Design of experiments
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1 Design of experiments CIRAD - UMR Ecologie des forêts de Guyane vivien.rossi@cirad.fr Master 2 - Ecologie des Forêts Tropicale AgroParisTech - Université Antilles-Guyane Kourou, novembre 2010
2 objectives of the course introduce Design of Experiment () 1 Basic Principles and Techniques 2 Problem formulation 3 Planning Experiments 4 Analysis data
3 outlines 1 Design Of Experiment: overview 2 Planning Experiments 3 Experimental designs 4 Examples 5 Conclusion
4 1 Design Of Experiment: overview 2 Planning Experiments 3 Experimental designs 4 Examples 5 Conclusion
5 experimental method Observation Selection of a proportion of the population and measurement or observation of the values of the variables in question for the selected elements Experimentation Manipulation of the values (or levels) of one or more (independent) variables or treatments and observation of the corresponding change in the values of one or more (dependent) variables or responses
6 Why experiment? To determine the causes of variation in the response To find conditions under which the optimal response is achieved To compare responses at different levels of controllable variables To develop a model for predicting responses
7 some definitions Treatments different combinations of conditions that we wish to test Treatment Levels the relative intensities at which a treatment will be set during the experiment Treatment Factor (or Factor) one of the controlled conditions of the experiment (these combine to form the treatments) Experimental Unit subject on which a treatment will be applied and from which a response will be elicited also called measurement or response units Experimental Design rule for assigning treatment levels to experimental units Observations outcomes that will be elicited from experimental units after treatments have been applied
8 design of experiment statement of 1 goals and condition of experiments 2 treatment factors and their levels 3 individuals, experimental units 4 observations and collect procedure 5 experimental design 6 data analysis ANOVA, regression,...
9 What characterizes a good experimental design? It avoids systematic error: systematic error leads to bias when estimating differences in responses between (i.e., comparing) treatments It allows for precise estimation: achieves a relatively small random error, which in turn depends on the random error in the responses the number of experimental units The experimental design employed It allows for proper estimation of error It has broad validity: the experimental units are a sample of the population in question
10 1 Design Of Experiment: overview 2 Planning Experiments 3 Experimental designs 4 Examples 5 Conclusion
11 Problem formulation what is the biological question? how to answer that? what is already known? what information is missing? problem formulation model of the biological system
12 Setting up an experiment what kind of data is needed to answer the question? how to collect the data? how much data is needed? biological and technical replicates pooling how to carry out the experiment (sample preparation, measurements)?
13 check list for planning experiment 1 Define the objectives of the experiment. 2 Identify all sources of variation, including: treatment factors and their levels, experimental units, blocking factors, noise factors, and covariates. 3 Choose a rule for assigning the experimental units to the treatments. 4 Specify the measurements to be made, the experimental procedure, and the anticipated difficulties. 5 Run a pilot experiment. 6 Specify the model. 7 Outline the analysis. 8 Calculate the number of observations that need to be taken. 9 Review the above decisions. Revise, if necessary.
14 1 Design Of Experiment: overview 2 Planning Experiments 3 Experimental designs 4 Examples 5 Conclusion
15 Complete factorial designs B3 B2 B1 one factor at a time B3 B2 B1 unable to assess interaction between factors A1 A2 A3 A1 A2 A3
16 Complete factorial designs B3 B2 B1 one factor at a time B3 B2 B1 unable to assess interaction between factors A1 A2 A3 A1 A2 A3 complete factorial designs (2 3 and 3 3 ) Facteur 2 Facteur 3 Facteur Facteur 3 able to assess interaction between factors need high number of treatments 1 +1 Facteur Facteur 1
17 Fractionnal factorial designs Case factors with 2 levels: A (ā, a), B ( b, b) and C ( c, c) 4 treatments: a b c, āb c, ā bc and abc c b Facteur A abc a Facteur C Facteur B (A,BC), (B,AC) and (C,AB) are aliases able to estimate principal effects if interactions are nulls
18 Fractionnal factorial designs Case factors with 3 levels: 1 (1, 2, 3), 2 (1, 2, 3) and 3 (1, 2, 3) 9 treatments: 111, 122, 133, 213, 221, 232, 312, 323, Facteur Facteur Facteur 1 (1,23), (2,13) and (3,12) are aliases able to estimate principal effects if interactions are nulls
19 Resolution of fractionnal factorial designs levels of resolution III able to estimate principal effects if interactions factors are nulls IV able to estimate principal effects if interactions between three or more factors are nulls V able to estimate principal effects and interactions between two factors if interactions between three or more factors are nulls case of 2 k designs Nb. Nb. tot. Nb. trait min factors trait III IV V
20 Experimental unit size: the smaller the better while keeping a meaning edge: avoid interferences path: close to the edge shape: square to reduce edge effects, but frame could be good in case of heterogeneity number of repetition: ensure the viability of the experiment mean estimation: n 4cv 2 /dr 2 with cv coefficient of variation and d r maximum relative error two means comparison: n 4cv 2 /δr 2 with δ r inter mean distance
21 Completely Randomized Designs goal avoid fluctuation from uncontrolled factors though time and space principle randomly affect treatment on experimental units: model: Response = constant + effect of treatment + error +/- +: very simple to implement -: may lead to abnormalities due to treatments concentration and heterogeneity
22 Randomized Blocks goal address fluctuation from uncontrolled factors by blocking homogeneous experimental units principle split experimental units into block randomly affect each treatment on experimental units into each block: Bloc 1 Bloc 2 Bloc 3 Bloc 4 Gradient model: Response = constant + effect of block + effect of treatment + error
23 Randomized Blocks: advantages very simple to implement more efficient than completely randomized design: experiment with p blocks and q treatments SSE b sum of squares relatives to blocks SSE tb sum of squares relatives to interaction treatments-blocks MSE tb = SSE tb /[(p 1)(q 1)] MSE r = (SSE b + SSE tb )/[(q 1) + (p 1)(q 1)] relative efficiency approximate by MSE r MSE tb = (p 1)( SSE b SSE tb + 1)/p. the higher SSE b the more efficient is the block design
24 Split-plot Designs goal study differently the effect of each factors principle case of 2 factors (6 and 3 levels) and 4 blocks: Bloc Bloc 2 Bloc Bloc 4
25 Split-plot Designs + allow to consider larger experimental units for first factors better accuracy for factor on the small experimental units (more repetitions) better accuracy for interaction allow to introduce a new factor during the experiment - lost in accuracy for factor on the larger experimental units (less repetitions) different number of degree of freedom
26 Cross-over Designs goal: control variability of experimental material principle latin square cross-over /- more efficient than random blocks because of double control low degree of freedom for residuals variability number of treatments = number of repetitions
27 fractionnal factorial experiment principle number total of treatments>number of experimental unit per block incomplete blocks completed by repetitions example 2 3 design with 4 experimental units per block ac bc ab (1) b a abc c a abc c b bc (1) ac ab c a b abc bc ab (1) ac 2 2 design with 2 experimental units per block b (1) b ab ab (1) a (1) (1) ab a ab a ab (1) a a b ab b a b (1) b +/- adapted to situations that need small blocks increase accuracy for a large number of treatments confounding effects data analysis is complex
28 1 Design Of Experiment: overview 2 Planning Experiments 3 Experimental designs 4 Examples 5 Conclusion
29 Completely Randomized Designs: Charcoal beech goal study the influence of the size and the moisture of the piece of wood on coal yield factors levels wood cube sizes: 2, 4 and 8 cm for edge moisture: 0%, 10%, 20% and 40% Experimental Design 36 experimental units 12 treatments (3 4) 3 repetitions
30 1 st representation of the data size moisture (cm) ,00 29,82 29,27 33,11 29,67 29,71 30,11 30,18 29,78 29,87 30,58 29, ,38 29,11 29,98 29,31 28,98 29,18 30,02 29,22 29,82 30,22 29,49 29, ,11 28,93 28,67 29,13 29,78 29,78 29,44 29,42 29,11 28,84 30,33 29,73
31 2 nd representation of the data s m k y , , , , , , , , , , , ,16 s m k y , , , , , , , , , , , ,93 s m k y , , , , , , , , , , , ,73
32 preliminary graphical data exploration
33 preliminary graphical data exploration input error for point 33.11? ANOVA...
34 Complete Block Design: Paracou question Can we possibly increase the production of timber within the managed areas without exhausting the resources? factor levels logging intensity: 3 increasing levels + control Experimental Design 12 experimental units 4 treatments 3 blocks
35 the Paracou experimental station Treatments description N/ha m 2 /ha m 3 /ha control T T T ,
36 fertilizers on wheat in Rwanda goal study the influence of fertilizers on wheat yield factors levels azote (N): 0 and 100 kg/ha potassium (K 2 O): 0 and 200 kg/ha phosphore (P 2 O 5 ): 0, 100, 200 and 300 kg/ha calcium (Ca): 1, 4.5 and 8 kg/ha Experimental Design 48 experimental units: weight of wheat (kg) for 18 m 2 (centred in the 25 m 2 plot) 16 treatments 3 blocks
37 fertilizers on wheat in Rwanda: Experimental Design Bloc 1 Bloc 2 Bloc 3
38 fertilizers on wheat in Rwanda: Experimental Design exp. unit N K 2 O P 2 O 5 Ca
39 fertilizers on wheat in Rwanda: anti-erosion hedges picture from P. Dagnelie
40 fertilizers on wheat in Rwanda: wheat yield exp.unit blocks means (kg/p) (t/ha) 1 0,00 0,04 0,06 0,03 0,02 2 0,14 0,22 0,35 0,24 0,13 3 0,42 0,45 0,44 0,44 0,24 4 0,44 0,28 0,84 0,52 0,29 5 0,28 0,49 0,33 0,37 0,20 6 1,09 1,17 0,84 1,03 0,57 7 0,79 0,94 0,82 0,85 0,47 8 1,30 0,80 2,01 1,37 0,76 9 2,05 2,37 2,52 2,31 1, ,07 2,60 2,25 2,31 1, ,36 2,71 (2,54) (1,41) 12 2,99 2,92 3,63 3,18 1, ,62 2,89 3,43 2,98 1, ,61 2,06 3,29 2,65 1, ,22 2,93 3,85 3,33 1, ,15 3,35 3,67 3,39 1,88
41 fertilizers on wheat in Rwanda: a plot with weak yield picture from P. Dagnelie
42 preliminary graphical data exploration ANOVA for exp. units 8 to 16 Df mse F P Phosphore 2 6,2949 3, ,9 *** 0,0000 Calcium 2 3,5022 1, ,2 *** 0,0000 Interaction 4 0,1525 0,0381 0,48 0,75 Blocs 2 1,9191 0,9596 residuals 15 1,1820 0,0788 totals 25 13,0507
43 Split Block criss cross: improvement of beef cattle goal compare different mixtures of forage associated with two doses of nitrogen fertilizer factor levels oats and vetch proportions: and oats variety: A,B and C dose of fertilizer: 30 and 60 N Experimental Design 32 experimental units, plots 8 20 m 16 treatments (8 mixtures 2 doses of fertilizer) 2 blocks
44 experimental design forage mixture design Mixture oats vetch A B C ,5 12,5 75 and yield t/ha ,79 8,67 7,97 7,61 8,69 10,61 7,72 8,78 6,68 9,61 3,55 4,83 4,32 7,25 5,30 3, ,03 7,16 4,92 4,63 7,70 6,36 6,14 5,79 5,52 5,81 5,07 8,16 9,12 8,85 5,57 6,19 Bloc 1 Bloc 2
45 Conclusion always plan an experiment write your : 1 objectives 2 experiment conditions 3 factors 4 treatments 5 experimental units 6 observations 7 experimental design 8 framework of data analysis
46 Some references Angela M. Dean & Daniel Voss, Design and Analysis of Experiments, Springer 2000 Box, G. E., Hunter,W.G., Hunter, J.S., Hunter,W.G., "Statistics for Experimenters: Design, Innovation, and Discovery", 2nd Edition, Wiley, 2005 Ghosh, S. and Rao, C. R., ed (1996). Design and Analysis of Experiments. Handbook of Statistics. 13. North-Holland. Jacques Goupy & Lee Creighton,Introduction aux plans d expérience Dunod/L usine nouvelle, 2006 Pierre Dagnelie,Principes d expérimentation: planification des expériences et analyse de leurs résultats, Presses agronomiques, Gembloux,
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