Design of Experiments An Introduction

Experiment Why? To uncover the truth. Truth: something that is not already known. We do it very often! Children learn about hot stove. Children test parents. Students test teachers.

Experiment How do you get your best listening satisfaction? Manipulate DENON DRA-685 Stereo Receiver How do you get your best listening satisfaction? Manipulate Volume Knob (low – high) Tone Knob (Low – High) Balance Knob (Left – Right)

Systematic Experiments Problem: How can we have the best sound effect for my Stereo system? Dependent Variable: How do we measure? – listening satisfaction Control (Independent) Variables: How we set them? Volume (low – high) Tone (Low – High) Balance (Left – Right Speakers)

True Experiment A true experiment is a study Independent variables are manipulated. Levels (values) of independent variables are assigned at random to the experiment units. Their effect on dependent variables is determined.

Three Steps Experiment Statement of Problem Choice of Response (dependent variable). Selection of factors (independent variables) to be varied. Choices of levels of these factors Design Number of observation to be taken Order of experimentation: Complete randomization. Mathematical model to describe the experiment. Hypothesis to be test. Analysis Data collection and processing. Computation of test statistics and preparation of graphics. Interpretation of results.

DOE Investigation Process Analysis and interpretation of experimental results Restate problem Perform Experiment Design of Experiment

Example-one factor A manufacturer wants to know if any of their four types fabrics, A, B, C, and D resists wear better.   There is a machine for wear testing that fabrics can go though a set length of wearing cycle to measure the fabric weight loss. Dependent variable: Weight loss of material in grams Independent variable: Fabric type, 4 levels: A, B, C, D 4 observations for each fabric. A total of 16 samples. Complete randomization

Experiment Data Table Factor Fabric Type Treatment (level #1, A)   Factor Fabric Type Treatment (level #1, A) Treatment (level #2, B) Treatment (level #3, C) Treatment (level #4, D) Observation #1 Observation #2 Observation #3 Observation #4

Hypothesis We want to see if different fabric makes difference?   Ho: no difference among the 4 fabrics. H1: At least one fabric is different.

Model Description An observation, Yij, for the ith observation and jth treatment may contain three parts:   Yij = m + tj + eij m: a common effect for the whole experiment, often fixed parameter tj: the effect of the jth treatment, tj = mj - m eij: random error present in the ith observation and jth treatment. NID

Analysis ANOVA Ho: t1 = t2 = t3 = t4   Ho: t1 = t2 = t3 = t4 H1: At least one treatment effect  0

Experimental Data on Weight Loss   Treatment Observation A B C D 1 1.93 2.55 2.4 2.33 2 2.38 2.72 2.68 3 2.2 2.75 2.31 2.28 4 2.25 2.7 How do we analyze the data and test if any of these fabric has better wear resistance? ANOVA

  1 2 3 4 Obs #1 Y11 Y12 Y13 Y14 Obs #2 Y21 Y22 Y23 Y24 Obs#3 Y31 Y32 Y33 Y34 Obs #4 Y41 Y42 Y43 Column total T.1 T.2 T.3 T.4 Number Mean

1 2 3 4 Obs #1 Y11 Y12 Y13 Y14 Obs #2 Y21 Y22 Y23 Y24 Obs#3 Y31 Y32   1 2 3 4 Obs #1 Y11 Y12 Y13 Y14 Obs #2 Y21 Y22 Y23 Y24 Obs#3 Y31 Y32 Y33 Y34 Obs #4 Y41 Y42 Y43 Mean  ~N(0,2) If 1= 2 = 3= = 4=0, e.1, e.2, e.3,e.4 should also follow  ~N(0,2)

If we test against If the ratio of them is 1,

One-way ANOVA Source SS df MS F p Treatment, t 0.5201 3 0.1734 8.53 0.0026 Error, e 0.2438 12 0.0203 Total 0.7639 f-statistic = 0.1734/0.0203 = 8.53; P(F(3,12) > 8.53) = 0.0026 Ho can be rejected for any  > 0.0026. There is a significant difference in the wear resistance among the 4 fabric. Factor fabric makes difference. Which (type of) fabric is better? Don’t know!