LISA Short Course SeriesBasics of Design of Experiments

Ana Maria Ortega-Villa

Fall 2014

LISA: DOE Fall 2014

LISA: R Basics Fall 2013LISA: 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

Laboratory for Interdisciplinary Statistical

Analysis

Collaboration:

Visit our website to request personalized statistical advice and assistance with:

Designing Experiments • Analyzing Data • Interpreting ResultsGrant 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.

All services are FREE for VT researchers. We assist with research—not class projects or homework.

LISA helps VT researchers benefit from the use of Statistics

www.lisa.stat.vt.edu

LISA also offers:

Educational Short Courses: Designed to help graduate students apply statistics in their researchWalk-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.

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

What are we doing?

LISA: DOE Fall 2014

Introduction to Design of Experiments

LISA: DOE Fall 2014

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

What is an Experiment?

LISA: DOE Fall 2014

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.

Why do we design experiments?

LISA: DOE Fall 2014

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.

What would be an alternative?

LISA: DOE Fall 2014

http://fluxicon.com/blog/wp-content/uploads/2012/02/observeandreport.jpg

• The researcher identifies and controls sources of variation that significantly impact the measured response.

• The researcher can gather evidence for causation.

Designed experiment

LISA: DOE Fall 2014

Correlation ≠ Causation

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.

But what are sources of variation?

LISA: DOE Fall 2014

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.

Rule of Thumb

LISA: DOE Fall 2014

Example: Impact of Exercise Intensity on Resting Heart Rate

LISA: DOE Fall 2014

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

How could we make it a designed expt?

LISA: DOE Fall 2014

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

…

What are our sources of variation?

LISA: DOE Fall 2014

Major Minor

Treatment

Nuisance Factor

Exercise intensity

Medication UseAir Temperature & Humidity

Location of measurementBody SizeBody Position

Minimum considerations:

• Response: Resting heart rate (beats per minute)

• Treatment: Exercise Program

o Low intensity

o Moderate intensity

o High intensity

Designing the experiment

LISA: DOE Fall 2014

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.

Designing the experiment

LISA: DOE Fall 2014

• 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.

Fundamentals of Design of Experiments

LISA: DOE Fall 2014

Three Basic Principles of Design of Experiments

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.

Randomization

LISA: DOE Fall 2014

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).

Randomization

LISA: DOE Fall 2014

1. Assignment of treatments to experimental units.

2. Order of the runs. Not applicable in this case since all participants are doing the experiment at the same time.

Randomization RHR Example

LISA: DOE Fall 2014

Participant Exercise Program

1 High

2 High

3 Low

4 Intermediate

5 Low

6 High

……. ……

Three Basic Principles of Design of Experiments

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.

# Replicates=# EUs/#Treatments

Replication

LISA: DOE Fall 2014

Participants 1, 2 and 6 can be considered as replicates of High intensity exercise treatment.

Replication in RHR Example

LISA: DOE Fall 2014

Participant Exercise Program

1 High

2 High

3 Low

4 Intermediate

5 Low

6 High

……. ……

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.

Pseudoreplication

LISA: DOE Fall 2014

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

Pseudoreplication

LISA: DOE Fall 2014

Three Basic Principles of Design of Experiments

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.

Local control of error

LISA: DOE Fall 2014

• 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.

RHR Local control of error

LISA: DOE Fall 2014

Common Designs:

LISA: DOE Fall 2014

Completely 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).

Complete Randomized Design (CRD)

LISA: DOE Fall 2014

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)

CRD Example: Plasma Etching Experiment

LISA: DOE Fall 2014

Energy is supplied by a

generator.

Chemical mixture gas is

is shot at a sample.

Plasma is generated in gap between

electrodes

CRD Example: Etching Process simplified

LISA: DOE Fall 2014

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.

CRD Example: Study

LISA: DOE Fall 2014

• 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.

CRD Example: Principles of DOE

LISA: DOE Fall 2014

CRD Example: Randomization Scheme

LISA: DOE Fall 2014

Run Treatment

1 3

2 4

3 1

4 2

5 2

6 2

7 4

8 3

9 3

10 1

Run Treatment

11 4

12 4

13 3

14 1

15 2

16 1

17 2

18 3

19 4

20 1

This run order was obtained using a random number generator.

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.

CRD Example: What is the question?

LISA: DOE Fall 2014

• 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.

CRD Example: Analysis

LISA: DOE Fall 2014

We will choose Rate as the Y response and Treatment as the X factor.

CRD Example: Analysis

LISA: DOE Fall 2014

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.

CRD Example: Visual Analysis

LISA: DOE Fall 2014

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.

CRD Example: ANOVA Table

LISA: DOE Fall 2014

Red Triangle: Compare Means -> Tukey HSD

At least two treatments are different, which ones?

CRD Example: Contrasts

LISA: DOE Fall 2014

• 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.

CRD: Summary

LISA: DOE Fall 2014

Common Designs:

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.

Randomized Complete Block Design (RCBD)

LISA: DOE Fall 2014

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)

RCBD Example: Vascular Graft Experiment

LISA: DOE Fall 2014

Response: Percentage of tubes that did not contain any flick.

Treatment: Extrusion Pressure (4 levels)

Block: Batch of resin (6 batches).

RCBD Example: Study

LISA: DOE Fall 2014

• 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.

RCBD Example: Principles of DOE

LISA: DOE Fall 2014

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.

RCBD Example: What is the question?

LISA: DOE Fall 2014

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.

RCBD Example: What is the question?

LISA: DOE Fall 2014

Analysis: Follow the same procedure.

Analyze->Fit Y by X.

RCBD Example: Analysis JMP

LISA: DOE Fall 2014

RCBD Example: Visual Analysis

LISA: DOE Fall 2014

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.

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.

RCBD Example: ANOVA Table

LISA: DOE Fall 2014

Significant differences between treatments 1 and 4, and 2 and 4.

RCBD Example: Contrasts

LISA: DOE Fall 2014

Common Designs:

LISA: DOE Fall 2014

Introduction to 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)

Factorial Designs

LISA: DOE Fall 2014

• 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).

Factorial Designs: Main Concepts

LISA: DOE Fall 2014

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)

Factorial Designs Example: Battery Design

LISA: DOE Fall 2014

Response: Battery life

Treatment: All combinations the factors:

Material: 3 levels (1, 2 and 3)

Temperature: 3 levels (15, 70 and 125 °F)

Factorial Design Example: Study

LISA: DOE Fall 2014

• 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.

Factorial Design Example: Principles of DOE

LISA: DOE Fall 2014

Factorial Design Example: Randomization

LISA: DOE Fall 2014

Mat Temp Run

1 15 6

1 15 11

1 15 26

1 15 31

1 70 22

1 70 34

1 70 1

1 70 33

1 125 15

1 125 21

1 125 28

1 125 19

Mat Temp Run

2 15 17

2 15 30

2 15 23

2 15 14

2 70 25

2 70 35

2 70 5

2 70 20

2 125 10

2 125 29

2 125 36

2 125 7

Mat Temp Run

3 15 13

3 15 9

3 15 32

3 15 24

3 70 27

3 70 8

3 70 12

3 70 2

3 125 3

3 125 4

3 125 18

3 125 16

Factorial Design Example: Randomization

LISA: DOE Fall 2014

You can create your own design in JMP:

DOE->Custom Design

Factorial Design Example: Analysis

LISA: DOE Fall 2014

Analyze->Fit Model

Red Triangle: Factor Profiling -> Interaction Plots

Factorial Design Example: Interaction

LISA: DOE Fall 2014

Factorial Design Example: ANOVA Theory

LISA: DOE Fall 2014

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.

Factorial Design Example: ANOVA

LISA: DOE Fall 2014

Model adequacy checking

LISA: DOE Fall 2014

Model Adequacy checking

LISA: DOE Fall 2014

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

Model Adequacy checking: Assumptions

LISA: DOE Fall 2014

• 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.

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

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)

Exercise:

LISA: DOE Fall 2014

Suppose you obtain this interaction plot, what would you interpret?

Exercise: Question 1

LISA: DOE Fall 2014

Conduct the factorial analysis in JMP, what can you conclude?

Exercise: Analysis

LISA: DOE Fall 2014

What can you say about the residuals?

Exercise: Analysis

LISA: DOE Fall 2014

Summary

LISA: DOE Fall 2014

• 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!– For short questions, attend our Walk-in Consulting hours– For research, come before you collect your data for

design help

Reference

LISA: DOE Fall 2014

Montgomery, Douglas C. Design and analysis of experiments. John Wiley & Sons, 2008.

LISA: DOE Fall 2014

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!