Designs with One Source of Variation

PhD seminar31/01/2014

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• Introduction• Randomization• Model for a Completely Randomized Design• Estimation of parameters• One-Way Analysis of Variance• Sample Sizes

Contents

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• Treatment: a level of a factor in a single factor experiment (or a combination in a factorial experiment).

Introduction (1)

Algorithms(Treatments)

Dataset(Experimental Unit)

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Introudction (2)

• Experimental Design

– the rule that determines the assignment of the experimental units to treatments.

– The simplest: completely randomized design

• One Source of Variation– The variation of the factor of interest (levels).

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• Randomization Design :– Consists of assigning the experimental units to the

treatments in a random manner.

– Reduces (and with a bit of luck, removes) the selection bias and/or accidental bias and tends to 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]

Randomized Design

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Model for a Completely Randomized Design

Cause X “” Effect Y

Observed

Y = X + ε

Controlled variable

•Independent Variable•Factor•Treatment

•Dependent Variable•Observations•Response variable

•External factors•Not controlled •Ignored

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Model for a Completely Randomized Design

Experimental Unit 1

y1

y2

yr

ObservationsTreatment 1

ε

Experimental Unit 2

y1

y2

yr

ObservationsTreatment 2

ε

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• Linear statistical model

Yit -is the observation of the t-th repetition of the i-th treatment

μi - is the mean of the i-th treatment

ϵit - experimental error

Model for a Completely Randomized Design

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• Full modelEach population defined by the treatment has his own mean

• Reduced model

There is no difference between the means of the populations

Figure from: [http://www.dpye.iimas.unam.mx/patricia/indexer/completamente_al_azar.pdf]

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Model for a Completely Randomized Design

• Linear statistical model

μ - is the general mean (common to all experimental units - homogeneous)

τi - effect of the i-th treatment

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• Knows as: One-way analysis of variance model– The model includes only one major source of

variation (treatment)– The analysis of data involves a comparition of

measures of variation.

Model for a Completely Randomized Design

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Estimation of parameters

Figure from: [Dean, A. Voss, D. ,Design and Analysis of Experiments]

• Least Squares (balanced model)

• Maximum-Likelihood Estimation (MLE) (unbalanced model)

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• In an experiment involving several treatments the treatments differ at all in terms of their effects on the response variable?

H0:{τ1= τ2=…= τv}

HA:{at least two of the τi’s differ}

One-Way Analysis of Variance

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• Compare the sum of squares for errors (ssE) between the full and reduced model.

H0 is false if ssE(fullModel) << ssE(reducedModel)

H0 is true if ssE(fullModel) ≈ ssE(reducedModel)

• ANOVA

One-Way Analysis of Variance

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• Before an experiment can be run, it is necessary to determine the number of observations that should be taken on each treatment.

• Consider time and money

• Two methods: – Specifying the desired length of confidence intervals– Specifying the power required of the analysis of

variance.

Sample sizes

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• Sample Sizes using power of a test– The power of a test at Δ, denoted π(Δ), is the

probability of rejecting H0 when the effects of two treatments differ by Δ.

Sample sizes

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• 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