Agricultural Statistics & Data Science · Sample surveys and official agricultural statistics · also Sample Surveys II, Advanced Sampling Techniques
The designs real agricultural surveys use: PPS sampling with and without replacement and its estimators, cluster and multistage sampling, double sampling, sampling on successive occasions, randomised response, non-response and imputation.
In the exams: AICE-JRF Ag Stat Unit 5 (heavy: each technique named); UPSC ISS Statistics III (heavy); ASRB NET Ag Stat (heavy)
This second sampling subject covers the designs and methods that real agricultural and household surveys use beyond simple random, stratified and systematic sampling: probability-proportional-to-size sampling with and without replacement and its estimators (Hansen-Hurwitz, Horvitz-Thompson, Des Raj, Murthy, Midzuno-Sen and Rao-Hartley-Cochran); cluster, two-stage and multistage sampling; double sampling for ratio, regression and stratification; sampling on successive occasions and rotation designs; randomised response for sensitive questions; non-response and its treatment by call-backs and weighting; imputation; variance estimation in complex surveys by linearisation, jackknife and bootstrap; and an introduction to inverse and adaptive sampling.
It builds directly on the Sampling Theory I subject (SRS, stratification, systematic sampling, ratio and regression estimation, inclusion probabilities) and supplies the methods used in the crop estimation surveys, agricultural statistics system and small area estimation subjects. Topics from Sampling Theory I are referred to rather than repeated.
In examinations, ICAR AICE-JRF/SRF Agricultural Statistics Unit 5 names cluster and multistage sampling with equal probability, sampling with unequal probabilities with and without replacement, inclusion probabilities proportional to size, double sampling, sampling on successive occasions, non-sampling errors, randomised response and imputation. The Indian Statistical Service papers (as reported) and ASRB NET in agricultural statistics also examine these topics. Typical questions ask for HH and HT estimates and variances, design effects, two-stage variances, double-sampling variances, randomised-response estimates, call-back estimates and multiple-imputation variances.
In many agricultural populations, units differ greatly in size: villages in crop area, farms in herd size, markets in arrivals. If the variable of interest is roughly proportional to a known size measure x, selecting units with probability proportional to size (PPS) gives large units a higher chance of selection and can reduce variance dramatically compared with simple random sampling (SRS), as described for equal-probability designs in the Sampling Theory I subject. The size measure must be known for every unit in the frame, for example last census crop area or the number of households.
In the cumulative total method, sizes are cumulated down the frame and a random number between 1 and the total X selects the unit whose cumulative range contains it. Lahiri's method avoids cumulation: choose a random unit i from 1 to N and a random number between 1 and M (where M is at least the largest size); accept unit i if the number does not exceed x_i, otherwise reject the pair and repeat. Both give selection probabilities p_i = x_i / X at each draw. With replacement, the same unit can be selected more than once, and draws are independent.
With n independent PPS draws with replacement, Hansen and Hurwitz (1943) estimated the population total by Y-hat_HH = (1/n) sum of y_i / p_i, the average of the n single-draw estimates y_i / p_i. It is unbiased, with variance V = (1/n) sum over the population of p_i (y_i / p_i - Y) squared, which is zero if y is exactly proportional to x. An unbiased variance estimator is v = sum of (y_i / p_i - Y-hat) squared / [n(n - 1)]. For example, with X = 200 and a unit of size 20 with y = 50, the single-draw estimate is 50 / 0.1 = 500; if a second draw gives 600, Y-hat = 550 and v = (50 squared + 50 squared)/(2 x 1) = 2500, a standard error of 50. The efficiency of PPS depends on how closely y is proportional to x: if y has a large intercept or is unrelated to x, PPS can be worse than SRS. A useful check from pilot data is to compare the variance of y_i / p_i with the variance of N y_i.
The best size measure is the variable most nearly proportional to the study variable: last census crop area for current crop production, livestock census numbers for milk production, population for household consumption. Outdated size measures still give unbiased estimates (the probabilities are known, even if not ideal), but efficiency falls as the size measure becomes less related to y. Units with zero size cannot be selected, so the size measure must be positive for all units in scope, or such units must be handled in a separate stratum.
With-replacement selection is rarely used for final samples, because duplicates waste effort, but its variance formula is simple and serves as a conservative approximation for without-replacement designs when the sampling fraction of PSUs is small. This is why many survey software packages compute variances 'with replacement' at the first stage.
For a population where y_i = beta x_i + e_i with small errors, PPS sampling estimates the total almost exactly, while SRS suffers from the full variation of y. When y has a large constant component unrelated to size, as with per-farm counts of household members, PPS by area can be less efficient than SRS. A pilot study or past data comparing the two variance formulas guides the choice.
Sampling without replacement avoids duplicate units and is usually more efficient than with replacement. Its general estimator is that of Horvitz and Thompson (1952): Y-hat_HT = sum over the sample of y_i / pi_i, where pi_i is the first-order inclusion probability of unit i, as introduced in Sampling Theory I. It is unbiased for any design with all pi_i positive. In a pi-PS design, inclusion probabilities are made proportional to size, pi_i = n x_i / X, which for n = 2 means pi_i = 2 p_i; such designs exist when no unit is so large that n x_i / X exceeds 1 (very large units are taken with certainty in a separate stratum).
The variance of the HT estimator is V = sum of [(1 - pi_i)/pi_i] y_i squared + sum over i not equal to j of [(pi_ij - pi_i pi_j)/(pi_i pi_j)] y_i y_j, which requires the second-order inclusion probabilities pi_ij. For fixed-size designs, the Sen-Yates-Grundy form V = sum over pairs i < j of (pi_i pi_j - pi_ij)(y_i/pi_i - y_j/pi_j) squared is equivalent, and its sample analogue v = sum over sample pairs of [(pi_i pi_j - pi_ij)/pi_ij](y_i/pi_i - y_j/pi_j) squared is unbiased and non-negative whenever pi_i pi_j >= pi_ij for all pairs (the Sen-Yates-Grundy condition). Computing the pi_ij is one of the practical difficulties of many unequal-probability schemes, which motivated the simpler schemes of chapter 3. The HT estimator can perform badly when y is not related to the inclusion probabilities, as in Basu's famous elephant example, where it gives absurd estimates; this is a reminder that the design should match the variable being estimated.
With n = 2 drawn from a population in which p1 = 0.1 and p2 = 0.2 are the relative sizes of the two sampled units, the inclusion probabilities in a pi-PS design are 0.2 and 0.4. If y1 = 50 and y2 = 120, the HT estimate of the total is 50/0.2 + 120/0.4 = 250 + 300 = 550. Its variance estimate requires pi_12, which depends on the selection scheme; for schemes such as Sampford's or Brewer's, formulas or approximations are available.
Because second-order probabilities are hard to compute, approximate variance estimators that use only first-order probabilities (for example the Hajek or Brewer approximations) are widely used, or the with-replacement formula is applied as a conservative approximation. Replication methods (chapter 11) also avoid explicit pi_ij.
In a pi-PS design, a unit whose size would give an inclusion probability above one is placed in a certainty stratum and always included, and the remaining sample is drawn from the other units. Large estates, cooperative farms or big markets are often treated this way, which removes their large contribution to variance.
When units are drawn PPS without replacement, one draw after another, the probabilities at later draws depend on earlier selections. Des Raj (1956) proposed an ordered estimator: for n = 2, t1 = y1/p1, t2 = y1 + y2 (1 - p1)/p2, and Y-hat = (t1 + t2)/2, which is unbiased with a simple unbiased variance estimator. Because it depends on the order of selection, it can be improved by averaging over the possible orders: Murthy's (1957) unordered estimator for n = 2 is Y-hat = [(1 - p2) y1/p1 + (1 - p1) y2/p2] / (2 - p1 - p2), which is more efficient than Des Raj's estimator.
In the Midzuno-Sen scheme, the first unit is drawn with probability proportional to size and the remaining n - 1 units by SRS without replacement from the rest. Under this scheme the probability of selecting a sample is proportional to its total size, which makes the ordinary ratio estimator unbiased, as noted in Sampling Theory I. The Rao-Hartley-Cochran (RHC, 1962) scheme randomly divides the population into n groups (of sizes as equal as possible), then draws one unit from each group with probability proportional to size within the group. The estimator is Y-hat = sum over groups of P_g y_i / p_i, where P_g is the sum of the p's in the group of the selected unit. It is unbiased, has a simple variance estimator and is always at least as efficient as PPS with replacement, which made it popular in large surveys where computing pi_ij for other schemes is impractical.
Two units are drawn in order with p1 = 0.1, y1 = 50 and p2 = 0.2, y2 = 120. Des Raj's estimates are t1 = 50/0.1 = 500 and t2 = 50 + 120 x 0.9/0.2 = 590, so Y-hat = 545. Murthy's estimator is [(0.8)(500) + (0.9)(600)]/(2 - 0.3) = (400 + 540)/1.7 = 553. The two differ because Des Raj depends on the order of selection; Murthy averages over orders and has smaller variance.
The RHC scheme is easy to implement (random grouping plus one PPS draw per group), its estimator and variance estimator are simple, it uses all the size information and it beats PPS with replacement. For these reasons it was used in Indian and other national surveys where frames were large and computing exact inclusion probabilities for other schemes was impractical.
For two units per stratum, schemes with exact and simple pi_ij (such as Durbin's or Brewer's methods) are convenient; for larger samples, RHC or systematic PPS from a randomly ordered list are practical. The choice balances efficiency, simplicity of selection, availability of exact variance estimators and the need for a fixed sample size.
In cluster sampling, the population is divided into groups (clusters) such as villages, households or segments, a sample of clusters is selected, and all units within selected clusters are observed. It saves travel and listing costs and is needed when no frame of individual units exists. With N clusters each of M units and an SRS of n clusters, the mean per unit is estimated by the mean of the sampled cluster means, and its variance is approximately [(1 - f)/(nM)] S squared [1 + (M - 1) rho], where S squared is the population variance among units and rho the intracluster correlation. The factor 1 + (M - 1) rho is the design effect: with clusters of 10 units and rho = 0.1, the design effect is 1.9, so the cluster sample is about half as efficient as an SRS of the same number of units. Because units in the same cluster tend to be alike (positive rho), many small clusters are usually better than a few large ones for the same total sample.
When clusters differ in size M_i, several estimators are available. The unbiased estimator of the total is (N/n) times the sum of sampled cluster totals, which is inefficient when cluster totals vary with size. The ratio estimator divides the total of sampled cluster totals by the total of sampled cluster sizes and multiplies by the known population size, giving a more stable estimate of the mean per unit with small bias. Selecting clusters with probability proportional to size and using the Hansen-Hurwitz estimator gives a self-weighting design for units. In agricultural surveys, villages are often selected with probability proportional to cultivated area or population for this reason.
Farms in the same village share soil, rainfall, markets and practices, so intracluster correlations for crop yield or input use are often substantial, making village clusters inefficient per farm compared with farms spread over many villages. Travel costs push the other way. The optimum cluster size balances these, using estimates of rho from earlier surveys.
A linear systematic sample is a single cluster chosen at random from k possible clusters, as noted in Sampling Theory I, which explains why it has no unbiased variance estimator from one sample and why its efficiency depends on the intraclass correlation within systematic samples.
In area-frame surveys, segments are clusters of fields. All fields in a selected segment are enumerated, so the segment total is the cluster total. Segment size is chosen to balance travel and enumeration costs against the intracluster correlation of crop areas, and segments are usually stratified by land use from satellite images before selection.
In two-stage sampling, a sample of first-stage units (FSUs, also called primary sampling units) is drawn, and within each a sample of second-stage units is selected, rather than enumerating them all. With N FSUs of M units each, an SRS of n FSUs and SRS of m units within each, the overall sample mean is unbiased and its variance is V = [(1 - f1)/n] S_b squared + [(1 - f2)/(nm)] S_w squared, where S_b squared is the variance between FSU means, S_w squared the variance within FSUs, f1 = n/N and f2 = m/M. When f1 is small, an unbiased variance estimator is simply s_b squared / n, computed from the sampled FSU means, because their variation includes the within-FSU sampling variation. This 'ultimate cluster' idea underlies variance estimation in most complex surveys.
With a cost function C = c1 n + c2 nm, the optimum number of second-stage units per FSU is approximately m = sqrt[c1 S_w squared / (c2 S_u squared)], where S_u squared = S_b squared - S_w squared / M is the component of variance between FSUs; the crop-cutting example of the crop estimation surveys subject applies exactly this. Multistage designs extend the idea to three or more stages (districts, villages, households, fields, plots), with probability-proportional-to-size selection at early stages and equal-probability selection later, often arranged to be self-weighting. Stratification at the first stage is almost universal in national surveys. Variance estimation for such designs uses the ultimate-cluster formula, linearisation or replication methods (chapter 11).
With S_b squared = 40 (variance among FSU means), S_w squared = 200 (within FSUs), n = 20 FSUs and m = 5 units per FSU, ignoring finite population corrections, V = 40/20 + 200/100 = 2 + 2 = 4. Doubling m to 10 reduces the second term to 1 and V to 3; doubling n to 40 halves both terms and gives V = 2. When between-FSU variation is substantial, adding FSUs is more effective than adding units within FSUs, subject to cost.
A design is self-weighting when every unit in the population has the same inclusion probability, so the unweighted sample mean is the estimate. Selecting FSUs with probability proportional to their size and then a fixed number of units within each FSU gives equal overall probabilities, which simplifies analysis and fieldwork (equal workloads per FSU). Outdated size measures make the design only approximately self-weighting, and weights then correct for the departures.
The crop-cutting design of villages, fields and plots is a three-stage design. Its variance has three components, between villages, between fields within villages and between plots within fields, and the optimum numbers at each stage follow from the components and costs. Because plots are usually one per field, the within-field component is merged with the field component in practice.
Ratio, regression and stratification gain precision from auxiliary information, but they require the population mean or distribution of the auxiliary variable. When this is unknown but cheap to observe, double sampling (two-phase sampling) takes a large first-phase sample of n' units in which only the auxiliary variable x is measured, and a smaller second-phase subsample of n units in which the expensive variable y is also measured. The first phase estimates what the population information would have provided.
The double-sampling regression estimator is y-bar_lr = y-bar + b(x-bar' - x-bar), where x-bar' is the first-phase mean and x-bar and y-bar are second-phase means; ignoring finite population corrections, its variance is approximately S_y squared (1 - rho squared)/n + rho squared S_y squared / n'. With S_y squared = 100, rho = 0.8, n = 50 and n' = 500, V = 100 x 0.36/50 + 0.64 x 100/500 = 0.72 + 0.128 = 0.848, compared with 2.0 for the second-phase mean alone. The ratio estimator (y-bar / x-bar) x-bar' is the analogue for proportional relations. Double sampling pays when x is much cheaper than y and highly correlated with it, for example eye estimates of yield or satellite indices against crop-cutting yields.
When stratum membership is unknown in the frame but easy to determine in the field, a large first-phase sample classifies units into strata, estimating stratum weights w_h = n'_h / n'; a second-phase subsample is then drawn within each stratum and the stratified estimator uses the estimated weights. The variance includes a term for the uncertainty of the weights. This design is used, for example, to stratify farms by size or irrigation status determined at a quick listing visit.
Double sampling is worthwhile when the gain from using x outweighs the cost of the first phase. With cost c' per first-phase unit and c per second-phase unit, the optimum allocation and the resulting variance can be compared with spending the whole budget on a single-phase sample of y. The gain is large when rho is high and c' is much smaller than c, as with remote-sensing indices (almost free per unit) against crop-cutting (expensive).
Eye estimates of yield on many fields with crop-cutting on a subsample, or satellite classifications of many segments with ground truth on a subsample, are two-phase designs. The regression estimator combines them, and its variance has the two components shown above: the second-phase residual variance and the first-phase variance of the auxiliary mean.
The variance estimator for a two-phase regression estimator combines the residual variance from the second phase and the variance of y estimated from the second-phase data scaled by the first-phase sample size. Replication methods that repeat both phases' estimation in each replicate are an alternative for complex two-phase designs.
Many agricultural surveys are repeated: monthly price surveys, annual crop surveys, periodic livestock surveys. The aims may be to estimate the current level, the change since the last occasion, or the average over occasions. Keeping the same units (a panel) maximises the correlation between occasions, which is best for estimating change; replacing all units gives independent samples, which is best for estimating the average over occasions; partial replacement, keeping some units (matched) and adding new ones (unmatched), is best for estimating the current level, because the matched part provides a regression estimate using the previous occasion and the unmatched part provides a fresh estimate.
Jessen (1942) gave the composite estimator for two occasions, combining the regression estimate from the matched part with the mean of the unmatched part, weighted inversely to their variances. Cochran showed that, for estimating the current mean, the optimum fraction of matched units is never more than one half, and that the gain over a fresh sample increases with the correlation between occasions. Rotation designs extend this to many occasions: each unit stays in the sample for a fixed number of occasions and then rotates out, so the sample is always partly matched and response burden is spread. The United States Current Population Survey's 4-8-4 rotation, in which a household is interviewed for four months, rests for eight and returns for four, is a well-known example. Rotation group bias (responses differing with the number of times a unit has been interviewed) is a non-sampling problem of such designs.
For the change between two occasions with the same sample on both, the variance of the difference of means is (S1 squared + S2 squared - 2 rho S1 S2)/n, which is much smaller than the variance with independent samples, 2S squared / n with equal variances, when rho is high: with rho = 0.8 the panel variance is 0.4 S squared / n, one fifth of the independent-sample variance. This is why price and stock surveys that focus on change use panels.
Panels suffer from attrition (units dropping out over time) and conditioning (responses changing because of repeated interviewing). Rotation limits both, and weighting adjusts for attrition under MAR assumptions. Agricultural panels of farm households, such as those used for monitoring incomes or technology adoption, need careful tracking of households that split or move.
Questions on sensitive topics (illegal pesticide use, unreported income, theft of produce) produce evasive or false answers. Warner (1965) proposed randomised response: each respondent privately uses a randomising device that, with probability p, presents the statement 'I belong to group A' and, with probability 1 - p, the statement 'I do not belong to group A', and answers yes or no truthfully without revealing which statement was presented. The probability of a yes is lambda = p pi + (1 - p)(1 - pi), where pi is the proportion in group A, so pi-hat = (lambda-hat - (1 - p))/(2p - 1), provided p is not 0.5. With p = 0.7 and 40% yes answers, pi-hat = (0.4 - 0.3)/0.4 = 0.25. The variance is pi(1 - pi)/n + p(1 - p)/[n(2p - 1) squared]: the second term is the price of privacy.
In the unrelated-question model (Greenberg and colleagues, 1969), the device presents either the sensitive question (with probability p) or an innocuous question with known yes-proportion pi_Y (such as 'Were you born in January to June?'). Then lambda = p pi + (1 - p) pi_Y and pi-hat = [lambda-hat - (1 - p) pi_Y]/p; with p = 0.8, pi_Y = 0.5 and lambda-hat = 0.3, pi-hat = 0.25. This model is usually more efficient than Warner's and easier for respondents to trust. Quantitative randomised response and list (item-count) techniques extend the idea. Randomised response works only if respondents understand and trust the procedure, and its variance is much larger than that of direct questioning, so larger samples are needed.
With n = 400 and pi = 0.25, direct questioning (if answers were truthful) has variance 0.25 x 0.75 / 400 = 0.00047. Warner's design with p = 0.7 adds p(1 - p)/[n(2p - 1) squared] = 0.21/(400 x 0.16) = 0.0033, making the total about eight times larger, so roughly eight times the sample is needed for the same precision. The unrelated-question design with p = 0.8 and pi_Y = 0.5 is considerably more efficient. The trade-off is privacy and honesty against sample size.
Randomised response has been used to estimate the prevalence of banned pesticide use, illegal water extraction, unreported sales, and veterinary drug misuse. In each case the gain depends on whether direct questioning would have produced serious under-reporting, which can sometimes be checked against external data.
The estimators assume that respondents follow the instructions and answer the presented question truthfully. If some respondents answer no regardless (self-protective behaviour), estimates are biased downward. Designs with cheating detection and comparisons with direct-question estimates help assess this.
Non-response occurs when selected units cannot be contacted, refuse or are unable to answer (unit non-response) or answer only some questions (item non-response). If non-respondents differ from respondents, estimates from respondents alone are biased, and the bias does not shrink with sample size. Rubin's classification guides treatment: data are missing completely at random (MCAR) if missingness is unrelated to anything; missing at random (MAR) if it depends only on observed variables; and missing not at random (MNAR) if it depends on the unobserved values themselves. Weighting and imputation correct bias under MAR, not under MNAR.
Hansen and Hurwitz (1946) proposed a two-phase approach for mail surveys: questionnaires are mailed to n units, n1 respond; from the n2 non-respondents a subsample of n2/k is interviewed personally, and the estimator is y-bar = (n1 y-bar_1 + n2 y-bar_2')/n, where y-bar_2' is the mean of the interviewed non-respondents. For example, with 500 mailed, 300 responding with mean 40 and 200 non-respondents of whom one in four (50) are interviewed with mean 30, y-bar = (300 x 40 + 200 x 30)/500 = 36; the respondents-only mean of 40 would have been biased upward. The variance includes an extra term for subsampling non-respondents, and the optimum k balances interview cost against precision.
Weighting-class adjustment divides the sample into classes with similar response propensities (by region, farm size, crop) and inflates the weights of respondents in each class by the inverse of the class response rate. Post-stratification and raking (iterative proportional fitting) adjust weights so that weighted totals match known population totals for several variables, and calibration estimators generalise this (small area estimation subject). Response-propensity models estimate each unit's probability of responding by logistic regression and weight by its inverse. All these remove bias only to the extent that the adjustment variables explain both non-response and the survey variable.
Large farms and absentee owners are often harder to reach, and refusal rates can differ by region and farm type. Because these characteristics relate to production, non-response can bias totals. Using auxiliary information known for all sampled units (frame size measures, census data, previous survey responses) in weighting classes or response-propensity models is the standard remedy, and response rates by class should be reported.
Prevention is better than correction: advance letters, local contacts, convenient timing (avoiding harvest peaks), short questionnaires, repeated call-backs and trained enumerators raise response rates. In agricultural censuses and surveys, legal obligations to respond exist in some countries. Monitoring response rates during fieldwork allows corrective action before data collection ends.
A high response rate does not guarantee small bias, and a low rate does not guarantee large bias: bias depends on how different non-respondents are. Comparing early and late respondents, or respondents with frame information on non-respondents, gives indirect evidence about non-response bias and should accompany published estimates.
Imputation fills in missing item values so that complete-data methods can be used. Mean imputation replaces missing values with the respondent mean (overall or within classes); it preserves the mean under MCAR but distorts distributions and understates variances. Hot-deck imputation replaces a missing value with an observed value from a similar respondent (a donor) in the same survey, chosen randomly within imputation classes or by nearest neighbour; it preserves distributions better. Cold-deck imputation uses values from another source such as a previous survey. Regression imputation predicts missing values from a model on observed covariates, optionally adding a random residual (stochastic regression imputation) to preserve variability. Ratio imputation, using the ratio of current to previous values, is common in repeated agricultural surveys.
Treating imputed values as real understates standard errors, because imputed values carry no new information about variability. Corrections include specific variance estimators for imputed data and replication methods that repeat the imputation in each replicate. Rubin's multiple imputation creates m completed data sets with different random imputations, analyses each, and combines the results: the point estimate is the average, and the total variance is T = W-bar + (1 + 1/m) B, where W-bar is the average within-imputation variance and B the variance between the m estimates. With m = 5, W-bar = 4 and B = 1, T = 4 + 1.2 = 5.2. Multiple imputation reflects imputation uncertainty and is widely used in survey and research data.
For item non-response in repeated agricultural surveys, ratio imputation using the unit's previous value adjusted by the change among respondents is natural; for cross-sectional surveys, hot-deck within classes defined by region and farm size is common. Imputation flags should be kept in the data so that analysts can see which values were imputed and assess sensitivity.
Five to twenty imputations are typical. The fraction of missing information, estimated from the ratio of between-imputation variance to total variance, shows how much the missing data affect each estimate. Multiple imputation assumes the imputation model includes the variables used in later analyses; otherwise relationships are biased towards zero.
In a survey with 10% item non-response, treating hot-deck imputed values as observed can understate variances noticeably for the affected items. Replication-based variance estimation that re-imputes within each replicate, or multiple imputation, gives more honest standard errors, especially for estimates of totals in small domains where imputed values form a large share of the data.
National agricultural and household surveys combine stratification, clustering, multistage selection, unequal probabilities and weighting. Standard errors computed as if the data were an SRS are usually too small. The design effect, deff = V(complex design) / V(SRS of the same size), summarises the combined effect; the effective sample size is n / deff. A survey of 1200 farms with deff = 2 has the precision of an SRS of 600. Design effects vary by variable and domain and are larger for variables that cluster strongly within PSUs.
Taylor-series linearisation approximates the variance of a non-linear estimator (a ratio, a regression coefficient, a proportion of a subgroup) by the variance of a linear approximation. For a ratio R-hat = y-hat / x-hat, the linearised variable is (y_i - R-hat x_i)/x-hat, whose design-based variance, computed with the ultimate-cluster formula from PSU totals within strata, estimates the variance of R-hat. Linearisation is the default in most survey software and is accurate for large samples.
Replication methods form many replicate estimates from subsets or reweighted versions of the sample and compute the variance from their spread. The jackknife deletes one PSU at a time (with re-weighting of the others in its stratum); balanced repeated replication (BRR) suits designs with two PSUs per stratum, forming half-samples balanced by a Hadamard matrix; the bootstrap for surveys resamples PSUs within strata (with rescaling, as in the Rao-Wu bootstrap). Replication handles non-linear statistics (medians, quantiles, complex models) and can incorporate weighting adjustments by repeating them in each replicate. Public-use survey files often supply replicate weights so that users can compute correct variances without access to design details.
A two-stage survey of 1200 farms in 120 villages (10 farms per village) has an intracluster correlation for fertiliser use of 0.12. The design effect from clustering alone is approximately 1 + 9 x 0.12 = 2.08, so the effective sample size is about 577. If unequal weights add a further factor of 1.1, the overall design effect is about 2.3. Planning sample sizes with design effects from earlier rounds of the survey avoids disappointing precision.
Linearisation is fast and standard for means, totals, ratios and regression coefficients. Replication is preferred for non-smooth statistics (medians, percentiles), for estimators involving complex weighting adjustments and for distributing public-use files. All methods require the design information (strata, PSUs, weights) to be present in the data file; analyses that ignore it understate standard errors.
Estimates for subgroups (domains) such as small farms or a particular district have smaller effective samples and often larger design effects. When domains cut across strata and PSUs, the domain size is random and ratio-type estimators are used; their variances are computed by linearisation or replication like any other ratio.
When an attribute is rare, a fixed sample may contain few or no units with it, giving an estimate of zero and useless precision. In inverse sampling, units are drawn until a predetermined number m of units with the attribute is found; the sample size n is random. Haldane showed that the unbiased estimator of the proportion is P-hat = (m - 1)/(n - 1): if 10 infected plants are found after inspecting 200, P-hat = 9/199 = 0.045. Inverse sampling guarantees a minimum relative precision for rare attributes, useful for disease prevalence and seed-lot contamination.
For rare, clustered populations such as pest outbreaks, weed patches or rare plants, adaptive cluster sampling (Thompson, 1990) starts with an initial probability sample of units; whenever a sampled unit satisfies a condition (for example containing at least one pest), its neighbours are added, and the process continues until no new units satisfy the condition. The final sample concentrates effort where the population is. Because inclusion probabilities depend on the population values, ordinary estimators are biased; modified Horvitz-Thompson and Hansen-Hurwitz estimators based on the networks of units are unbiased. Adaptive designs can be much more efficient than conventional designs for highly clustered rare populations but give less control over final sample size and cost.
In testing seed lots for contamination by off-types or weed seeds, inspection can continue until a fixed number of contaminants is found, giving an estimate with controlled relative precision. If none or very few are found within a maximum sample, a fixed-sample upper confidence bound is reported instead. The same logic applies to disease surveillance where cases are rare.
Surveys of aphid colonies, locust swarms or invasive weeds find most individuals in a few patches. An initial sample of grid cells, expanded around cells where the pest is found, captures the patches efficiently. Practical rules cap the total sample size and define the neighbourhood and condition in advance, and the network-based estimators must be used for unbiased totals.
Students meet these methods in derivation-heavy examinations and in theses based on survey data, where weights, clustering and non-response must be handled correctly. They must compute HH and HT estimates and variances, design effects and two-stage variances, and explain when each design or estimator is appropriate.
A five-year survey statistician designs multistage samples with PPS selection, computes weights and non-response adjustments, imputes missing items, and estimates standard errors with linearisation or replicate weights. They advise on rotation for repeated surveys, use double sampling with satellite or eye-estimate data, and occasionally design randomised-response modules for sensitive items.
A senior survey methodologist sets the architecture of national surveys: frames, stages, selection probabilities, rotation patterns, weighting and imputation rules, variance estimation and quality reporting. They decide when to adopt new methods such as adaptive sampling for pests or calibration to administrative totals.
A PhD with long experience develops sampling theory and methods (new unequal-probability schemes, variance estimators, non-response models), teaches them and advises statistical offices on methodology.
Variance formulas are given in their standard forms without full proofs; second-order inclusion probabilities for specific PPS schemes, the full variance of Hansen-Hurwitz call-back estimators and Jessen's composite estimator are described, not derived. Small area estimation and calibration are in their own subject. Adaptive and inverse sampling are introduced only. The rotation pattern example is illustrative.
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. Every numerical answer was computed with a script; the values needed are stated in the question.
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