2Designing Engineering Experiments Factor: Parameter of interestLevels: Possible value of Factors.Analysis of Variance (ANOVA): Analysis the effects of different factor levels to the response.Randomization: Random running order.Controllable variables: Other parameters that involve in the experimentExperiment activities:ConjectureExperimentAnalysisConclusion
3Completely Randomized Single-Factor Experiment Ex: A manufacturer of paper is interested in improving the tensile strength of the product. The product engineer thinks that tensile strength is a function of the hardwood concentration in the paper. The range of hardwood concentrations of practical interest is between 5 and 20% . The engineers design to investigate four levels of hardwood concentration: 5% 10% 15% and 20%. They design to make up six test specimens at each concentration level.
4Completely Randomized Single-Factor Experiment Randomizing order: randomly select the order for each run to reduce the effect of nuisance variable such as the warm-up effect.Box plot: Represent the variability within a treatmentand the variability between treatments.
5Analysis of Variance Typical data for single-factor experiment Linear Statistical ModelFor each treatmentFor each treatment, yij is normal distribution with µi andEij has a normal distribution with mean 0 and sd =
6If there are no differences between treatments => 1=2 Analysis of VarianceWe are interested in the equality of the a treatment means µ1, µ2,…, µaIf H0 is true, each observation consists of the overall mean µ plus the random error Eij => Changing the level of the factor has no effect on the mean response.ANOVA partitions the total variability in the sample data into two components1. The variation between treatments2. The variation within treatmentIf there are no differences between treatments => 1=2
7Analysis of VarianceTotal variation is described by the total sum of squares (SST)
8Analysis of Variance Degree of freedom Mean square for treatment Mean square for errorTo verify hypothesis, we compareBy using F test statistic
9Analysis of VarianceIf H0 is reject => MStreatment > MSE => H1 should be upper-tail testReject H0 ifComputing formulas for ANOVAANOVA Table
10Analysis of VarianceEx. Consider the tensile strength. We can use ANOVA to test the hypothesis that different hardwood concentrations do not affect the mean tensile strength of the paper. Use α=0.01For at least one i
11Analysis of Variance f0 = 19.60 >f0.01,3,20 =4.94 , reject H0 Conclusion: hardwood concentration significantly affects the mean strength of the paper
12Analysis of VarianceConfidence interval on the mean of the ith treatmentEx. Find the 95% CI of the mean strength of 20% hardwood concentration?
13Analysis of VarianceConfidence interval on a difference in treatment meansEx. Find a 95% CI on the difference in means µ3-µ2
14Multiple comparisons following ANOVA When is rejected in ANOVA, we know that some of the treatment are different.ANOVA doesn’t identify which means are different.Methods for investigating this issue is called multiple comparisons methods.Fisher LSD : compares all pairs of the means (µi and µj)H0: µi = µj with test-statisticThe pair of means µi and µj would be declared significantly different if
15Multiple comparisons following ANOVA Ex. Apply the Fisher LSD to the hardwood concentration experiment.a= 4 levels, n=6 replicates, and t0.025,20 = 2.086The treatment means are:The value of LSD,Compare the difference for every pairs of treatments and LSD,5% % 15% 20%
16Model Adequacy Checking Residual Analysis and Model CheckingResidual VS time: Test independence assumption of errorResidual VS fitted values: Test constant variance assumption of errorNormality Plot: test the normal distribution assumption of error.
17Randomized Complete Block Design An extension of the paired t-test but with more than 2 treatments.Reduce the nuisance factor.Ex. 3 methods could be used to evaluate the strength reading on steel plate girders. If there are 4 plates and each plate is large enough to hold all the treatment, the experimental design would be appear as Figure.
18Randomized Complete Block Design ANOVA Sums of square
19Randomized Complete Block Design Computing Formulas for ANOVA randomized block
20Randomized Complete Block Design ANOVA for a Randomized Complete Block Design
21Randomized Complete Block Design Ex. Fabric Strength Data from a randomized complete block design can be shown in table. We want to test the effect of chemical type to the strength of fabric by using α=0.01H0:all types provide identical strengthH1: not equal strength
22Randomized Complete Block Design f0=75.13 > f0.01,3,12 =5.95, reject Ho and conclude that there is a significant difference in the chemical types
23Randomized Complete Block Design Multiple comparisonSimilar to simple ANOVA butEx. Refer to previous example, use Fisher’s LSD method to analyze the difference between each pair of treatment.type 4 results in significantly different strengthsthan the other three types .types 2 and 3 do not differ, and types 1 and 3 do notdiffer.There may be a small difference in strength between types 1 and 2.
24ExerciseA civil engineer is interested in determining whether four different methods of estimating flood flow frequency produce equivalent estimates of peak discharge when applied to the same watershed. Each procedure is used six times on the watershed and the resulting discharge data are shown in tableEstimation methodObservation10.3126.96.36.1991.7520.912.942.142.362.864.5536.318.379.756.099.827.24417.1511.8210.9517.2014.3516.82
25Exercise2. An experiment was conducted to investigate leaking current in a SOS MOSFETS device. The purpose of the experiment was to investigate how leakage current varies as the channel length changes. Four channel lengths were selected. For each channel length, five different widths were also used, and width is to be considered a nuisance factor. The data are as follows:
26Exercise3. An article in the American Industrial Hygiene Association Journal (Vol. 37, 1976, pp. 418– 422) describes a field test for detecting the presence of arsenic in urine samples. The test has been proposed for use among forestry workers because of the increasing use of organic arsenics in that industry. The experiment compared the test as performed by both a trainee and an experienced trainer to an analysis at a remote laboratory. Four subjects were selected for testing and are considered as blocks. The response variable is arsenic content (in ppm) in the subject’s urine. The data are as follows:
27Exercise4. An article in the Food Technology Journal (Vol. 10, 1956, pp. 39–42) describes a study on the protopectin content of tomatoes during storage. Four storage times were selected, and samples from nine lots of tomatoes were analyzed. The protopectin content (expressed as hydrochloric acid soluble fraction mg/kg) is in the following table.
28Exercise5. An article in the IEEE Transactions on Components, Hybrids, and Manufacturing Technology (Vol.15, No. 2, 1992, pp. 146–153) describes an experiment in which the contact resistance of a brake-only relay was studied for three different materials (all were silver-based alloys). The data are as follows.
29Exercise6. An article in Lubrication Engineering (December 1990) describes the results of an experiment designed to investigate the effects of carbon material properties on the progression of blisters on carbon face seals. The carbon face seals are used extensively in equipment such as air turbine starters. Five different carbon materials were tested, and the surface roughness was measured. The data are as follows:
30Exercise7. article in Communications of the ACM (Vol. 30, No. 5, 1987) studied different algorithms for estimating software development costs. Six algorithms were applied to eight software development projects and the percent error in estimating the development cost was observed. The data are in the table at the bottom of the page.