Bailey, Rosemary

Brien, Chris

Correll, Ray

Harch, Bronwyn

Queensland University of Technology ePrints Archive

Principles in the design of multiphase experiments with a later laboratory phase: orthogonal designsChris Brien1, Bronwyn Harch2, Ray Correll2& Rosemary Bailey31University of South Australia, 2CSIRO Mathematics, Informatics & Statistics, 3Queen Mary University of LondonChris.brien@unisa.edu.auhttp://chris.brien.name/multitierOutline1. Primary experimental design principles2. Factor-allocation description for standard designs.3. Principles for simple multiphase experiments.4. Principles leading to complications, even with orthogonality.5. Summary21) Primary experimental design principles Principle 1 (Evaluate designs with skeleton ANOVA tables)¾ 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.3A 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 differences¾ intensities are only included to observe the surfaces over a range of intensities. Testing is to be conducted over 4 Months: ¾ In each month, 3 endurance athletes are to be recruited.¾ 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.¾ A split-unit design, employing Principles 2, 3 and 4(iii).4Peeling 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:¾ treatments are whatever are allocated;¾ units are what treatments are allocated to;¾ treatments and units each referred to as a set of objects; Often do by randomization using a permutation of the units.¾ 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:¾ Unit or unallocated factors (indexing units);¾ Treatment or allocated factors (indexing treatments). Represent the allocation using factor-allocation diagrams that have a panel for each set of objects with:¾ 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 experiment6 One allocation (randomization): ¾ a set of training conditions to a set of tests.3 Intensities3 Surfaces9 training conditions4 Months3 Athletes in M3 Tests in M, A36 tests The set of factors belonging to a set of objects forms a tier:¾ they have the same status in the allocation (randomization): ¾ {Intensities, Surfaces} or {Months, Athletes, Tests} ¾ 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):¾ Months, Months∧Athletes, Months∧Athletes∧Tests;¾ Intensities, Surfaces, Intensities∧Surfaces. Corresponding types of entities (groupings of objects):¾ month, athlete, test (last two are Eus);¾ intensity, surface, training condition (intensity-surface combination). Corresponding sources (in an ANOVA):¾ Months, Athletes[M], Tests[M∧A];¾ Intensities, Surfaces, Intensities#Surfaces.73 Intensities3 Surfaces9 training conditions4 Months3 Athletes in M3 Tests in M, A36 testsSkeleton ANOVA Intensities is confounded with the more-variable Athletes[M] & Surfaces with Tests[M^A]. 8training conditions tier source df   Intensities 2 Residual 6 Surfaces 2 I#S 4 Residual 18  tests tier source df Months 3 Athletes[M] 8   Tests[M∧A] 24      3 Intensities3 Surfaces9 training conditions4 Months3 Athletes in M3 Tests in M, A36 testsE[MSq] 2 2 2MAT MA Mσ σ σ  1 3 9  ( )I1 3 q μ  1 3  ( )S1 q μ  ( )IS1 q μ  1   3) Principles for simple multiphase experiments Suppose in the athlete training experiment:¾ in addition to heart rate taken immediately upon completion of atest,¾ the free haemoglobin is to be measured using blood specimens taken from the athletes after each test, and ¾ the specimens are transported to the laboratory for analysis. The experiment is two phase: testing and laboratory phases.¾ The outcome of the testing phase is heart rate and a blood specimen.¾ The outcome of the laboratory phase is the free haemoglobin. How to process the specimens from the first phase in the laboratory phase?9Some 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.¾ 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): ¾ Confound big first-phase sources with big laboratory sources, provided no confounding of treatment with first-phase sources.10A simple two-phase athlete training experiment Simplest is to randomize specimens from a test to locations (in time or space) during the laboratory phase.113 Intensities3 Surfaces9 training conditions4 Months3 Athletes in M3 Tests in M, A36 tests36 Locations36 locationstraining conditions tier source df   Intensities 2 Residual 6 Surfaces 2 I#S 4 Residual 18 tests tier source df Months 3 Athletes[M] 8   Tests[M∧A] 24     locations tier source df Locations 35           E[MSq] 2 2 2 2L MAT MA Mσ σ σ σ  1 3 91  ( )I1 3 q1 μ  1 31  ( )S1 q1 μ  ( )IS1 q1 μ  11  Composed randomizationsA 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. 12training conditions tier source df   Intensities 2 Residual 6 Surfaces 2 I#S 4 Residual 18  tests tier source df Months 3 Athletes[M] 8   Tests[M∧A] 24      locations tier source df Locations 35            E[MSq] 2 2 2 2L MAT MA Mσ σ σ σ  1 1 3 9  ( )I1 1 3 q μ  1 1 3  ( )S1 1 q μ  ( )IS1 1 q μ  1 1   A simple two-phase athlete training experiment (cont’d) Simplest is to align lab-phase and first-phase blocking.133 Intensities3 Surfaces9 training conditions4 Months3 Athletes in M3 Tests in M, A36 tests4 Batches9 Locations in B36 locationstraining conditions tier source df   Intensities 2 Residual 6 Surfaces 2 I#S 4 Residual 18  tests tier source df Months 3 Athletes[M] 8   Tests[M∧A] 24      locations tier source df Batches 3 Locations[B] 32          E[MSq] 2 2 2 2 2BL B MAT MA Mσ σ σ σ σ  1 1 3 99  ( )I1 1 3 q μ  1 1 3( )S1 1 q μ  ( )IS1 1 q μ  1 1 Note Months confounded with Batches (i.e. Big with Big).Composed randomizationsThe 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. 14training conditions tier source df   Intensities 2 Residual 6 Surfaces 2 I#S 4 Residual 18  tests tier source df Months 3 Athletes[M] 8   Tests[M∧A] 24      locations tier source df Batches 3 Locations[B] 32          E[MSq] 2 2 2 2 2BL B MAT MA Mσ σ σ σ σ  1 9 1 3 9  ( )I1 1 3 q μ  1 1 3( )S1 1 q μ  ( )IS1 1 q μ  1 1 DF for first phase sources unaffected.4) Principles leading to complications, even with orthogonality Principle 9 (Use pseudofactors): ¾ An elegant way to split sources (as opposed to introducing grouping factors unconnected to real sources of variability). Principle 10 (Compensating across phases):¾ Sometimes, if something is confounded with more variable first-phase source, can confound with less variable lab source. Principle 11 (Laboratory replication): ¾ Replicate laboratory analysis of first-phase units if lab variability much greater than 1st-phase variation;¾ 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): ¾ Sometimes treatments are introduced in the laboratory phase and this involves extra randomization. 15165) Summary 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?17References 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., 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, available online. Littell, R. C., G. A. Milliken, et al. (2006). SAS for Mixed Models. Cary, N.C., SAS Press. Peeling, P., B. Dawson, et al. (2009). Training Surface and Intensity: Inflammation, Hemolysis, and Hepcidin Expression. Medicine & Science in Sports & Exercise, 41, 1138-1145. Searle, S. R., G. Casella, et al. (1992). Variance components. New York, Wiley.Web address for link to Multitiered experiments site: http://chris.brien.name/multitier

Principles in the design of multiphase experiments with a later laboratory phase: Orthogonal designs

Name not available

http://eprints.qut.edu.au/78546/1/brienc.pdf

Principles in the design of multiphase experiments with a later laboratory phase: Orthogonal designs

Abstract

Similar works

Full text

Available Versions

Queensland University of Technology ePrints Archive

Name not available