Designs with One Source of Variation

1. DesignswithOneSource of Variation PhDseminar 31/01/2014

2. Contents • Introduction • Randomization • Modelfor a CompletelyRandomizedDesign • Estimation of parameters • One-WayAnalysis of Variance • SampleSizes

4. Introduction (1) • Treatment:a level of a factor in a single factor experiment (or a combination in a factorial experiment). Dataset (Experimental Unit) Algorithms (Treatments)

5. Introudction (2) • Experimental Design • the rule that determines theassignment of the experimental unitstotreatments. • Thesimplest: completelyrandomizeddesign • OneSource of Variation • Thevariation of the factor of interest (levels).

6. RandomizedDesign • RandomizationDesign : • Consists of assigningthe experimental unitstothetreatments in a randommanner. • Reduces (and with a bit of luck, removes) theselectionbiasand/oraccidental biasand tendsto produce comparable groups[Suresh KP 2011] • It is best used when relatively homogeneous experimental units are available • This design will normally be analysed using a one-way analysis of variance [Michael FW Festing]

7. Modelfor a CompletelyRandomizedDesign Controlled variable Observed Cause X “”Effect Y • Dependent Variable • Observations • Response variable • Independent Variable • Factor • Treatment Y = X + ε • Externalfactors • Notcontrolled • Ignored

8. Modelfor a CompletelyRandomizedDesign ε Observations Treatment 1 y1 y2 Experimental Unit 1 yr ε Observations Treatment 2 y1 y2 Experimental Unit 2 yr

9. Modelfor a CompletelyRandomizedDesign • Linear statisticalmodel Yit-istheobservation of thet-threpetition of the i-thtreatment μi - isthe mean of thei-thtreatment ϵit - experimental error

10. Fullmodel • Each population defined by the treatment has his own mean • Reducedmodel • There is no difference between the means of the populations Figure from: [http://www.dpye.iimas.unam.mx/patricia/indexer/completamente_al_azar.pdf]

11. Modelfor a CompletelyRandomizedDesign • Linear statisticalmodel • μ - isthe general mean (commontoall experimentalunits - homogeneous) τi - effect of thei-thtreatment

12. Modelfor a CompletelyRandomizedDesign • Knows as: One-wayanalysis of variancemodel • Themodelincludesonlyonemajorsource of variation (treatment) • Theanalysis of data involves a comparition of measures of variation.

13. Estimation of parameters • LeastSquares (balancedmodel) • Maximum-LikelihoodEstimation (MLE) (unbalancedmodel) Figure from: [Dean, A. Voss, D. ,Design and Analysis of Experiments]

14. One-WayAnalysis of Variance • In anexperimentinvolvingseveraltreatmentsthetreatmentsdiffer at all in terms of theireffectsonthe response variable? H0:{τ1= τ2=…= τv} HA:{at leasttwo of theτi’sdiffer}

15. One-WayAnalysis of Variance • Compare thesum of squaresforerrors (ssE) betweenthe full and reducedmodel. H0isfalseifssE(fullModel) << ssE(reducedModel) H0istrueifssE(fullModel) ≈ ssE(reducedModel) • ANOVA

16. Samplesizes • Beforeanexperiment can berun, itisnecessarytodetermine thenumber of observationsthatshouldbetakenoneachtreatment. • Considertime and money • Twomethods: • Specifyingthedesiredlength of confidenceintervals • Specifyingthepowerrequired of theanalysis of variance.

17. Samplesizes • SampleSizesusingpower of a test • Thepower of a test at Δ, denotedπ(Δ), istheprobability of rejectingH0 whentheeffects of twotreatmentsdifferbyΔ.

18. Dean, A.M., Voss, D., Design and Analysis of Experiments, Spring-Verlag, 1999 • http://www.3rs-reduction.co.uk/html/about.html • http://www.ugr.es/~bioestad/guiaspss/practica7/ArchivosAdjuntos/EfectosFijos.pdf • http://www.dpye.iimas.unam.mx/patricia/indexer/completamente_al_azar.pdf • http://www.ugr.es/~bioestad/guiaspss/practica7/ArchivosAdjuntos/EfectosFijos.pdf