Dissenys de Plackett-Burman
Els dissenys de Plackett-Burman són dissenys experimentals presentats el 1946 per Robin L. Plackett i J. P. Burman mentre treballaven al Ministeri d'Abastaments britànic.[1] El seu objectiu era trobar dissenys experimentals que permetessin investigar la dependència d'alguna quantitat mesurada (resposta) respecte a un nombre d'independents variables (factors), cadascun d'ells amb L nivells, d'una manera tal que la variància de les estimacions d'aquestes dependències sigui mínima i utilitzant un nombre limitat d'experiments.
| Experiment | X1 | X₂ | X₃ | X₄ | X₅ | X₆ | X₇ | X₈ | X9 | X10 | X11 |
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| 1 | + | + | + | + | + | + | + | + | + | + | + |
| 2 | − | + | − | + | + | + | − | − | − | + | − |
| 3 | − | − | + | − | + | + | + | − | − | − | + |
| 4 | + | − | − | + | − | + | + | + | − | − | − |
| 5 | − | + | − | − | + | − | + | + | + | − | − |
| 6 | − | − | + | − | − | + | − | + | + | + | − |
| 7 | − | − | − | + | − | − | + | − | + | + | + |
| 8 | + | − | − | − | + | − | − | + | − | + | + |
| 9 | + | + | − | − | − | + | − | − | + | − | + |
| 10 | + | + | + | − | − | − | + | − | − | + | − |
| 11 | − | + | + | + | − | − | − | + | − | − | + |
| 12 | + | − | + | + | + | − | − | − | + | − | − |
Totes les interaccions entre els factors es consideren insignificants. La solució a aquest problema és trobar un disseny experimental on cada combinació de nivells per a qualsevol parell de factors aparegui en el disseny el mateix nombre de vegades en el conjunt d'experiments a realitzar. Un disseny factorial complet podria satisfer aquest criteri, però la idea era trobar dissenys més petits.
Si cada factor té de dos nivells (L = 2), Plackett i Burman van emprar el mètode trobat el 1933 per Raymond Paley per a la generació de matrius ortogonals en les que tots els elements són +1 o -1 (matrius de Hadamard). El mètode de Paley podria ser usat per trobar aquest tipus de matrius de mida N per a tot N igual a un múltiple de 4. En particular van treballar per a valors de N fins a 100 excepte N = 92. Si N és una potència de 2, el disseny resultant és idèntic a un disseny factorial fraccionat, de manera que els dissenys Plackett-Burman s'utilitzen sobretot quan N és un múltiple de 4 però no una potència de 2 (és a dir, N = 12, 20, 24, 28, 36 ...).[3] En el cas de voler estimar menys de N paràmetres (incloent la mitja general), simplement cal utilitzar un subconjunt de les columnes de la matriu o bé considerar factors ficticis (“dummy”).
En cas de més de dos nivells, Plackett i Burman varen redescobrir els dissenys que ja havien estat descrits per Raj Chandra Bose i K. Kishen de l'Institut d'Estadística de la India.[4] Plackett i Burman recopilaren els dissenys que tenen un nombre d'experiments iguals el nombre de nivells L, per a L = 3, 4, 5, o 7.
Quan les interaccions entre els factors no són menyspreables, en els dissenys Plackett-Burman es confonen amb els efectes principals, el que significa que els dissenys no permeten distingir entre efectes principals i efectes de les interaccions. Això es coneix amb el nom de “aliasing” o confusió.
Construcció i aplicacions
[modifica]Per construir un disseny de Plackett-Burman un cop establert el nombre d'experiments a realitzar o el nombre de factors a estudiar hom pot trobar a la bibliografia els codis de nivell de l'experiment base (filera de la matriu de disseny).[1][5]
Nombre Nombre
d'experiments de Factors Experiment base
4 3 + + –
8 7 + + + – + – –
12 11 + + – + + + – – – + –
16 15 + + + + – + – + + – – + – – –
20 19 + + – – + + + + – + – + – – – – + + –
24 23 + + + + + – + – + + – – + + – – + – + – – – –
El procediment és molt senzill. Per exemple, per a 7 factors (8 experiments) s'escriu la primera fila amb els codis de l'experiment base. La segona fila s'omple amb els codis de la primera desplaçats cap a la dreta (o l'esquerra, és indiferent) en una posició. El procediment es repeteix fins a la setena fila i la vuitena filera que s'omple amb tots els codis negatius.
Els dissenys de Plackett-Burman són extraordinàriament útils i econòmics quan es vol investigar un gran nombre d'efectes principals, assumint que els efectes de les interaccions són poc importants comparats amb el dels factors. Una aplicació immediata és el cribratge inicial de factors en qualsevol recerca experimental.
El 1993, Dennis Lin va descriure un mètode de construcció de dissenys a partir de dissenys de Plackett-Burman utilitzant una columna per prendre la meitat de la resta de les columnes.[6] La matriu resultant, menys aquesta columna, és un "disseny supersaturat" per trobar efectes significatius de primer ordre, sota el supòsit que existeixen pocs.[7]
Matrius de Plackett-Burman de 4 fins a 48 experiments ordenades per mostrar les mitges fraccions
[modifica]N = 4 experiments; F = 3 Factors
1a fracció 2ª fracció
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N = 24 experiments; F = 23 Factors
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N = 28 experiments; F = 27 Factors
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N = 44 experiments; F = 43 Factors
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– – – – + + – + – + + – – + – – + – + – – + + + – + + + + + – – – + – + + + – – – – +
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N = 48 experiments; F = 47 Factors
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Referències
[modifica]- ↑ 1,0 1,1 Plackett, R. L.; Burman, J. P «The Design of Optimum Multifactorial Experiments». Biometrika, 33, (4), 1946, pàg. 305 - 325. DOI: 10.1093/biomet/33.4.305.
- ↑ Plackett–Burman designs NIST/SEMATECH e-Handbook of Statistical Methods.
- ↑ Ledolter, J.; Swersey, A. J. Testing 1-2-3: experimental design with applications in marketing and service operations. Palo Alto (CA): Stanford University Press, 2007. ISBN 978-0-8047-5612-9.
- ↑ Bose, R. C:; Kishen, K «On the problem of confounding in the general symmetrical factorial design». Sankhya, 5, 1940, pàg. 21 - 36. JSTOR: 25047628.
- ↑ Walpole, R. E.; Myers, R. H.; Myers, S. L.; Ye, K. Probabilidad y Estadística para Ingeniería y Ciencias. 8ª Ed.. Naucalpan de Juárez, México: Pearson Educacion, 2007. ISBN 978-970-26-0936-0.
- ↑ Lin, D. K. J «A new class of supersaturated designs». Technometrics, 35, 1993, pàg. 28 - 31.
- ↑ Gupta, V. K.; Rajender, Parsad; Basudev, Kole; et al. «Supersaturated Designs». [Consulta: 11 gener 2016].