Unsorted Treatments Random Numbers 1 0.533 1 0.683 2 0.702 2 0.379 3 0.411 3 0.962 3 0.139 Sorted Sorted Experimental Treatments Random Units Numbers.

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Presentation transcript:

Unsorted Treatments Random Numbers Sorted Sorted Experimental Treatments Random Units Numbers Randomization

Bread Rise Experiment 1. Mix The Dough 2. Divide the dough into 12 small loaves of the same size. 3. Randomly assign 4 loaves to rise 35 minutes, 4 to rise 40 minutes, etc. 4. After allowing each loaf to rise the specified time, measure the height of the loaf.

Model for CRD Design Cell Means Model

Alternate Model Effects Model

Notation Sample means Grand Mean

Least Squares Estimates Cell Means Choose estimates to minimize

Matrix Notation for Alternate Model LS Estimators are solution to Problem is singular

SAS proc glm Non-singular

is a generalized inverse for Biased Estimates

Bread Rise Experiment

= ^

:

Matrix Notation for Estimable Functions is an unbiased estimator for when the rows of L are linear combination of the rows of for example

is a linear

ssE must represent varation in experimental units not subsamples, repeated measures or duplicates Teaching Example (illustration of problems) Classes randomized to different teaching methods experimental unit=class No replicate classes no way to compute ssE Teaching method confounded with difference in classes Use of student to student variability (i.e. subsamples) to calculate ssE Could be totally misleading

● Independence of error terms ε ij ● Equality of variance across levels of treatment factor ● Normal distribution of ε ij

Check equal variance assumption 1. plot data vs treatment factor level 2. plot residuals vs predicted values or cell means

Check normality with normal plot of residuals

ods graphics on; proc glm data=bread plots=diagnostics; class time; model height=time/solution; run; ods graphics off;

λ =

Solutions►Analysis►Design of Experiments Two-Level Factorial Response Surface MixtureMixed-Level Factorial Optimal Design Split-Plot Design General Factorial

Define Variables►Add> ►Add qualitative factorial variable

Customize…►Replicate Runs Edit Responses… Design►Randomize Design …

Fit …

Model ► Fit Details…Model►Check Assumptions►Perform Residual AnalysisModel►Check Transformation►Box-Cox Plot

Teaching Experiment Objective: Compare student satisfaction between 3 different teaching methods Experimental unit: class Two replicate classes for each teaching method. Response: rating given by each student, summarized over class as multinomial vector of counts

power, 1-β practical significance

power Size of a practical difference

3. H 0 : μ 3 = μ 4 Does a mix of artificial fertilizers enhance yield? Is there a difference in plowed and broadcast? Does Timing of Application change Yield?

^^ Option 1Option 2

Option 1

Option 2

Review Important Concepts Experimental Unit Randomization Replication Practical Difference Determining the number of replicates by calculating the power