Statistical Design and Analysis of Experiments

Mason, Robert; Gunst, Richard; Hess, James (2003). Statistical Design and Analysis of Experiments. John Wiley and Sons. 

Overview

Book is broken into parts:

1. fundamental statistical concepts

- chapters 1-3

2. design and analysis with factorial structure

- chapters 4-8

3. design and analysis with random effects

- chapters 9-13

4. design and analysis with quantitative predictors and factors

- chapters 14-19

Approach: figure out the difference between each chapter (i.e. what is factorial structure, what are random effects, what are quantitative predictors/factors?)

Then go into detail in each chapter

Definitions

• factor - experimental variable influencing the response
• block - group of homogeneous experimental units
• homogeneous experimental units - units that are as uniform as possible on all characteristics that could affect the response
• unit - entity on which a measurement or observation is made (sometimes refers to the actual measurement or observation)

blocks: e.g. manufacturer of an item, or field plot, etc.; there are known differences between different blocks, but experiments are run on each block to eliminate or control that variation as much as possible

Chapter 4: Statistical principles in experiment design

Selecting a statistical design (p138):

Design criteria:

• Consideration of objectives
  • nature of anticipated conclusions
  • definition of concepts
  • determination of observable variables
• factor effects
  • elimination of systematic error
  • measurement of covariates
  • identification of relationships
  • exploration of the entire experimental region
• precision
  • estimation of variability (uncertainty)
  • blocking
  • repeat tests, replication
  • adjustment for covariates
• efficiency
  • multiple factors
  • screening experiments
  • fractional factorials
• randomization

objectives: necessary in all cases

factor effects: what are likely effects of factors? important to include all relevant factors to ensure uncontrolled systematic variation of these factors doesn't create bias experimental design should allow fitting of general model so features of response (and relationship with factors) can be identified

Chapter 5: Factorial experiments in completely randomized designs

especially useful for evaluating joint factor effects

appropriate when no restrictions on order of testing

chapter organization:

• sections 1-2: characterization of randomized designs and joint factor effects
• sections 3-4: calculation of factor effects, interpretation of influence of factor levels on response

factorial experiments: include all possible factor-level combinations

Characterization of randomized designs

in completely randomized designs, all factor-level combinations are randomly assigned to experimental units or to sequence of test runs

randomization doesn't prevent experimental problems that can cause bias, but it provides some protection against bias by averaging bias effects over all levels of factors in experiment

definition: randomization is a procedure whereby factor-level combinations are (a) assigned to experimental units or (b) assigned to a test sequence in such a way that every factor-level combination has equal chance of being assigned to any experimental unit or position in the test sequence

construction of factorial experiments in randomized designs:

1. enumerate all factor-level combinations

2. number factor-level combinations sequentially, 1 to N

3. from a random number table or sequence of random numbers, obtain random sequence of 1 to N

4. assign factor-level combinations to the experimental units in order specified by random number sequence

example: pipe flow... 2 types of pipe, 3 types of fittings, 4 types of valves random set of experiments of different combinations of each (but this requires a FULL exploration of response surface...)

advantages:

• inclusion of each factor level with variety of levels for other factors means effects of each factor on response investigated under variety of conditions
• randomization protects against unknown biases (repeat tests can also give estimate of experimental error)
• ability to investigate joint factor effects

definition: interactions exist among two or more factors if the effect of one factor on a response depends on the levels of other factors

example: yield as function of temperature for two different types of catalysts, if yield vs. temperature curves are parallel, then no interaction if yield vs. temperature curves are not parallel, i.e. one crosses the other, then there are interaction effects

Calculation of factor effects

effects coding of factor levels:

1. designate one level of each factor as -1 and the other level as +1

2. lay out a table with column headings for each of the factors A, B, ..., K

3. let n = 2^k, where k is the number of factors

4. set the first n/2 of the levels for factor A equal to -1 and the last n/2 equal to +1. Set the first n/4 levels of factor B equal to -1, the next n/4 equal to +1, the next n/4 equal to -1, etc. Continue in this fashion until the last column (for factor K) has alternating -1 and +1 sign

this is the effects representation for a two-factorial experiment

definitions:

main effect: the difference between the average responses at the two levels of a factor two-factor interaction: half the difference between the main effects of one factor at the two levels of a second factor three-factor interaction: half the difference between the two-factor interaction effects at the two levels of a third factor

calculation of effects for two-level factors:

1. construct effects representation for each main effect and each interaction

2. calculate linear combinations of the average responses using the signs in the effects column for each main effect and for each interaction

3. divide respective linear combinations of average responses by 2^{k-1}, where k is number of factors in experiment

r = number of observations

example of how to do this yield as function of temperature, concentration, catalyst

Chapter 6: analysis of completely randomized designs

• analysis of variance decomposition of observed quantity's variance into different components with different sources/causes
• estimation of model parameters for qualitative/quantitative factors
• statistical tests of factor effects
• multiple comparison techniques for comparing means/groups of means

Definitions

fixed effects: influence of factor levels that affect the mean of the response

random effects: influence of factor levels that affect the variance of the response

Multifactor Experiments

Factor Effects

Fixed factor effects: factors have fixed effects if the levels chosen for inclusion in the experiments are the only ones for which inferences are desired

Fixed factor effects are specifically selected because they are the only ones for which inferences are desired

Factor levels are assumed to exert constant influences on response

Factor effects are modeled by unknown parameters (in statistical models) that relate mean of response variable to factor levels

Fixed factor levels are not randomly selected; they are intentionally chosen for investigation

Example: pilot-plant chemical yield study... temperature, conc., catalyst type specifically chosen and only ones for which inferences can be made (e.g. can't make inferences about yield based on temperatures not included in the fixed factor experiment, unless you make additional assumptions)

Analysis of Variance (ANOVA) Models

ANOVA procedures separate variation observable in response variables into two components:

1. variation due to assignable causes
2. uncontrolled/random variation

Assignable causes: variation due to changes in experimental factors or measured covariates

Random variation: variation due to uncontrolled effects (chance causes, measurement errors)

Note that assignable causes will correspond to factors, and can be either fixed effects or random effects

Statistical model for pilot-plant chemical yield example:

$y_{ijkl}=\mu _{ijkl}+e_{ijkl}$

where $i,j,k,l$ represent the factors, $\mu _{ijkl}$ represents assignable causes of variation, $e_{ijkl}$ represents random error effects

This model is intended to draw a connection between multifactor experiments with fixed effects, and sampling from a number of populations or processes that differ only in their means

The assignable cause can be further decomposed into main effects and interaction effects due to different factors:

$\mu _{ijkl}=\mu +\alpha _{i}+\beta _{j}+\gamma _{k}+(\alpha \beta )_{ij}+(\beta \gamma )_{jk}+(\alpha \beta \gamma )_{ijk}$

with i = temperature, j = concentration, k = catalyst, and l = repeat test number.

Problem: right hand side is not unique... e.g. if you have

$\mu _{2}=\mu _{1}+c$

and

$\alpha _{2}=\alpha _{1}-c$

then

$\mu _{1}+\alpha _{1}=\mu _{2}+\alpha _{2}$

In order to bypass this problem, subject the parameters to constraints; most common:

${\begin{array}{rcl}\sum _{i}\alpha _{i}&=&0\\\sum _{j}\beta _{j}&=&0\\\sum _{k}\gamma _{k}&=&0\\\sum _{i}(\alpha \beta )_{ij}&=&0\\\sum _{j}(\alpha \beta )_{ij}&=&0\\{\mbox{etc.}}&&\end{array}}$

using these constraints, effects parameters can be expressed in terms of the averages of the model means, e.g.

${\begin{array}{rcl}\mu &=&{\overline {\mu }}_{\centerdot \centerdot \centerdot }\\&&\\\alpha _{i}&=&{\overline {\mu }}_{i\centerdot \centerdot }-{\overline {\mu }}_{\centerdot \centerdot \centerdot }\\\beta _{j}&=&{\overline {\mu }}_{\centerdot j\centerdot }-{\overline {\mu }}_{\centerdot \centerdot \centerdot }\\\gamma _{k}&=&{\overline {\mu }}_{\centerdot \centerdot k}-{\overline {\mu }}_{\centerdot \centerdot \centerdot }\\&&\\(\alpha \beta )_{ij}&=&{\overline {\mu }}_{ij\centerdot }-{\overline {\mu }}_{i\centerdot \centerdot }-{\overline {\mu }}_{j\centerdot \centerdot }+{\overline {\mu }}_{\centerdot \centerdot \centerdot }\\{\mbox{etc.}}&&\\&&\\(\alpha \beta \gamma )_{ijk}&=&\mu _{ijk}-{\overline {\mu }}_{ij\centerdot }-{\overline {\mu }}_{i\centerdot k}-{\overline {\mu }}_{\centerdot jk}+{\overline {\mu }}_{i\centerdot \centerdot }+{\overline {\mu }}_{\centerdot j\centerdot }+{\overline {\mu }}_{\centerdot \centerdot k}-{\overline {\mu }}_{\centerdot \centerdot \centerdot }\end{array}}$

Another way to express this:

Main effect parameters are differences between the model means for one level of a factor, and the overall mean

Two-factor interaction parameters are differences in the main effects for one level of one of the factors at a fixed level of a second and the main effect of the first factor:

$(\alpha \beta )_{ij}=({\overline {\mu }}_{ij\centerdot }-{\overline {\mu }}_{\centerdot j\centerdot })-({\overline {\mu }}_{i\centerdot \centerdot }-{\overline {\mu }}_{\centerdot \centerdot \centerdot })$

also expressed as:

$(\alpha \beta )_{ij}=({\mbox{main effect for A at level j of B}})-({\mbox{main effect for A}})$

Three-factor interaction parameters are, similarly, difference in the two-factor interaction between any two of the factors (say, A and B) at fixed level of third factor (say, C), and the two-factor interaction between A and B:

$(\alpha \beta \gamma )_{ijk}=(\mu _{ijk}-{\overline {\mu }}_{i\centerdot k}-{\overline {\mu }}_{\centerdot jk}+{\overline {\mu }}_{\centerdot \centerdot k})-({\overline {\mu }}_{ij\centerdot }-{\overline {\mu }}_{i\centerdot \centerdot }-{\overline {\mu }}_{\centerdot j\centerdot }+{\overline {\mu }}_{\centerdot \centerdot \centerdot })$

also expressed as:

$(\alpha \beta \gamma )_{ijk}=({\mbox{A x B interaction at level k of C}})-({\mbox{A x B interaction}})$

Hierarchical model: a statistical model is hierarchical if an interaction term involving k factors is included only when the main effects and all lower-order interaction terms involving the k factors are also included in the model

Analysis of Variance (ANOVA) Tables

Example given for three-factor experiment

• A with a levels
• B with b levels
• C with c levels
• r repeats

Previous sections of book: sample variance (or its square root, standard deviation) used to measure variability

in ANOVA procedure, first step is to define measure of variation for which partitioning into different assignable causes and random variations can be done

Measure of variability: total sum of squares (TSS)

${\mbox{TSS}}=\sum _{i=1}^{n}(y_{i}-{\overline {y}})^{2}$

For three-factor experiment, adjust for the overall average by subtracting it from each individual response

Thus it is also called the total adjusted sum of squares TSS(adj)

TSS(adj) consists of squared differences of observations $y_{ijkl}$ from overall oaverage response ${\overline {y}}_{\centerdot \centerdot \centerdot \centerdot }$:

${\mbox{TSS}}=\sum _{i}\sum _{j}\sum _{k}\sum _{l}(y_{ijkl}-{\overline {y}}_{\centerdot \centerdot \centerdot \centerdot })^{2}$

and to partition into assignable causes and random variation, add/subtract factor-level ocmbination averages ${\overline {y}}_{ijk\centerdot }$ from each term in the summation for the total sum of squares:

${\begin{array}{rcl}{\mbox{TSS}}&=&\sum _{i}\sum _{j}\sum _{k}\sum _{l}(y_{ijkl}-{\overline {y}}_{ijk\centerdot }+{\overline {y}}_{ijk\centerdot }-{\overline {y}}_{\centerdot \centerdot \centerdot \centerdot })^{2}\\&=&r\sum _{i}\sum _{j}\sum _{k}({\overline {y}}_{ijk\centerdot }-{\overline {y}}_{\centerdot \centerdot \centerdot \centerdot })^{2}\\&&+\sum _{i}\sum _{j}\sum _{k}\sum _{l}(y_{ijkl}-{\overline {y}}_{ijk\centerdot })^{2}\\&=&{\mbox{MMS}}+{\mbox{SS}}_{E}\end{array}}$

where the first component of the above equation is the contribution to the total sum of squares of all the variability attributable to assignable cuases (model sum of squares):

${\mbox{MMS}}=r\sum _{i}\sum _{j}\sum _{k}(y_{ijk\centerdot }-{\overline {y}}_{\centerdot \centerdot \centerdot \centerdot })^{2}$

and it can be partitioned into the following components:

${\mbox{MSS}}={\mbox{SS}}_{A}+{\mbox{SS}}_{B}+{\mbox{SS}}_{C}+{\mbox{SS}}_{AB}+{\mbox{SS}}_{AC}+{\mbox{SS}}_{BC}+{\mbox{SS}}_{ABC}$

where the first three terms measure individual factor contributions (main effects of 3 experimental factors)

the next three terms measure contributiuons of two-factor joint effects

and the last term measures contribution of the joint effects of all three factos

Calculation of individual factor contributions:

${\mbox{SS}}_{A}=bcr\sum _{i=1}^{a}({\overline {y}}_{i\centerdot \centerdot \centerdot }-{\overline {y}}_{\centerdot \centerdot \centerdot \centerdot })^{2}$

Calculation of interaction effects:

difference between (a) main effects of one factor at a particular level of a second factor, and (b) the overall main effect of the factor

e.g.: Calculation of interaction effects for A and B:

${\begin{array}{rcl}({\mbox{main effect for A at level j of B}})-({\mbox{main effect for A}})&&\\&=&({\overline {y}}_{ij\centerdot \centerdot }-{\overline {y}}_{\centerdot j\centerdot \centerdot })-({\overline {y}}_{i\centerdot \centerdot \centerdot }-{\overline {y}}_{\centerdot \centerdot \centerdot \centerdot })\\&=&{\overline {y}}_{ij\centerdot \centerdot }-({\overline {y}}_{i\centerdot \centerdot \centerdot }-{\overline {y}}_{\centerdot j\centerdot \centerdot }+{\overline {y}}_{\centerdot \centerdot \centerdot \centerdot })\end{array}}$

and if you reverse the roles, you get the same expression

Combining these, you can calculate an overall sum of squares for the interaction effects of factors A and B:

${\mbox{SS}}_{AB}=cr\sum _{i=1}^{a}\sum _{j=1}^{b}\left({\overline {y}}_{ij\centerdot \centerdot }-{\overline {y}}_{i\centerdot \centerdot \centerdot }-{\overline {y}}_{\centerdot j\centerdot \centerdot }+{\overline {y}}_{\centerdot \centerdot \centerdot \centerdot }\right)^{2}$

So to construct an ANOVA table:

Source of Variation	Degrees of freedom (df)	Sum of Squares (SS)	Mean sum of square (SS/df)	F-value $MS_{i}/MS_{E}$
$A$	$a-1$	$SS_{A}$	$MS_{A}=SS_{A}/df(A)$	$F_{A}=MS_{A}/MS_{E}$
$B$	$b-1$	$SS_{B}$	$MS_{B}=SS_{B}/df(B)$	$F_{B}=MS_{B}/MS_{E}$
$C$	$c-1$	$SS_{C}$	$MS_{C}=SS_{C}/df(C)$	$F_{C}=MS_{C}/MS_{E}$
$AB$	$(a-1)(b-1)$	$SS_{AB}$	$MS_{AB}=SS_{AB}/df(AB)$	$F_{AB}=MS_{AB}/MS_{E}$
$AC$	$(a-1)(c-1)$	$SS_{AC}$	$MS_{AC}=SS_{AC}/df(AC)$	$F_{AC}=MS_{AC}/MS_{E}$
$BC$	$(b-1)(c-1)$	$SS_{BC}$	$MS_{BC}=SS_{BC}/df(BC)$	$F_{BC}=MS_{BC}/MS_{E}$
$ABC$	$(a-1)(b-1)(c-1)$	$SS_{ABC}$	$MS_{ABC}=SS_{ABC}/df(ABC)$	$F_{ABC}=MS_{ABC}/MS_{E}$
${\mbox{Error}}$	$abc(r-1)$	$SS_{E}$	$MS_{E}=SS_{E}/df({\mbox{Error}})$

Analysis of Variance (ANOVA) Calculations

Total sum of squares:

$TSS=\sum _{i}\sum _{j}\sum _{k}\sum _{l}y_{ijkl}^{2}-SS_{M}$

Sum of squares for factor A:

$SS_{A}=\sum _{i}{\frac {y_{i\centerdot \centerdot \centerdot }^{2}}{bcr}}-SS_{M}$

Sum of squares for factor B:

$SS_{B}={\frac {\sum _{j}y_{\centerdot j\centerdot \centerdot }^{2}}{bcr}}-SS_{M}$

Sum of squares for factor C:

$SS_{C}={\mbox{etc}}$

Sum of squares for AB:

$SS_{AB}=\sum _{i}\sum _{j}{\frac {y_{ij\centerdot \centerdot }^{2}}{cr}}-SS_{M}-SS_{A}-SS_{B}$

Sum of squares for BC:

$SS_{BC}=\sum _{j}\sum _{k}{\frac {y_{\centerdot jk\centerdot }^{2}}{ar}}-SS_{M}-SS_{B}-SS_{C}$

Sum of squares for AC:

$SS_{AC}={\mbox{etc}}$

Sum of squares for ABC:

${\begin{array}{rcl}SS_{ABC}&=&\sum _{i}\sum _{j}\sum _{k}{\frac {y_{ijk\centerdot }^{2}}{r}}-SS_{M}-SS_{A}-SS_{B}-SS_{C}-SS_{AB}-SS_{AC}-SS_{BC}\\&=&\sum _{i=1}^{a}\sum _{j=1}^{b}\sum _{k=1}^{c}\left({\overline {y}}_{ijk\centerdot }-{\overline {y}}_{ij\centerdot \centerdot }-{\overline {y}}_{i\centerdot k\centerdot }+{\overline {y}}_{i\centerdot \centerdot \centerdot }+{\overline {y}}_{\centerdot j\centerdot \centerdot }+{\overline {y}}_{\centerdot \centerdot k\centerdot }-{\overline {y}}_{\centerdot \centerdot \centerdot \centerdot }\right)^{2}\end{array}}$

Sum of squares for M (mean?):

$SS_{M}={\frac {y_{\centerdot \centerdot \centerdot \centerdot }^{2}}{abcr}}$

Error sum of squares:

$SS_{E}=\sum _{i=1}^{a}\sum _{j=1}^{b}\sum _{k=1}^{c}\sum _{l=1}^{r}\left(y_{ijkl}-{\overline {y}}_{ijk\centerdot }\right)^{2}$

Averaging of response:

$y_{\centerdot \centerdot \centerdot \centerdot }=\sum _{i}\sum _{j}\sum _{k}\sum _{l}y_{ijkl}$

$y_{i\centerdot \centerdot \centerdot }=\sum _{j}\sum _{k}\sum _{l}y_{ijkl}$

$y_{ij\centerdot \centerdot }=\sum _{k}\sum _{l}y_{ijkl}$

$y_{ijk\centerdot }=\sum _{l}y_{ijkl}$

Chapter 7: fractional factorial experiments

• properties of fractional factorial experiments that determine which factor-level combinations are to be included in experiment
• construction of fractional factorial experiments in randomized designs
• efficient fractional factorials for experiments w/ two-level, three-level, and combination two-level/three-level factors
• use of fractional factorial experiments in saturated/supersaturated screening experiments
• sequentially building experiments using fractional factorials

fractional factorials used when there are limitations on experiments that can be done

can also be used in situations with many factor-levels since not necessary/possible to test all possible factor-level combinations

example: corrosion test in reactor

• six factors
• unknown if any factors would have effect on corrosion rate
• 64 tests required to investigate all combinations of all levels
• shutdown of plant required, so only gross factor effects identified

outline:

• sections 1-2: confounding and design resolution
• sections 3-5: two/three level fractional factorial w/ completely randomized designs
• section 6: use of fractional experiments in saturated/supersaturated screening experiments

Confounding of effects

fractional factorial experiments: some effects always confounded with each other

goal of fractional factorial design: ensure effects of primary interest unconfounded with others (or confounded with effects of small magnitude)

to design a fraction factorial design, utilize planned confounding of factor effects

definition: confounding: situation in which an effect cannot be unambiguously attributed to single main effect or interaction

section 5.3: factor effects defined in terms of differences/contrasts (linear combinations whose coefficients sum to zero) of response averages

expanded definition: confounded effects: two or more experimental effects are confounded (aliased) if their calculated values can only be attributed to their combined influence on the response and not to their unique individual influences. two or more effects are confounded if the calculation of one effect uses the same (apart from sign) difference/contrast of response averages as calculation of the other effects

when designing an experiment, designer must know which effects are confounded with which

example: plant with three factors, temperature/concentration/catalyst; table of effect representation (Section 5.3)

inclusion of column of coefficients for calculation of constant effect (I) (???)

constant effect signs are all +

constant effect: overall response average (???)

obtained by summing the individual factor-leel responses (r=1), and dividing by total number of factor level combinations ($2^{k}=2^{3}=8$)

all other effects obtained by taking contrasts of responsess indicated in respective columns, and dividing by half the number of factor level combinations $2^{k-1}=4$ (see Section 5.3)

example of chemical plant again...

• suppose table 7.2 has entries for observations in absence of bias due to which shift is performing the experiments (night shift/day shift)
• first four factor-level combinations listed in table 7.2 are taken during first shift
• last four factor-level combinations listed in table are taken during second shift

• first four combinations: temperature is at low level (-1)
• second four combinations: temperature is at upper level (+1)

• to account for influence of operating conditions and operators on each shift:
  • add constant amount, $s_{1}$ for first shift, $s_{2}$ for second shift, to each shift's observation
  • average responses for each shift are thereby increased by the same amounts

• to confirm confounding of main effect for temperature and main effect due to shifts: observe that these two main effects have same effect representation

• main effect for temperature: difference between first four and last four factor-level combinations (i.e. in notation of book ${\overline {y}}_{2\cdot \cdot }-{\overline {y}}_{1\cdot \cdot }$)
• but this is the same difference in averages that one woul dtake to calculate the effect of the shifts
• main effect representation for temperature in table 7.3 is also main effect representation for the shift effect (except, changing all of the signs)
  • i.e. for first shift, all temperatures are same sign, and shift is all same sign too
  • for second shift, all temperatures are same sign, and shift is all same sign too

• the calculated temperature effect is increased by an amount equal to the shift effect, $s_{2}-s_{1}$

• the other two main effects (concentration and catalyst) have equal number of positive and negative signs associated with responses, so the shift effect will cancel out
• half of responses will add shift, half of responses will subtract shift
• i.e.:

${\begin{array}{rcl}M(B)&=&{\frac {1}{4}}[(y_{121}+s_{1})+(y_{122}+s_{1})+(y_{221}+s_{2})\\&&+(y_{222}+s_{2})-(y_{111}+s_{1})-(y_{112}+s_{1})-(y_{211}+s_{2})-(y_{212}+s_{2})]\\&=&{\overline {y}}_{\cdot 2\cdot }-{\overline {y}}_{\cdot 1\cdot }\end{array}}$

• in the example of the chemical plant, the factorial experiment is purposefully partitioned so that half the complete factorial conducted during each of two shifts
• preferred approach: perform pilot study, and the fraction chosen for each shift would be a factor with little influence on response

when designing fractional factorial experiments, one seeks to confound:

• effects known to be small relative to uncontrolled experimental error variation
• (or, if you don't know) high-order interactions (those involving 3 or more factors)

frequently high-order interactions don't exist or are negligible compared to main effects

in the absence of information provided by a pilot-plant study... confound three-factor interaction in manufacturing plant experiment, rather than a main (temperature) effect

one would do this by assigning all factor-level combinations with one sign to a shift

all factor-level combinations with ABC=+1 (combos 1, 4, 6, 7) would be part of shift 1

all factor-level combinations with ABC=-1 (combos 2, 3, 5, 8) would be part of shift 2

confounding occurs when complete factorial experiment is conducted in groups of test runs (shifts)

confounding also cocurs when only portion of possible factor-level combinations included in an experiment

• occurs because two or more effects representations (apart from change in sign) are all the same
• calculated effect represents combined influence of the effects (???)

planned confounding (confounding in which important effects unconfounded or only confounded with negligible effects) is basis for statistical construction of fractional factorial experiments

same example... but now only 4 experiments can be performed

effects representation of only first four experiments being run

• grouped by pairs: | I A | B AB | C AC | BC ABC | Yield
• effects representations grouped in pairs because effects in each pair are calculated from same linear combination of responses
• each effect in a pair is the alias of the other effect

example: main effect for B and interaction AB are:

$M(B)={\frac {1}{2}}\left(-y_{111}-y_{112}+y_{121}+y_{122}\right)={\overline {y}}_{\cdot 2\cdot }-{\overline {y}}_{\cdot 1\cdot }$

and

$I(AB)={\frac {1}{2}}\left(+y_{111}+y_{112}-y_{121}-y_{122}\right)=-M(B)$

and the negative sign means the two effects representations are identical but of opposite signs

what does this mean? it means that the fractional factorial experiment makes each experimental effect aliased with one other experimental effect

from the fact that temperature is aliased with constant, $I=-A$, that tells us that no information is available on the temperature effect (b/c only one temperature level included in design)

every choice of four factor-level combinations wouold result in different compounding patterns

e.g. choosing combinations 2, 3, 5, 8 would create confounding patterns:

• $I=ABC$
• $A=BC$
• $B=AC$
• $C=AB$

all information is lost on the three-factor interaction, b/c all factor-level cominations in the design have the same sign on the effects representation for the ABC interaction

this example is not a good one, because it is known that both B (concentration) and B (joint catalyst type-temperature effect) both have strong influence on yield

this would be a good design if it was shown all interactions were zero or negligible

remainder of chapter: construction of fractional factorial experiments in completely randomized designs

focus on planning confounding of effcts to make most interesting effects either unconfounded or confounded with negligible interactions

Two-level fractional factorial experiments

can do half fractions, or quarter/smaller fractions

Half fractions

defining contrast: the effect that will be aliased with the constant effect

normally: effect for defining contrast is highest-order interaction among factors

pick (random coin flip) whether to include positive or negative effects

back to acid-plant corrosion rate (6 factors)

defining contrast: six-factor interaction ABCDEF

• effects representation for this defining contrast determined by multiplying all 6 elements (+1/-1) together
• this results in 32 factor-level combinations to be included in experiment

Design Resolution

• experimental design of resolution R has all effects containing s or fewer factors unconfounded with any effects containing fewer than R - s factors

Designing Half Fractions of Two-Level Factorial Experiments in Completely Randomized Designs

1. Choose defining contrast (effect to be aliased with constant effect)
2. Randomly decide wehther experiment will contain factor-level combinations with positive or negative signs in effects representation of defining contrast
3. Form table containing effects representations for main effects for each factor; add a column containing effects representation of defining contrast
4. select factor-level combinations that have chosen sign in the effects representation of the defining contrast; if applicable, pick any repeat tests
5. randomize assignment of factor-level combinations to experimental units or to the test sequence

Confounding Pattern for Effects in Half Frations of Two-Level Complete Factorial Experiments

1. write defining equation of fractional factorial experiment
2. symbolically multiply both sides of defining equation by one of factor effects
3. reduce both sides of equation by using algebraic convention

$X\cdot I=X,X\cdot X=X^{2}=I$

1. repeat steps 2 and 3 until each factor is listed in either left or right side of one of the equations

symbolic multiplication of two effects is equivalent to muitplying individual elements in the effects representations of the two effects

constant effect: all elements equal to 1

any effect multiplying the constant effect is unchanged

any effect mulitplying itself results in column of 1's (since $1^{2}=(-1)^{2}=1$)

manufacturing plant (concentration, temperature, catalyst -> yield):

• half-fraction with defining equation $I=ABC$
• this results in design of resolution 3

acid-corrosion (6 factors):

• half-fraction with defining equation $I=ABCDEF$
• this results in design of resolution 6

for defining equation $I=ABCDEF$, the above procedure can be used to determine confounding pattern of main effects:

$A=-AABCDEF=-BCDEF$

$B=-ABBCDEF=-ACDEF$

etc... and similar procedure for second- and third-order interactions

Quarter/smaller fractions

main difference between these and half-fraction designs is more than one defining contrast is needed

quarter-fraction design for 6 factors:

• 16 of 64 runs
• e.g.: two defining contrasts are ABCDEF and ABC
• results in resolution 3 design (smallest number of factors appearing in defining contrasts is 3)
• defining equation can be multiplied to determine main effects equations:

$I=ABCDA=BCD=ACDEF=BEFB=ACD=BCDEF=AEF$

etc...

Three-level fractional factorial experiments

more than 2 levels can lead to burdensome/inefficient sets of runs

4-factor experiment, 3 levels, 3^4 = 81 test runs

6-factor experiment, 3 levels, 3^6 = 729 test runs

large number of test runs used to estimate high-order interactions (not of primary interest)

48 degrees of freedom for three- and four-factor interactions in 3^4 complete factorial design (48 out of 81 test runs are for estimating these high-order interactions)

656 of 729 test runs in 3^6 complete factorial used to estimate third- and higher-order interactions

fractional factorial designs can lead to significant benefits, but more difficult to construct designs with 3, 5, 6, 7, etc. levels

to facilitate construction of fractional factorials of 3^k experiments, use tables of orthogonal arrays

Chapter 7 Appendix: fractional factorial experiments for factors with number of levels equal to power of 2

Chapter 17: two-level complete/fractional factorial designs with quantitative factor levels augmented with test runs at one or more specified additional levels (experiments include central composite designs and Box-Behnken designs); these are usually more efficient than 3^k designs

Designing One-Third Fractions of Three-Level Factorial Experiments in Completely Randomized Designs

1. Let $x_{j}$ denote coded level for factor j, wwhere levels coded as 0, 1, 2
2. Write defining contrast as $T=a_{1}x_{1}+a_{2}x_{2}+\dots +a_{k}x_{k}$, where $a_{j}=0,1,2$ is the power on factor j in the algebraic expression for defining contrast
3. Randomly decide whether T = 0 mod(3), T =1 mod(3), or T = 2 mod(3) in the defining relation; T = r mod(3) only if (T-r)/3 is an integer
4. Form table listing factor-level combinations in coded form; add column for values of T in defining contrast
5. Select test runs that hae the chosen value of T; randomly add repeat tests if applicable
6. Randomize assignment of factor-level combinations to experiment units or to test sequence, as appropriate

Confounding pattern for Effects in One-Third Fractions of Three-Level Complete Factorial Experiments

1. Write defining equation of fractional factorial experiment
2. Symbolically multiply both sides by one of factor effects components
3. Reduce RHS of eqn using following conventions:

$X\cdot I=X,X\cdot X=X^{2},X\cdot X\cdot X=X^{3}=I$

1. By convention, force exponent on first letter of multiplied effect to equal one and reduce remaining exponents of effect to mod(3) units; if exponent on first letter does not equal 1, square entire term and reduce exponents mod(3) to obtain exponent of 1
2. repeat steps 2-4 until each factor effect listed in either LHS or RHS of one of equations

Othogonal tables: can be used for experiment design... "pre-designed"

• highly efficient with respect to number of test runs, effective when fractional factorials of resolution R = 3 can be conducted
• Orthogonal arrays consist of very few test runs; design layouts are saturated (can be used with up to one fewer factors than test runs)
• key design properties:
  • resulting designs are only resolution 3
  • presence of interaction effects will bias main effects

definition: saturated design: an experimental design is saturated if degrees of freedom for all events (constant, main effects, interactions) equal the number of unique factor-level combinations included in the experiment; saturated 2^k experiment in which only main effects are analyzed has resolution n = k+1;

Levels Power of 2

if number of levels is power of 2, this is special case of number of levels > 2

it is important to understand nature of effects for factors having more than 2 levels (Chapter 5 appendix)

example: three factor design

• A = four levels
• B, C = two levels

Following Chapter 5 appendix, define

A = A1 and A2, each having two levels, and use defining equation:

$I=A_{1}A_{2}BC$

design is resolution 3, NOT resolution 4, because the two As represent a single factor

confounding pattern:

A1 = A2 B C A2 = A1 B C A1 A2 = B C

the last one shows that the main effect A1 A2 is confounded with two-factor interaction B C

Chapter 8: analysis of fractional factorial designs

• analysis of unbalanced, completely randomized designs with fixed effects
• analysis of two- and three-level fractional factorial experiments
• analysis of fractional experiments containing mix of two-level and three-level factors
• analysis of screening experiments

having some trouble understanding the opening 2 sections

Analysis from two-level fractional factorial experiments

no repeat runs = no estimate of error standard deviation = no analysis-of-variance table to determine significance of factor effects

can use a normal quantile plot (Section 5.4?) to determine which effects are dominant

Chapter 11: Nested designs

experimental situations sometimes require unique levels of one factor occur within each level of a second factor

this nesting of factors can also arise when experimental procedure restricts randomization of factor-level combinations

emphasis this chapter on:

• distinction between crossed/nested factors
• designs for hierarchically nested factors
• staggered nested designs in which number of levels of factor can vary within stage of the nesting
• split-plot designs, including both nested and crossed factors