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

Outline 1.Primary experimental design principles. 2.Factor-allocation description for standard designs. 3.Single set description. 4.Principles for simple multiphase experiments. 5.Principles leading to complications, even with orthogonality. 6.Summary. 2

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 (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

A standard athlete training example 9 training conditions — combinations of 3 surfaces and 3 intensities of training — to be investigated. Assume the prime interest is in surface differences  intensities are only included to observe the surfaces over a range of intensities. Testing is to be conducted over 4 Months:  In each month, 3 endurance athletes are to be recruited.  Each athlete will undergo 3 tests, separated by 7 days, under 3 different training conditions. On completion of each test, the heart rate of the athlete will be measured. Randomize 3 intensities to 3 athletes in a month and 3 surfaces to 3 tests in an athlete.  A split-unit design, employing Principles 2, 3 and 4(iii). 4 Peeling et al. (2009)

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)

Factor-allocation diagram for the standard athlete training experiment 6 One allocation (randomization):  a set of training conditions to a set of tests. 3Intensities 3Surfaces 9 training conditions 4Months 3Athletes in M 3Tests in M, A 36 tests The set of factors belonging to a set of objects forms a tier:  they have the same status in the allocation (randomization):  {Intensities, Surfaces} or {Months, Athletes, Tests}  Textbook experiments are two-tiered. A crucial feature is that diagram automatically shows EU and restrictions on randomization/allocation.

Some derived items Sets of generalized factors (terms in the mixed model):  Months, Months  Athletes, Months  Athletes  Tests;  Intensities, Surfaces, Intensities  Surfaces. Corresponding types of entities (groupings of objects):  month, athlete, test (last two are Eus);  intensity, surface, training condition (intensity-surface combination). Corresponding sources (in an ANOVA):  Months, Athletes[M], Tests[M  A];  Intensities, Surfaces, Intensities#Surfaces. 7 3Intensities 3Surfaces 9 training conditions 4Months 3Athletes in M 3Tests in M, A 36 tests

Skeleton ANOVA Intensities is confounded with the more-variable Athletes[M] & Surfaces with Tests[M^A]. 8 3Intensities 3Surfaces 9 training conditions 4Months 3Athletes in M 3Tests in M, A 36 tests

Mixed model Take generalized factors derived from factor-allocation diagram and assign to either fixed or random model: Intensities + Surfaces + Intensities  Surfaces | Months + Months  Athletes + Months  Athletes  Tests Corresponds to the mixed model: Y = X I  I + X S  S + X I  S  I  S + Z M u M + Z M  A u M  A + Z M  A  T u M  A  T. where the Xs and Zs are indicator variable matrices for the generalized factors in its subscript, and  s and us are fixed and random parameters, respectively, with 9 This is an ANOVA model, equivalent to the randomization model, and is also written: Y = X I  I + X S  S + X I  S  I  S + Z M u M + Z M  A u M  A + e. To fit in SAS must set the DDFM option of MODEL to KENWARDROGER. Brien & Demétrio (2009).

3)Single-set description Single set of factors that uniquely indexes observations:  {Months, Intensities, Surfaces} (Athletes and Tests omitted). What are the EUs in the single-set approach?  A set of units that are indexed by Months-Intensities combinations and another set by the Months-Intensities-Surfaces combinations.  Of course, Months-Intensities(-Surfaces) are not actual EUs, as Intensities (Surfaces) are not randomized to those combinations.  They act as a proxy for the unnamed units. Mixed model is: I + S + I  S | M + M  I + M  I  S.  Previous model: I + S + I  S | M + M  A + M  A  T.  Former more economical as A and T not needed.  In SAS, default DDFM option of MODEL ( CONTAIN ) works. However, M  A and M  I are different sources of variability:  inherent variability vs block-treatment interaction. This "trick" is confusing, false economy and not always possible. 10 e.g. Searle, Casella & McCulloch (1992); Littel et al. (2006).

Single-set description ANOVA Single set of factors that uniquely indexes observations:  {Months, Intensities, Surfaces} Use factors to derive skeleton ANOVA 11 Confounding not exhibited, and need E[MSq] to see that M#I is denominator for Intensities.

Summary of factor-allocation versus single-set description Factor-allocation description is based on the tiers and so is multi-set. It has a specific factor for the EUs and so their identity not obscured. Single-set description factors are a subset of those identified in the factor-allocation description and so more economical. Skeleton ANOVA from factor-allocation description shows:  The confounding resulting from the allocation;  The origin of the sources of variation more accurately (Athletes[Months] versus Months#Intensities). 12

4)Principles for simple multiphase experiments Suppose in the athlete training experiment:  in addition to heart rate taken immediately upon completion of a test,  the free haemoglobin is to be measured using blood specimens taken from the athletes after each test, and  the specimens are transported to the laboratory for analysis. The experiment is two phase: testing and laboratory phases.  The outcome of the testing phase is heart rate and a blood specimen.  The outcome of the laboratory phase is the free haemoglobin. How to process the specimens from the first phase in the laboratory phase? 13

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. 14

A simple two-phase athlete training experiment Simplest is to randomize specimens from a test to locations (in time or space) during the laboratory phase. 15 3Intensities 3Surfaces 9 training conditions 4Months 3Athletes in M 3Tests in M, A 36 tests 36Locations 36 locations Composed randomizations

A simple two-phase athlete training experiment (cont’d) No. tests = no. locations = 36 and so tests sources exhaust the locations source. Cannot separately estimate locations and tests variability, but can estimate their sum. But do not want to hold blood specimens for 4 months. 16

A simple two-phase athlete training experiment (cont’d) Simplest is to align lab-phase and first-phase blocking. 17 3Intensities 3Surfaces 9 training conditions 4Months 3Athletes in M 3Tests in M, A 36 tests 4Batches 9Locations in B 36 locations Note Months confounded with Batches (i.e. Big with Big). Composed randomizations

A simple two-phase athlete training experiment – mixed model Form generalized factors and assign to fixed or random:  I + S + I  S | M + M  A + M  A  T + B + B  L. ANOVA shows us i) there will be aliasing and ii) model without lab terms will fit and be sufficient – a “model of convenience”. 18 3Intensities 3Surfaces 9 training conditions 4Months 3Athletes in M 3Tests in M, A 36 tests 4Batches 9Locations in B 36 locations

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. 19 DF for first phase sources unaffected.

Factor-allocation in multiphase experiments While multiphase experiments will often involve multiple allocations, not always:  A two-phase experiment will not if the first phase involves a survey i.e. no allocation e.g. tissues sampled from animals of different sexes.  A two-phase experiment may include more than two allocations: e.g. a grazing trial in the first phase that involves two composed randomizations. Factor-allocation description is particularly helpful in understanding multiphase experiments with multiple allocations. 20

5)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. 21

A replicated two-phase athlete training experiment Suppose duplicates of free haemoglobin to be done:  2 fractions taken from each specimen;  One fraction taken from 9 specimens for the month and analyzed;  Then, the other fraction from 9 specimens analyzed. 22 3Intensities 3Surfaces 9 training conditions 4Months 2Fractions in M, A, T 3Athletes in M 3Tests in M, A 72 fractions 4Batches 2Rounds in B 9Locations in B, R 72 locations 2F 1 in M Problem: 18 fractions in a month to assign to 2 rounds in a batch:  Use F 1 to group the 9 fractions to be analyzed in the same round.  An alternative is to introduce the grouping factor FGroups,  but, while in the analysis, not an anticipated variability source.

A replicated two-phase athlete training experiment (cont’d) 23 3Intensities 3Surfaces 9 training conditions 4Months 2Fractions in M, A, T 3Athletes in M 3Tests in M, A 72 fractions 2F 1 in M Note split source 4Batches 2Rounds in B 9Locations in B, R 72 locations

A replicated two-phase athlete training experiment (cont’d) 24 Last line measures laboratory variation. Again, no. fractions = no. locations = 72 and so fractions sources exhaust locations sources.  Consequently, not all terms could be included in a mixed model.  Pseudofactors not needed in mixed model (cf. grouping factors).

A replicated two-phase athlete training experiment – mixed model 25 Form generalized factors and assign to fixed or random:  I + S + I  S | M + M  A + M  A  T + M  A  T  F + B + B  R + B  R  L. Removing aliased terms gives one “model of convenience”:  I + S + I  S | M  A + M  A  T + B + B  R + B  R  L (M, M  A  T  F omitted).  Another model would result from removing B and B  R  L (need B  R).

Single-set description for example Single set of factors that uniquely indexes observations:  {4 Months x 2 Rounds x 3 Intensities x 3 Surfaces}. 26 3Intensities 3Surfaces 9 training conditions 4Months 2Fractions in M, A, T 3Athletes in M 3Tests in M, A 72 fractions 2F 1 in M 4Batches 2Rounds in B 9Locations in B, R 72 locations How do:  the locations sources;  Fractions; affect the response? Similar to model of convenience

27 6)Summary Factor-allocation, rather than single-set, description used, being more informative and particularly helpful in multiphase experiments. Multiphase experiments usually have multiple allocations. Have provided 4 standard principles and 8 principles specific to orthogonal, multiphase designs. In practice, will be important to have some idea of likely sources of laboratory variation. Are laboratory treatments to be incorporated? Will laboratory replicates be necessary?

28 References Brien, C. J. (1983). Analysis of variance tables based on experimental structure. Biometrics, 39, 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. J. Agr. Biol. Env. Stat., 14, Brien, C.J., Harch, B.D., Correll, R.L. and Bailey, R.A. (2010) Multiphase experiments with laboratory phases subsequent to the initial phase. I. Orthogonal designs. Journal of Agricultural, Biological and Environmental Statistics, submitted. Littell, R. C., G. A. Milliken, et al. (2006). SAS for Mixed Models. Cary, N.C., SAS Press. Nelder, J. A. (1965). The analysis of randomized experiments with orthogonal block structure. Proceedings of the Royal Society of London, Series A, 283(1393), , Peeling, P., B. Dawson, et al. (2009). Training Surface and Intensity: Inflammation, Hemolysis, and Hepcidin Expression. Medicine & Science in Sports & Exercise, 41, Searle, S. R., G. Casella, et al. (1992). Variance components. New York, Wiley.

29 Web address for link to Multitiered experiments site