Agricultural Statistics & Data Science · Design and analysis of experiments · also Response Surface Methodology, Optimal Designs
Finding the best dose or combination: first- and second-order response surfaces, central composite and Box-Behnken designs, rotatability, fertiliser response optimisation, mixture designs for blends and feed formulations, and the alphabetic optimality criteria.
In the exams: Not an explicit unit in AICE-JRF Ag Stat; standard MSc/PhD course in agricultural statistics programmes; ASRB NET Ag Stat (reported)
Response surface methodology asks a different question from most of this track's design subjects: not 'which treatments differ?' but 'which combination of quantitative factor levels gives the best response, and what does the response look like near it?'. This subject covers first- and second-order response surface models, central composite and Box-Behnken designs, rotatability and orthogonal blocking, steepest ascent and canonical analysis of the fitted surface, fertiliser response functions and the economic optimum dose, mixture experiments with simplex lattice and simplex centroid designs, the alphabetic optimality criteria (A, D, E, G), computer-generated optimal designs, and an introduction to Taguchi's robust parameter design.
It builds on the basic and factorial design subjects (2^k factorials and fractions are the starting point of RSM designs) and on the linear models subject (all response surfaces are fitted by least squares and checked by residual analysis and lack-of-fit tests). The economic side of the fertiliser optimum is treated in the Agribusiness track's production economics subject, and robustness across environments in the multi-environment trials subject.
Response surface designs are not an explicit unit of the ICAR AICE-JRF/SRF Agricultural Statistics syllabus but are a standard MSc and PhD course in agricultural statistics and are drawn on in ASRB NET and ARS papers and interviews (as reported). Typical questions test the number of runs and parameters, the rotatable alpha, the location and nature of a stationary point, the economic optimum dose, the number of points in mixture designs, and the meaning of the optimality criteria.
Response surface methodology (RSM), developed by Box and Wilson (1951) for chemical processes, studies how a response depends on several quantitative factors and seeks the combination of factor levels that gives the best response. In agriculture the factors may be fertiliser rates, irrigation amounts, plant density, temperature and time in a drying process, or the proportions of ingredients in a feed. Unlike the factorial experiments of the factorial subject, whose aim is usually to test effects and interactions, RSM fits a smooth polynomial approximation to the response and uses it to locate an optimum or to describe the shape of the response near it. Factors are expressed in coded units, x = (X - centre) / half-range, so that the experimental region is centred at zero and the factorial levels are -1 and +1. Coding makes coefficients comparable, reduces correlation between linear and quadratic terms and simplifies the design formulas.
A first-order model, y = b0 + b1 x1 + ... + bk xk + e, is a plane. It is adequate far from the optimum, where the response changes roughly linearly, and is used to find the direction of improvement. A first-order model with interaction terms adds bij xi xj. A second-order (quadratic) model adds the squared terms bii xi squared and all two-factor interactions: it can represent a maximum, a minimum, a saddle or a ridge, and is used near the optimum. A full second-order model in k factors has (k + 1)(k + 2)/2 parameters: 6 for two factors, 10 for three and 15 for four. Both models are linear models in the sense of the linear models subject: they are fitted by least squares, and their adequacy is checked by residual analysis and, when the design contains replicated runs (usually centre points), by a lack-of-fit test comparing the residual mean square with pure error. A significant lack of fit means the polynomial is too simple for the region studied: the region may be too large, a higher-order term or a transformation may be needed.
First-order designs need two levels per factor. The 2^k factorial and its fractions are orthogonal and give minimum-variance estimates of the linear effects; Plackett-Burman designs screen many factors in few runs; simplex designs use k + 1 runs. Adding centre points (replicated runs at x = 0) gives a pure-error estimate and a test of curvature: if the mean response at the centre differs significantly from the mean at the factorial points, the surface is curved and a second-order model is needed. Centre points do not change the estimates of the linear effects in an orthogonal design.
A drying experiment studies temperature from 50 to 70 degrees Celsius and time from 4 to 8 hours. The centres are 60 degrees and 6 hours and the half-ranges 10 degrees and 2 hours, so a run at 65 degrees for 7 hours has coded values x1 = (65 - 60)/10 = 0.5 and x2 = (7 - 6)/2 = 0.5. A fitted coefficient of 3.0 for x1 means that moving from 60 to 70 degrees (one coded unit) raises the response by 3.0 units at the centre of time, which is directly comparable with the coefficient for x2 even though the natural units differ. Results are converted back to natural units only at the end, for reporting the optimum.
Suppose a 2^2 factorial gives a mean response of 40.0 at the four factorial points and five centre points give a mean of 44.0 with a standard deviation of 1.5. The curvature contrast is 44.0 - 40.0 = 4.0, with standard error 1.5 x square root of (1/4 + 1/5) = 1.01, so t = 4.0/1.01, about 4.0 on 4 degrees of freedom, clear evidence of curvature. A first-order model would be inadequate, and axial points should be added to fit a second-order model. Note that the curvature test detects the presence of quadratic effects but cannot say which factor is responsible; that requires the axial points.
The ANOVA of a fitted second-order model splits the regression sum of squares into linear, quadratic and interaction parts (each tested against the residual) and splits the residual into lack of fit and pure error. A typical sequence is to check lack of fit first; if it is not significant, the residual mean square is used to test the model terms and to compute standard errors. The coefficient of determination of a response surface should be read with the number of runs in mind: with 13 runs and 6 parameters, R squared is easily high. Adjusted R squared and the prediction R squared based on PRESS are more informative.
The central composite design (CCD) of Box and Wilson is the standard second-order design. It combines three parts: the 2^k factorial (or a resolution V fraction) with points at plus or minus 1; 2k axial (star) points at plus or minus alpha on each axis with the other factors at zero; and n0 centre points. The total number of runs is 2^k + 2k + n0 when the full factorial is used: for k = 2 with five centre points, 4 + 4 + 5 = 13; for k = 3 with six centre points, 8 + 6 + 6 = 20. The CCD is built sequentially in a natural way: a first-order factorial with centre points is run first and, if curvature is found, the axial points are added. With alpha different from 1, each factor takes five levels (-alpha, -1, 0, 1, alpha). The face-centred CCD uses alpha = 1, giving only three levels, which is convenient when factor levels are hard to set or the cube is the region of interest, at the cost of rotatability.
Box and Behnken (1960) proposed three-level second-order designs formed by combining two-level factorials with incomplete block designs: each block of the incomplete block design names the factors set at plus or minus 1 in a 2^2 (or larger) factorial, with the remaining factors at zero, and centre points are added. For three factors the design has 12 runs at the midpoints of the edges of the cube plus centre points, typically 15 runs in all; for four factors, 24 runs plus centre points. Box-Behnken designs have no runs at the corners of the cube, which helps when extreme combinations of all factors at once are impossible, unsafe or costly (for example, highest temperature with longest time). They are rotatable or nearly so for some k and require fewer runs than CCDs for three factors.
The 3^k factorial also supports a second-order model but needs 27 runs for three factors and 81 for four, many more than necessary. Hybrid designs, small composite designs and Doehlert designs are alternatives with particular properties. When the region is irregular or the number of runs is fixed by resources, computer-generated optimal designs (chapter 8) are used.
For four factors, a full second-order model has 15 parameters. A CCD with the full 2^4 factorial, 8 axial points and 6 centre points has 30 runs, leaving 15 residual degrees of freedom, of which 5 are pure error from the centre points and 10 are for lack of fit. A CCD based on a half-fraction 2^(4-1) has only 8 factorial points, but a 2^(4-1) of resolution IV aliases two-factor interactions with each other, which is not acceptable for a second-order model; a resolution V fraction is needed for interactions to be estimated cleanly, which for four factors means the full factorial. For five factors, the resolution V half-fraction 2^(5-1) with 16 points, 10 axial points and centre points gives a CCD of about 32 runs for 21 parameters.
The CCD is preferred when sequential experimentation is planned (first-order design first, axial points later), when five levels per factor are acceptable and when the corners of the region are feasible. The Box-Behnken design is preferred when only three levels can be set, when extreme corners must be avoided and, for three or four factors, when fewer runs are wanted. Both support the full second-order model and both should include several centre points.
Centre points provide pure error at one location only. When variability may change across the region, or when the experiment is run in the field with large plot-to-plot variation, replicating the whole design (or selected points) in blocks gives a better error estimate and protects the analysis against non-constant variance.
A design is rotatable if the variance of the predicted response depends only on the distance of the point from the design centre, not on the direction. Rotatability means that the precision of predictions is the same in all directions, which is attractive when the location of the optimum is unknown. A CCD is rotatable when alpha equals the fourth root of the number of factorial points F: alpha = F^(1/4), giving 1.414 for k = 2 (F = 4), 1.682 for k = 3 (F = 8) and 2.0 for k = 4 (F = 16). With a half-fraction of a 2^5 (F = 16), alpha = 2.0 as well. The number of centre points affects how the prediction variance behaves near the centre. Box and Hunter suggested numbers of centre points giving uniform precision, so that the variance is roughly constant within a unit distance of the centre; in practice three to six centre points are common. Exact rotatability is less important than good overall prediction properties, and near-rotatable designs are acceptable.
When all runs cannot be made under homogeneous conditions (different days, batches, fields), the design is divided into blocks. A design blocks orthogonally if the block effects are orthogonal to the model coefficients, so that block differences do not bias the estimates of the polynomial. A CCD blocks naturally into a factorial block (factorial points plus some centre points) and an axial block (axial points plus some centre points); orthogonal blocking requires particular choices of alpha and the numbers of centre points in each block. For k = 2, for example, alpha = 1.414 with three centre points in each block gives both rotatability and orthogonal blocking. The factorial block can be further split into fractions when it is large. Sequential experimentation, running the factorial block first and the axial block later, fits naturally with blocking.
With too few centre points, the prediction variance of a rotatable CCD is larger at the centre than at a distance of one coded unit, which is undesirable because the optimum is often expected near the centre. With uniform-precision numbers of centre points (for example five for k = 2, six for k = 3), the variance is approximately constant within the unit sphere. Variance dispersion graphs, which plot the maximum, minimum and average prediction variance against distance from the centre, are used to compare designs, including non-rotatable ones.
In field experiments a CCD can be run over two seasons, the factorial block in the first and the axial block in the second, with centre points in both. Orthogonal blocking ensures that the season difference does not bias the estimates of the polynomial, although any interaction between season and the factor effects remains unmodelled and may inflate the residual. If interactions with season are likely, the whole design should be replicated in each season instead.
The rotatable alpha is not the only sensible choice. Alpha = 1 (face-centred) keeps all runs inside the cube when the cube is the region of interest or when levels outside it are impossible. Alpha equal to the square root of k puts the axial points on the same sphere as the factorial points (a spherical design), suited to spherical regions. Alpha values chosen for orthogonal blocking may differ slightly from the rotatable value. The choice depends on the region of interest, the feasibility of extreme levels and the blocking needs, more than on exact rotatability.
When the current region is far from the optimum, a first-order model is fitted and the experimenter moves along the path of steepest ascent (or descent, for minimising), the direction of the gradient, proportional to the estimated first-order coefficients (b1, b2, ..., bk) in coded units. If b1 = 3 and b2 = 1.5, then for each unit step in x1 the path moves 0.5 units in x2. Runs are made at successive points along the path until the response stops improving; a new first-order design is then centred near the best point, and the process repeats until curvature appears, when a second-order design is used. The path depends on the coding (the scales of the factors), so it is a direction of steepest ascent in coded units. Practical constraints often limit how far one can move, and in field research, where each run is a season, steepest ascent is used less than in laboratory and process work.
For a fitted second-order model y = b0 + x'b + x'Bx, where b is the vector of linear coefficients and B the symmetric matrix with the quadratic coefficients on the diagonal and half the interaction coefficients off the diagonal, the stationary point is x_s = -(1/2) B^(-1) b, and the predicted response there is y_s = b0 + (1/2) x_s'b. For one factor this reduces to x = -b1 / (2 b11). For example, y = 80 + 4x1 + 2x2 - 2x1 squared - x2 squared has its stationary point at x1 = 1, x2 = 1, with predicted response 80 + (4 + 2)/2 = 83. With an interaction, y = 50 + 2x1 + 3x2 - x1 squared - 2x2 squared + x1x2 gives B with diagonal -1 and -2 and off-diagonal 0.5, and x_s = (1.57, 1.14), with predicted response 53.3.
Canonical analysis rotates the axes to the principal axes of the fitted surface, writing it as y = y_s + lambda_1 w1 squared + ... + lambda_k wk squared, where the lambda_i are the eigenvalues of B and the w_i are coordinates along its eigenvectors. If all eigenvalues are negative, the stationary point is a maximum; if all are positive, a minimum; if they have mixed signs, a saddle point (minimax). If one or more eigenvalues are close to zero, the surface has a ridge: a line or plane along which the response changes little, so there are many near-optimal combinations, which is useful because the cheapest or most convenient combination on the ridge can be chosen. In the example with interaction above, the eigenvalues are about -0.79 and -2.21, both negative, so the stationary point is a maximum. A stationary point far outside the experimental region should not be trusted; the polynomial is only an approximation within the region.
Suppose canonical analysis of a two-factor fertiliser surface gives eigenvalues -2.5 and -0.05. The second eigenvalue is close to zero, so the surface is a rising or stationary ridge: along the direction of its eigenvector the response hardly changes. Agronomically, many combinations of the two nutrients give nearly the same maximum yield, and the cheapest combination along the ridge can be recommended. Ridge analysis, which finds the best response on spheres of increasing radius around the centre, is a related tool when the stationary point lies outside the region.
The estimated stationary point is a non-linear function of the coefficients and has sampling uncertainty that can be large, especially when the surface is flat. Confidence regions for the location of the optimum can be computed (for example by the method of Box and Hunter or by bootstrapping), and they often show that a broad region of factor combinations is statistically indistinguishable from the estimated optimum. Reporting this region is more honest than reporting a single point to several decimal places.
Because the path of steepest ascent is defined in coded units, it must be translated back to natural units to run the experiments. If x1 is nitrogen coded with a half-range of 40 kg/ha and x2 is plant density coded with a half-range of 10,000 plants/ha, then a step of 1 coded unit in x1 and 0.5 in x2 corresponds to adding 40 kg/ha of nitrogen and 5,000 plants/ha. Step sizes are chosen so that the response is likely to change detectably between successive runs, and runs continue until two successive points show a decline.
Fertiliser trials relate yield to the rate of a nutrient. Common functional forms are the quadratic y = a + bx - cx squared, the square-root model y = a + b root(x) + cx, the Mitscherlich (exponential) model y = A[1 - exp(-c(x + b))] with an asymptote A, and linear-plateau and quadratic-plateau models in which yield rises to a plateau and stays there. Different forms fitted to the same data can give similar fits within the range of rates but quite different estimated optima; quadratic models tend to overestimate the optimum when the true response has a sharp plateau, and linear-plateau models tend to underestimate it. The choice should be guided by agronomy, residuals and, where possible, by results over many sites.
For the quadratic y = a + bx - cx squared, the rate giving maximum yield is x_max = b / (2c). The economic optimum is the rate at which the value of the last unit of fertiliser equals its cost: the marginal physical product dy/dx = b - 2cx equals the price ratio r = (price of fertiliser per kg) / (price of produce per kg), giving x_opt = (b - r) / (2c), which is always below x_max when fertiliser has a positive price. For example, with y = 2000 + 30x - 0.1x squared (kg/ha, x in kg N/ha), x_max = 30/0.2 = 150 kg N/ha with predicted yield 4,250 kg/ha; if nitrogen costs 15 per kg and grain is worth 10 per kg, r = 1.5 and x_opt = (30 - 1.5)/0.2 = 142.5 kg N/ha. This is the factor-product relationship of production economics (marginal value product equals marginal factor cost), treated from the economic side in the Agribusiness track. With several nutrients, a second-order response surface in N, P and K is fitted and the economic optimum found by setting each partial derivative equal to its price ratio. Because response varies between sites and years, recommendations based on pooled response functions, or on site-specific models with soil-test covariates, are more robust than single-trial optima.
Suppose two sites give quadratic responses y = 1800 + 28x - 0.10x squared and y = 2200 + 32x - 0.08x squared. With a price ratio of 1.5, the economic optima are (28 - 1.5)/0.20 = 132.5 and (32 - 1.5)/0.16 = 190.6 kg N/ha, quite different. A recommendation based on the average response function, or on a model that relates the response parameters to soil tests, is more useful than either single-site value. The flatness of the profit function near its maximum also matters: being 20 kg/ha away from the optimum often costs little, so a rounded recommendation is reasonable.
In a linear-plateau model, y = a + bx for x below a critical rate x0 and y = a + bx0 above it; the critical rate is estimated by non-linear least squares. Linear-plateau models often fit cereal nitrogen responses well and give lower optimum rates than quadratics. Because the economic optimum of a linear-plateau model is at the join point whenever the price ratio is below the slope b, the estimated x0 is the recommendation. Comparing models by residual sum of squares and by agronomic plausibility is good practice.
The square-root model y = a + b root(x) + cx, with c negative, rises steeply at low rates and flattens more gradually than a quadratic. Its maximum is at root(x) = -b/(2c), and its economic optimum satisfies b/(2 root(x)) + c = r. It has been used widely for nitrogen and phosphorus responses because it fits the typical diminishing-returns shape with few parameters and remains a linear model that can be fitted by ordinary least squares.
In a mixture experiment the factors are proportions of components that sum to one: x1 + x2 + ... + xq = 1 with each xi >= 0. The response depends on the proportions, not on the total amount. Examples are feed rations, fertiliser blends, potting media, food formulations and pesticide combinations. Because of the constraint, the experimental region is a simplex (a triangle for three components, a tetrahedron for four), and ordinary factorial designs and polynomials cannot be used directly.
Scheffe (1958) introduced the simplex lattice design {q, m}, whose points have proportions 0, 1/m, 2/m, ..., 1 for each component, in all combinations that sum to one; it has C(q + m - 1, m) points. The {3, 2} lattice has 6 points (the three pure components and the three 50:50 binary blends); the {3, 3} lattice has 10. The simplex centroid design (Scheffe, 1963) has 2^q - 1 points: the q pure components, all binary 50:50 blends, all ternary equal blends, and so on up to the overall centroid; for q = 3 it has 7 points. Augmented designs add interior points (such as axial blends) to check model adequacy.
Mixture models omit the intercept and squared terms because of the constraint. The Scheffe canonical linear model is y = sum of beta_i xi; the quadratic model is y = sum of beta_i xi + sum over i < j of beta_ij xi xj, with q(q + 1)/2 terms (6 for q = 3); the special cubic adds beta_123 x1 x2 x3. In the quadratic model, beta_i is the expected response to the pure component i, and a positive beta_ij indicates synergism (the blend does better than the average of its parts), a negative one antagonism. When components have lower and upper bounds (for example a feed must contain at least 20% and at most 60% maize), the region is an irregular polytope; extreme vertices designs (McLean and Anderson, 1966) or computer-generated designs are then used. Mixture-amount experiments combine proportions with the total amount applied.
A three-component feed blend is studied with a simplex centroid design of seven points. Suppose the fitted quadratic Scheffe model is y = 10 x1 + 12 x2 + 8 x3 + 6 x1x2 - 4 x1x3 + 0 x2x3. The pure components give expected responses 10, 12 and 8. The 50:50 blend of components 1 and 2 gives 10(0.5) + 12(0.5) + 6(0.25) = 12.5, higher than either pure component's average of 11, showing synergism; the 50:50 blend of 1 and 3 gives 9 - 1 = 8, below the average of 9, showing antagonism. The optimum blend is found by maximising the fitted model on the simplex, usually with constraints on the proportions.
Because x1 + x2 + x3 = 1, a column of ones in the model matrix is the sum of the component columns, so an intercept is redundant; likewise x1 squared = x1(1 - x2 - x3), so squared terms are absorbed into linear and cross-product terms. The Scheffe forms remove this redundancy, and their coefficients have direct meaning in terms of pure components and blending effects. Fitting an ordinary polynomial with an intercept to mixture data gives a singular model matrix or coefficients that cannot be interpreted.
When the total amount of a blend matters as well as its composition, as with a fertiliser blend applied at different total rates, mixture-amount designs cross a mixture design with levels of the total amount, and the model combines Scheffe terms with amount terms and their interactions. When process variables such as temperature are also varied, mixture-process designs cross a mixture design with a factorial in the process variables. These designs grow quickly in size, which is another reason for computer-generated designs.
Optimal design theory, developed by Kiefer and others, chooses design points to optimise a criterion based on the information matrix X'X of the intended model. D-optimality maximises the determinant of X'X, which minimises the generalised variance of the parameter estimates and the volume of their joint confidence ellipsoid. A-optimality minimises the trace of (X'X)^(-1), the average variance of the parameter estimates. E-optimality maximises the smallest eigenvalue of X'X, minimising the variance of the worst-estimated linear combination of parameters. G-optimality minimises the maximum variance of the predicted response over the region, and I-optimality (also called V- or IV-optimality) minimises the average prediction variance over the region. The Kiefer-Wolfowitz equivalence theorem shows that, for continuous (approximate) designs, D-optimality and G-optimality are equivalent, and the maximum standardised prediction variance of a D-optimal design equals the number of parameters p. For straight-line regression on the interval from -1 to 1 with four runs, putting two runs at each end gives X'X = diag(4, 4) with determinant 16, whereas runs at -1, 0, 0 and 1 give diag(4, 2) with determinant 8: the end-point design is D-optimal, though it gives no check of curvature.
The D-efficiency of a design compares its determinant with that of the D-optimal design, scaled by the number of parameters: (det of X'X for the design / det for the optimum)^(1/p). Designs for estimating effects precisely favour D- or A-optimality; designs for prediction and optimisation favour I- or G-optimality. Optimal designs depend on the assumed model, and a design optimal for a quadratic model may leave no degrees of freedom to test lack of fit. Good practice adds centre points or extra distinct points so that model adequacy can be checked.
For a quadratic model in one factor on the interval from -1 to 1, the D-optimal continuous design puts equal weight on -1, 0 and 1. With six runs, two at each of these points is D-optimal; a design with six equally spaced points is less efficient for estimating the quadratic coefficients but gives more information about lack of fit. This illustrates the general tension between optimality for an assumed model and robustness to model inadequacy, which practitioners resolve by adding a few extra distinct points to an optimal design.
The joint confidence region for the parameters is an ellipsoid whose volume is proportional to the square root of the determinant of (X'X)^(-1). D-optimality minimises this volume; A-optimality minimises the sum of the squared semi-axis lengths (the average variance); E-optimality minimises the longest semi-axis. A design can be good on one criterion and poor on another, so the criterion should reflect the purpose of the experiment.
When the purpose of the experiment is to predict the response well over the whole region, as in optimisation, I-optimal designs, which minimise the average prediction variance, are often preferred to D-optimal ones. For second-order models, I-optimal designs typically place more runs near the centre than D-optimal designs, which concentrate runs at the edges of the region. Software usually offers both criteria, and comparing the two designs' prediction-variance profiles helps choose.
Standard designs assume a regular region (a cube or sphere), quantitative factors and a flexible number of runs. Real problems often break these assumptions: some factor combinations are infeasible, factors are a mix of quantitative and qualitative (a variety factor with three levels alongside two fertiliser rates), the number of runs is fixed by resources, some runs are already done, or the model of interest is non-standard. Computer-generated designs handle these cases by searching over a set of candidate points for the subset that optimises a chosen criterion.
Fedorov's exchange algorithm starts from a random design of the required size and repeatedly exchanges a design point with a candidate point when the exchange improves the criterion, until no improving exchange exists. Coordinate-exchange algorithms change one coordinate of one run at a time and do not need an explicit candidate list, which helps with many factors. Because these algorithms find local optima, they are run from many random starts and the best design kept. Software reports the D-efficiency or other measures, and the analyst should also check the prediction-variance profile over the region (for example with a fraction-of-design-space plot) and whether degrees of freedom remain for lack-of-fit and pure error. In agricultural research, computer-generated designs are used for constrained mixture problems (feed and media formulations), for trials where the land or number of plots is fixed, and for augmenting existing data with new runs; the same algorithmic ideas generate the alpha and row-column designs of the incomplete block designs subject.
A potting-medium experiment uses three components (soil, compost, sand) with constraints: soil at least 30%, compost between 10% and 50%, sand at most 40%. The feasible region is a polygon inside the simplex. A D-optimal design for a quadratic Scheffe model chosen from the vertices, edge midpoints and centroid of this polygon gives an efficient 10- to 12-run design, which would be impossible to obtain from standard lattice designs. The same approach handles a variety factor crossed with two continuous factors, or a trial in which only 14 plots are available.
When some runs have already been done, perhaps from an earlier screening experiment, an optimal design algorithm can choose additional runs that, together with the existing ones, best support the new model. This is efficient and avoids discarding information, provided conditions have not changed between the two sets of runs; a block term for the two phases protects against such changes.
Before running a computer-generated design, check its properties: the correlations among the model terms (ideally small), the variance inflation factors of the coefficients, the prediction variance over the region, the number of distinct points and replicated points, and whether any run is impractical. An optimal design for an assumed model can have every run at the extremes of the region, which may be undesirable for practical or safety reasons; constraints can be added to the candidate set.
Genichi Taguchi's approach to quality engineering distinguishes control factors, which the producer sets (machine settings, formulation, management practice), from noise factors, which vary in use and are hard or expensive to control (raw material variation, ambient temperature, farmers' practices, weather). The aim of robust parameter design is to choose control-factor settings that make the response insensitive to the noise factors while keeping its mean on target. In agriculture the same idea appears as choosing practices or varieties that perform well across uncontrollable environmental conditions, the concern of the multi-environment trials subject.
Taguchi used orthogonal arrays (such as L8, L9 and L18) for the control factors (the inner array), crossed with an orthogonal array for the noise factors (the outer array), and summarised the responses for each control setting by signal-to-noise ratios: larger-the-better, smaller-the-better and nominal-the-best. The control setting with the best signal-to-noise ratio is chosen, and an adjustment factor is used to bring the mean to target. Statisticians welcomed the robust-design idea but criticised several methods: signal-to-noise ratios confound location and dispersion and are not always appropriate; crossed arrays can be inefficient; interactions among control factors are often ignored; and the analysis of marginal means can mislead. The response-surface alternative uses combined arrays with control and noise factors in one design and models the response as a function of both, finding control settings where the control-by-noise interactions make the response flat with respect to noise. This combined-array approach, due to Welch, Shoemaker, Myers and others, is now widely recommended.
In crop management, the control factors might be sowing date, seed rate and nitrogen timing, and the noise factors rainfall pattern, soil type and pest pressure, which the farmer cannot control. A robust recommendation is a combination of control settings whose yield varies little across noise conditions while remaining high. Multi-environment variety trials pursue the same goal for genotypes, using stability analysis rather than Taguchi's arrays, and simulation with crop models can supply the noise variation when field trials are too limited.
For a larger-the-better response, Taguchi's signal-to-noise ratio is -10 log10 of the mean of 1/y squared over the noise conditions; for smaller-the-better it is -10 log10 of the mean of y squared; for nominal-the-best one common form is 10 log10 of (mean squared / variance). Larger values are better in all cases. Critics note that these summaries mix location and spread; modelling the mean and the variance (or log variance) separately, as functions of the control factors, gives clearer conclusions.
Robust-design thinking appears in agro-processing (choosing machine settings that keep product quality stable despite variable raw material), in input formulation (choosing blends whose efficacy is insensitive to weather), and in crop management under variable rainfall. The statistical tools are the same response-surface and mixed models used elsewhere in this track, with noise factors deliberately varied or simulated.
Students meet RSM in theses that optimise a process or a nutrient combination: drying temperature and time for a food product, NPK rates, media for tissue culture, feed formulations. They must choose a CCD or Box-Behnken design, code the factors, fit and check a second-order model, find and classify the stationary point and, for fertiliser work, compute the economic optimum.
A five-year statistician advises process engineers, food technologists, agronomists and animal nutritionists on optimisation studies: choosing designs that respect infeasible combinations, adding centre points for pure error, blocking runs over days, fitting response functions and mixture models, generating D-optimal designs for constrained problems, and translating fitted surfaces into recommendations with uncertainty.
A senior statistician sets institutional approaches to fertiliser recommendations (which response functions, how site variation and prices enter), to product and process optimisation in agro-processing units, and to the use of optimal-design software. They review whether optimisation claims rest on adequate designs and models.
A PhD with long experience develops response surface and optimal design methodology for agricultural problems (designs for mixtures with process variables, robust designs across environments, Bayesian optimal designs) and advises on complex optimisation programmes.
Proofs of rotatability conditions, of the equivalence theorem and of the canonical form are outlined only. The detailed conditions for orthogonal blocking of CCDs for all k, ridge analysis, Bayesian and non-linear optimal designs, and the analysis of mixture-process-variable experiments are not covered. Taguchi methods are introduced, not taught in full. Optimum fertiliser rates in the examples are illustrative; real recommendations depend on site, season and prices.
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.
These notes are free to read here, with no account.
What the Laisha app adds:
Free for ten days, then a subscription. We would rather write that here than have you find it later.