index of dispersion (variance-to-mean ratio), I = s²/x̄, which is about 1 for random, above 1 for clumped and below 1 for regular patterns. Under randomness, (n − 1)s²/x̄ follows approximately a chi-square distribution with n − 1 degrees of freedom, giving a test: with 25 quadrats, mean 4 and variance 10, I = 2.5 and the statistic is 24 × 2.5 = 60, far above the upper 2.5% point of chi-square with 24 degrees of freedom (about 39.4), so the pattern is significantly clumped. The negative binomial parameter k, estimated by moments as k = x̄²/(s² − x̄), measures aggregation inversely: small k means strong clumping, and k → ∞ approaches the Poisson; for the data above, k = 16/6 ≈ 2.7. Other indices include Lloyd's mean crowding, Morisita's index (1 for random, larger for clumped), which is less sensitive to quadrat size, and Taylor's power law, s² = a x̄ᵇ, which relates variance to mean across many samples and is widely used in pest sampling: b > 1 indicates aggregation, and the fitted relationship sets sample sizes and transformations. For mapped points, distance-based methods such as nearest-neighbour statistics (the Clark–Evans ratio, 1 for random, below 1 for clumped and up to about 2.15 for perfectly regular hexagonal spacing) and Ripley's K function describe pattern at several scales.
Aggregation increases the variance of quadrat counts, so more samples are needed for a given precision, and estimates from few quadrats are unreliable. Many small quadrats are generally more efficient than a few large ones for clumped populations. Variance-stabilising transformations depend on pattern: square root for Poisson counts and log(x + k/2) or similar for negative binomial counts. Sequential sampling plans for pest management, which decide after each sample whether to stop and treat, not treat or keep sampling, are built on the fitted dispersion relationship, so that sampling effort adapts to density.
Quadrat sampling counts organisms in plots of known area placed by a probability design, randomly, systematically or in strata, and estimates density as the mean count per unit area: 120 plants in 30 quadrats of 1 m² give 4 plants per m², or 40,000 per hectare. Frequency (the proportion of quadrats containing the species) and cover (the proportion of ground covered) are other measures; frequency depends strongly on quadrat size. For random patterns, density and frequency are linked through the Poisson probability of a zero count. Strip transects count all individuals within a fixed distance w either side of a line of length L, so density is n/(2wL); they assume that every individual in the strip is detected, which is often false for animals and plants hidden in vegetation, especially near the edges of the strip.
Distance sampling relaxes the assumption of complete detection. An observer walks a line of length L and records each detected object and its perpendicular distance from the line. The probability of detection declines with distance according to a detection function g(x), with g(0) = 1, meaning objects on the line are detected with certainty. Fitting g(x), commonly a half-normal or hazard-rate model, to the distribution of distances gives the effective strip half-width μ = ∫g(x)dx, the distance within which as many objects would be detected if detection were perfect as are actually detected over all distances. Density is then D = n/(2μL), or equivalently D = n f(0)/(2L), where f(0) = 1/μ is the probability density of detected distances at zero. With 48 detections along 4 km of transect and an effective strip half-width of 20 m, D = 48/(2 × 20 × 4,000) = 0.0003 per m², or 3 per hectare. Point transects record radial distances from points instead and are common for birds. The key assumptions are that objects on the line are always detected, that objects are detected at their initial positions before responding to the observer, that distances are measured accurately, and that lines are placed independently of the objects' distribution. Clusters such as flocks are handled by recording cluster size.
Transect lines should be placed randomly or systematically with a random start across the study area, not along roads or tracks, which are rarely representative. Stratifying by habitat improves precision. Enough detections, often 60 to 80 or more, are needed to model the detection function reliably. Variance of density estimates comes from three sources: the encounter rate (variation in n between lines, usually the largest component), the detection function and, for clusters, cluster size; replicate lines provide the empirical estimate of encounter-rate variance.
Capture–recapture estimates the size of a population whose members cannot all be seen. In the simplest two-sample design, M animals are captured, marked and released; later, C animals are captured, of which R are marked. If marked animals mix fully and are as catchable as unmarked ones, the proportion marked in the second sample estimates the proportion marked in the population, R/C ≈ M/N, giving the Lincoln–Petersen estimator N̂ = MC/R. With M = 100, C = 80 and R = 20, N̂ = 400. The estimator is biased upwards for small samples, and Chapman's modification, N̂ = (M + 1)(C + 1)/(R + 1) − 1, is nearly unbiased when M + C ≥ N; with the same data it gives about 389. The assumptions are that the population is closed (no births, deaths, immigration or emigration between samples), that marks are not lost, overlooked or harmful, and that all animals have equal probability of capture in each sample. Unequal catchability, such as trap-happy or trap-shy behaviour or heterogeneity among individuals, biases the estimate: trap-shy animals (low recapture) inflate N̂, and trap-happy ones deflate it. With several capture occasions, the Schnabel estimator generalises Lincoln–Petersen, and models in the closed-population family (M₀ with constant capture probability, Mt with time variation, Mb with behavioural response, Mh with individual heterogeneity, and combinations) are fitted by maximum likelihood and compared. Mark–recapture logic also underlies estimates of the size of human populations from overlapping lists, in which it is called multiple-systems estimation.
Over longer periods populations gain and lose members, and open-population models estimate survival and recruitment as well as abundance. The Cormack–Jolly–Seber (CJS) model uses the capture histories of marked individuals, sequences of 1s and 0s over occasions, to estimate apparent survival φ between occasions and capture probability p on each occasion; apparent survival combines true survival and permanent emigration. The Jolly–Seber model adds estimates of population size and recruitment by also modelling unmarked animals. Robust designs combine closed sessions within short primary periods with open-model intervals between them, separating capture probability from survival and estimating temporary emigration. Assumptions include homogeneous survival and capture probabilities within groups, independence of individuals and instantaneous sampling relative to the interval. Goodness-of-fit tests check for transience and trap dependence, and overdispersion is handled with a variance inflation factor. Capture–recapture now uses natural marks (photographs of stripes and spots, genetic identification from hair or faeces) and spatially explicit models that use where animals were detected to estimate density directly.
When individuals cannot be marked or counted reliably, the proportion of sites occupied by a species is often a more practical state variable. Site-occupancy models, developed by MacKenzie and colleagues, use repeated presence–absence surveys of each site within a season to separate two probabilities: ψ, the probability that a site is occupied, and p, the probability of detecting the species on one visit when it is present. A species missed on every visit to an occupied site produces a false absence, so the naive proportion of sites with detections underestimates occupancy; with three visits and p = 0.5, an occupied site is missed entirely with probability 0.5³ = 0.125. Repeat visits, by different observers or methods, provide the information to estimate p, which may depend on weather, observer and habitat covariates, while ψ may depend on site covariates. Multi-season models estimate colonisation and local extinction rates, linking occupancy to metapopulation dynamics. Camera traps, acoustic recorders and environmental DNA surveys generate exactly the repeated detection data these models need.
Biodiversity within a community (alpha diversity) has two components: species richness, the number of species, and evenness, how equally individuals are distributed among them. The Shannon index H′ = −Σpᵢ ln pᵢ, where pᵢ is the proportion of individuals in species i, increases with both; for proportions 0.5, 0.3 and 0.2 it is about 1.03, and for four equally common species it is ln 4 ≈ 1.39, the maximum for four species. Pielou's evenness J = H′/ln S compares it with that maximum. Simpson's index D = Σpᵢ² is the probability that two individuals drawn at random (with replacement) belong to the same species; for the same three proportions, D = 0.25 + 0.09 + 0.04 = 0.38. Because D decreases as diversity increases, it is usually reported as 1 − D (0.62, the Gini–Simpson index) or 1/D (about 2.63, the inverse Simpson). For finite samples, the unbiased form uses Σnᵢ(nᵢ − 1)/[N(N − 1)]. Shannon's index is more sensitive to rare species and Simpson's to dominant ones.
Hill numbers unify these measures as effective numbers of species: the number of equally common species that would give the same diversity. Of order q, ᑫD = (Σpᵢ^q)^(1/(1−q)); order 0 is species richness, order 1 is exp(H′), the exponential of Shannon's index, and order 2 is the inverse Simpson 1/Σpᵢ². For the proportions above, the three are 3, about 2.80 and about 2.63. Effective numbers behave intuitively: two equally diverse, completely distinct communities combined have twice the diversity, which is not true of H′ or 1 − D themselves, so Hill numbers are preferred for comparing and partitioning diversity into alpha, beta (turnover between sites) and gamma (regional) components.
Observed species richness depends strongly on sampling effort: the more individuals or plots sampled, the more species found. Fair comparison therefore uses rarefaction, which computes the expected number of species in a random subsample of a standard size, or extrapolation of species accumulation curves. Non-parametric estimators predict total richness, including undetected species, from the numbers of rare species: the Chao1 estimator, based on abundance data, is S_obs + F₁²/(2F₂), where F₁ and F₂ are the numbers of species represented by exactly one and exactly two individuals (singletons and doubletons); with 20 observed species, 6 singletons and 3 doubletons, Chao1 = 20 + 36/6 = 26. Chao2 and the jackknife estimators use incidence (presence–absence) data across sampling units. Many singletons signal that many species remain undetected. Sample coverage, the estimated proportion of the community's individuals belonging to detected species, about 1 − F₁/N, provides a standard for comparing communities at equal completeness.
Diversity between sites, or beta diversity, measures how much species composition changes from place to place. Whittaker's multiplicative definition, β = γ/ᾱ, where γ is regional diversity and ᾱ the mean diversity of individual sites, gives the effective number of distinct communities; with Hill numbers this partitioning is coherent for any order q. Pairwise dissimilarity indices compare two sites: the Jaccard index J = a/(a + b + c), where a is the number of shared species and b and c the species unique to each site, and the Sørensen index 2a/(2a + b + c), for presence–absence data, and the Bray–Curtis dissimilarity for abundance data. Beta diversity can be split into turnover (replacement of species) and nestedness (loss of species from richer to poorer sites), which have different implications for conservation: high turnover argues for protecting many sites, high nestedness for protecting the richest.
Communities typically contain a few very common species and many rare ones. Species-abundance distributions describe this pattern, displayed as rank–abundance (Whittaker) plots of log abundance against rank, or as frequency histograms of species in abundance classes. Several models have been proposed. The geometric series, in which each species takes a constant fraction of the remaining resources, produces a straight line on a rank–abundance plot and describes species-poor, harsh or early successional communities dominated by a few species. Fisher's log-series, introduced in 1943 for moth catches, has many singletons and a diversity parameter α that is fairly independent of sample size, and has been widely used as a diversity index. Preston's lognormal model, in which abundances are lognormally distributed and grouped into doubling classes called octaves, fits many large, species-rich communities, with the left part of the curve hidden behind a 'veil line' of species too rare to be sampled. MacArthur's broken-stick model, in which a resource is divided at random points, gives the most even distribution and fits some narrowly defined guilds. Fitting these models is descriptive; similar shapes can arise from different mechanisms, including neutral models of random drift, so fit alone rarely identifies process.
The species–area relationship, S = cAᶻ, describes the increase of species with area; on log–log axes it is a straight line with slope z, commonly 0.1 to 0.2 for nested samples within continuous habitat and about 0.2 to 0.35 for isolated islands or habitat fragments. With z = 0.25, a tenfold increase in area raises species by about 78%. The relationship is used to compare sampling efforts, to extrapolate richness to larger areas and, more controversially, to predict extinctions after habitat loss; such predictions overestimate immediate losses because some species survive in the modified matrix, and because extinction lags behind habitat loss. Species accumulation curves plot species found against effort (number of samples or individuals); their slope at the end shows whether sampling is nearly complete, and asymptotic models (such as the Michaelis–Menten form) or Chao estimators predict total richness.
In exponential growth, dN/dt = rN, a population grows without limit at intrinsic rate r, N(t) = N₀e^(rt), with doubling time ln 2/r; the discrete form Nₜ₊₁ = λNₜ has finite rate λ = e^r. Real populations are limited by resources, and the logistic model dN/dt = rN(1 − N/K) adds a carrying capacity K, giving an S-shaped curve N(t) = K/(1 + [(K − N₀)/N₀]e^(−rt)). Growth rate rN(1 − N/K) is greatest at N = K/2, where it equals rK/4. This is the basis of the surplus-production idea in harvest management: harvesting the surplus at N = K/2 gives the maximum sustainable yield, MSY = rK/4; with r = 0.4 per year and K = 10,000, MSY = 1,000 per year. In practice, MSY is uncertain, recruitment varies and depensation (reduced growth at low density) can make populations collapse, so modern management uses precautionary reference points below MSY levels. Discrete-time logistic models, such as the Ricker model Nₜ₊₁ = Nₜ exp[r(1 − Nₜ/K)], can show damped oscillations, cycles and chaos at high r, behaviour seen in some insect populations. Stochastic models add environmental and demographic variability, giving extinction probabilities for small populations, the basis of population viability analysis.
Age-structured populations are modelled with the Leslie matrix. The population is divided into age classes; the first row of the matrix holds age-specific fecundities Fₓ (offspring produced per individual in class x that survive to the first class), the subdiagonal holds survival probabilities Pₓ from class x to x + 1, and all other entries are zero. The population vector advances as nₜ₊₁ = Lnₜ. Over time the population converges to a stable age distribution, given by the right eigenvector of L for its dominant eigenvalue λ, and grows at rate λ per time step: λ > 1 means growth, λ = 1 stability and λ < 1 decline. For two age classes with fecundities 0 and 2 and survival 0.5, the characteristic equation λ² = 1 gives λ = 1; with fecundities 1 and 2, λ² − λ − 1 = 0 gives λ ≈ 1.618. The left eigenvector gives reproductive values. Lefkovitch matrices generalise the model to stage classes (seed, seedling, juvenile, adult), more useful for plants, trees and long-lived animals whose fate depends on size rather than age. Sensitivity and elasticity analysis shows how λ responds to changes in each vital rate, and so which life stages management should target: for long-lived species such as turtles and many trees, adult survival typically has the largest elasticity.
Small populations face risks that deterministic models ignore. Demographic stochasticity is chance variation in births and deaths among individuals, important when numbers are small; environmental stochasticity is year-to-year variation in vital rates caused by weather, disease or food supply, which affects populations of any size; catastrophes are rare, severe events such as droughts or fires; and genetic factors such as inbreeding depression reduce fitness in very small populations. Population viability analysis (PVA) simulates many possible futures of a population with these sources of variation, using matrix or individual-based models, to estimate quantities such as the probability of falling below a threshold or of extinction within a given time, and the expected time to extinction. Its results are sensitive to uncertain parameters, particularly the variance of vital rates and density dependence, so PVA is most useful for comparing management options and identifying which vital rates matter rather than for predicting exact extinction probabilities.
Fish grow throughout life at a declining rate, commonly described by the von Bertalanffy growth function L(t) = L∞[1 − e^(−K(t − t₀))], where L∞ is the asymptotic length, K the growth coefficient (how quickly L∞ is approached) and t₀ the theoretical age at zero length. With L∞ = 100 cm, K = 0.2 per year and t₀ = 0, a five-year-old fish is expected to be 100(1 − e^(−1)) ≈ 63 cm. Weight is related to length by W = aLᵇ with b often close to 3. Growth parameters are estimated from length-at-age data, where age is read from annual rings on otoliths, scales or spines, from tagging, or from length-frequency analysis, which follows modes of cohorts through time; the Ford–Walford plot of Lₜ₊₁ against Lₜ is a straight line whose slope is e^(−K).
Fish die from natural causes and from fishing, and in stock assessment both are expressed as instantaneous rates: total mortality Z = F + M, where F is fishing and M natural mortality. A cohort declines exponentially, Nₜ = N₀e^(−Zt), so the annual survival is S = e^(−Z) and the annual death fraction is 1 − e^(−Z); with Z = 0.5, S ≈ 0.61. The Baranov catch equation gives the catch from a cohort over a period as C = (F/Z)N(1 − e^(−Z)): of 1,000 fish with F = 0.3 and M = 0.2, about 393 die in the year, of which F/Z = 60%, about 236, are caught. Total mortality is estimated from catch curves, plots of the logarithm of numbers caught at each age against age, whose descending slope is −Z for ages fully vulnerable to the gear, assuming constant recruitment and mortality. Natural mortality is hard to measure and is often taken from empirical relationships with growth, maximum age or water temperature, an important source of uncertainty.
Yield-per-recruit models, such as Beverton and Holt's, combine growth and mortality to calculate the yield in weight obtained from each recruit over its lifetime, as a function of fishing mortality and the age (or size) at first capture. With very low F, few fish are caught; with very high F, fish are caught before they grow, called growth overfishing. The F that maximises yield per recruit is F_max, but because it can be high and may ignore effects on reproduction, the more conservative F₀.₁, the F at which the slope of the yield-per-recruit curve falls to 10% of its slope at the origin, is often used. Yield per recruit does not consider whether enough fish survive to spawn; spawning-stock-biomass-per-recruit analysis and stock–recruitment relationships (Beverton–Holt and Ricker forms) address recruitment overfishing. Full stock assessments use surplus-production models fitted to catch and abundance-index series, virtual population analysis that reconstructs cohort numbers backwards from catches at age, and statistical catch-at-age and integrated models that combine catch, survey, length and age data in one likelihood. Catch per unit effort is used as an abundance index, but it can stay high while stocks decline if fishers concentrate on remaining aggregations, a phenomenon called hyperstability.
Forest inventory estimates the area, growing stock, growth and condition of forests by sampling. In fixed-area plot sampling, all trees above a minimum diameter are measured in circular, square or rectangular plots of fixed size, placed by systematic or stratified random designs, and per-plot totals are multiplied by an expansion factor, the reciprocal of plot area in hectares, to give per-hectare values: each tree in a 0.05 ha plot represents 20 trees per hectare. The sample size needed to estimate the mean within a relative error E with confidence given by Student's t is approximately n = t²CV²/E², where CV is the coefficient of variation between plots in per cent: with CV = 30%, E = 10% and t ≈ 2, n ≈ 36 plots, adjusted by the finite population correction when the sampled fraction is large. Stratifying by forest type or stand age reduces CV within strata and so the number of plots needed. National forest inventories use permanent systematic grids of plots remeasured at intervals, combined with remote sensing in two-phase or model-assisted designs.
In point sampling, also called variable-radius plot sampling, angle-count sampling or Bitterlich sampling, the observer at a sample point sights every tree at breast height through an angle gauge, prism or relascope that projects a fixed angle, and counts the trees whose stem appears wider than the angle. Each tree is included with probability proportional to its basal area, because larger trees are counted from further away, so basal area per hectare is simply the basal area factor (BAF) multiplied by the count: with BAF 2 m² per ha and nine trees counted, basal area is 18 m² per ha. Each counted tree represents BAF/gᵢ trees per hectare, where gᵢ is its own basal area, which converts counts into stem numbers and volumes. Point sampling is efficient because large, valuable trees are sampled more intensively, but borderline trees must be checked by measurement, slope corrections applied, and boundary effects at stand edges handled.
Tree volume is estimated from diameter at breast height (dbh, conventionally 1.3 m above ground in most countries, 1.37 m in some) and height. Basal area is g = πd²/4; a tree of 40 cm dbh has g ≈ 0.126 m². Volume can be written V = f × g × h, where the form factor f, typically around 0.4 to 0.6 for stem volume, corrects for taper: with f = 0.5 and h = 20 m, V ≈ 1.26 m³. Sections of felled logs are measured with Huber's formula (cross-sectional area at the middle times length), Smalian's formula (mean of the end areas times length) or Newton's formula, which combines both ends and the middle and is exact for common solids of revolution. Volume equations or tables, regressions of measured volume on dbh and height fitted to sample trees for each species and region, are applied to inventory trees; log-transformed models such as ln V = a + b ln d + c ln h are common, with a bias correction when back-transforming. Biomass and carbon are estimated from volume with wood density and expansion factors or from direct allometric equations, the basis of forest carbon reporting.
Environmental monitoring measures change in quantities such as water quality, soil carbon, air pollutants, pest densities or bird populations, and must be designed to detect changes that matter. A good design states its objectives and the size of change to be detected, chooses indicators, sampling units, locations, timing and frequency, and plans the analysis and power before data are collected. Probability-based designs, such as stratified random or spatially balanced designs, allow results to be generalised to the whole area; sites chosen for convenience do not. Revisit designs balance precision of status estimates against detection of trends: revisiting the same sites (panels) improves estimates of change because site effects cancel, while new sites improve coverage. Composite sampling, physically mixing several samples before analysis, reduces laboratory costs when the mean is the target. Measurements below a laboratory's detection limit are censored, not zero, and require methods for censored data rather than substitution of arbitrary values.
To assess the effect of an intervention or impact, such as a new irrigation scheme, a pesticide ban or an effluent discharge, a design is needed that separates the impact from natural variation. The before–after–control–impact (BACI) design measures both an impact site and one or more control sites before and after the event; the impact is estimated by the interaction, the change at the impact site minus the change at the controls. Measuring only after the event, or only at the impact site, confounds the impact with pre-existing differences or with general trends. Because a single impact site gives no replication of the impact, multiple controls and multiple sampling times before and after (beyond-BACI designs) are used to estimate natural variability, and statistical models must account for temporal autocorrelation in repeated measurements.
Trend tests ask whether a quantity has changed systematically over time. Linear regression on time is simple but assumes linearity, normal errors and independence. The non-parametric Mann–Kendall test counts concordant and discordant pairs of observations in time order and tests for a monotonic trend without assuming a distribution, and Sen's slope, the median of all pairwise slopes, estimates the trend's magnitude robustly. The seasonal Kendall test handles seasonal data by comparing values within the same season across years. Serial correlation inflates the apparent significance of trends, so it must be modelled or corrected, for example by prewhitening or by adjusting the effective sample size. Power analysis for monitoring programmes, often by simulation, shows how many years and sites are needed to detect a given annual percentage change; detecting small trends against large year-to-year variation can take a decade or more, which should be known before a programme starts. Change-point methods detect abrupt shifts, and smoothers such as generalised additive models describe non-linear trends.
Designing culverts, spillways, levees and irrigation structures, and pricing crop insurance against floods and droughts, requires estimates of how often extreme events occur. The return period T of an event is the average interval between events at least that large: an event with annual exceedance probability p has T = 1/p, so the '100-year flood' has a 1% chance of being exceeded in any year. It is not an event that occurs once every hundred years: the probability of at least one exceedance in n years is 1 − (1 − 1/T)ⁿ, so over 30 years the chance of at least one 100-year flood is 1 − 0.99³⁰ ≈ 26%. This calculation is the basis of design risk: a structure designed for the 100-year event and expected to last 30 years has about a one-in-four chance of facing that event.
Extreme value theory describes the distribution of maxima. The Fisher–Tippett–Gnedenko theorem states that the maximum of a large number of independent observations, suitably normalised, follows one of three types, unified as the generalised extreme value (GEV) distribution with location μ, scale σ and shape ξ: ξ = 0 gives the Gumbel distribution (light tail, as for maxima of normal or exponential variables), ξ > 0 the Fréchet (heavy tail, common for rainfall) and ξ < 0 the reversed Weibull (bounded upper tail). In the block-maxima approach, the largest value in each year (annual maximum daily rainfall or peak flow) is fitted by a GEV. Under a Gumbel model with location μ and scale β, the T-year return level is x_T = μ − β ln[−ln(1 − 1/T)]; with μ = 50 mm and β = 10 mm, the 100-year daily rainfall is about 50 + 10 × 4.60 = 96 mm. The peaks-over-threshold approach uses all values above a high threshold, fitted by the generalised Pareto distribution, and makes better use of data than taking one value per year. Parameters are estimated by maximum likelihood, probability-weighted moments or L-moments; the last are favoured in hydrology because they are robust with short records.
Empirical return periods are estimated from ranked annual maxima with plotting positions: the Weibull formula assigns the m-th largest of n values an exceedance probability m/(n + 1). Extrapolating far beyond the record length is uncertain, so return levels must be reported with confidence intervals, which become very wide for return periods much longer than the record; a 30-year record says little about the 500-year event. Regional frequency analysis pools records from similar sites to reduce this uncertainty. Extreme value methods assume stationarity, but climate change, land-use change and river regulation alter extremes, so non-stationary models with parameters depending on time or covariates are increasingly used. Droughts are analysed with indices such as the standardised precipitation index and with the frequency of minima, and the same methods apply to heat waves, frost and wind damage.
A student uses this subject to analyse field and laboratory exercises: testing whether insect counts in quadrats are clumped, estimating fish numbers in a pond by mark–recapture, fitting a detection function to transect data, comparing the diversity of plots with rarefied richness and Hill numbers, projecting a population with a Leslie matrix, fitting von Bertalanffy curves to length-at-age data, computing basal area from a prism count, and estimating return levels from rainfall records. In examinations, the student must apply formulas such as Lincoln–Petersen, Chao1, MSY = rK/4, survival e^(−Z) and T = 1/p correctly and state the assumptions behind them.
A statistician five years into practice works with fisheries, forestry, wildlife or environmental agencies: designing transect and capture–recapture surveys, analysing them with appropriate models, preparing stock assessments, designing forest inventories and computing plot numbers, setting up water-quality or biodiversity monitoring with enough power to detect trends, and estimating design rainfall and flood magnitudes. Typical decisions are which assumptions are plausible in the field, how many plots, lines or years are needed, and how to report uncertainty to managers. They also explain to managers which assumptions are most likely to fail.
A senior statistician with twenty-five years of experience leads assessment and monitoring programmes, chooses the models behind harvest quotas, forest resource reports and flood-design standards, reviews the assumptions and uncertainty of assessments before they are used in policy, and explains to managers why a precautionary catch limit or a wider confidence band is necessary. They also judge when long-term monitoring designs should be changed and how to keep series comparable. They decide which surveys are worth continuing when budgets shrink.
A researcher with a PhD and long experience develops methods for ecological and environmental statistics: hierarchical and spatially explicit capture–recapture and occupancy models, integrated population models, state-space stock assessment models, design-based and model-based inventory with remote sensing, and non-stationary extreme value models under climate change. Such researchers advise international bodies on assessment methods, review stock and biodiversity assessments, supervise doctoral students and examine candidates. They also train field staff so that survey protocols are followed consistently.
This manual introduces each method with its main estimator and assumptions; it does not cover occupancy models, integrated population models, Bayesian hierarchical models, spatial capture–recapture or full statistical catch-at-age assessment in detail. Parameter values in examples are illustrative; real values of growth, mortality, form factors, detection functions and extreme-value parameters vary by species, region and data and must be estimated locally. Standards such as breast height and inventory protocols differ between countries.
Original practice questions written to the syllabus and pattern of the exams named in each question's examTags. They are not past papers and do not reproduce any board's questions.
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