Agricultural Statistics & Data Science · Design and analysis of experiments · also Design of Experiments II, Factorial Experiments
Studying several factors at once: 2^n, 3^n and asymmetrical factorials, main effects and interactions by Yates' method, total and partial confounding, fractional factorials, and the split-plot, split-split-plot and strip-plot layouts that field operations such as irrigation and tillage force on agronomists.
In the exams: AICE-JRF Ag Stat Unit 4 (heavy: 2^n, 3^n, asymmetrical, confounding, split, strip and crossover named); AICE-JRF Agronomy Unit 6 (heavy); ASRB NET Ag Stat (heavy)
Most agricultural questions involve more than one factor at a time: varieties and nitrogen, irrigation and tillage, feed and breed. This subject covers the treatment structures and designs that make such experiments possible: factorial experiments with their main effects and interactions, 2^n factorials and Yates' algorithm, 3^n and asymmetrical factorials, confounding to keep blocks small, fractional factorials for many factors, the split-plot, split-split-plot and strip-plot designs used when some factors need large plots, factorials with extra control treatments, and crossover designs used in animal research.
It builds directly on the basic designs subject: the CRD, RCBD and Latin square remain the layouts into which factorial treatments are placed, and the analysis of variance is extended by partitioning treatment sums of squares and by recognising more than one error term.
In examinations the subject is heavy. ICAR AICE-JRF/SRF Agricultural Statistics Unit 4 names 2^n, 3^n and asymmetrical factorials, confounding, split- and strip-plot designs and crossover designs, and the statistics unit of AICE-JRF Agronomy names split-plot, strip-plot, factorial and simple confounding designs. Typical numerical questions ask for effect estimates and sums of squares, Yates' algorithm entries, degrees of freedom for each error term, the confounded effects or block contents, aliases in a fraction, and the right standard error for a comparison in a split plot. ASRB NET and ARS papers in agricultural statistics draw on the same material.
Incomplete block designs, response surfaces, multi-location trials and mixed-model analysis of repeated measures have their own subjects.
A factorial experiment tests every combination of the levels of two or more factors. A trial with three nitrogen rates and two varieties has 3 x 2 = 6 treatment combinations, and a trial with two levels each of nitrogen, phosphorus and potassium has 2 x 2 x 2 = 8. The alternative, studying one factor at a time while holding the others fixed, answers a narrower question and cannot detect interactions. Factorials have three advantages. They show whether factors interact, that is whether the effect of one factor depends on the level of another: a new variety may respond to nitrogen far more than an old one. They are efficient: each main effect is estimated from all the plots, because every plot carries information on every factor (sometimes called hidden replication). And conclusions about one factor hold over a range of levels of the others, which widens the inductive basis of recommendations. Their drawback is that the number of combinations grows quickly, blocks become large and heterogeneous, and interpretation of high-order interactions can be difficult; confounding, fractional replication and split-plot arrangements address these problems.
Take two factors A and B at two levels each, written low and high, with treatment combinations denoted (1) (both low), a (A high), b (B high) and ab (both high); the symbols also stand for treatment totals or means. The simple effect of A at the low level of B is a - (1), and at the high level of B it is ab - b. The main effect of A is the average of the two simple effects, A = [(ab - b) + (a - (1))]/2, using means. The interaction AB is half the difference between the simple effects, AB = [(ab - b) - (a - (1))]/2. With treatment totals over r replications, each effect is the contrast divided by 2r, for example A = [ab + a - b - (1)]/(2r). If the simple effects are equal there is no interaction and the main effect describes the factor fully. If they differ, the main effect is an average that may hide opposite responses, and the simple effects must be reported. Interaction is symmetrical: the AB interaction measures equally how the effect of B changes with A. Graphically, lines joining the means of one factor at each level of the other are parallel when there is no interaction.
A factorial experiment is a treatment structure, not a design: the combinations are laid out in a CRD, RCBD or Latin square like any set of treatments. In an RCBD with r blocks and ab combinations, the ANOVA first splits the total into blocks (r - 1 df), treatments (ab - 1 df) and error ((r - 1)(ab - 1) df), and then splits the treatment sum of squares into A (a - 1 df), B (b - 1 df) and AB ((a - 1)(b - 1) df). The A sum of squares is computed from the A totals over all levels of B (each based on rb plots), the B sum of squares from the B totals (each over ra plots), and the AB sum of squares as the treatment sum of squares minus those of A and B. Means are compared accordingly: two A means differ by more than t times the square root of (2 MSE / rb) to be significant, two B means by t times the square root of (2 MSE / ra), and two treatment-combination means by t times the square root of (2 MSE / r). When the interaction is significant, comparisons of one factor are made at each level of the other.
Suppose a 2 x 2 factorial of nitrogen (A) and irrigation (B) is run in four replications and the treatment totals over the four plots are (1) = 40, a = 60, b = 48 and ab = 80 kg. The contrast for A is ab + a - b - (1) = 80 + 60 - 48 - 40 = 52, so the main effect of nitrogen is 52/(2 x 4) = 6.5 kg per plot and its sum of squares is 52 squared/(4 x 4) = 169. The contrast for B is ab + b - a - (1) = 28, giving an effect of 3.5 and a sum of squares of 49. The interaction contrast is ab - a - b + (1) = 12, an interaction effect of 1.5 and a sum of squares of 9. The simple effects of nitrogen are (60 - 40)/4 = 5 kg without irrigation and (80 - 48)/4 = 8 kg with it; the main effect 6.5 is their average and the interaction 1.5 is half their difference. Whether the interaction is significant depends on the error mean square from the RCBD analysis: with 4 blocks and 4 treatments the error has 9 degrees of freedom, and each single-degree-of-freedom effect is tested by F = SS(effect)/MSE on 1 and 9 degrees of freedom. If the interaction were significant, the report would show the nitrogen response separately with and without irrigation, rather than only the average.
In a 2^n factorial each of n factors has two levels. Treatment combinations are written in standard order: (1), a, b, ab, c, ac, bc, abc, d, ad, and so on, each new letter multiplying all the combinations already listed. Every main effect and interaction is a single-degree-of-freedom contrast with coefficients plus or minus one. The sign of a combination in the contrast for an effect is found by multiplying, for each letter of the effect, plus if the letter is present in the combination and minus if it is absent. For example in a 2^3, the contrast for AC has sign plus for (1), b, ac and abc, and minus for a, c, ab and bc. With treatment totals over r replications, the effect estimate is the contrast divided by r 2^(n - 1), and the sum of squares of the effect is the contrast squared divided by r 2^n. In a 2^2 with r replications: A = [ab + a - b - (1)]/(2r), SS(A) = [ab + a - b - (1)] squared / (4r).
Yates devised a quick tabular method for all effect totals at once. List the treatment totals in standard order. Form a new column whose first half consists of the sums of successive pairs (first plus second, third plus fourth, and so on) and whose second half consists of the differences of the same pairs (second minus first, fourth minus third, and so on). Repeat this operation on the new column, n times in all. In the final column the first entry is the grand total and the remaining entries are the effect totals (contrasts) in standard order: A, B, AB, C, AC, BC, ABC, and so on. Dividing each by r 2^(n - 1) gives the effects, and squaring and dividing by r 2^n gives the sums of squares. A check is that the sum of squares of the entries in the final column equals 2^n times the sum of squares of the original totals. Yates' algorithm was the working method of agricultural statisticians before computers and remains a favourite of examiners because it is mechanical and easy to test.
Two-level factorials are screening tools: they show which factors matter and whether they interact, but with only two levels they cannot reveal curvature. In fertiliser trials, the classical NPK 2^3 factorial asks whether each nutrient raises yield and whether, say, the response to phosphorus depends on nitrogen. With many factors, high-order interactions are usually assumed negligible, which is the basis of confounding them with blocks or of fractional replication. When an experiment has a single replicate, the high-order interactions are often pooled to form an error estimate, or effects are judged by a normal probability plot of the estimated effects.
Take a 2^3 with two replications and treatment totals in standard order (1) = 20, a = 26, b = 24, ab = 34, c = 22, ac = 30, bc = 28, abc = 40. The first column of sums of pairs is 46, 58, 52, 68, and of differences (second minus first) 6, 10, 8, 12. Applying the same operation to 46, 58, 52, 68, 6, 10, 8, 12 gives sums 104, 120, 16, 20 and differences 12, 16, 4, 4. A third application gives 224, 36, 28, 8 and 16, 4, 4, 0. The final column is the grand total 224 followed by the effect totals: A = 36, B = 28, AB = 8, C = 16, AC = 4, BC = 4, ABC = 0. Each effect is its total divided by r 2^(n - 1) = 2 x 4 = 8, so the main effect of A is 4.5, and each sum of squares is the total squared over r 2^n = 16, so SS(A) = 81, SS(B) = 49 and SS(C) = 16. The check holds: the sum of squares of the final column, 224 squared plus the squares of the effects, equals 8 times the sum of squares of the original totals. Students should practise the algorithm until the order of the final column (A, B, AB, C, AC, BC, ABC) is automatic, because examiners often ask for one particular entry.
In a 3^n factorial each factor has three levels, often coded 0, 1 and 2 for low, medium and high doses. A 3^2 has nine combinations and a 3^3 has 27. Each main effect has 2 degrees of freedom, each two-factor interaction 4, each three-factor interaction 8. With equally spaced quantitative levels, each main effect can be split into a linear component, contrast coefficients (-1, 0, 1), and a quadratic component, coefficients (1, -2, 1), each with one degree of freedom. The sum of squares of a component is the contrast squared divided by (number of observations in each level total) times (the sum of squared coefficients). Two-factor interactions split into products of the components (A_L B_L, A_L B_Q, A_Q B_L, A_Q B_Q), which describe how the linear and quadratic responses of one factor change with the other. Three-level factorials therefore suit response studies where curvature and an optimum dose are expected, such as nitrogen response curves.
An alternative partition of the 4 degrees of freedom of the A x B interaction in a 3^2 uses the two orthogonal Latin-square-type components AB and AB^2 (also called the I and J components), each with 2 degrees of freedom. The level of AB for a combination (x1, x2) is x1 + x2 modulo 3, and of AB^2 is x1 + 2x2 modulo 3. This partition has no direct physical meaning but is the basis for confounding in 3^n experiments: in a 3^2 laid out in blocks of three plots, one of the 2-df components, say AB^2, is confounded with blocks, and the blocks are formed by the three groups of combinations with equal value of x1 + 2x2 modulo 3. In 3^3 experiments, blocks of nine plots are formed by confounding a 2-df component of the three-factor interaction, such as ABC^2. Because three-level designs need many plots (27 for three factors before replication), response surface designs such as central composite designs are often more economical when the aim is to fit a quadratic surface.
Factors need not have the same number of levels. A trial with four varieties, three nitrogen rates and two weed-control methods is a 4 x 3 x 2 asymmetrical (mixed-level) factorial with 24 combinations. Its analysis follows the general rule: main effects have (levels - 1) degrees of freedom, and each interaction has the product of the degrees of freedom of its factors, so V x N has 3 x 2 = 6 df, V x W has 3, N x W has 2 and V x N x W has 6; the total treatment df is 23. Sums of squares are computed from the marginal tables of totals, subtracting lower-order sums of squares for interactions. Quantitative factors are partitioned into polynomial components and qualitative ones into planned contrasts. Confounding in asymmetrical factorials is more complicated than in symmetrical ones because interactions cannot always be confounded without loss of information on several effects; balanced confounded designs for 3 x 2^2 and 3^2 x 2 were developed by Yates and others, and partial confounding over replicates is common. Asymmetrical factorials are frequent in agricultural research because natural levels of factors such as varieties, seasons or irrigation regimes rarely match.
For a 4 x 3 factorial of varieties (V) and nitrogen rates (N) in an RCBD with three replications, the analysis begins with the 12 treatment totals arranged in a 4 x 3 two-way table. The correction factor uses the grand total over all 36 plots. The V sum of squares is the sum of the squared variety totals (each over 9 plots) divided by 9 minus the correction factor; the N sum of squares is the sum of the squared rate totals (each over 12 plots) divided by 12 minus the correction factor; the treatment sum of squares uses the 12 cell totals divided by 3; and the V x N sum of squares is the treatment sum of squares minus those of V and N. The degrees of freedom are 3 for V, 2 for N, 6 for V x N, 2 for blocks and 22 for error. With three equally spaced nitrogen rates, the N sum of squares splits into linear and quadratic components, and the V x N interaction into V x N-linear (3 df) and V x N-quadratic (3 df), which asks whether varieties differ in the slope or curvature of their nitrogen response. That partition is usually far more informative than the overall interaction F.
As the number of combinations grows, a complete replicate in one block becomes too large to be homogeneous; a 2^5 needs 32 plots per block. Confounding divides each replicate into smaller blocks so that a chosen effect, usually a high-order interaction believed negligible, is deliberately made indistinguishable from block differences. The other effects are estimated within blocks with the smaller error of the smaller blocks. In a 2^3 in blocks of four, confounding ABC puts the combinations with a plus sign in the ABC contrast, a, b, c and abc, in one block and those with a minus sign, (1), ab, ac and bc, in the other. The block containing (1) is the principal block; the other blocks can be obtained by multiplying its members by any combination not in it, with squares of letters set to one. When two effects are confounded to form four blocks in a 2^n, their generalised interaction (the product with squared letters removed) is automatically confounded too: confounding AB and AC also confounds (AB)(AC) = A^2BC = BC.
If the same effect is confounded in every replicate, the confounding is total and no information on that effect is available from the experiment. In the ANOVA of a 2^3 with ABC totally confounded in r replicates of two blocks each, the blocks take 2r - 1 degrees of freedom (including the ABC contrast), the six unconfounded effects take one each, and the error has 8r - 1 - (2r - 1) - 6 = 6(r - 1) degrees of freedom. Total confounding is chosen when the confounded effect is of no interest or very unlikely to be real.
If different effects are confounded in different replicates, the confounding is partial: each confounded effect is estimated from the replicates in which it is not confounded. In a 2^3 with four replicates, confounding AB in the first, AC in the second, BC in the third and ABC in the fourth, each interaction is estimated from three of the four replicates, so its relative information is 3/4 compared with an unconfounded effect, while main effects keep full information. The sums of squares for the partially confounded interactions are computed only from the replicates where they are unconfounded. The ANOVA has 31 total df, 7 for blocks, 7 for treatments (all effects remain estimable) and 17 for error. Partial confounding spreads the loss of information over several effects instead of losing one entirely, and it is preferred when all interactions are of some interest. The standard error of a comparison involving a partially confounded effect must use the reduced number of replicates.
To build the blocks for a 2^4 in four blocks of four plots, choose two interactions to confound, for example ABC and BCD; their generalised interaction AD is then confounded as well. The principal block contains the combinations that have an even number of letters in common with both ABC and BCD: (1), bc, acd and abd. Multiplying the principal block by a combination not in it, such as a, gives a second block: a, abc, cd, bd. Multiplying by b gives b, c, abcd, ad, and multiplying by ab gives the fourth block: ab, ac, bcd, d. Each block contains four combinations, and the 16 are all used once. A careless choice can confound a main effect: confounding AB and ABC, for instance, also confounds their product C. The rule of thumb is to confound high-order interactions whose generalised interactions are also high-order. With 2^5 in four blocks of eight, confounding ABC and CDE also confounds ABDE, losing only three- and four-factor interactions.
With many factors, even one complete replicate may be too large, and most high-order interactions are negligible. A fractional factorial runs a chosen fraction of the combinations: a half replicate of a 2^k, written 2^(k - 1), uses 2^(k - 1) runs; a quarter replicate, 2^(k - 2), uses 2^(k - 2). The fraction is defined by a defining relation: the half replicate of a 2^4 defined by I = ABCD consists of the eight combinations with a plus sign (or, for the alternative fraction, a minus sign) in the ABCD contrast. The price is aliasing: each effect is confounded with one or more other effects, its aliases, and the experiment estimates only their sum. The aliases of an effect are found by multiplying it by each word in the defining relation and deleting squared letters. With I = ABCD, A is aliased with BCD, B with ACD, AB with CD, AC with BD and AD with BC.
The resolution of a fraction is the length of the shortest word in its defining relation, written in Roman numerals. In a resolution III design (for example the 2^(3 - 1) with I = ABC), main effects are aliased with two-factor interactions. In a resolution IV design (the 2^(4 - 1) with I = ABCD), main effects are aliased only with three-factor interactions, but two-factor interactions are aliased with each other. In a resolution V design (the 2^(5 - 1) with I = ABCDE), main effects and two-factor interactions are aliased only with interactions of three or more factors. Higher resolution protects more effects but needs more runs. Fractional factorials are widely used in industrial and laboratory screening and, more cautiously, in agricultural experiments with many factors, such as nutrient omission trials, media composition studies and sensory work. If an important aliased pair is found, a second fraction (the fold-over) can separate them.
A half replicate of a 2^4 defined by I = ABCD consists of (1), ab, ac, ad, bc, bd, cd and abcd, the combinations with an even number of letters. Its eight runs give seven contrasts, each estimating a pair of aliased effects: A + BCD, B + ACD, C + ABD, D + ABC, AB + CD, AC + BD and AD + BC. If three-factor interactions are negligible, the main effects are estimated cleanly, but the two-factor interactions remain confounded in pairs, and subject-matter knowledge or a follow-up fraction is needed to decide which member of a significant pair is real. With no replication, the error is taken from effects assumed negligible or from a normal plot of the effects.
In a split-plot design one factor, the main-plot factor, is applied to large plots, and each main plot is divided into sub-plots to which the levels of a second factor, the sub-plot factor, are applied. Main-plot treatments are randomised to main plots within each block (in an RCBD) and sub-plot treatments are randomised separately within each main plot. The design is used when one factor needs large plots for practical reasons (irrigation methods, tillage, sowing dates, land preparation or machinery), when a new factor is added to an experiment already in progress, or when greater precision is wanted for one factor and the interaction than for the other. Because main plots are larger and fewer, the main-plot factor is estimated less precisely than the sub-plot factor and the interaction. The factor whose comparisons matter most, and which can be applied to small plots, should therefore be put in the sub-plots.
For a split plot with a main-plot levels in r blocks and b sub-plot levels, the ANOVA has two error terms. The main-plot part: blocks (r - 1 df), A (a - 1 df) and error (a) ((r - 1)(a - 1) df). The sub-plot part: B (b - 1 df), A x B ((a - 1)(b - 1) df) and error (b) (a(r - 1)(b - 1) df). The total is rab - 1. A is tested against error (a); B and A x B are tested against error (b). Error (a) is usually larger than error (b), because main plots are farther apart and larger. The sums of squares for blocks, A and error (a) come from the table of main-plot totals (block x A), exactly as in an RCBD of the main plots; B and A x B come from the A x B table; error (b) is found by subtraction.
Four kinds of comparison arise, each with its own standard error, where Ea and Eb are the error (a) and error (b) mean squares. Between two main-plot means (averaged over sub-plots): the square root of 2Ea/(rb). Between two sub-plot means (averaged over main plots): the square root of 2Eb/(ra). Between two sub-plot levels at the same main-plot level: the square root of 2Eb/r. Between two main-plot levels at the same or different sub-plot levels: the square root of 2[(b - 1)Eb + Ea]/(rb). The last comparison involves both errors, so its critical t is a weighted value, t' = [(b - 1) Eb t_b + Ea t_a] / [(b - 1) Eb + Ea], where t_a and t_b are tabulated t at the error (a) and error (b) degrees of freedom. Getting the right error term for each comparison is a frequent source of mistakes in published work.
Suppose four irrigation levels (A) are in main plots and three varieties (B) in sub-plots, in three blocks. The ANOVA then has 2 df for blocks, 3 for A and (3 - 1)(4 - 1) = 6 for error (a); 2 for B, 6 for A x B and 4 x 2 x 2 = 16 for error (b); 35 in total. If Ea = 12 and Eb = 4, the standard error of the difference between two irrigation means is the square root of (2 x 12/9) = 1.63; between two variety means the square root of (2 x 4/12) = 0.82; between two varieties under the same irrigation the square root of (2 x 4/3) = 1.63; and between two irrigation levels for the same variety the square root of 2[(3 - 1) x 4 + 12]/9 = 2.11. The example shows why the arrangement matters: the irrigation comparisons, with only 6 error degrees of freedom and a large error mean square, are the least precise, while variety comparisons are the most precise. If irrigation responses were the main question, an experimenter would need more blocks or a different design.
A split-split-plot design adds a third factor by dividing each sub-plot into sub-sub-plots. With a main-plot levels in r blocks, b sub-plot levels and c sub-sub-plot levels, there are three error terms. The ANOVA has: blocks (r - 1), A (a - 1), error (a) ((r - 1)(a - 1)); B (b - 1), A x B ((a - 1)(b - 1)), error (b) (a(r - 1)(b - 1)); C (c - 1), A x C ((a - 1)(c - 1)), B x C ((b - 1)(c - 1)), A x B x C ((a - 1)(b - 1)(c - 1)), error (c) (ab(r - 1)(c - 1)). Precision increases from the main plots to the sub-sub-plots. A typical agronomic example puts irrigation levels in main plots, varieties in sub-plots and nitrogen rates in sub-sub-plots. The design is efficient for the sub-sub-plot factor and its interactions but complex to lay out, and the main-plot factor may be tested with very few error degrees of freedom; the experimenter should check that error (a) has enough degrees of freedom to be useful.
Repeated harvests from the same plots (cuttings of a forage crop, pickings of cotton, years of a perennial crop) are sometimes analysed as a split plot in time, with harvests as a sub-plot factor. This is only approximately valid, because harvests cannot be randomised and measurements close in time are more correlated than distant ones; the sphericity assumption behind the split-plot F tests may fail. Modern practice analyses such data with mixed models and suitable covariance structures, covered in the mixed-models subject.
In a strip-plot design, also called a split-block design, each block is divided into horizontal strips, to which the levels of factor A are randomised, and vertical strips, to which the levels of factor B are randomised independently. Each A x B combination occupies the intersection plot. The design suits experiments in which both factors need large plots or continuous strips, such as irrigation methods crossed with tillage methods, or row spacing crossed with machinery that must run across the block. Compared with a split plot, both main effects are now applied to large strips and are estimated less precisely, while the interaction, estimated from the small intersection plots, gains precision. It is therefore chosen when the interaction is the main interest or when practical constraints force it.
For a horizontal factor with a levels, a vertical factor with b levels and r blocks, the ANOVA has: blocks (r - 1); A (a - 1) with error (a) ((r - 1)(a - 1)); B (b - 1) with error (b) ((r - 1)(b - 1)); A x B ((a - 1)(b - 1)) with error (c) ((r - 1)(a - 1)(b - 1)). A, B and A x B are each tested against their own error. Standard errors for comparing two A means are the square root of 2Ea/(rb), for two B means the square root of 2Eb/(ra); comparisons within the interaction require weighted combinations of the error mean squares, as in the split plot.
If only one factor needs large plots, a split plot is the natural choice. If both factors need strips, for example because sowing machinery and irrigation channels must each run the full length or width of the block, the strip plot is the practical answer. The cost is that both main effects are estimated from strips with few error degrees of freedom; the gain is precision for the interaction. A strip plot laid out simply for convenience, when a split plot or RCBD would have worked, gives away precision on the main effects for no reason.
A factorial set of treatments is often accompanied by one or more extra treatments, typically an untreated control. For example a 3 x 3 factorial of three fertiliser sources and three rates, plus a no-fertiliser control, gives ten treatments. The control does not fit the factorial structure (zero of each source is the same thing), so it cannot be treated as an extra level of either factor. The usual analysis lays out all ten treatments in an RCBD and splits the 9 treatment degrees of freedom into: control versus the factorial combinations (1 df); source (2 df); rate (2 df); source x rate (4 df). The sum of squares for control versus rest is T_c^2/r + T_f^2/(rk) - (T_c + T_f)^2/[r(k + 1)], where T_c is the control total, T_f the total of all factorial plots and k the number of factorial combinations. The factorial sums of squares are computed from the factorial plots only, as in an ordinary factorial.
The control-versus-rest comparison tells whether fertiliser of any kind raised yield; the factorial components tell how sources and rates differ. Comparing each combination with the control uses Dunnett's test or the standard error of the difference between two treatment means. A design with too many extra treatments should be reconsidered: if each extra is of real interest the factorial structure may be the wrong model.
Some experiments add more than one extra treatment, such as an untreated control and a farmers' practice check alongside a factorial. The treatment degrees of freedom are then split into comparisons among the extras, extras versus the factorial group, and the factorial effects. Each split should correspond to a question the experiment was designed to answer. When the factorial includes a zero level of a factor, as in a nitrogen rate factor with 0, 40 and 80 kg, no extra control is needed, because the zero rate is already part of the factorial.
In a crossover (changeover) design, each experimental unit receives a sequence of treatments in successive periods, so that treatments are compared within units. It is common in animal nutrition (dairy cows, sheep), veterinary pharmacology and human trials, where between-animal variation is large and the response returns to baseline after a treatment is withdrawn. The simplest is the two-period two-treatment design, with half the animals receiving A then B and half B then A. Latin squares are the standard layout for more treatments: with t treatments, t animals and t periods, each animal receives each treatment once and each treatment occurs once in each period, so animal and period effects are removed from the error. Several squares are used to obtain enough error degrees of freedom; a single 4 x 4 square has only 6.
The main difficulty is carry-over (residual) effects: the response in one period may be influenced by the treatment of the previous period. Washout or rest periods between treatments reduce carry-over. When carry-over cannot be excluded, designs balanced for first-order residual effects are used: in Williams designs every treatment is preceded by every other treatment equally often, which for an even number of treatments needs one suitably arranged Latin square and for an odd number a pair of squares. Direct and residual effects are then estimated by least squares. In lactation trials, milk yield declines through lactation, so the trend within each period is part of the design; switchback (reversal) designs such as A-B-A and B-A-B compare treatments against the trend. Crossover designs are inappropriate when treatments permanently change the animal (such as a curative treatment or a growth-stage effect), or when periods would be too long relative to the animal's productive cycle.
For four rations tested on four cows over four periods in a Latin square, the ANOVA has 3 df each for cows, periods and rations, and (4 - 1)(4 - 2) = 6 for error, out of 15 total. Because 6 error df is small, nutrition trials usually use two or three squares, adding squares to the analysis as a further classification. If residual effects are suspected, the model adds a carry-over term for the treatment of the previous period, which is estimable when the design is balanced for carry-over. Data from the first part of each period are often discarded as an adaptation period, which is a practical complement to a formal washout.
Students meet factorials in every agronomy and breeding thesis that tests more than one factor, and split plots whenever irrigation, tillage or sowing date is a factor. In examinations they must run Yates' algorithm, identify confounded effects and block contents, write the split-plot ANOVA with both error terms and choose the right standard error for each comparison. Understanding why the main-plot factor is less precise helps them design their own trials sensibly.
A station statistician five years in advises scientists daily on these designs: whether a three-factor trial fits in a full factorial or needs confounding, which factor goes into main plots, whether a strip plot is needed because both factors need machinery, how to add an untreated control to a factorial, and how to analyse a cow feeding trial as a changeover design with washout periods. They also catch the common analysis errors, such as testing a main-plot factor against the sub-plot error.
A senior head of a statistics division sets institutional guidance on multi-factor trials: when split plots are acceptable, minimum error degrees of freedom for main-plot factors, standard reporting of interactions and the correct standard errors, and the move from classical split-plot-in-time analyses to mixed models. They review large coordinated trial programmes whose treatment structures are factorial and decide how they should be pooled and reported.
A PhD with long experience develops and teaches the theory of factorial and confounded designs, constructs efficient fractional and asymmetrical designs, and works on designs balanced for carry-over effects. As a reviewer they judge whether a published factorial or split-plot analysis used the right model and error terms.
Confounding in asymmetrical factorials and in 3^3 experiments is described, not worked in full. Construction of optimal fractional factorials beyond the simple half and quarter fractions, Taguchi designs, response surfaces and mixed-model analysis of repeated measures are left to other subjects. Formulas for standard errors in strip plots for interaction comparisons are not given in full. Statistical tables are not reproduced; questions state the values they need.
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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