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**Factorial Experiments: -Blocking,**

-Confounding, and -Fractional Factorial Designs. Emanuel Msemo Wednesday, July 30, 2014 4:30pm – 6:30 pm 1020 Torgersen Hall

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ABOUT THE INSTRUCTOR Graduate student in Virginia Tech Department of Statistics B.A. ECONOMICS AND STATISTICS (UDSM,TANZANIA) MSc. STATISTICS (VT,USA) LEAD/ASSOCIATE COLLABORATOR IN LISA “If your experiment needs a statistician, you need a better experiment.” Ernest Rutherford

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**MORE ABOUT LISA www.lisa.vt.edu What?**

Laboratory for Interdisciplinary Statistical Analysis Why? Mission: to provide statistical advice, analysis, and education to Virginia Tech researchers How? Collaboration requests, Walk-in Consulting, Short Courses Where? Walk-in Consulting in GLC and various other locations Collaboration meetings typically held in Sandy 312 Who? Graduate students and faculty members in VT statistics department

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**HOW TO SUBMIT A COLLABORATION REQUEST**

Go to Click link for “Collaboration Request Form” Sign into the website using VT PID and password Enter your information ( , college, etc.) Describe your project (project title, research goals, specific research questions, if you have already collected data, special requests, etc.) Contact assigned LISA collaborators as soon as possible to schedule a meeting

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**LISA helps VT researchers benefit from the use of Statistics**

Collaboration: Visit our website to request personalized statistical advice and assistance with: Experimental Design • Data Analysis • Interpreting Results Grant Proposals • Software (R, SAS, JMP, SPSS...) LISA statistical collaborators aim to explain concepts in ways useful for your research. Great advice right now: Meet with LISA before collecting your data. Short Courses: Designed to help graduate students apply statistics in their research Walk-In Consulting: M-F 1-3 PM GLC Video Conference Room; 11 AM-1 PM Old Security Building Room 103 For questions requiring <30 mins All services are FREE for VT researchers. We assist with research—not class projects or homework.

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**COURSE CONTENTS: 1. INTRODUCTION TO DESIGN AND ANALYSIS OF EXPERIMENTS**

1.2 Basic Principles 1.3 Some standard experimental designs designs 2. INTRODUCTION TO FACTORIAL DESIGNS 2.1 Basic Definitions and Principles 2.2 The advantage of factorials 2.3 The two-Factor factorial designs 2.4 The general factorial designs 2.5 Blocking in a factorial designs

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**4. BLOCKING AND CONFOUNDING IN THE 2K FACTORIAL DESIGNS**

3.1 Introduction 3.2 The 22 and 23 designs and the General 2k designs 3.3 A single replicate of the 2k designs 4. BLOCKING AND CONFOUNDING IN THE 2K FACTORIAL DESIGNS 4.1 Introduction 4.2 Blocking a replicated 2k factorial design. 4.3 Confounding in the 2k factorial designs. 5. TWO LEVEL FRACTIONAL FACTORIAL DESIGNS 5.1 Why do we need fractional factorial designs? 5.2 The one-half Fraction of the 2k factorial design 5.3The one-quarter Fraction of the 2k factorial design

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**INTRODUCTION TO DESIGN AND ANALYSIS OF EXPERIMENTS**

Questions: What is the main purpose of running an experiment ? What do one hope to be able to show? Typically, an experiment may be run for one or more of the following reasons: 1. To determine the principal causes of variation in a measured response 2. To find conditions that give rise to a maximum or minimum response 3. To compare the response achieved at different settings of controllable variables 4. To obtain a mathematical model in order to predict future responses

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**An Experiment involves the manipulation of one **

or more variables by an experimenter in order to determine the effects of this manipulation on another variable. Much research departs from this pattern in that nature rather than the experimenter manipulates the variables. Such research is referred to as Observational studies This course is concerned with COMPARATIVE EXPERIMENTS These allows conclusions to be drawn about cause and effect (Causal relationships)

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**Independent Variables**

Sources of Variation A source of variation is anything that could cause an observation to be different from another observation Independent Variables The variable that is under the control of the experimenter. The terms independent variables, treatments, experimental conditions, controllable variables can be used interchangeably

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**Dependent variable The dependent variable (response) reflects**

any effects associated with manipulation of the independent variable Now Sources of Variation are of two types: Those that can be controlled and are of interest are called treatments or treatment factors Those that are not of interest but are difficult to control are nuisance factors

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**PROCESS ……. ……. Uncontrollable factors Controllable factors**

Z1 Z2 ZP ……. PROCESS INPUTS OUTPUT (Response) ……. X1 X2 XP Controllable factors Adapted from Montgomery (2013) The primary goal of an experiment is to determine the amount of variation caused by the treatment factors in the presence of other sources of variation

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**The objective of the experiment may include the following;**

Determine which conditions are most influential on the response Determine where to set the influential conditions so that the response is always near the desired nominal value Determine where to set the influential conditions so that variability in the response is small Determine where to set the influential conditions so that the effects of the uncontrollable Variables are minimized

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**EXAMPLE; Researchers were interested to see the food consumption of**

albino rats when exposed to microwave radiation “If albino rats are subjected to microwave radiation, then their food consumption will decrease”

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**Independent variable? Dependant variable? Nuisance factor (s)? TRY!**

………………………. Dependant variable? ………………………. Nuisance factor (s)? ……………………….

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**BASIC PRINCIPLES Randomization Replication**

The three basic principles of experimental designs are; Randomization The allocation of experimental material and the order in which the individual runs of the experiment are to be performed are randomly determined Replication Independent repeat run of each factor combination Number of Experimental Units to which a treatment is assigned

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**Blocking A block is a set of experimental units sharing a**

common characteristics thought to affect the response, and to which a separate random assignment is made Blocking is used to reduce or eliminate the variability transmitted from a nuisance factor

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**OR SOME STANDARD EXPERIMENTAL DESIGNS**

The term experimental design refers to a plan of assigning experimental conditions to subjects and the statistical analysis associated with the plan. OR An experimental design is a rule that determines the assignment of the experimental units to the treatments.

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**Completely Randomized design**

Some standard designs that are used frequently includes; Completely Randomized design A completely randomized design (CRD) refer to a design in which the experimenter assigns the EU’s to the treatments completely at random, subject only to the number of observations to be taken on each treatment. The model is of the form; Response = constant + effect of a treatment + error

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Block designs This is a design in which experimenter partitions the EU’s in blocks, determines the allocation of treatments to blocks, and assigns the EU’s within each block to the treatments completely at random The model is of the form Response = Constant + effect of a block + effect of treatment + error

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**Designs with two or more blocking factors**

These involves two major sources of variation that have been designated as blocking factors. The model is of the form Response = Constant + effect of row block + effect of column block + effect of treatment + error

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**INTRODUCTION TO FACTORIAL DESIGNS**

Experiments often involves several factors, and usually the objective of the experimenter is to determine the influence these factors have on the response. Several approaches can be employed to deal when faced with more than one treatments Best – guess Approach Experimenter select an arbitrary combinations of treatments, test them and see what happens

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**One - Factor - at - a - time (OFAT)**

Consists of selecting a starting point, or baseline set of levels, for each factor, and then successively varying each factor over its range with the other factors held constant at the baseline level.

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**FACTORIAL EXPERIMENT The valuable approach to dealing with**

several factors is to conduct a FACTORIAL EXPERIMENT This is an experimental strategy in which factors are varied together, instead of one at a time

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**In a factorial design, in each complete trial **

or replicate of the experiment, all possible combination of the levels of the factors are investigated. e.g. If there are a levels of factor A and b levels of factor B, each replicate contains all ab treatment combinations The model is of the form Response = Constant + Effect of factor A + Effect of factor B + Interaction effect + Error term

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**Consider the following example (adapted from Montgomery, 2013) **

of a two-factors (A and B) factorial experiment with both design factors at two levels (High and Low) B High A Low B High A High 30 52 B Low A Low B Low A High 20 40

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**Main effect : Change in response produced by a **

change in the level of a factor Factor A _ Main Effect = 2 2 = 21 ,Increasing factor A from low level to high level, causes an average response increase of 21 units Factor B = ? Main Effect

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**Interaction A High B High A Low B High 40 12 50 A Low B Low 20 A High**

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**The effect of A depends on the level chosen for factor B**

At low level of factor B The A effect = 50 – 20 = 30 At high level of factor B The A effect = = -28 The effect of A depends on the level chosen for factor B

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**1 “If the difference in response between the levels of one**

factor is not the same at all levels of the other factors then we say there is an interaction between the factors” (Montgomery 2013) The magnitude of the interaction effect is the average difference in the two factor A effects AB (-28 – 30) = 2 = -29 A effect 1 = In this case, factor A has an effect, but it depends on the level of factor B be chosen

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**Interaction Graphically**

B High B High B Low Response Response B Low Low High Low High Factor A Factor A A factorial experiment without interaction A factorial experiment with interaction

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**Factorial designs has several advantages;**

They are more efficient than One Factor at a Time A factorial design is necessary when interactions may be present to avoid misleading conclusions Factorial designs allow the effect of a factor to be estimated at a several levels of the other factors, yielding conclusions that are valid over a range of experimental conditions

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**The two factor Factorial Design**

The simplest types of factorial design involves only two factors. There are a levels of factor A and b levels of factor B, and these are arranged in a factorial design. There are n replicates, and each replicate of the experiment contains all the ab combination.

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Example An engineer is designing a battery for use in a device that will be subjected to some extreme variations in temperature. The only design parameter that he can select is the plate material for the battery. For the purpose of testing temperature can be controlled in the product development laboratory (Montgomery, 2013) Life (in hours) Data Temperature Material Type 15 70 125 130 74 150 159 138 168 155 180 188 126 110 160 34 80 136 106 174 150 40 75 122 115 120 139 20 82 25 58 96 70 58 45 104 60 1 2 3

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**The design is a completely Randomized Design**

The design has two factors each at three levels and is then regarded as 32 factorial design. The design is a completely Randomized Design The engineer wants to answer the following questions; 1. What effects do material type and temperature have on the life of the battery? 2 .Is there a choice of material that would give uniformly long life regardless of temperature? Both factors are assumed to be fixed, hence we have a fixed effect model

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**Analysis of Variance for Battery life (in hours)**

Source DF Seq SS Adj SS Adj MS F P-value Material Type Temperature Material Type*Temperature Error Total We have a significant interaction between temperature and material type.

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Interaction plot Significant interaction is indicated by the lack of parallelism of the lines,Longer life is attained at low temperature, regardless Of material type

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**The General Factorial Design**

The results for the two – factor factorial design may be extended to the general case where there are a levels of factor A, b levels of factor B, c levels of factor C, and so on, arranged in a factorial experiment.

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**Sometimes, it is not feasible or practical **

to completely randomize all of the runs in a factorial. The presence of a nuisance factor may require that experiment be run in blocks. The model is of the form Response = Constant + Effect of factor A + Effect of factor B + interaction effect + Block Effect + Error term

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**The 2K Factorial designs**

This is a case of a factorial design with K factors, each at only two levels. These levels may be quantitative or qualitative. A complete replicate of this design requires 2K observation and is called 2K factorial design. Assumptions 1. The factors are fixed. 2. The designs are completely randomized. 3. The usual normality assumptions are satisfied.

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**- - The design with only two factors each at two levels is**

called 22 factorial design The levels of the factors may be arbitrarily called “Low” and “High” Factor A B Treatment Combination - + - + A Low, B Low A High, B Low A Low, B High A High, B High (1) a b ab The order in which the runs are made is a completely randomized experiment

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**The four treatment combination in the design can be**

represented by lower case letters The high level factor in any treatment combination is denoted by the corresponding lower case letter The low level of a factor in a treatment combination is represented by the absence of the corresponding letter The average effect of a factor is the change in the response produced by a change in the level of that factor averaged over the levels of the other factor

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**The symbols (1), a, b, ab represents the total**

of the observation at all n replicates taken at a treatment combination A main effect = 1/2n[ab + a – b – (1)] B main effect = 1/2n[ab +b - a – (1)] AB effect = 1/2n{[ab + (1) – a – b]

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**In experiments involving 2K designs, it is**

always important to examine the magnitude and direction of the factor effect to determine which factors are likely to be important Effect Magnitude and direction should always be considered along with ANOVA, because the ANOVA alone does not convey this information

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**We can write the treatment combination in the order **

We define; Contrast A = ab + a – b – (1) = Total effect of A We can write the treatment combination in the order (1), a, b, ab. Also called the standard order (or Yates order) Factorial Effect Treatment Combination I A B AB (1) a b ab + + + + - + - + + + - The above is also called the table of plus and minus signs

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**Suppose that three factors, A ,B and C, each at two levels**

are of interest. The design is referred as 23 factorial design Treatment Combination Factorial Effects I A B AB C AC BC ABC - + + + (1) a b ab c ac bc abc - + - + + + + - - + - + + - - + + A contrast = [ab + a + ac + abc – (1) – b – c - bc B contrast = ?

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**In General; The design with K factors each at two levels is**

called a 2K factorial design The treatment combination are written in standard order using notation introduced in a 22 and 23 designs

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**A single replicate of the 2K Designs**

For even a moderate number of factors, the total number of treatment combinations in a 2K factorials designs is large. 25 design has 32 treatment combinations 26 design has 64 treatment combinations Resources are usually limited, and the number of replicates that the experimenter can employ may be restricted Frequently, available resources only allow a single replicate of the design to be run, unless the experimenter is willing to omit some of the original factors

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**An analysis of an unreplicated factorials assume**

that certain high –order interaction are negligible and combine their means squares to estimate the error This is an appeal to sparsity of effect principle, that is most systems are dominated by some of the main effect and low – order interactions, and most high – order interactions are negligible When analyzing data from unreplicated factorial designs, its is suggested to use normal probability plot of estimates of the effects

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**Example A chemical product is produced in a pressure vessel.**

A factorial experiment is carried out in the pilot plant to study the factors thought to influence filtration rate of this product. The four factors are Temperature (A), pressure (B), concentration of formaldehyde (C), and string rate (D). Each factor is present at two levels. The process engineer is interested in maximizing the filtration rate. Current process gives filtration rate of around 75 gal/h. The process currently uses the factor C at high level. The engineer would like to reduce the formaldehyde concentration as much as possible but has been unable to do so because it always results in lower filtration rates (Montgomery, 2013)

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**The design matrix and response data obtained from single **

replicate of the 24 experiment Factors Treatment Combination A B C D Response (1) a b ab c ac bc abc d ad bd abd cd acd bcd abcd

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**The Normal probability plot is given below**

The important effects that emerge from this analysis are the main effects of A,C and D and the AC and AD interactions

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**The main effect plot for Temperature**

The plot indicate that its better to run the Temperature at high levels

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**The main effect plot for Concentration of Formaldehyde**

The plot indicate that its better to run the concentration of formaldehyde at high levels

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**The main effect plot for Stirring rate**

The plot indicate that its better to run the stirring rate at high levels

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**However, its necessary to examine any interactions that are important**

The best results are obtained with low concentration of formaldehyde and high temperatures

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**The AD interaction indicate that stirring rate D has little effect**

at low temperatures but a very positive effects at high temperature Therefore best filtration rates would appear to be obtained when A and D are at High level and C is at low level. This will allow Formaldehyde to be reduced to the lower levels

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**Factor effect Estimates and sums of squares for the 24 Design**

Model Term Effect Estimates Sum of Squares Percent Contribution A 21.63 32.64 B 3.13 39.06 0.68 C 9.88 390.06 6.81 D 14.63 855.56 14.93 AB 0.13 0.063 1.091E-003 AC -18.13 22.93 AD 16.63 19.29 BC 2.38 22.56 0.39 BD -0.38 0.56 9.815E-003 CD -1.12 5.06 0.088 ABC 1.88 14.06 0.25 ABD 4.13 68.06 1.19 ACD -1.62 10.56 0.18 BCD -2.63 27.56 0.48 ABCD 1.38 7.56

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**ANOVA for A, C and D Source DF Seq SS Adj SS AdjMS F P**

C*D <1 A*C*D <1 Residual Error Total

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**Blocking and Confounding in the 2K factorial designs**

There are situations that may hinder the experimenter to perform all of the runs in a 2K factorial experiment under homogenous conditions A single batch of raw material might not be large enough to make all of the required runs An experimenter with a prior knowledge, may decide to run a pilot experiment with different batches of raw materials The design technique used in this situations is Blocking

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**Blocking a Replicated 2K Factorial design**

Suppose that the 2K factorial design has been replicated n times With n replicates, then each set of homogenous conditions defines a block, and each replicate is run in one of the blocks The run in each block (or replicate) will be made in random order

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**Confounding in the 2K Factorial designs**

Many situations it is impossible to perform a complete replicate of a factorial design in one block Confounding is a design technique for arranging a complete factorial experiment in blocks, where the block size is smaller than the number of treatment combinations in one replicate The technique causes information about certain treatment effects (usually) higher order interactions) to be indistinguishable from or confounded with blocks

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**Confounding the 2K Factorial design in two Blocks**

Suppose we want to run a single replicate of the 22 design Each of the 22 = 4 treatment combination requires a quantity of raw material Suppose each batch of raw material is only large enough for two treatment combination to be tested, thus two batches of raw material are required If batches of raw materials are considered as blocks, then we must assign two of the four treatment combinations to each block

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**Consider table of plus and minus signs for the 22 design**

Treatment Combination Factorial Effect I A B AB Block (1) a b ab - + - + + + - 1 2 Block 1 Block 2 The order in which the treatment combination are run within a block are randomly Determined (1) ab a b The block effect and the AB interaction are identical. That is, AB is confounded with blocks.

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**This scheme can be used to confound any 2K design into two blocks**

Consider a 23 design run into two blocks Suppose we wish to confound the ABC interaction with blocks Factorial Effect Treatment Combination I A B AB C AC BC ABC Block (1) a b ab c ac bc abc - + - + + - + + - - + - + + - - + + + 1 2

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**Again, we assign treatment combinations that are **

minus on ABC to Block 1 and the rest to block 2 The treatment combinations within a block are run in a random order Block 1 Block 2 (1) ab ac bc a b c abc ABC is confounded with blocks

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**Alternative method for constructing the block**

The method uses the linear combination; L = a1x1 + a2x akxk This is called a defining contrast For the 2K ,xi = 0 (low level) or xi = 1 (high level), ai = 0 or 1 Treatment combination that produces the same value of L (mod 2) will be placed in the same block The only possible values of L (mod 2) are 0 and 1, hence we will have exactly two blocks

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**This approach is called partial confounding**

If resources are sufficient to allow the replication of confounded designs, it is generally better to use a slightly different method of designing the blocks in each replicate We can confound different effects in each replicate so that some information on all effects is obtained This approach is called partial confounding

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**Consider our previous example;**

Two modification; The 16 treatment combination cannot all be run using one batch of raw material. Experimenter will use two batches of raw material, hence two blocks each with 8 runs 2. Introduce a block effect, by considering one batch as of poor quality, such That all the responses will be 20 units less in this block Experimenter will confound the highest order interaction ABCD The defining contrast is; L = x1 +x2 + x3

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**The two resulting blocks are;**

(1) ab ac bc ad bd cd abcd a b c d abc bcd acd abd

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**The half Normal plot for the blocked design**

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**Source DF Seq SS Adj SS Adj MS F P-Value**

Blocks A <0.0001 C <0.0001 D <0.0001 A*C <0.0001 A*D <0.0001 Residual Error Total

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**NOTE: Similar methods can be used to confound the 2K designs**

to four blocks, and so on, depending on requirement NOTE: Blocking is a noise reduction technique. If we don’t block, then the added variability from the nuisance variable effect ends up getting distributed across the other design factors

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**Two – level Fractional Factorial Designs**

As the number of factors in a 2K factorial designs increases, the number of runs required for a complete replicate of the design rapidly outgrows the resources of most experimenters If the experimenter can reasonably assume that certain high-order interactions are negligible, information on the main effects and lower order interactions may be obtained by running a fraction of a complete factorial experiment Fractional factorials designs are widely used for product and process designs, process improvement and industrial/business experimentation Fractional factorials are used for screening experiments

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**1. The sparsity of effect principle**

The successfully use of Fractional factorials designs is based on three key ideas; 1. The sparsity of effect principle 2. The projection property 3. Sequential experimentation

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**The one – half Fraction of the 2K Design**

Suppose an experimenter has two factors, each at two levels but cannot afford to run all 23 = 8 treatment combinations They can however afford four runs This suggests a one – half fraction of a 23 design A one – half fraction of the 23 design is often called a 23-1 design

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**Recall the table of plus and minus signs for a 23 design**

Suppose we select those treatment combinations that have a plus in the ABC column to form 23-1 design, then ABC is called a generator of this particular design Usually a generator such as ABC is referred as a WORD The identity column is always plus, so we call; I = ABC , The defining relation for our design

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**We say the effects are aliased**

Now, It is impossible to differentiate between A and BC, B and AC, and C and AB We say the effects are aliased The alias structure may be easily determined by using a defined relation by multiplying any column by the defining relation A *I = A * ABC = A2BC = BC A = BC B*I = B * ABC = AB2C = AC B = AC This half fraction with I = ABC is called the Principal fraction

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**Design Resolution A design is of resolution R if no p-factor effect is**

aliased with another effect containing less than R-p factors Roman numeral subscript are usually used to denote design resolutions Designs of resolution III, IV and V are particularly important

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**Resolution III designs**

These are designs in which no main effects are aliased with any other main effects, but main effects are aliased with two factor interactions and some two factor interactions may be aliased with each other e.g. the 23-1 design with I = ABC is of resolution III Resolution IV designs No main effects is aliased with any other main effect or with any two factor interactions, but two factor interactions are aliased with each other e.g. A 24-1 design with I = ABCD is a resolution IV design Resolution V designs No main effect or two factor interactions is aliased with any other main effect or two factor interaction, but two factor interactions are aliased with three factor interactions e.g. A 25-1 design with I = ABCDE is a resolution V design

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**Construction of One half Fraction**

A one half fraction of the 2K design is obtained by writing down a basic design consisting of the runs for the full 2K-1 factorials and then adding the kth factor by identifying its plus and minus levels with the plus and minus signs of the highest order interactions ABC..(K-1) The 23-1 resolution III design is obtained by writing down the full 22 factorials as the basic design and then equating C to the AB interactions

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**One half fraction of the 23 design**

Full 22 Factorial (Basic Design) Resolution III, I = ABC Run A B A B C = AB 1 2 3 4 - + - + - + - + + -

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**Consider the filtration rate example;**

We will simulate what would happen if a half – fraction of the 24 design had been run instead of the full factorial We will use the 24-1 with I = ABCD, As this will generate the highest resolution possible We will first write down the basic design, which is 23 design The basic design has eight runs but with three factors To find the fourth factor levels, we solve I = ABCD for D D * I = D * ABCD = ABCD2 = ABC

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**The resolution IV design with I = ABCD**

Basic Design Treatment Combination Run A B C D = ABC (1) ad bd ab cd ac bc abcd 1 2 3 4 5 6 7 8 - + - + - + - +

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**Estimates of Effects Term Effect A 19.000 B 1.500 C 14.000 D 16.500**

A*C A*D A ,C and D have large effects, and so is the interactions involving them

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Thank you

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Reference Montgomery, D.C (2013). Design and analysis of experiments. Wiley, New York.

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