ITK-226 Statistika & Rancangan Percobaan Dicky Dermawan

DOX 6E Montgomery 2 Introduction to DOX  An experiment is a test or a series of tests  Experiments are used widely in the engineering world  Process characterization & optimization  Evaluation of material properties  Product design & development  Component & system tolerance determination  “All experiments are designed experiments, some are poorly designed, some are well- designed”

DOX 6E Montgomery 3 Engineering Experiments  Reduce time to design/develop new products & processes  Improve performance of existing processes  Improve reliability and performance of products  Achieve product & process robustness  Evaluation of materials, design alternatives, setting component & system tolerances, etc.

DOX 6E Montgomery 4 Four Eras in the History of DOX  The agricultural origins, 1918 – 1940s  R. A. Fisher & his co-workers  Profound impact on agricultural science  Factorial designs, ANOVA  The first industrial era, 1951 – late 1970s  Box & Wilson, response surfaces  Applications in the chemical & process industries  The second industrial era, late 1970s – 1990  Quality improvement initiatives in many companies  Taguchi and robust parameter design, process robustness  The modern era, beginning circa 1990

DOX 6E Montgomery 5 The Basic Principles of DOX  Randomization  Running the trials in an experiment in random order  Notion of balancing out effects of “lurking” variables  Replication  Sample size (improving precision of effect estimation, estimation of error or background noise)  Replication versus repeat measurements? (see page 13)  Blocking  Dealing with nuisance factors

DOX 6E Montgomery 6 What If There Are More Than Two Factor Levels?  The t-test does not directly apply  There are lots of practical situations where there are either more than two levels of interest, or there are several factors of simultaneous interest  The analysis of variance (ANOVA) is the appropriate analysis “engine” for these types of experiments – Chapter 3, textbook  The ANOVA was developed by Fisher in the early 1920s, and initially applied to agricultural experiments  Used extensively today for industrial experiments

DOX 6E Montgomery 7 An Example (See pg. 60)  An engineer is interested in investigating the relationship between the RF power setting and the etch rate for this tool. The objective of an experiment like this is to model the relationship between etch rate and RF power, and to specify the power setting that will give a desired target etch rate.  The response variable is etch rate.  She is interested in a particular gas (C2F6) and gap (0.80 cm), and wants to test four levels of RF power: 160W, 180W, 200W, and 220W. She decided to test five wafers at each level of RF power.  The experimenter chooses 4 levels of RF power 160W, 180W, 200W, and 220W  The experiment is replicated 5 times – runs made in random order

DOX 6E Montgomery 8 The Analysis of Variance (Sec. 3-2, pg. 63)  In general, there will be a levels of the factor, or a treatments, and n replicates of the experiment, run in random order…a completely randomized design (CRD)  N = an total runs  We consider the fixed effects case…the random effects case will be discussed later  Objective is to test hypotheses about the equality of the a treatment means

DOX 6E Montgomery 9 The Analysis of Variance  The name “analysis of variance” stems from a partitioning of the total variability in the response variable into components that are consistent with a model for the experiment  The basic single-factor ANOVA model is

DOX 6E Montgomery 10 Models for the Data There are several ways to write a model for the data:

DOX 6E Montgomery 11 The Analysis of Variance  Total variability is measured by the total sum of squares:  The basic ANOVA partitioning is:

DOX 6E Montgomery 12 The Analysis of Variance  A large value of SS Treatments reflects large differences in treatment means  A small value of SS Treatments likely indicates no differences in treatment means  Formal statistical hypotheses are:

DOX 6E Montgomery 13 The Analysis of Variance  While sums of squares cannot be directly compared to test the hypothesis of equal means, mean squares can be compared.  A mean square is a sum of squares divided by its degrees of freedom:  If the treatment means are equal, the treatment and error mean squares will be (theoretically) equal.  If treatment means differ, the treatment mean square will be larger than the error mean square.

DOX 6E Montgomery 14 The Analysis of Variance is Summarized in a Table  Computing…see text, pp  The reference distribution for F 0 is the F a-1, a(n-1) distribution  Reject the null hypothesis (equal treatment means) if

DOX 6E Montgomery 15 An Example (See pg. 60)  An engineer is interested in investigating the relationship between the RF power setting and the etch rate for this tool. The objective of an experiment like this is to model the relationship between etch rate and RF power, and to specify the power setting that will give a desired target etch rate.  The response variable is etch rate.  She is interested in a particular gas (C2F6) and gap (0.80 cm), and wants to test four levels of RF power: 160W, 180W, 200W, and 220W. She decided to test five wafers at each level of RF power.  The experimenter chooses 4 levels of RF power 160W, 180W, 200W, and 220W  The experiment is replicated 5 times – runs made in random order

DOX 6E Montgomery 16 An Example (See pg. 62)  Does changing the power change the mean etch rate?  Is there an optimum level for power? Power (W)Etch Rate (A/min)

DOX 6E Montgomery 17 ANOVA Table Example 3-1

DOX 6E Montgomery 18 The Reference Distribution:

A product development engineer is interesed in investigating the tensile strength of a new synthetic fiber that will be used to make clothe for men’s shirt. The engineer knows from previous experience that the strength is affected by the weight percent of cotton used in the blend of materials for the fiber. Furthermore, she also knows that cotton content should range between 10% - 40% if the final product is to have other quality characteristics that are desired (such as the ability to take a permanent-press finishing treatment). The engineer decided to test speciments at 5 levels of cotton weight percent: 15%, 20%, 25%, 30%, and 35%. She also decided to test 5 speciments at each level of cotton weight. Exercise: Analysis of Variance

Exercise: Analysis of Variance (cont) 5 levels = 5 treatments : 15%, 20%, 25%, 30%, and 35%. 5 speciments at each level : 5 replicates

Exercise: Analysis of Variance (cont) Randomization

Exercise: Analysis of Variance (cont) Summary of Experimental Results a. Draw the Box & Whisker Plot b. Make the appropriate analysis of variance

DOX 6E Montgomery 23 Model Adequacy Checking in the ANOVA Text reference, Section 3-4, pg. 75  Checking assumptions is important  Normality  Constant variance  Independence  Have we fit the right model?  Later we will talk about what to do if some of these assumptions are violated

DOX 6E Montgomery 24 Model Adequacy Checking in the ANOVA  Examination of residuals (see text, Sec. 3-4, pg. 75)  Design-Expert generates the residuals  Residual plots are very useful  Normal probability plot of residuals

DOX 6E Montgomery 25 Other Important Residual Plots

Statistical Test for Equality of Variance: Bartlett’s Test

1. Tukey’s Test 2. Fisher Least Significant Difference (LSD) Method 3. Duncan’s Multiple Range Test 4. Neuman-Keul’s Test We have concluded that at least one of the treatment means is different, but….. Which one(s) ? To use the Fisher LSD procedure, we simply compare the observed difference between each pair of averages to the corresponding LSD. If the absolute difference > LSD, we conclude that the population means  i and  j differ.

Unbalanced Data

Randomized Complete Block Design Is one of the most widely used experimental design. The models:

The ANOVA

RBCD