1. Principles in the design of multiphase experiments with a later laboratory phase: orthogonal designs Chris Brien 1, Bronwyn Harch 2, Ray Correll 2 & Rosemary Bailey 3 1 University of South Australia, 2 CSIRO Mathematics, Informatics & Statistics, 3 Queen Mary University of London Chris.brien@unisa.edu.au http://chris.brien.name/multitier

2. Outline 1.Primary experimental design principles 2.Factor-allocation description for standard designs. 3.Principles for simple multiphase experiments. 4.Principles leading to complications, even with orthogonality. 5.Summary 2

3. 1)Primary experimental design principles Principle 1 (Evaluate designs with skeleton ANOVA tables) Use whether or not data to be analyzed by ANOVA. Principle 2 (Fundamentals): Use randomization, replication and blocking or local control. Principle 3 (Minimize variance): Block entities to form new entities, within new entities being more homogeneous; assign treatments to least variable entity-type. Principle 4 (Split units): confound some treatment sources with more variable sources if some treatment factors: i. require larger units than others, ii. are expected to have a larger effect, or iii. are of less interest than others. 3

4. A standard agricultural example 8 treatments combinations of 4 harvest dates and 2 different fertilizers – are to be investigated for a plant crop. During harvest the yield is to be measured. Assume differences between harvest dates expected to be larger than between fertilizers. A split-plot design is to be used: It will have 3 blocks each containing 4 main plots, with each main plot split into 2 subplots; Randomize 4 harvest dates to 4 main plots in a block and 2 fertilizers to 2 subplots in a main plot. This design employs Principles 2, 3 and 4(ii). 4

5. 2) Factor-allocation description for standard designs Standard designs involve a single allocation in which a set of treatments is assigned to a set of units: treatments are whatever are allocated; units are what treatments are allocated to; treatments and units each referred to as a set of objects; Often do by randomization using a permutation of the units. More generally treatments are allocated to units e.g. using a spatial design or systematically Each set of objects is indexed by a set of factors: Unit or unallocated factors (indexing units); Treatment or allocated factors (indexing treatments). Represent the allocation using factor-allocation diagrams that have a panel for each set of objects with: a list of the factors; their numbers of levels; their nesting relationships. 5 (Nelder, 1965; Brien, 1983; Brien & Bailey, 2006)

6. Cf. single set of factors uniquely indexing observations: {Blocks, Harvests, Fertilizers}. A crucial feature of diagram is that it shows EU and restrictions on randomization/allocation. Factor-allocation diagram for the standard agricultural experiment 6 One allocation (randomization): a set of treatments to a set of subplots. 4Harvests 2Fertilizers 8 treatments 3Blocks 4Mainplots in B 2Subplots in B, M 24 subplots The set of factors belonging to a set of objects forms a tier: {Harvests, Fertilizers} or {Blocks, Mainplots, Subplots}; they have the same status in the allocation (randomization); Textbook experiments are two-tiered. (e.g. Searle et al., 1992)

7. Some derived items Sets of generalized factors (terms in the mixed model): Blocks, Blocks Mainplots, Blocks Mainplots Subplots; Harvests, Fertilizers, Harvests Fertilizers. Corresponding types of entities (groupings of objects): block, main plot, subplot (last two are EUs); harvest, fertilizer, treatment (harvest-fertilizer combination). Corresponding sources (in an ANOVA): Blocks, Mainplots[B], Subplots[B M]; Harvests, Fertilizers, Harvests#Fertilizers. 7 4Harvests 2Fertilizers 8 treatments 3Blocks 4Mainplots in B 2Subplots in B, M 24 subplots

8. Skeleton ANOVA Harvests is confounded with the more-variable Mainplots[B] & Fertilizers with Subplots[B^M]. 8 4Harvests 2Fertilizers 8 treatments 3Blocks 4Mainplots in B 2Subplots in B, M 24 subplots

9. 3)Principles for simple multiphase experiments Suppose in the agricultural experiment: in addition to yield measured during harvest, the amount of Na in the plant is to be measured for each subplot, and the harvested material is transported to the laboratory for drying and analysis. The experiment is two phase: field and laboratory phases. The outcome of the field phase is yield and batches of plant material. The outcome of the laboratory phase is the amount of Na. How to process the specimens from the first phase in the laboratory phase? 9

10. Some principles Principle 5 (Simplicity desirable): assign first-phase units to laboratory units so that each first-phase source is confounded with a single laboratory source. Use composed randomizations with an orthogonal design. Principle 6 (Preplan all): if possible. Principle 7 (Allocate all and randomize in laboratory): always allocate all treatment and unit factors and randomize first-phase units and lab treatments. Principle 8 (Big with big): Confound big first-phase sources with big laboratory sources, provided no confounding of treatment with first-phase sources. 10

11. A simple two-phase agricultural experiment Simplest is to randomize batches from a subplot to locations (in time or space) during the laboratory phase. 11 4Harvests 2Fertilizers 8 treatments 3Blocks 4Mainplots in B 2Subplots in B, M 24 subplots 24Locations 24 locations Composed randomizations (Brien & Bailey, 2006)

12. A simple two-phase agricultural experiment (contd) No. subplots = no. locations = 24 and so subplots sources exhaust locations sources. Cannot separately estimate locations and subplots variability, but can estimate their sum. However want to block the laboratory phase. 12

13. A simple two-phase agricultural experiment (contd) Simplest is to align lab-phase and first-phase blocking. 13 4Harvests 2Fertilizers 8 treatments 3Blocks 4Mainplots in B 2Subplots in B, M 24 subplots 3Intervals 8Locations in I 24 locations Note Blocks confounded with Intervals (i.e. Big with Big). Composed randomizations

14. The multiphase law DF for sources from a previous phase can never be increased as a result of the laboratory-phase design. However, it is possible that first-phase sources are split into two or more sources, each with fewer degrees of freedom than the original source. 14 DF for first phase sources unaffected.

15. 4)Principles leading to complications, even with orthogonality Principle 9 (Use pseudofactors): An elegant way to split sources (as opposed to introducing grouping factors unconnected to real sources of variability). Principle 10 (Compensating across phases): Sometimes, if something is confounded with more variable first- phase source, can confound with less variable lab source. Principle 11 (Laboratory replication): Replicate laboratory analysis of first-phase units if lab variability much greater than 1 st -phase variation; Often involves splitting product from the first phase into portions (e.g. batches of harvested crop, wines, blood specimens into aliquots, drops, lots, samples and fractions). Principle 12 (Laboratory treatments): Sometimes treatments are introduced in the laboratory phase and this involves extra randomization. 15

16. A three-phase agricultural experiment with laboratory treatments Suppose that material from a harvest date cannot be stored and must be analysed before the next harvest. Also, 3 different methods are to be used to measure Na. For this, the material from each subplot is divided into 3 portions and the Methods randomized to them. For each harvest, there are 18 portions to be dried and analysed. There are three ovens to be used simultaneously for the drying: they differ in speed of drying and can only fit 6 portions for drying blocks will be randomized to ovens. Piepho et al. (2003)

17. A three-phase agricultural experiment with laboratory treatments – drying phase Problem: 12 main plots to assign to 4 dates: Use M 1 to group the 4 main plots to be dried on the same date. An alternative is to introduce a grouping factor, say MainGroups, o but, while in the analysis, not an anticipated variability source. Dashed arrow indicates systematic allocation. The blocks are randomized to ovens (Big with Big). The 2 subplots x 3 portions in a main plot are randomized to the 6 locations in an oven on a date. The methods are randomized to portions. 17 4Harvests 2Fertilizers 8 field treatments 4Mainplots in B 3Blocks 2Subplots in B, M 3Portions in B, M, S 72 portions 4Dates 3Ovens in D 6Locations in D, O 72 locations 4M14M1 3Methods 3 lab treatments

18. A three-phase agricultural experiment with laboratory treatments – lab phase After drying, the analysis to measure Na must be done. In what order will this be done? All 18 dried portions for a harvest will be done together. Then, could randomize the order of analysis for these or could group according to ovens. The latter, so that analyses are performed in batches of 6. 18 4Harvests 2Fertilizers 8 field treatments 4Mainplots in B 3Blocks 2Subplots in B, M 3Portions in B, M, S 72 portions 4Dates 3Ovens in D 6Locations in D, O 72 locations 4M14M1 3Methods 3 lab treatments 4Times 3Intervals in T 6Analyses in T, I 72 analyses

19. A three-phase agricultural experiment with laboratory treatments (contd) 19 Note split source 4Harvests 2Fertilizers 8 field treatments 4Mainplots in B 3Blocks 2Subplots in B, M 3Portions in B, M, S 72 portions 4Dates 3Ovens in D 6Locations in D, O 72 locations 4M14M1 3Methods 3 lab treatments 4Times 3Intervals in T 6Analyses in T, I 72 analyses

20. A three-phase agricultural experiment with laboratory treatments (contd) 20 Harvests confounded with Times & Dates. No. analyses = no. locations = no. portions = 72: so portions sources exhaust locations sources exhaust analyses sources. Consequently, not all terms are estimable.

21. Mixed models Mixed models can be easily obtained from the factor- allocation description using Brien & Demétrios (2009) method: In each panel, form terms as the set of generalized factors for that panel all combinations of the factors, subject to nesting restrictions; For each term from each panel, add to either fixed or random model. 21

22. Mixed model for 3-phase experiment Full mixed model is: Harvests + Fertilizers + Harvests Fertilizers + Methods | Blocks + Blocks Mainplots + Blocks Mainplots Subplots + Blocks Mainplots Subplots Portions + Dates + Dates Ovens + Dates Ovens Locations + Times + Times Intervals + Times Intervals Analyses. Pseudofactors not needed in mixed model (cf. grouping factors). 22 4Harvests 2Fertilizers 8 field treatments 4Mainplots in B 3Blocks 2Subplots in B, M 3Portions in B, M, S 72 portions 4Dates 3Ovens in D 6Locations in D, O 72 locations 4M14M1 3Methods 3 lab treatments 4Times 3Intervals in T 6Analyses in T, I 72 analyses

23. Mixed model of convenience (for fitting) Full mixed model is: Harvests + Fertilizers + Harvests Fertilizers + Methods | Blocks + Blocks Mainplots + Blocks Mainplots Subplots + Blocks Mainplots Subplots Portions + Dates + Dates Ovens + Dates Ovens Locations + Times + Times Intervals + Times Intervals Analyses. 23 Model will not fit because of confounding (see ANOVA). Mixed model of convenience (a term for each line in the ANOVA): Harvests + Fertilizers + Harvests Fertilizers + Methods | Blocks Mainplots + Blocks Mainplots Subplots + Blocks Mainplots Subplots Portions. This model of convenience does not reflect all the sources of variability in the experiment. For example, the Residual is not just portions variability, but has locations and analyses variability contributing to it also.

24. 24 5)Summary Have provided 4 standard principles and 8 principles specific to orthogonal, multiphase designs. Extension to nonorthogonal designs is coming.

25. 25 References Brien, C. J. (1983). Analysis of variance tables based on experimental structure. Biometrics, 39, 53–59. Brien, C.J., and Bailey, R.A. (2006) Multiple randomizations (with discussion). J. Roy. Statist. Soc., Ser. B, 68, 571–609. Brien, C.J. and Demétrio, C.G.B. (2009) Formulating mixed models for experiments, including longitudinal experiments. Journal of Agricultural, Biological and Environmental Statistics, 14, 253–80. Brien, C.J., Harch, B.D., Correll, R.L. and Bailey, R.A. (2011) Multiphase experiments with laboratory phases subsequent to the initial phase. I. Orthogonal designs. Journal of Agricultural, Biological and Environmental Statistics, 16, 422–450. Nelder, J. A. (1965). The analysis of randomized experiments with orthogonal block structure. Proceedings of the Royal Society of London, Series A, 283(1393), 147-162, 163–178. Piepho, H.P., Büchse, A. and Emrich, K. (2003) A hitchhikers guide to mixed models for randomized experiments. Journal of Agronomy and Crop Science, 189, 310–322. Searle, S. R., Casella, G. & Mcculloch, C. E. (1992) Variance components. New York, Wiley. Web address for link to Multitiered experiments site: http://chris.brien.name/multitier