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LISA Short Course Series Basics of Design of Experiments Ana Maria Ortega-Villa Fall 2014 LISA: DOE Fall 2014 About me Home country Colombia. 5th year PhD student in Statistics Ms. Statistics, Virginia Tech Ms. Operations research, Universidad de los Andes, Colombia. • Instructor: STAT 4705 Probability and Statistics for Engineers. • Contact: [email protected] • • • • LISA: LISA:DOE R Basics Fall Fall 2014 2013 Laboratory for Interdisciplinary Statistical Analysis LISA helps VT researchers benefit from the use of Statistics Collaboration: Visit our website to request personalized statistical advice and assistance with: Designing Experiments • Analyzing Data • Interpreting Results Grant Proposals • Software (R, SAS, JMP, Minitab...) LISA statistical collaborators aim to explain concepts in ways useful for your research. Great advice right now: Meet with LISA before collecting your data. LISA also offers: Educational Short Courses: Designed to help graduate students apply statistics in their research Walk-In Consulting: Available Monday-Friday from 1-3 PM in the Old Security Building (OSB) for questions <30 mins. See our website for additional times and locations. All services are FREE for VT researchers. We assist with research—not class projects or homework. www.lisa.stat.vt.edu What are we doing? 1. Introduction to Design of Experiments 2. DOE main principles • Randomization • Replication • Local control of error 3. Complete Randomized Design 4. Randomized Complete Block Design 5. Introduction to factorial Designs LISA: DOE Fall 2014 Introduction to Design of Experiments LISA: DOE Fall 2014 What is an Experiment? An experiment can be thought of as a test or series of tests in which we make controlled changes to the input variables of a process or a system, in order to determine how they change the output of interest. LISA: DOE https://weakinteractions.files.wordpress.com/2009/08/s1e1.jpg?w=450 Fall 2014 Why do we design experiments? MAXIMIZE: • Probability of having a successful experiment. • Information gain: the results and conclusions derived depend on the way information was collected. MINIMIZE • Unwanted effects from other sources of variation. • Cost of experiment if results are limited. LISA: DOE Fall 2014 What would be an alternative? Observational study: • The researcher has little to no control over sources of variation and simply observes what is happening. • The researcher can only determine information about how our inputs are related to the outputs… we cannot determine causation. Examples: • Surveys • Weather Patterns • Stock market price • etc. LISA: DOE http://fluxicon.com/blog/wp-content/uploads/2012/02/observeandreport.jpg Fall 2014 Designed experiment • The researcher identifies and controls sources of variation that significantly impact the measured response. • The researcher can gather evidence for causation. Correlation ≠ Causation LISA: DOE Fall 2014 But what are sources of variation? Sources of variation are anything that could cause an observation to be different from another observation. Two main types: • Those that can be controlled and are of interest are called treatments or treatment factors. • Those that can influence the experimental response but in which we are not directly interested are called nuisance factors. LISA: DOE Fall 2014 Rule of Thumb List all major and minor sources of variation before collecting the data, classifying them as either a treatment or a nuisance factor. • We want our design to minimize the impact of minor sources of variation, and to be able to separate effects of nuisance factors from treatment factors • We want the majority of the variability of the data to be explained by the treatment factors. LISA: DOE Fall 2014 Example: Impact of Exercise Intensity on Resting Heart Rate Suppose a researcher surveys a sample of individuals to obtain information about their intensity of exercise each week and their resting heart rate. Subject Reported Intensity of Resting Heart Rate Exercise each week 1 2 3 … What type of study is this? LISA: DOE http://karmajello.com/postcont/2014/02/What-ExerciseCan-Heart-Patients-Undertake-e1352999185475.jpg Observational Study Fall 2014 How could we make it a designed expt? The researcher finds a sample of individuals, enrolls groups in exercise programs of different intensity levels, and then measures their resting heart rate. LISA: DOE Subject Intensity level of exercise each week 1 2 3 … Fall 2014 Resting Heart Rate What are our sources of variation? Major Treatment Exercise intensity Nuisance Factor Medication Use Air Temperature & Humidity LISA: DOE Minor Location of measurement Body Size Body Position Fall 2014 Designing the experiment Minimum considerations: • Response: Resting heart rate (beats per minute) • Treatment: Exercise Program LISA: DOE o Low intensity o Moderate intensity o High intensity Fall 2014 Designing the experiment Basic Design: • 36 participants, 18 male and 18 female under the conditions listed previously. • Every person is assigned to one of the three 8week exercise programs. • Resting heart rate is measured at the beginning and end of 8 weeks. LISA: DOE Fall 2014 Fundamentals of Design of Experiments • An experimental unit (EU) is the “material” to which treatment factors are assigned. o For the resting heart rate example, the participants are the EU. o We want EUs to be as similar as possible, but that isn’t always realistic. • A block is a group of EUs similar to each other, and different from other groups. o In the resting heart rate example, women are physiologically similar to each other and different from men. • A blocking factor is the characteristic used to create the blocks. o In the resting heart rate example, gender is a blocking factor. LISA: DOE Fall 2014 Three Basic Principles of Design of Experiments Randomization LISA: DOE Fall 2014 Randomization Randomization consists of randomly assigning: • the experimental treatments to experimental units. • the order in which the independent runs will be performed (when applicable). Purpose: • Often we assume an independent, random distribution of observations and errors – randomization validates this assumption. • Averages out the effects of extraneous/lurking variables. • Reduces bias and accusations of bias. LISA: DOE Fall 2014 Randomization The way you randomize depends on your experiment, what is important here is to remember there are two levels of randomization. 1. Assignment of treatments to experimental units 2. Order of the runs (when applicable). LISA: DOE Fall 2014 Randomization RHR Example 1. Assignment of treatments to experimental units. Participant Exercise Program 1 High 2 High 3 Low 4 Intermediate 5 Low 6 High ……. …… 1. Order of the runs. Not applicable in this case since all participants are doing the experiment at the same time. LISA: DOE Fall 2014 Three Basic Principles of Design of Experiments Replication LISA: DOE Fall 2014 Replication Replication consists of independently repeating runs of each treatment. Purpose: • Improves precision of effect estimation. • Decreases Variance. • Allows for estimation of experimental error. This error will later become a unit of measurement to determine whether observed differences are really statistically significant. Note: Try to have the same amount of replicates for each treatment assignment. LISA: DOE # Replicates=# EUs/#Treatments Fall 2014 Replication in RHR Example Participant Exercise Program 1 High 2 High 3 Low 4 Intermediate 5 Low 6 High ……. …… Participants 1, 2 and 6 can be considered as replicates of High intensity exercise treatment. LISA: DOE Fall 2014 Pseudoreplication What is pseudoreplication? Occurs when there is more than one observation per EU and they are treated as replicates. In our RHR example it would be like taking measurements in different locations (wrist, side of the neck and foot) of the same person and treating them as separate observations. LISA: DOE Fall 2014 Pseudoreplication A way to deal with multiple measurements per EU is to average them over and work with the new value. Consequences: • Underestimation of error • Potentially exaggerate differences LISA: DOE the Fall 2014 true treatment Three Basic Principles of Design of Experiments Local Control of Error LISA: DOE Fall 2014 Local control of error Local control of error is taking any means of improving the accuracy of measuring treatment effects in the design. Purpose: • Removes or minimizes sources of nuisance. • Improves the precision with which comparisons among factors are made. Note: There are several ways of doing this. One could control as much as possible all the previously listed sources of variation. Often this is done by the use of blocking or more advanced designs such as ANCOVA. LISA: DOE Fall 2014 RHR Local control of error • We will be monitoring the participant’s exercise program throughout the study (not relying on selfreporting). • We will only consider participants that are not taking any medication that might alter their heart rate. • We will take all measurements on the same location of the body: the wrist. • We will take all measurements with the participant on the same position: standing. • We will only accept participants with a body mass index within the normal range. • We will measure all participants on the same day at the beginning and the end of the study. LISA: DOE Fall 2014 Common Designs: Completely Randomized Design (CRD) LISA: DOE Fall 2014 Complete Randomized Design (CRD) The CRD is the simplest design. It assumes all EUs are similar and the only major sources of variation are the treatments. In this design all treatment-EU assignments are randomized for the specified number of treatment replications. If you are equally interested in comparisons of all treatments get as close as possible to equally replicating the treatments. (Balanced design). LISA: DOE Fall 2014 CRD Example: Plasma Etching Experiment Etching is a process in which unwanted material is removed from circuit wafers in order to obtain circuit patterns, electrical interconnects and areas in which diffusions or metal depositions are to be made. * Example from Montgomery (2009) LISA: DOE Fall 2014 CRD Example: Etching Process simplified LISA: DOE Energy is supplied by a generator. Chemical mixture gas is is shot at a sample. Fall 2014 Plasma is generated in gap between electrodes CRD Example: Study An engineer is interested in investigating the relationship between the generator power setting and the etch rate for the tool. Response: Etch rate Treatment: Generator power setting (4 levels to consider) Experimental Unit: Circuit Wafer Possible sources of variation: • Generator power setting • Chemical mixture gas (the gases affect the plasma behavior) • Size of the gap between the electrodes. LISA: DOE Fall 2014 CRD Example: Principles of DOE • Replication We will consider 5 EUs for each treatment level (generator power setting) • Randomization Since all EUs are considered to be identical, we will randomize the running order. • Local control of error In order to minimize variability we will use the same chemical mixture (C2F6) and size of gap (0.8 cm) for all runs of the experiment. LISA: DOE Fall 2014 CRD Example: Randomization Scheme Run Treatment Run Treatment 1 3 11 4 2 4 12 4 3 1 13 3 4 2 14 1 5 2 15 2 6 2 16 1 7 4 17 2 8 3 18 3 9 3 19 4 10 1 20 1 This run order was obtained using a random number generator. LISA: DOE Fall 2014 CRD Example: What is the question? We are interested in testing the equality of the treatment means: If we reject the null hypothesis, then this would mean there is a difference between at least two of the means, which translates to a significant different between the treatments. LISA: DOE Fall 2014 CRD Example: Analysis • We want to enter the data such that each each response has its own row, with the corresponding treatment type. • We then choose Analyze -> Fit Y by X. LISA: DOE Fall 2014 CRD Example: Analysis We will choose Rate as the Y response and Treatment as the X factor. LISA: DOE Fall 2014 CRD Example: Visual Analysis From the red triangle: Display Options ->Boxplot Remarks: • These box plots show that the etch rate increases as the power setting increases. • From this graphical analysis we suspect: 1. Generator power settings affects the etch rate. 2. Higher power settings result in increased etch rate. LISA: DOE Fall 2014 CRD Example: ANOVA Table From red triangle select means and ANOVA. ANOVA partitions total independent pieces: variability into three separate MSTrt: Variability due to treatment differences. MSE: Variability due to experimental error. If MSTrt>MSE then treatments likely have different effects. LISA: DOE Fall 2014 CRD Example: Contrasts Red Triangle: Compare Means -> Tukey HSD At least two treatments are different, which ones? LISA: DOE Fall 2014 CRD: Summary • CRD has one overall randomization. • Try to equally replicate all the treatments. • Plot your data in a meaningful way to help visualize analysis. • Use ANOVA to test for an overall difference. • Look at specific contrasts of interest to better understand the relationship between treatments. LISA: DOE Fall 2014 Common Designs: Randomized Complete Block Design (RCBD) LISA: DOE Fall 2014 Randomized Complete Block Design (RCBD) The RCBD is a design in which there are one or more nuisance factors that are known and controllable. This design systematically eliminates the effect of these nuisance factors on the statistical comparisons among treatments. The block size equals the number of treatments. Basic Idea: Compare treatments within blocks to account for the source of variation. LISA: DOE Fall 2014 RCBD Example: Vascular Graft Experiment Vascular grafts (artificial veins) are produced by extruding billets of polytetrafluoroethylene (PFTE) resin combined with a lubricant into tubes. Sometimes these tubes contain defects known as flicks. These defects are cause for rejection of the unit. The product developer suspects that the extrusion pressure affects the occurrence of flicks. An engineer suspects that there may be significant batch-to-batch variation from the resin. * Example from Montgomery (2009) LISA: DOE Fall 2014 RCBD Example: Study Response: Percentage of tubes that did not contain any flick. Treatment: Extrusion Pressure (4 levels) Block: Batch of resin (6 batches). LISA: DOE Fall 2014 RCBD Example: Principles of DOE • Replication Each treatment (extrusion pressure) is replicated once in each block. • Randomization The treatments (extrusion pressure) are randomized inside each block. • Local control of error In order to minimize variability we will use Blocking and keeping all other possible controllable nuisance factors controlled. LISA: DOE Fall 2014 RCBD Example: What is the question? We are interested in testing the equality of the treatment means: If we reject the null hypothesis, then this would mean there is a difference between at least two of the means. LISA: DOE Fall 2014 RCBD Example: What is the question? We are interested in testing the equality of the treatment means: If we reject the null hypothesis, then this would mean there is a difference between at least two of the means. LISA: DOE Fall 2014 RCBD Example: Analysis JMP Analysis: Follow the same procedure. Analyze->Fit Y by X. LISA: DOE Fall 2014 RCBD Example: Visual Analysis Boxplot: From this graphical analysis we suspect: 1. Extrusion pressure affects the response. 2. Higher pressure settings seem to result in decreased no flicks percentages. 3. These results can be potentially affected by the resin batch. LISA: DOE Fall 2014 RCBD Example: ANOVA Table According to this analysis, we reject the null hypothesis. This means that there is a significant effect by the treatments. Software is going to give you a p-value for Block, but only use this to gauge how much we reduced experimental error. Do not test the blocks using this p-value. LISA: DOE Fall 2014 RCBD Example: Contrasts Significant differences between treatments 1 and 4, and 2 and 4. LISA: DOE Fall 2014 Common Designs: Introduction to Factorial Designs LISA: DOE Fall 2014 Factorial Designs In this type of design we want to study the effect of two or more factors. Here, we have that in each complete trial or replication of the experiment, all possible combinations of the levels of the factors are investigated. Basic idea: Treatments are a combination of multiple factors with different levels (i.e. settings) LISA: DOE Fall 2014 Factorial Designs: Main Concepts • The effect of factor is defined to be as the change in the response produced by a change in the level of the factor (main effect). • Interaction between factors is present when the difference in response between the levels of one factor is not the same at all levels of the other factors (i.e. the effect of factor A depends on the level chose for factor B). LISA: DOE Fall 2014 Factorial Designs Example: Battery Design An engineer is designing a battery that will be used in a device that will be subject to extreme variations in temperature. She is interested in examining three different materials for this battery at three different temperatures (15, 70 and 125 °F) in order to determine how battery life is affected by these conditions. * Example from Montgomery (2009) LISA: DOE Fall 2014 Factorial Design Example: Study Response: Battery life Treatment: All combinations the factors: LISA: DOE Material: 3 levels (1, 2 and 3) Temperature: 3 levels (15, 70 and 125 °F) Fall 2014 Factorial Design Example: Principles of DOE • Replication Each treatment (combination of levels of factors) is replicated 4 times. • Local control of error In order to minimize variability we will keep everything else in the testing lab constant throughout the experiment. LISA: DOE Fall 2014 Factorial Design Example: Randomization Mat Temp Run Mat Temp Run Mat Temp Run 1 15 6 2 15 17 3 15 13 1 15 11 2 15 30 3 15 9 1 15 26 2 15 23 3 15 32 1 15 31 2 15 14 3 15 24 1 70 22 2 70 25 3 70 27 1 70 34 2 70 35 3 70 8 1 70 1 2 70 5 3 70 12 1 70 33 2 70 20 3 70 2 1 125 15 2 125 10 3 125 3 1 125 21 2 125 29 3 125 4 1 125 28 2 125 36 3 125 18 1 125 19 2 125 7 3 125 16 LISA: DOE Fall 2014 Factorial Design Example: Randomization You can create your own design in JMP: DOE->Custom Design LISA: DOE Fall 2014 Factorial Design Example: Analysis Analyze->Fit Model LISA: DOE Fall 2014 Factorial Design Example: Interaction Red Triangle: Factor Profiling -> Interaction Plots LISA: DOE Fall 2014 Factorial Design Example: ANOVA Theory Here the ANOVA table is partitioned: SST= SSModel+SSError And SSModel is partitioned: SSModel=SSTemp+SSMat+SSInt SSTemp: Compares Temperature level means to overall mean. SSMat: Compares Material level means to overall mean. SSInt: Looks at differences between changes depending on material. LISA: DOE Fall 2014 temperature LISA: DOE Factorial Design Example: ANOVA Fall 2014 Model adequacy checking LISA: DOE Fall 2014 Model Adequacy checking It is recommended to check the adequacy of the model by examining the residuals (difference between the true values and the ones predicted by the model. These residuals should be structureless, which means they should not contain an obvious pattern. To save the residuals from Fit Model (Not fit Y by X): Red triangle: Save columns -> Residuals LISA: DOE Fall 2014 Model Adequacy checking: Assumptions • Residuals should be normally distributed Can inspect with a normal probability plot: Analyze-> Distribution. Red triangle: Normal Quantile plot • Plot Residuals vs fitted values and check for patterns In the effect analysis window, red triangle: Row diagnostics • Plot Residuals by treatment, can do it with saved residuals using the graph builder. LISA: DOE Fall 2014 LISA: DOE Model Adequacy checking: Battery Fall 2014 Model Adequacy checking: Plasma Etching LISA: DOE Fall 2014 Model Adequacy checking: Vascular Graft LISA: DOE Fall 2014 Exercise LISA: DOE Fall 2014 Exercise: A soft drink bottler is interested in obtaining more uniform fill heights in the bottles produced by his manufacturing process. The process engineer can control three variables during the filling process: • Percent carbonation • Operating pressure in the filler • Line speed. The engineer can control carbonation at three different levels (10, 12 and 14%), two levels for pressure (25 and 30 psi) and two levels for line speed (200 and 250 bpm). She designs to run two replicates of a factorial design in these factors, with all runs taken in random order. The response variable is the average deviation from the target fill height observed in a production run of bottles at each set of conditions. How many factors do we have? How many runs would we need to perform? * Example from Montgomery (2009) LISA: DOE Fall 2014 Exercise: Question 1 Suppose you obtain this interaction plot, what would you interpret? LISA: DOE Fall 2014 Exercise: Analysis Conduct the factorial analysis in JMP, what can you conclude? LISA: DOE Fall 2014 Exercise: Analysis What can you say about the residuals? LISA: DOE Fall 2014 Summary • Remember to randomize! – Randomize run order, and treatments • Remember to replicate! – Use multiple EUs for each treatment– it will help you be more accurate in estimating your effects • Remember to block! – In the case where you suspect some inherent quality of your experimental units may be causing variation in your response, arrange your experimental units into groups based on similarity in that quality • Remember to contact LISA! LISA: DOE – For short questions, attend our Walk-in Consulting hours – For research, come before you collect your data for design help Fall 2014 Reference Douglas C. Design and experiments. John Wiley & Sons, 2008. Montgomery, LISA: DOE Fall 2014 analysis of LISA: DOE Please don’t forget to fill the sign in sheet and to complete the survey that will be sent to you by email. Thank you! Fall 2014