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DESIGN OF EXPERIMENTS by R. C. Baker How to gain 20 years of experience in one short week! 1 Role of DOE in Process Improvement • DOE is a formal mathematical method for systematically planning and conducting scientific studies that change experimental variables together in order to determine their effect of a given response. • DOE makes controlled changes to input variables in order to gain maximum amounts of information on cause and effect relationships with a minimum sample size. 2 Role of DOE in Process Improvement • DOE is more efficient that a standard approach of changing “one variable at a time” in order to observe the variable’s impact on a given response. • DOE generates information on the effect various factors have on a response variable and in some cases may be able to determine optimal settings for those factors. 3 Role of DOE in Process Improvement • DOE encourages “brainstorming” activities associated with discussing key factors that may affect a given response and allows the experimenter to identify the “key” factors for future studies. • DOE is readily supported by numerous statistical software packages available on the market. 4 BASIC STEPS IN DOE • • • • • Four elements associated with DOE: 1. The design of the experiment, 2. The collection of the data, 3. The statistical analysis of the data, and 4. The conclusions reached and recommendations made as a result of the experiment. 5 TERMINOLOGY • Replication – repetition of a basic experiment without changing any factor settings, allows the experimenter to estimate the experimental error (noise) in the system used to determine whether observed differences in the data are “real” or “just noise”, allows the experimenter to obtain more statistical power (ability to identify small effects) 6 TERMINOLOGY • .Randomization – a statistical tool used to minimize potential uncontrollable biases in the experiment by randomly assigning material, people, order that experimental trials are conducted, or any other factor not under the control of the experimenter. Results in “averaging out” the effects of the extraneous factors that may be present in order to minimize the risk of these factors affecting the experimental results. 7 TERMINOLOGY • Blocking – technique used to increase the precision of an experiment by breaking the experiment into homogeneous segments (blocks) in order to control any potential block to block variability (multiple lots of raw material, several shifts, several machines, several inspectors). Any effects on the experimental results as a result of the blocking factor will be identified and minimized. 8 TERMINOLOGY • Confounding - A concept that basically means that multiple effects are tied together into one parent effect and cannot be separated. For example, • 1. Two people flipping two different coins would result in the effect of the person and the effect of the coin to be confounded • 2. As experiments get large, higher order interactions (discussed later) are confounded with lower order interactions or main effect. 9 TERMINOLOGY • Factors – experimental factors or independent variables (continuous or discrete) an investigator manipulates to capture any changes in the output of the process. Other factors of concern are those that are uncontrollable and those which are controllable but held constant during the experimental runs. 10 TERMINOLOGY • Responses – dependent variable measured to describe the output of the process. • Treatment Combinations (run) – experimental trial where all factors are set at a specified level. 11 TERMINOLOGY • • Fixed Effects Model - If the treatment levels are specifically chosen by the experimenter, then conclusions reached will only apply to those levels. Random Effects Model – If the treatment levels are randomly chosen from a population of many possible treatment levels, then conclusions reached can be extended to all treatment levels in the population. 12 PLANNING A DOE • Everyone involved in the experiment should have a clear idea in advance of exactly what is to be studied, the objectives of the experiment, the questions one hopes to answer and the results anticipated 13 PLANNING A DOE • Select a response/dependent variable (variables) that will provide information about the problem under study and the proposed measurement method for this response variable, including an understanding of the measurement system variability 14 PLANNING A DOE • Select the independent variables/factors (quantitative or qualitative) to be investigated in the experiment, the number of levels for each factor, and the levels of each factor chosen either specifically (fixed effects model) or randomly (random effects model). 15 PLANNING A DOE • Choose an appropriate experimental design (relatively simple design and analysis methods are almost always best) that will allow your experimental questions to be answered once the data is collected and analyzed, keeping in mind tradeoffs between statistical power and economic efficiency. At this point in time it is generally useful to simulate the study by generating and analyzing artificial data to insure that experimental questions can be answered as a result of conducting your experiment 16 PLANNING A DOE • Perform the experiment (collect data) paying particular attention such things as randomization and measurement system accuracy, while maintaining as uniform an experimental environment as possible. How the data are to be collected is a critical stage in DOE 17 PLANNING A DOE • Analyze the data using the appropriate statistical model insuring that attention is paid to checking the model accuracy by validating underlying assumptions associated with the model. Be liberal in the utilization of all tools, including graphical techniques, available in the statistical software package to insure that a maximum amount of information is generated 18 PLANNING A DOE • Based on the results of the analysis, draw conclusions/inferences about the results, interpret the physical meaning of these results, determine the practical significance of the findings, and make recommendations for a course of action including further experiments 19 SIMPLE COMPARATIVE EXPERIMENTS • Single Mean Hypothesis Test • Difference in Means Hypothesis Test with Equal Variances • Difference in Means Hypothesis Test with Unequal Variances • Difference in Variances Hypothesis Test • Paired Difference in Mean Hypothesis Test • One Way Analysis of Variance 20 CRITICAL ISSUES ASSOCIATED WITH SIMPLE COMPARATIVE EXPERIMENTS • How Large a Sample Should We Take? • Why Does the Sample Size Matter Anyway? • What Kind of Protection Do We Have Associated with Rejecting “Good” Stuff? • What Kind of Protection Do We Have Associated with Accepting “Bad” Stuff? 21 Single Mean Hypothesis Test • After a production run of 12 oz. bottles, concern is expressed about the possibility that the average fill is too low. • Ho: m = 12 • Ha: m <> 12 • level of significance = a = .05 • sample size = 9 • SPEC FOR THE MEAN: 12 + .1 22 Single Mean Hypothesis Test • • • • • • Sample mean = 11.9 Sample standard deviation = 0.15 Sample size = 9 Computed t statistic = -2.0 P-Value = 0.0805162 CONCLUSION: Since P-Value > .05, you fail to reject hypothesis and ship product. 23 Single Mean Hypothesis Test Power Curve Power Curve alpha = 0.05, sigma = 0.15 1 Power 0.8 0.6 0.4 0.2 0 11.8 11.9 12 12.1 12.2 True Mean 24 Single Mean Hypothesis Test Power Curve - Different Sample Sizes 25 DIFFERENCE IN MEANS - EQUAL VARIANCES • • • • • Ho: m1 = m2 Ha: m1 <> m2 level of significance = a = .05 sample sizes both = 15 Assumption: s1 = s2 ***************************************************** • Sample means = 11.8 and 12.1 • Sample standard deviations = 0.1 and 0.2 • Sample sizes = 15 and 15 26 DIFFERENCE IN MEANS - EQUAL VARIANCES Can you detect this difference? 27 DIFFERENCE IN MEANS - EQUAL VARIANCES 28 DIFFERENCE IN MEANS - unEQUAL VARIANCES • Same as the “Equal Variance” case except the variances are not assumed equal. • How do you know if it is reasonable to assume that variances are equal OR unequal? 29 DIFFERENCE IN VARIANCE HYPOTHESIS TEST • • • • • • • • • Same example as Difference in Mean: Sample standard deviations = 0.1 and 0.2 Sample sizes = 15 and 15 ********************************** Null Hypothesis: ratio of variances = 1.0 Alternative: not equal Computed F statistic = 0.25 P-Value = 0.0140071 Reject the null hypothesis for alpha = 0.05. 30 DIFFERENCE IN VARIANCE HYPOTHESIS TEST Can you detect this difference? 31 DIFFERENCE IN VARIANCE HYPOTHESIS TEST -POWER CURVE 32 PAIRED DIFFERENCE IN MEANS HYPOTHESIS TEST • Two different inspectors each measure 10 parts on the same piece of test equipment. • Null hypothesis: DIFFERENCE IN MEANS = 0.0 • Alternative: not equal • Computed t statistic = -1.22702 • P-Value = 0.250944 • Do not reject the null hypothesis for alpha = 0.05. 33 PAIRED DIFFERENCE IN MEANS HYPOTHESIS TEST - POWER CURVE Power Curve alpha = 0.05, sigma = 3.866 1 Power 0.8 0.6 0.4 0.2 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 Difference in Means 34 ONE WAY ANALYSIS OF VARIANCE • Used to test hypothesis that the means of several populations are equal. • Example: Production line has 7 fill needles and you wish to assess whether or not the average fill is the same for all 7 needles. • Experiment: sample 20 fills from each of the 9 needles and test at 5% level of sign. • Ho: m1 = m2 = m3= m4 = m5 = m6 = m7 35 RESULTS: ANALYSIS OF VARIANCE TABLE Analysis of Variance --------------------------------------- -----------------------------------Source Sum of Squares D f Mean Square F-Ratio P-Val --------------------------------------- -----------------------------------Between groups 1.10019 6 0.183364 18.66 0.00 Within groups 1.30717 13 3 0.00982837 --------------------------------------- -----------------------------------Total (Corr.) 2.40736 13 9 36 SINCE NEEDLE MEANS ARE NOT ALL EQUAL, WHICH ONES ARE DIFFERENT? • Multiple Range Tests for 7 Needles Method: 95.0 percent LSD Col_2 Count Mean Homogeneous Groups -------------------------------------------------------------------------------N7 20 11.786 X N2 20 11.9811 X N1 20 11.9827 X N6 20 11.9873 X N3 20 11.9951 X N5 20 11.9953 X N4 20 12.11 X 37 VISUAL COMPARISON OF 7 NEEDLES Box-and-Whisker Plot N1 Col_2 N2 N3 N4 N5 N6 N7 11.5 11.7 11.9 12.1 12.3 Col_1 38 FACTORIAL (2k) DESIGNS • Experiments involving several factors ( k = # of factors) where it is necessary to study the joint effect of these factors on a specific response. • Each of the factors are set at two levels (a “low” level and a “high” level) which may be qualitative (machine A/machine B, fan on/fan off) or quantitative (temperature 800/temperature 900, line speed 4000 per hour/line speed 5000 per hour). 39 FACTORIAL (2k) DESIGNS • Factors are assumed to be fixed (fixed effects model) • Designs are completely randomized (experimental trials are run in a random order, etc.) • The usual normality assumptions are satisfied. 40 FACTORIAL (2k) DESIGNS • Particularly useful in the early stages of experimental work when you are likely to have many factors being investigated and you want to minimize the number of treatment combinations (sample size) but, at the same time, study all k factors in a complete factorial arrangement (the experiment collects data at all possible combinations of factor levels). 41 FACTORIAL (2k) DESIGNS • As k gets large, the sample size will increase exponentially. If experiment is replicated, the # runs again increases. k 2 3 4 5 6 7 8 9 10 # of runs 4 8 16 32 64 128 256 512 1024 42 FACTORIAL (2k) DESIGNS (k = 2) • Two factors set at two levels (normally referred to as low and high) would result in the following design where each level of factor A is paired with each level of factor B. Generalized Settings RUN Factor A Factor B RESPONSE Orthogonal Settings RUN Factor A Factor B RESPONSE 1 low low y1 1 -1 -1 y1 2 high low y2 2 +1 -1 y2 3 low high y3 3 -1 +1 y3 4 high high y4 4 +1 +1 y4 43 FACTORIAL (2k) DESIGNS (k = 2) • Estimating main effects associated with changing the level of each factor from low to high. This is the estimated effect on the response variable associated with changing factor A or B from their low to high values. ( y 2 y 4 ) ( y1 y3 ) Factor A Effect = 2 2 ( y3 y4 ) ( y1 y2 ) Factor B Effect = 2 2 44 FACTORIAL (2k) DESIGNS (k = 2): GRAPHICAL OUTPUT • Neither factor A nor Factor B have an effect on the response variable. 45 FACTORIAL (2k) DESIGNS (k = 2): GRAPHICAL OUTPUT • Factor A has an effect on the response variable, but Factor B does not. 46 FACTORIAL (2k) DESIGNS (k = 2): GRAPHICAL OUTPUT • Factor A and Factor B have an effect on the response variable. 47 FACTORIAL (2k) DESIGNS (k = 2): GRAPHICAL OUTPUT • Factor B has an effect on the response variable, but only if factor A is set at the “High” level. This is called interaction and it basically means that the effect one factor has on a response is dependent on the level you set other factors at. Interactions can be major problems in a DOE if you fail to account for the interaction when designing your experiment. 48 EXAMPLE: FACTORIAL (2k) DESIGNS (k = 2) • A microbiologist is interested in the effect of two different culture mediums [medium 1 (low) and medium 2 (high)] and two different times [10 hours (low) and 20 hours (high)] on the growth rate of a particular CFU [Bugs]. 49 EXAMPLE: FACTORIAL (2k) DESIGNS (k = 2) • Since two factors are of interest, k =2, and we would need the following four runs resulting in Generalized Settings RUN Medium Time Growth Rate 1 low low 17 2 high low 15 3 low high 38 4 high high 39 50 EXAMPLE: FACTORIAL (2k) DESIGNS (k = 2) • Estimates for the medium and time effects are • Medium effect = [(15+39)/2] – [(17 + 38)/2] = -0.5 • Time effect = [(38+39)/2] – [(17 + 15)/2] = 22.5 51 EXAMPLE: FACTORIAL (2k) DESIGNS (k = 2) 52 EXAMPLE: FACTORIAL (2k) DESIGNS (k = 2) • A statistical analysis using the appropriate statistical model would result in the following information. Factor A (medium) and Factor B (time) Type III S ums of Square s ---- ------ ------ ------- ------ ------ ------- ------ ------ ------- ------ ------ ------ ---Sour ce Su m of S quares Df Mea n Squar e F -Ratio P- Valu ---- ------ ------ ------- ------ ------ ------- ------ ------ ------- ------ ------ ------ ---FACT OR A 0.25 1 0.2 5 0.11 0 .795 FACT OR B 506.25 1 506.2 5 225.00 0 .042 Resi dual 2.25 1 2.2 5 ---- ------ ------ ------- ------ ------ ------- ------ ------ ------- ------ ------ ------ ---Tota l (cor rected ) 508.75 3 All F-rati os are based on the resid ual mea n squa re err or. 53 EXAMPLE: CONCLUSIONS • In statistical language, one would conclude that factor A (medium) is not statistically significant at a 5% level of significance since the p-value is greater than 5% (0.05), but factor B (time) is statistically significant at a 5 % level of significance since this pvalue is less than 5%. 54 EXAMPLE: CONCLUSIONS • In layman terms, this means that we have no evidence that would allow us to conclude that the medium used has an effect on the growth rate, although it may well have an effect (our conclusion was incorrect). 55 EXAMPLE: CONCLUSIONS • Additionally, we have evidence that would allow us to conclude that time does have an effect on the growth rate, although it may well not have an effect (our conclusion was incorrect). 56 EXAMPLE: CONCLUSIONS • In general we control the likelihood of reaching these incorrect conclusions by the selection of the level of significance for the test and the amount of data collected (sample size). 57 2k DESIGNS (k > 2) • As the number of factors increase, the number of runs needed to complete a complete factorial experiment will increase dramatically. The following 2k design layout depict the number of runs needed for values of k from 2 to 5. For example, when k = 5, it will take 25 = 32 experimental runs for the complete factorial experiment. 58 Interactions for 2k Designs (k = 3) • Interactions between various factors can be estimated for different designs above by multiplying the appropriate columns together and then subtracting the average response for the lows from the average response for the highs. 59 Interactions for 2k Designs (k = 3) a -1 +1 -1 +1 -1 +1 -1 +1 b -1 -1 +1 +1 -1 -1 +1 +1 c -1 -1 -1 -1 +! +1 +1 +1 ab 1 -1 -1 1 1 -1 -1 1 ac 1 -1 1 -1 -1 1 -1 1 bc 1 1 -1 -1 -1 -1 1 1 abc -1 1 1 -1 1 -1 -1 1 60 2k DESIGNS (k > 2) • Once the effect for all factors and interactions are determined, you are able to develop a prediction model to estimate the response for specific values of the factors. In general, we will do this with statistical software, but for these designs, you can do it by hand calculations if you wish. 61 2k DESIGNS (k > 2) • For example, if there are no significant interactions present, you can estimate a response by the following formula. (for quantitative factors only) Y = (average of all responses) + =Y ( factorEFFECT [( ) * ( factorLEVEL)] 2 A )* A ( B )*B 2 2 62 ONE FACTOR EXAMPLE Plot of Fitted Model GRADE 95 85 75 65 55 10 12 14 16 #HRS STUDY 18 20 63 ONE FACTOR EXAMPLE • The output shows the results of fitting a general linear model to describe the relationship between GRADE and #HRS STUDY. The equation of the fitted general model is • GRADE = 29.3 + 3.1* (#HRS STUDY) • The fitted orthogonal model is • GRADE = 75 + 15 * (SCALED # HRS) 64 Two Level Screening Designs • Suppose that your brainstorming session resulted in 7 factors that various people think “might” have an effect on a response. A full factorial design would require 27 = 128 experimental runs without replication. The purpose of screening designs is to reduce (identify) the number of factors down to the “major” role players with a minimal number of experimental runs. One way to do this is to use the 23 full factorial design and use interaction columns for factors. 65 Note that * Any factor d effect is now confounded with the a*b interaction * Any factor e effect is now confounded with the a*c interaction * etc. * What is the d*e interaction confounded with???????? a -1 +1 -1 +1 -1 +1 -1 +1 b -1 -1 +1 +1 -1 -1 +1 +1 c -1 -1 -1 -1 +! +1 +1 +1 d = ab 1 -1 -1 1 1 -1 -1 1 e = ac 1 -1 1 -1 -1 1 -1 1 f = bc 1 1 -1 -1 -1 -1 1 1 g = abc -1 1 1 -1 1 -1 -1 1 66 Problems that Interactions Cause! • Interactions – If interactions exist and you fail to account for this, you may reach erroneous conclusions. Suppose that you plan an experiment with four runs and three factors resulting in the following data: Run 1 2 3 4 Factor A +1 +1 -1 -1 Factor B +1 -1 +1 -1 Results 10 5 5 10 67 Problems that Interactions Cause! • Factor A Effect = 0 • Factor B Effect = 0 • In this example, if you were assuming that “smaller is better” then it appears to make no difference where you set factors A and B. If you were to set factor A at the low value and factor B at the low value, your response variable would be larger than desired. In this case there is a factor A interaction with factor B. 68 Problems that Interactions Cause! Interaction Plot RESPONSE 10 FACT OR B -1 1 9 8 7 6 5 -1 1 FACTOR A 69 Resolution of a Design • • • Resolution III Designs – No main effects are aliased with any other main effect BUT some (or all) main effects are aliased with two way interactions Resolution IV Designs – No main effects are aliased with any other main effect OR two factor interaction, BUT two factor interactions may be aliased with other two factor interactions Resolution V Designs – No main effect OR two factor interaction is aliased with any other main effect or two factor interaction, BUT two factor interactions are aliased with three factor interactions. 70 Common Screening Designs • Fractional Factorial Designs – the total number of experimental runs must be a power of 2 (4, 8, 16, 32, 64, …). If you believe first order interactions are small compared to main effects, then you could choose a resolution III design. Just remember that if you have major interactions, it can mess up your screening experiment. 71 Common Screening Designs • Plackett-Burman Designs – Two level, resolution III designs used to study up to n-1 factors in n experimental runs, where n is a multiple of 4 ( # of runs will be 4, 8, 12, 16, …). Since n may be quite large, you can study a large number of factors with moderately small sample sizes. (n = 100 means you can study 99 factors with 100 runs) 72 Other Design Issues • May want to collect data at center points to estimate non-linear responses • More than two levels of a factor – no problem (multi-level factorial) • What do you do if you want to build a nonlinear model to “optimize” the response. (hit a target, maximize, or minimize) – called response surface modeling 73 Response Surface Designs – Box-Behnken RUN F1 F2 F3 Y100 1 10 45 60 11825 2 30 45 40 8781 3 20 30 40 8413 4 10 30 50 9216 5 20 45 50 9288 6 30 60 50 8261 7 20 45 50 9329 8 30 45 60 10855 9 20 45 50 9205 10 20 60 40 8538 11 10 45 40 9718 12 30 30 50 11308 13 20 60 60 10316 14 10 60 50 12056 15 20 30 60 10378 74 Response Surface Designs – Box-Behnken Regression coeffs. for Var_3 ----------------------------------- ----------------------------------constant = 2312.5 A:Factor_A = 36.575 B:Factor_B = 200.067 C:Factor_C = 3.85 AA = 9.09875 AB = -9.81167 AC = -0.0825 BB = 0.117222 BC = -0.311667 CC = 1.10875 75 Response Surface Designs – Box-Behnken Contours of Estimated Response Surface Factor_C=60.0 60 Factor_B 55 50 45 40 35 30 10 14 18 22 Factor_A 26 30 Var_3 9300.0 9500.0 9700.0 9900.0 10100.0 10300.0 10500.0 10700.0 10900.0 11100.0 11300.0 11500.0 11700.0 76 CLASSROOM EXERCISE • • • • STUDENT IN-CLASS EXPERIMENT: Collect data for experiment to determine factor settings (two factors) to hit a target response (spot on wall). Factor A – height of shaker (low and high) Factor B – location of shaker (close to hand and close to wall) Design experiment – would suggest several replications 77 CLASSROOM EXERCISE • Conduct Experiment – student holds 3 foot “pin the tail on the donkey” stick and attempts to hit the target. An observer will assist to mark the hit on the target. • Collect data – students take data home for week and come back with what you would recommend AND why. • YOU TELL THE CLASS HOW TO PLAY THE GAME TO “WIN”. 78 CLASSROOM EXERCISE 79 CLASSROOM EXERCISE MARKER VERTICAL 1ST OBS 2ND OBS 3RD OBS 4TH OBS STICK POLE MEAN STANDARD DEVIATION L L -2.750 -4.500 -4.750 -5.000 -4.250 1.021 H L -12.500 -6.750 -4.625 -4.000 -6.969 3.871 L H 3.000 3.250 3.875 6.250 4.094 1.484 H H 4.625 11.250 12.625 14.000 10.625 4.155 MARKER STICK L = VERTICAL POLE WAS CLOSE TO WALL (MARKER END OF STICK H=VERTICAL POLE WAS CLOSE TO HAND VERTICAL POLE L=SHAKING DEVICE LOCATED LOW ON VERTICAL POLE H=SHAKING DEVICE LOCATED HIGH ON VERTICAL POLE 80 Contour Plots for Mean and Std. Dev. 81