A Conceptual Basis and an Approach to the
             Planning of Experiments

                         João Gilberto Cor...
Approach

 Experiment - Method of scientific research for inferences on causal
  relations of characteristics of the units...
­   Procedure: In each one of 3 farms of the region of production of meat
    lambs, 24 animals of the local race are plac...
Treatment – A level of a treatment factor in the sample.
    Experimental condition – A level of an experimental factor in...
­   Procedures: 1) control of experimental techniques, 2) local control, 3)
    statistical control and 4) randomization.
...
Control of experimental techniques is restricted - it must be effected
  until the point where it does not harm the repres...
­   Wide range technological experiments – the local control considers the
    classification of the observation units in ...
Local control and statistical control have restrictions - they imply loss
  of information on the experimental error § the...
Randomization of the treatments – Random association of the
     experimental units to the treatments § experimental units...
Classification of the extraneous characteristics according to the
  procedures of experimental control that affect them:
 ...
Stratum of the experimental error - Fraction of the experimental
  error that corresponds to a unit factor.
    Example – ...
Origins of the experimental condition structure and of the unit
  structure:



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A Conceptual Basis and an Approach to the Planning of Experiments

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Texts of Experimental Statistics emphasize the statistical analysis of experiments and make only references to the conceptual and methodological foundations of the experimental research. Basic concepts are defined in a vague, incoherent and incomplete way, which drives to the incomprehension of their meanings. That is the case, for instance, of the concepts of experimental material, experimental unit and experimental error. The lack of understanding of the foundations of the experimental research is an important origin of flaws of the planning of experiments that imply inadequate analysis and inefficiency of many researches. This article proposes a reformation of important concepts with the purpose of establishing a rational, coherent and complete conceptual, methodological and computational basis for the experimental research. The approach to the generation of the experimental design is based on the separate definitions of the structures of the experimental factors and of the unit factors, and the association between these two structures determined by randomization. It leads to a clear identification of the confounding of effects of these two structures and of the appropriated errors for inferences about the experimental factors. Algebraic representation through the operators of Wilkinson & Rogers and graphical representations through Hasse diagrams are used to describe the structures. A classic example is considered to contextualize the proposed approach.

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A Conceptual Basis and an Approach to the Planning of Experiments

  1. 1. A Conceptual Basis and an Approach to the Planning of Experiments João Gilberto Corrêa da Silva Universidade Federal de Pelotas - E-mail: jgcs@ufpel.edu.br Introduction The texts and the teaching of Experimental Statistics: ­ emphasize the statistical analysis of experiments, ­ consider superficially the conceptual and methodological basis of the planning of experiments. Definitions of basic concepts are inaccurate, incomplete and ambiguous: • experimental material • experimental factor • experimental unity • experimental error • unity factor • experimental design ³ Consequences: • misunderstanding of their meaning and incorrect application, • flaws in the planning and analysis of experiments, • biased inferences, • waste of resources. ­ The dissatisfaction with these concepts is salient in the literature. For example, Brien & Demétrio (1998) revise and discuss the diversity of opinions regarding the identification of the experimental unit in grazing experiments. ­ Silva (1997) suggests a conceptual basis for the experimental research, founded on a revision of the concepts that are presented in the literature. . ­ Silva (1999) discusses the flaws that can originate from the usual concepts and presents procedures for inferences based on Silva (1997).. ­ Purpose of this presentation: revision of concepts and methods that are basic for the experimental research and particularly for the design of experiments.
  2. 2. Approach Experiment - Method of scientific research for inferences on causal relations of characteristics of the units of a target population: Á Relations between response characteristics and explanatory characteristics, in the presence of extraneous characteristics. ­ Difficulty for the inferences: the units of the sample, as well as the unities of the target population, are complex systems of characteristics that interact dynamically in the space and the time. ³ Confounding of the effects of explanatory characteristics with the effects of extraneous characteristics. ­ Requirement: concepts must consider the complexity of the relation of characteristics that is the object of the inferences. ³ Complete identification of the extraneous characteristics of the sample. ­ Strategy: identification of the aggregates of these characteristics and their successive decomposition, until the identification of the relevant ones. Example - Experiment: Effect of energetic diet and of growth promoter on the corporal development of male lambs between weaning and slaughter. Response characteristics - characteristics of the animal after the beginning of the experimental period. Important ones: partial body weight and partial body weight gain at intervals of 14 days from wean to slaughter, final body weight, body weight gain and carcass yield. Explanatory characteristics: • diet – levels in the target population: levels of metabolizable energy of the interval [2,4; 2,8 Mcal/kg DM]; levels in the sample: 2,4, 2,6 and 2,8 Mcal/kg DM; • antibiotic – levels in the target population and in the sample: with and without antibiotic. • race – levels in the target population and in the sample: Ideal, Texel and Suffolk. Extraneous characteristics of the sample – the characteristics of the sample, excepted the response and explanatory ones: characteristics of the animal at the beginning of the experimental period (genotype, phenotype, health, age, weight, etc.), the environment (weather, illnesses, parasites, predators, etc.), the management (provision of water and ration, protection against pests, illnesses predators, etc.) and the process of measurement. 2
  3. 3. ­ Procedure: In each one of 3 farms of the region of production of meat lambs, 24 animals of the local race are placed in 12 pens for 2 animals. Four pens (each with 2 animals) are assigned to each of the 3 diets, and the 2 animals of each pen are assigned to the 2 antibiotic levels. The body weight of the animals is measured at the beginning and at intervals of 14 days of the experimental period; the carcass weight is measured immediately after slaughter. Conceptual Basis Experimental material – The sample, which is composed of the three classes of characteristics: response, explanatory and extraneous. Experimental factor – An explanatory characteristic whose levels in the sample have the following properties: x they are chosen and defined for each unit; y they are a small set and they are repeated in these units; z they constitute a partition of these units; and { their relations with the levels of the other experimental factors constitute a significant structure for the objectives of the experiment. Treatment factor – An experimental factor whose levels are associated to the sample unities by a random process. ­ Attributes of a treatment factor: x its levels can be associated to any of the experimental units for the factor; y they are associated to these units by a random process; z they are explicit distinct stimulations. Intrinsic factor – An experimental factor inherent to the sample units: • it does not satisfy to the first and second attributes of a treatment factor - it is not subjected to randomization. Example – Experimental factor – treatment factors: diet and antibiotic; intrinsic factor: race. 3
  4. 4. Treatment – A level of a treatment factor in the sample. Experimental condition – A level of an experimental factor in the sample. Experimental unit for an experimental factor – The smallest fraction of the experimental material that is associated to a level of this factor, independently of the other fractions. • Association is determined by randomization – for treatment factor, or inherent to the experimental units – for intrinsic factor. Example – Experimental unit for the factor diet: a pen and the respective animals with the levels of the characteristics of the experimental material that correspond to them: the levels of the explanatory characteristics: the race of the animals and the diet and the antibiotics that they receive; the levels of the extraneous characteristic: of the animals, the environment, the management and the measurement; and the levels of the response characteristics; for the antibiotic factor: an animal with the corresponding levels of the characteristic of the experimental material; for the factor race: a site, that is, the set of pens and respective animals of a race, with the levels of the characteristics of the experimental material that correspond to them. ­ Identification of the experimental unit: • complete - description of all the characteristics of the fraction of the experimental material that correspond to the unit. • abbreviate - by means of its basic component: pen, animal, site,... Formation of experimental units – The set of the experimental units for an experimental factor. Number of replications of an experimental condition – The number of experimental units with this experimental condition. Unit of observation for a response characteristic – The fraction of the experimental material where an individual measurement of this characteristic is made, independently of the other fractions. Experimental error – The effect of the extraneous characteristics on the response characteristics – variation of the observed values of the response variable that is due to the extraneous characteristics. ³ Consequences for the inferences: imprecision and bias. Experimental control – The set of the actions exerted for the control of the experimental error. ­ Purpose – to diminish and to render unbiased the confounding of the effects of experimental factors with the effects of extraneous characteristics. 4
  5. 5. ­ Procedures: 1) control of experimental techniques, 2) local control, 3) statistical control and 4) randomization. ­ Requirement - identification of the extraneous characteristics of the sample that can cause relevant confounding. ­ Strategy – begin with the listing of the great aggregates of the extraneous characteristics and decompose progressively each of these aggregates, until the identification of the potentially relevant characteristics. Control of experimental techniques – The set of the actions exerted for the control of extraneous characteristic with the intention of diminishing their effects on the response characteristics. ­ Implications for the sample: the fractions of the extraneous characteristic that are controlled are excluded from the sample. ³ The control of experimental techniques molds the sample - it can harm the representativeness of the sample. Example – Choice of animals of each race with similar characteristics and use of uniform pens in each site, homogenization of the management techniques, and use of uniform and accurate measuring processes.
  6. 6. Control of experimental techniques is restricted - it must be effected until the point where it does not harm the representativeness of the sample. ­ Applicable procedures to the extraneous characteristics whose control by experimental techniques is not appropriate: local control and statistical control. Local control – Classification of the observation units according to the levels of one or more relevant extraneous characteristics and association of the observation units to the experimental conditions in such a way that the effects of these extraneous characteristics are not confounded with important effects of the experimental factors. ³ The effects of the extraneous characteristics that are controlled are separated from the experimental error that affects important effects of the experimental factors. ­ Extraneous characteristics proper for local control – characteristics that correspond to natural or convenient classifications of the observation units that can constitute relevant sources of variation. ­ Efficient local control considers the largest gamma of the relevant extraneous characteristics with the smaller number of classifications. 5
  7. 7. ­ Wide range technological experiments – the local control considers the classification of the observation units in the space and the time, and the additional relevant classifications in each section of the space and the time. Example – If the amplitude of the dates of weaning of the lambs of each race is considerable and the pens of each farm are heterogeneous, it is convenient to proceed the local control of these characteristics in each site by a single formation of blocks, as follows: a) classification of the 24 animals in 4 groups of 6 animals of next dates of weaning and classification of the 12 pens in 4 groups of 3 homogeneous pens; b) constitution of blocks of animals and pens by the allocation of the 4 groups of animals to the 4 groups of pens, in such a way that in each block results 3 pens of 2 animals; c) assignment of the 3 pens of each block with the respective animals to the 3 diets, and of the 2 animals of each pen to the 2 levels of the antibiotic factor. Statistical control – Register of the observed values of one or more variables that express relevant extraneous characteristics, which are called extraneous co-variables, and adjustment of the observed values of response variables for the elimination of the variation due to these co-variables. Example – If the amplitude of the dates of weaning of the lambs of each race is not considerable, the local control of this extraneous characteristic may not be appropriate. The statistical control of the initial corporal weight can be more convenient. ­ Local control and statistical control eliminate from the effects of the experimental factors and separate from the experimental error that affects these effects the effect due to the controlled extraneous characteristics.
  8. 8. Local control and statistical control have restrictions - they imply loss of information on the experimental error § they can control few extraneous characteristics of the sample. ­ Resource to prevent the bias from the remaining confounding: randomization. Randomization - Random association of the extraneous characteristics to the experimental conditions. ­ Purpose - to prevent bias from the confounding of the effects of the extraneous characteristics not controlled by local control and statistical control with the effects of the experimental factors. ­ Randomization can be achieved by: randomization of the treatments and randomization of experimental techniques. 6
  9. 9. Randomization of the treatments – Random association of the experimental units to the treatments § experimental units have equal probability of allocation to any of the treatments § Treatments have equal probability of being favoured or disfavoured by the effects of the extraneous characteristics that are randomized. ³ The confounding of the treatment effects with the effects of the randomized characteristic is not biased. ­ Intrinsic factor is not subject to randomization – the confounding of its effects with effects of extraneous characteristic can be biased. Randomization of experimental techniques – Execution of experimental techniques in random order § the experimental conditions have equal probability of being favoured or disfavoured by the order of implementation. ³ The confounding of the effects of experimental factors with the effects of the randomized extraneous characteristics is not biased. ­ Randomization is restricted by the local control, to guarantee the arrangement of the treatments in the experimental units that it determines, and by the formations of experimental units. Example – The randomization must consider the classification of the observation units according to the blocks of pens and the respective animals and take into account the formations of experimental units for the experimental factors: diet, antibiotic and race. Then, for each site, the 3 pens of each block are associated randomly to the 3 levels of the factor diet, separately and independently for each block; and the 2 animals of each pen are associated randomly to the 2 levels of the antibiotic factor, separately and independently for each pen.
  10. 10. Classification of the extraneous characteristics according to the procedures of experimental control that affect them: • controlled ← local control or statistical control; • randomized ← randomization; • potentially disturbing ← not controlled and not randomized: • irrelevant – behaviour as randomized; • relevant - disturbing. Unit factor – Extraneous characteristic whose levels have the following properties: x they form a partition of the set of the observation units; y their relations with the levels of the other unit factors constitute a structure that expresses the relevant effects of the extraneous characteristics. ­ A unity factor stratifies the experimental error. 7
  11. 11. Stratum of the experimental error - Fraction of the experimental error that corresponds to a unit factor. Example – Unit factors: lamb, pen, block and site. Experimental error that affects an effect of experimental factors – The effect of the extraneous characteristics that is confounded with this effect of experimental factors and that is not controlled by local control and statistical control. ­ The experimental error that affects effects of experimental factors is reduced by the local control and the statistical control. • Composition of this experimental error: randomized and potentially disturbing extraneous characteristics. • Basic assumption: absence of disturbing extraneous characteristics - random experimental error. Generation of the Experimental Design Structure of the experimental conditions – Organization of the experimental conditions that expresses the relations between the experimental factors. ³ It is derived from the objective of the experiment. ­ Process of generation: 1 – choice of the experimental factors; 2 – choice of the levels of these factors; 3 – choice of the combinations of levels; 4 – choice of additional treatments. ­ Kinds of structures: unifatorial, crossed factorial, nested or hierarchic factorial and mixed factorial. Example – Crossed factorial structure of three factors: D - diet, A - antibiotic and R - race: D*A*R. Structure of the units - Organization of the observation units that expresses the relations between the unit factors. ³ It is derived from the available experimental material. ­ Kinds of structures: the same as the structures of the experimental conditions. Example – Unit structure for body weight, body weight gain and carcass yield: hierarchic factorial structure of four factors: S - site, B - block, P - pen and L - lamb: S/B/P/L.
  12. 12. Origins of the experimental condition structure and of the unit structure: 8
  13. 13. Structure of the experiment – experimental design - Relation between the structure of the experimental conditions and the structure of the units, determined by the randomization of the levels of the treatment factors and the manifestation of the levels of the intrinsic factors: ­ Recommendable procedure to correctly express the structure of the experiment: separate planning of the structure of the experimental conditions and of the structure of the units. ­However, these two structures are interdependent: • the structure of the experimental conditions is conditional to the availability of experimental material and • the structure of the units must be appropriate for the structure of the experimental conditions.
  14. 14. A strategy for the generation of the experimental design: x elaborate the structure of the experimental conditions; y consider the alternative unit structures for this structure of experimental conditions; z choose between these unit structures the one that, associated to the structure of the experimental conditions, provides the most efficient inferences about the relevant effects of the experimental factors; { if a satisfactory unit structure is not found, reconsider the sequence of steps x, y and z. 9
  15. 15. ­ Steps x and y can lead to several structures of experiment. ­ General rule - choose the design that: • provides the maximum information that is relevant to the objective of the experiment with the minimum cost; • more adequately satisfies the basic principles of the experimental design: replication, local control, randomization, orthogonality, balance, confounding and efficiency.
  16. 16. Randomization establishes a correspondence between the levels of the experimental factors and the levels of the unit factors. Experimental units for an experimental factor – The levels of the unit factor that is associated with this experimental factor in the structure of the experiment. Experimental error that affects an effect of experimental factors – The extraneous variation between the experimental units for this experimental factor, excluded the variation that is controlled by local control and statistical control. ­An intrinsic experimental factor is associated with a unit factor by a one to one correspondence of their levels - their effects are completely confounded - they are equivalent factors.
  17. 17. Factorial structures are convenient represented: by symbols and by structure diagrams. Example – Structure of the experimental conditions: crossed factorial: D*A*R; structure of the units: hierarchic factorial: S/B/P/L. These two factorial structures are orthogonal. The randomization of the treatments associates the 3 diets to the 3 pens within each block and the 2 antibiotics to the 2 lambs within each pen; and the races are associated one to one to the sites. So, the structure of the experimental design is orthogonal. ­ Effects in the strata of the experiment: The orthogonality of the design implies that the effect of each experimental factor is confined to the corresponding stratum: 10
  18. 18. Effect of Stratum Unit factor Experimental factor S≡R S R B^S B[S] P^B^S≅R^D P[B^S] D, R*D L^P^B^S≅R^D^A L[P^B^S] A, R*A, D*A, R*D*A ­ Origin of the information on the effects of race: stratum S≡R. As race and farm are equivalent factors, there is no source for the estimation of the experimental error that affects the effects of the factor race. ­ Origin of the information on the effects of diet and the interaction of diet and race: stratum P^B^S≅R^D. These effects of experimental factors are confounded with the effect of the experimental error P[B^S]. However, as there are 12 replications for each level of D and 4 for each level of R^D, the experimental error that affects these effects can be estimated from the residual in this stratum. This kind of confounding is denoted by P^B^S≅R^D. ­ Origin of the information on the effects of antibiotic and its interactions with race, diet and with the interaction of race and diet: stratum L^P^B^S≅R^D^A. These effects are confounded with the effect of the experimental error L[P^B^S]. As there are replications for the factors A, R^A, D^A, R^D^A, the experimental error that affects these effects can be estimated from the residual in this stratum.
  19. 19. Estimates of the fractions of the experimental error that affect the effects of the experimental factors diet and antibiotic are provided by the residuals in the corresponding strata: • Residual in the pen stratum: variation between pens within blocks that is due to the extraneous characteristics: characteristic inherent to the animals, the environment, the management and the measurement. • Residual in the lamb stratum: variation between animals within pens that is due to the extraneous characteristics: characteristic inherent to the animals, the environment, the management and the measurement. 11
  20. 20. Conclusions • The units of the sample, as well as the units of the target population, are generally complex systems of innumerable characteristics that interact dynamically in the space and the time. • The planning of the experiment, particularly the formulation of the experimental design, must be based on concepts that take into consideration and express this complexity correctly. • This approach requires the identification of the characteristics of the sample according to their role in the causal relation that is the object of the inferences and its sufficiently complete description for the identification of the important characteristics. • The formulation of the experimental design based on the separate derivations of the structure of the experimental conditions and the structure of the units allows the clear identification of the origin of the confounding which involve the effects of experimental factors and the determination of the fractions of the experimental error that affect these effects. Acknowledgements • To the National Council of Scientific and Technological Development (CNPq- Brazil) by the support to the research project that originated this presentation. Bibliography • BAILEY, R.A. Design of comparative experiments. Cambridge, UK: Cambridge University Press, 2008. 330p. • BRIEN, C.J.; DEMÉTRIO, C.G.B. Using the randomization in specifying the ANOVA model and table for grazing trials. 1998. Available in: http://chris.brien.name/multitier/grazall.ps. Access on: 10 jun. 2007. • SILVA, J.G.C. Estatística Experimental: Planejamento de experimentos. Versão preliminar. Pelotas: Universidade Federal de Pelotas, 1997. 216p. • SILVA, J.G.C. A consideração da estrutura das unidades em inferências derivadas do experimento. Pesquisa Agropecuária Brasileira, v.34, n.6, 911-925, 1999. 12

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