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TM 720 - Lecture 12

TM 720 - Lecture 12. Introduction to Designed Experiments. Assignment:. Reading: Chapter 12 Start reading Assignment: None. What is an Experiment?. Montgomery (2001):

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TM 720 - Lecture 12

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  1. TM 720 - Lecture 12 Introduction to Designed Experiments TM 720: Statistical Process Control

  2. Assignment: • Reading: • Chapter 12 • Start reading • Assignment: • None. TM 720: Statistical Process Control

  3. What is an Experiment? • Montgomery (2001): • A test or series of tests in which purposeful changes are made to the input variables of a process or system so that we may observe or identify the reasons for changes that may be observed in the output response. • Strategic manipulation of a system in order to observe and understand its’ response. • Usually, sequential experiments are better: • One at a time – misses interactions • All variable combinations – too expensive TM 720: Statistical Process Control

  4. Some Examples of Elements in Experimentation • Purpose: • Characterizing • Screening • Optimizing • Strategy: • One-factor-at-a-time • Comprehensive • Sequential • Design: • Simple Comparison • Response Surface methods • Factorial • Fractional Factorial TM 720: Statistical Process Control

  5. Experimental Factors • Design Factors • Design (varied) Factors • Constant (held-constant) Factors • Allowed to Vary Factors • Nuisance Factors • Controllable • Uncontrollable • Noise TM 720: Statistical Process Control

  6. Identifying Factors & Ranges • Experience • Team Approach • Fishbone Diagrams • Four M’s and an E • Man • Material • Machine • Method • Environment • Trial/Pilot Runs TM 720: Statistical Process Control

  7. Controlled Experimental Observation • Control • Blocking • Randomization • Replication • Replication vs. Repeated Measures • Observation • Main Effects • Interaction Effects • Estimation • Location • Variation TM 720: Statistical Process Control

  8. Can do in any order 7 Steps of Designed Experiments • Statement of problem • Selection of response variable • Choice of factors, levels, and ranges • Choice of experimental design • Perform the experiment protocol • Statistical analysis of the data • Conclusions & recommendations TM 720: Statistical Process Control

  9. Example: Eye Drop Effectiveness • Purpose: Determine better of two eye drops • Blocking Variable: Patients • Why Block • Variation (due to patients) is great, perhaps greater than effect of medication • To Block • Assign one medicine to an eye, and the other medicine to the other eye • Randomized variables? (left vs. right eye,…) TM 720: Statistical Process Control

  10. Example: Gas Mileage (Octane) • Purpose: Mileage w/ Fuel Quality • Desired Blocking Variable: Time of Day (Traffic Load) • Why Repeated Measures, Not Replications • Can’t empty tank and replace octane immediately • Variation (due to sequence of trips) is just giving information on measurement accuracy for traffic • To Replicate • Repeat trip conditions (time, etc.) with both octanes • Lurking variables? (summer vs. academic year,…) TM 720: Statistical Process Control

  11. Effects: Main • A Main Effect is the difference between responses at different levels of a Design Factor • Example: Intelligence Drug • Design Factors • Drug • Student Yes 100% 85% No 75% 90% Avg Good TM 720: Statistical Process Control

  12. Effects: Interactions • An Interaction is the failure of a factor to produce the same effect on the response at different levels • Example: Intelligence Drug • Design Factors • Drug • School Yes 100% 75% No 45% 99% HSU SDSMT TM 720: Statistical Process Control

  13. Some Terms • Primary Factors = Design Factors - manipulated levels • Treatments = Levels • Blocking - making comparisons under homogeneous conditions • Replications - all actions required to set the experimental conditions are taken for each observation. • Repeated Measures - observations that cannot be randomized in order. TM 720: Statistical Process Control

  14. Introduction to Comparisons • Comparisons usually look for an effect that is comparably large with respect to the variation present • Visual Inference Testing • Dot Diagrams / Barcode Plots • Applications • Small Data Sets • Subjective? • Statistical Inference Testing • Applications • Large Data Sets • More powerful to find smaller effects • Objective? TM 720: Statistical Process Control

  15. Visual Tests of Comparison • Stragglers are: • Left stragglers are observations less than the larger of the two minima • Right stragglers are observations greater than the smaller of the two maxima • Total number of stragglers = Left stragglers + Right stragglers • Tukey’s Quick TestTukey, J. W. (1959) • If the total number of stragglers is 8 or more, then the locations can be judged statistically significant at the .05 level • Significance level is about .035 for larger sample sizes TM 720: Statistical Process Control

  16. Visual Tests of Comparison • Three-Straggler RuleLenth, R. V. (1994) • If there are at least 3 left stragglersand at least 3 right stragglers, then the locations can be judged statistically significant at the .05 level • Should have at least 5 observations in each set • Significance is about .035 for larger sample sizes • Modified Quick TestLenth, R. V. (1994) • Conclude a statistical difference in location if the total number of stragglers is 8 or more, or if there are at least 3 stragglers at each end. • Significance level is almost exactly .05 TM 720: Statistical Process Control

  17. Ex 1: Popcorn Brand - Method • Purpose: • Determine the best process for popping corn seeds • Response Variable: • Number of un-popped seeds (50% unbroken by flower) • Factors: • Brand (design, two discrete levels) • Orville Redenbacher - Regular • Jolly Time - Yellow • Method (design, three discrete levels) • Microwave Bowl • Hot Air • Oil Skillet • Time (constant, continuous 2:00 minutes, except as noted) TM 720: Statistical Process Control

  18. Experiment Data * Time was 2:30 (min:sec); otherwise, time was 2:00 min TM 720: Statistical Process Control

  19. Run Sequence * Time was 2:30 (min:sec); otherwise, time was 2:00 min TM 720: Statistical Process Control

  20. Basic Statistical Concept • Noise results in variation = Experimental Error • Should be unavoidable, certainly uncontrolled, and indicates that the measured value is a Random Variable (abbreviated r.v.). TM 720: Statistical Process Control

  21. Definitions • Analysis-of-variance (ANOVA) is a statistical method used to test hypotheses regarding more than two sample means. • For a one-factor experiment the hypothesis tested is: TM 720: Statistical Process Control

  22. Definitions • The strategy in an analysis of variance is to compare the variability between sample means to the variability within sample means. If they are the same, the null hypothesis is accepted. If the variability between is bigger than within, the null hypothesis is rejected. NullHypothesis Alternative Hypothesis TM 720: Statistical Process Control

  23. Definitions • An experimental unitis the item measured during an experiment. The errors in these measurements are described by random variables. • It is important that the error in measurement be the same for all treatments (random variables must be independent and have the same distribution). • The easiest way to assure the error is the same for all treatments is to randomly assign experimental units to treatment conditions. TM 720: Statistical Process Control

  24. Definitions • The variable measured in an experiment is called the dependent variable. • The variable manipulated or changed in an experiment is called the independent variable. • Independent variables are also called factors, and the sample means within a factor are called levelsortreatments. TM 720: Statistical Process Control

  25. Definitions • Random samples of size n are selected from each of k different populations. The k different populations are classified on the basis of a single criterion or factor. (one-factor and k treatments) • It is assumed that the k populations are independent and normally distributed with means µ1, µ2, ... , µk, and a common variance 2. • Hypothesis to be tested is TM 720: Statistical Process Control

  26. Definitions • A fixed effects modelassumes that the treatments have been specifically chosen by the experimenter, and our conclusions apply only to the levels chosen • Fixed Effect Statistical Model:where eij is a iid N(0,s2). • A random effects model assumes the treatments are random samples from a larger population, and our conclusions apply to the larger population in general. • Because the fixed effects model assumes that the experiment is performed in a random manner, a one-way ANOVA with fixed effects is often called a completely randomized design. TM 720: Statistical Process Control

  27. Definitions • For a fixed effects model, if we restrict: • Thenis equivalent to: TM 720: Statistical Process Control

  28. . . . . . . . . . . . . Analysis of the Fixed Effects Model TM 720: Statistical Process Control

  29. Sum of Squares Errors (SSE) Factor level 1 Factor level 2 Factor level 3 X3· X1· X2· Sum of Squares Treatments (SSTr) Analysis of the Fixed Effects Model • Sum of Squares Treatments:The sum of squares treatments is a measure of the variability between the factor levels. • Error Sum of Squares:The error sum of squares is a measure of the variability within the factor levels. TM 720: Statistical Process Control

  30. BASE Larger SSTr Smaller SSTr Larger SSE Smaller SSE Analysis of the Fixed Effects Model • P-values:The believability of the null hypothesis (that the factor level means are all equal)depends upon the relative size of the sum of squares for treatments (SSTr) tothe sum of squares for error (SSE). TM 720: Statistical Process Control

  31. Analysis of the Fixed Effects Model • Sum of Squares Partition for One Factor Layout:In a one factor layout the total variability in the data observations is measured by the total sum of squares SST which is defined to be Total Sum of Squares SST Treatment Sum of Squares SSTr Error Sum of Squares SSE TM 720: Statistical Process Control

  32. Analysis of the Fixed Effects Model • Sum of Squares Partition for One Factor Layout:This can be partitioned into two componentsSST = SSTr + SSE,where the sum of squares for treatments (SSTr) measures the variability between the factor levels, and the sum of squares for error (SSE) measures the variability within the factor levels. TM 720: Statistical Process Control

  33. Analysis of the Fixed Effects Model • Sum of Squares Partition for One Factor Layout:On an intuitive level, the plausibility of the null hypothesis that the factor level means µi are all equal depends upon the relative size of the sum of squares for treatments (SSTr) to the sum of squares for error (SSE). TM 720: Statistical Process Control

  34. Analysis of the Fixed Effects Model • F-Test for One Factor Layout:In a one factor layout with k levels and n replications gives a total sample size kn = N, the treatments are said to have k - 1 degrees of freedom and the error is said to have N - k degrees of freedom. Mean squares are obtained by dividing a sum of squares by its respective degrees of freedom so thatand TM 720: Statistical Process Control

  35. Analysis of the Fixed Effects Model • F-Test for One Factor Layout:A p-value for the null hypothesis that the factor level means µi, are all equal is calculated as p-value = P(X ³ F), where the F-statistic isand the random variable X has an Fk-1, N -k distribution. TM 720: Statistical Process Control

  36. Analysis of the Fixed Effects Model TM 720: Statistical Process Control

  37. ANOVA Example • The tensile strength of a synthetic fiber used to make cloth for men’s shirts is of interest to a manufacturer. It is suspected that strength is affected by the percentage of cotton in the fiber. • Five levels of cotton percentage are of interest: 15%, 20%, 25%, 30%, and 35%. • Five observations are to be taken at each level of cotton percentage and the 25 total observations are to be run in random order. TM 720: Statistical Process Control

  38. ANOVA Example RANDOMIZATIONPROCEDURE . . . . . . . . . TM 720: Statistical Process Control

  39. ANOVA Example Tensile Strength of Synthetic Fiber (lb/in2) TM 720: Statistical Process Control

  40. ANOVA Example 118.94/8.06 = 14.75 475.76/4 = 118.94 5 -1= 4 24 -4= 20 161.20/20 = 8.06 5*5 -1= 24 TM 720: Statistical Process Control

  41. Critical Points for the F-Distribution Alpha = 0.05 TM 720: Statistical Process Control

  42. ANOVA Example TM 720: Statistical Process Control

  43. ANOVA Example TM 720: Statistical Process Control

  44. Assumptions of ANOVA Models • Analysis of Variance models make the following assumptions with regard to the underlying structure of the data: • The error variance is a Normal random variable with mean equal to zero and variance equal to s2. • The error variance is the same (homogeneous) for all conditions. • The error variance is independent from trial to trial. • Violation of these assumptions can have only minor effects or very large effects depending on the data set and the assumption. TM 720: Statistical Process Control

  45. Residuals • Violations in the assumptions of ANOVA models are most often uncovered through examining the residuals: TM 720: Statistical Process Control

  46. Normality Assumption • The normality assumption can be evaluated by comparing residuals with values that would be expected from a Normal distribution. • If fewer residuals are available (more typical), then normal probability plots can be used. A good approximation to the expected value of the kth smallest observation in a random sample of size n is: • Not much can be done to correct for violations of this assumption. However, ANOVA’s are very robust with respect to this assumption. TM 720: Statistical Process Control

  47. Equal Variance Assumption • The equal variance assumption is usually checked by plotting the residuals versus the predicted or fitted value. Characteristic patterns that indicate unequal variance are cone-shaped: Residual Residual Fitted Value Fitted Value TM 720: Statistical Process Control

  48. Equal Variance Example TM 720: Statistical Process Control

  49. Factorial Experiments • Experiments are often performed to investigate the effects of two or more independent variables on a single dependent variable. • The simplest experimental design to accomplish this is called the factorialor full factorial experiment. When employing this design, each complete trial or replication is done at every possible combination of the independent variables. • Factors arranged a full factorial design are often said to be crossed. TM 720: Statistical Process Control

  50. Factor B B1 B2 A1 Factor A A2 Factor B B1 B2 A1 Factor A A2 Main Effects and Interactions Factor B Factor B TM 720: Statistical Process Control

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