Lecture 4

Recapitulation from the last lecture - one factor,
several levels
Blocking and Factorial Experimental Design
Chapter 4, Section 1, Chapter 5, Chapter 6 of
the text
Farrukh Javed
Department of Statistics
Lund University
April 7, 2014
More than one factor - The Blocking Principle
An engineer is interested in investigating the relationship
between the RF power setting and the etch rate for this tool.
In this example, we have one factor (RF power) with several
levels.
The Blocking Principle
Sometimes we are interested in testing whether a factor (say A) has an
effect, but another factor (say B) also influences the experiment.
We call such factors as nuisance factors and Blocking is a technique for
dealing with such factors.
If the nuisance factor is known and controllable, blocking can be use to
eliminate its effect on the experiment result.
A nuisance factor (say B) is a factor that probably has some effect on
the response, but its of no interest to the experimenter however, the
variability it transmits to the response needs to be minimized
If the nuisance factor is unknown and uncontrollable, that is, we do not
know the factor exists and it may have some influence on the
experiment. Randomization is a design technique used to guard against
such factors.
For example, an experiment is designed to test a new drug on patients.
There are two levels of the treatment, drug, and placebo, administered
to male and female patients. The gender of the patient is a blocking
factor accounting for treatment variability between males and females.
Typical nuisance factors include batches of raw material, operators,
pieces of test equipment, time (shifts, days, etc.), different experimental
units
Many industrial experiments involve blocking (or should) because it
reduces sources of variability
If the nuisance factor is known but uncontrollable, the analysis of
covariance can be used to compensate for it.
Failure to block is a common flaw in designing an experiment
(consequences?)
More than one factor - blocking (nuisance) factors
The Hardness Testing Example
We call such a design –> the randomized complete block
design or the RCBD
We wish to determine whether 4 different tips produce different
(mean) hardness reading on a Rockwell hardness tester
Randomized: within each block the order of measurements
should be randomized.
Assignment of the tips to an experimental unit; that is, a test
coupon
Complete: all combinations of levels in A and B should be
measured.
One can consider a completely randomized experiment
Block: B is a block and is typically a nuisance factor, i.e.
we are not interested in whether the influence of this is
significant or not.
Alternatively, the experimenter may want to test the tips across
coupons of various hardness levels
Extension of the ANOVA to the RCBD
The Hardness Testing Example
To conduct this experiment as a RCBD, assign all 4 tips to each
coupon
The test coupons are a source of nuisance variability
There is now a need for blocking
The Hardness Testing Example
Suppose that we use b = 4 blocks:
Each coupon is called a block; that is, its a more homogeneous
experimental unit on which to test the tips
Variability between blocks can be large, variability within a block
should be relatively small
In general, a block is a specific level of the nuisance factor
A complete replicate of the basic experiment is conducted in
each block
A block represents a restriction on randomization
All runs within a block are randomized
Notice the two-way structure of the experiment
Once again, we are interested in testing the equality of treatment
means, but now we have to remove the variability associated
with the nuisance factor (the blocks)
Extension of the ANOVA to the RCBD
ANOVA Display for the RCBD
Suppose that there are a treatments (factor levels) and b blocks.
A statistical model (effects model) for the RCBD is
yij = µ + τi + βj + ij , i = 1, . . . , a, j = 1, . . . , b,
Null hypothesis
H0 : τ1 = · · · = τa = 0.
Once again, we are interested in testing the equality of treatment
means, but now we have to remove the variability associated
with the nuisance factor (the blocks)
Model
Connection between projections and sums of squares
Suppose that there are a treatments (factor levels) and b blocks.
A statistical model (effects model) for the RCBD is
yij = µ + τi + βj + ij , i = 1, . . . , a, j = 1, . . . , b,
Relations
SST = kyk2 − kP1 yk2 = ky − P1 yk2 ,
SSE = SSRes = kyk2 − kPX yk2 = ky − PX yk2 ,
SSTreat = kPX yk2 − kPXred yk2 = kPX y − PXred yk2 ,
Null hypothesis
H0 : τ1 = · · · = τa = 0.
SSBlocks = kPXred yk2 − kP1 yk2 = kPXred y − P1 yk2 .
Formulas for manual computations
The test statistics can be again analyzed through the regression
model.
The test statistic takes the form
F =
kPX y − PXred yk2 /(a − 1)
ky − PX yk2 /(ab − a − b + 1)
Can you write down the design matrices: X and Xred ?
Vascular Graft Experiment - Description
Vascular Graft Experiment - Data
To conduct this experiment as a RCBD, assign all 4 pressures to
each of the 6 batches of resin
Each batch of resin is called a “block”; that is, its a more
homogeneous experimental unit on which to test the extrusion
pressures
Vascular Graft Experiment - ‘Manual’ Computations
Vascular Graft Experiment - Results
Vascular Graft Experiment - Incorrect Analysis
Residual Analysis for the Vascular Graft Example
Residual Analysis for the Vascular Graft Example
Conclusions from Residual Analysis for the Vascular
Graft Example
Basic residual plots indicate that normality, constant variance
assumptions are satisfied
No obvious problems with randomization
No patterns in the residuals vs. block
No patterns in the residuals versus the pressure (residuals by
factor)
These plots provide information about the constant variance
assumption, possible outliers
Multiple Comparisons for the Vascular Graft Example
Which Pressure is Different?
Factorial Experiments
General principles of factorial experiments
The two-factor factorial
The ANOVA for factorials
Extensions to more than two factors
Quantitative and qualitative factors: response curves and
surface
Factors without Interaction
In some experiments, the difference of response between
the levels of one factor is not the same at all the levels of the other factors.
At the low level of factor B (B − ), the A effect is, A = 40 − 20 = 20
At the high level of factor B (B + ), the A effect is, A = 52 − 30 = 22
The magnitude of interaction effect is the average difference in these
two A effects.
AB = (22 − 20)/2 = 1
Factors with Interaction
In some experiments, the difference of response between the levels of
one factor is not the same at all the levels of the other factors.
At the low level of factor B (B − ), the A effect is , A = 50 − 20 = 30
At the high level of factor B (B + ), the A effect is, A = 12 − 40 = −28
The magnitude of interaction effect is the average difference in these
two A effects.
AB = (−28 − 30)/2 = −29
The Battery Life Experiment
The General Two-Factor Factorial Experiment
Statistical (effects) model:
Factor A = Material type;
Factor B = Temperature (A quantitative variable)
What effects do material type and temperature have on life?
Is there a choice of material that would give long life regardless
of temperature (a robust product)?
Extension of the ANOVA to Factorials
SS decomposition:
SST = SSA + SSB + SSAB + SSE
DF decomposition
[abn − 1] = [a − 1] + [b − 1] + [(a − 1)(b − 1)] + [ab(n − 1)]
yijk = µ + τi + βj + (τ β)ij + ijk
where i = 1, . . . , a, j = 1, . . . , b, k = 1, . . . , n and
X
X
X
X
τi =
βj =
(τ β)ij =
(τ β)ij = 0
i
j
i
j
Other models (means model, regression models) can be useful
ANOVA Table and Interaction Plot
Model adequacy checking - Residual Analysis
Residual Analysis vs. Factors
Factorials with More Than Two Factors
Three Factors
Basic procedure is similar to the two-factor case; all abckn
treatment combinations are run in random order
ANOVA identity is also similar:
SST = SSA +SSB +· · ·+SSAB +SSAC +· · ·+SSABC +· · ·+SSAB···K +SSE
Example of three factorial design - the soft drink
bottling problem
Data
A soft drink bottler is interested in obtaining more uniform fill
heights in the bottles.
The process engineer can control three variables during the
filling process: the percent carbonation (A), the operating
pressure in the filler(B), and the bottles produced per minute or
the line speed (C).
For the purpose of an experiment, the engineer controls
carbonation at three levels: 10, 12, and 14 percent, pressure at
two levels: 25 and 30psi and two levels for line speed: 200 and
250bpm.
Analysis of variance table
23 Factorial Design
An engineer is interested in investigating the relationship between the
RF power setting and the etch rate for this tool. The objective of an
experiment like this is to model the relationship between etch rate and
RF power, and to specify the power setting that will give a desired target
etch rate.
The response variable is etch rate.
She is interested in a particular gas (C2F6) for which two flow rates has
been chosen: 125, 200; electrode gap at two levels: 0.8 cm and 1.2 cm,
and power: 275W, and 325W.
The experiment is replicated twice
Diagram of plasma-wafer etching tool
Quantitative and Qualitative Factors
The basic ANOVA procedure treats every factor as if it were
qualitative
Sometimes an experiment will involve both quantitative and
qualitative factors
This can be accounted for in the analysis to produce regression
models for the quantitative factors at each level (or combination
of levels) of the qualitative factors
These response curves and/or response surfaces are often a
considerable aid in practical interpretation of the results
Data and Analysis of variance

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