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: anaorte@vt.edu LISA: DOE Fall 2014 LISA: R Basics Fall 2013

Collaboration: 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? Introduction to Design of Experiments DOE main principles Randomization Replication Local control of error Complete Randomized Design Randomized Complete Block Design Introduction to factorial Designs LISA: DOE Fall 2014

Introduction to Design of Experiments LISA: DOE Fall 2014

https://weakinteractions.files.wordpress.com/2009/08/s1e1.jpg?w=450 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. https://weakinteractions.files.wordpress.com/2009/08/s1e1.jpg?w=450 LISA: DOE 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. http://fluxicon.com/blog/wp-content/uploads/2012/02/observeandreport.jpg LISA: DOE 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 Exercise each week Resting Heart Rate 1 2 3 … http://karmajello.com/postcont/2014/02/What-Exercise-Can-Heart-Patients-Undertake-e1352999185475.jpg What type of study is this? Observational Study LISA: DOE 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. Subject Intensity level of exercise each week Resting Heart Rate 1 2 3 … LISA: DOE Fall 2014

What are our sources of variation? Major Minor Treatment Nuisance Factor Exercise intensity Medication Use Air Temperature & Humidity Location of measurement Body Size Body Position LISA: DOE Fall 2014

Designing the experiment Minimum considerations: Response: Resting heart rate (beats per minute) Treatment: Exercise Program Low intensity Moderate intensity High intensity LISA: DOE 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 8-week 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. For the resting heart rate example, the participants are the EU. 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. 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. 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. Assignment of treatments to experimental units Order of the runs (when applicable). LISA: DOE Fall 2014

Randomization RHR Example Assignment of treatments to experimental units. Order of the runs. Not applicable in this case since all participants are doing the experiment at the same time. Participant Exercise Program 1 High 2 3 Low 4 Intermediate 5 6 ……. …… LISA: DOE Fall 2014

Three Basic Principles of Design of Experiments Replication LISA: DOE Fall 2014

# Replicates=# EUs/#Treatments 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. # Replicates=# EUs/#Treatments LISA: DOE Fall 2014

Replication in RHR Example Participants 1, 2 and 6 can be considered as replicates of High intensity exercise treatment. Participant Exercise Program 1 High 2 3 Low 4 Intermediate 5 6 ……. …… 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 the true treatment differences LISA: DOE Fall 2014

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 self-reporting). 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

Completely Randomized Design (CRD) 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) lksjfklsnf LISA: DOE Fall 2014

CRD Example: Etching Process simplified Energy is supplied by a generator. Chemical mixture gas is is shot at a sample. Plasma is generated in gap between electrodes lksjfklsnf LISA: DOE Fall 2014

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 1 3 2 4 5 6 7 8 9 10 Run Treatment 11 4 12 13 3 14 1 15 2 16 17 18 19 20 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: Generator power settings affects the etch rate. 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 variability into three separate independent pieces: 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

Randomized Complete Block Design (RCBD) 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) lksjfklsnf 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: Extrusion pressure affects the response. Higher pressure settings seem to result in decreased no flicks percentages. 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

Introduction to Factorial Designs 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: Material: 3 levels (1, 2 and 3) Temperature: 3 levels (15, 70 and 125 °F) LISA: DOE 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 1 15 6 11 26 31 70 22 34 33 125 21 28 19 Mat Temp Run 2 15 17 30 23 14 70 25 35 5 20 125 10 29 36 7 Mat Temp Run 3 15 13 9 32 24 70 27 8 12 2 125 4 18 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 temperature changes depending on material. LISA: DOE Fall 2014

Factorial Design Example: ANOVA LISA: DOE 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

Model Adequacy checking: Battery LISA: DOE 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: LISA: DOE Fall 2014 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! Remember to replicate! 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! For short questions, attend our Walk-in Consulting hours For research, come before you collect your data for design help LISA: DOE Fall 2014

Reference Montgomery, Douglas C. Design and analysis of experiments. John Wiley & Sons, 2008. LISA: DOE Fall 2014

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