STAT/ME 424 Homework
Due Dec 15 in lecture
1. Split a 25 factorial into four blocks of eight, confounding BCD and ACE with
block eects. Write out the factor-level combinations that go into each block.
2. Suppose that we divide a 28 factorial into 16 block
1 Practice Exam 2 Statistics 424
1 Given the two replicates in an experiment where Factor A has 2 levels and Factor B has 3 levels Factor A 1 1 1 2 2 2 Factor B 1 2 3 1 2 3 Rep. 1 Rep. 2 19 25 5 7 4 12 5 3 17 15 6 14
(a) Find the sums of squares re
Stat/ME 424, Spring 2008 Problems and Solutions for Practice Exam for Midterm I
This is a closed book exam, except for three pages of notes. You are allowed to use calculators. There are four problems in this exam. In order to get full credit, pl
Notes for Stat/ME 424 Design and Analysis of Experiments, Spring 2008
Instructor : Peter Qian E-mail : [email protected] Department of Statistics University of Wisconsin-Madison
Text book : Experiments : Planning, Analysis, and Parameter Desig
Design Analysis of Experiments
The Longest Putt
4 Components:
1 Design what will the experiment consist of, what are you testing
2 Modeling Regression Analysis of a 24 Factorial design
3 Analysis Minitab output
4 Scientific Inference Conclusions
Design/Ba
Unit 4: Full Factorial Experiments at Two Levels
Source : Chapter 4 (sections 4.1-4.4, 4.6-4.12, 4.15).
An Epitaxial Layer Growth Experiment (Section 4.1).
Basic concepts for 2k designs (Section 4.2).
Factorial effects and plots (Section 4.3).
Using R
Unit 6: Robust Parameter Design
Source : Chapter 11 (sections 11.1 - 11.6, part of sections 11.7 - 11.8 and 11.9).
Revisiting two previous experiments.
Strategies for reducing variation.
Types of noise factors.
Variation reduction through robust param
Effects
R
Yates
Minitab
2.594
2.594
Coefficients
R
Yates
Minitab
1.297
1.297
A
x1
1.204
1.204
1.204
B
x2
0.216
0.216
0.216
C
x3
-0.256
0.099
-0.256
D
x4
0.308
-0.256
0.308
E
x5
-0.018
-0.266
-0.018
AB
x1:x2
0.099
0.071
0.099
AC
x1:x3
-0.266
0.039
-0.266
B
Notes for Stat/ME 424
Design and Analysis of Experiments
Instructor : Peter Qian
Department of Statistics
University of Wisconsin-Madison
Text book : Experiments : Planning, Analysis, and Parameter Design
Optimization
(by Wu and Hamada; Wiley, 2009, 2nd E
Chapter 10 of Experiments (2nd)
by Wu and Hamada
Response Surface Methodology
To study the relationship between the response and the input
factors.
To optimize the response or to understand the underlying
mechanism.
The input factors are quantitative.
1 /
STAT 424
Discussion 13
TA: Donggyu Kim
1. Robust Parameter Design
Two types of factors in a system (product/process):
control factors: once chosen, values remain fixed.
noise factors: hard-to-control during normal process or usage
Robust Parameter des
STAT 424
Discussion 3
TA: Donggyu Kim
1. One-Way ANOVA
Model:
y = + i + ij ,
i = 1, . . . , k, j = 1, . . . , ni
ij N (0,
2
),
where yij the jth observation with treatment i, and i is the ith treatment eect
Model fit:
yij
where y =
Pk 1
i=1
ni
Pk
i=1
A
STAT 424
Discussion 5
TA: Donggyu Kim
1. Two-way layout
Model:
yijl = + i + j + ij + ijl ,
i = 1, . . . , I, j = 1, . . . , J, l = 1, . . . , n,
where yijl = observation for the lth replicate of the ith level of factor A and the jth level of factor B
i =
STAT 424
Discussion 10
TA: Donggyu Kim
1. Fractional Factorial Experiments
A 2kp design has k factors, 2kp runs, and it is a 2p th fraction of the 2k design. The fraction is
defined by p independent defining words. The group formed by these p words is ca
STAT 424
Discussion 4
TA: Donggyu Kim
1. One-way random effects model
Model:
y = + i + ij , i = 1, . . . , k, j = 1, . . . , ni ,
where ij N (0, 2 ), i N (0, 2 ), and ij and i are independent.
Test:
H0 : 2 = 0
vs.
HA = > 0
Under H0, the F test and the A
STAT 424
Discussion 8
TA: Donggyu Kim
1. 2k factorial design
Main effect A: M E(A) = Z(A+)
Z(A)
Conditional main effect of B at + level of A: M E(B|A+) = Z(B|A+)
Z(B|A+)
Two-factor interaction effect between A and B:
IN T (A, B) =
=
=
1
[M E(B|A+) M
STAT 424
Discussion 2
TA: Donggyu Kim
1. Simple Regression
Model:
N (0, 2 )
y = 0 + 1 x + ,
Least Squares estimates
1 =
PN
)(yi
i=1 (xi x
PN
)2
i=1 (xi x
0 = y 1 x
,
where x
=
1
N
PN
i=1 xi
and y =
1
N
y)
var(1 ) = PN
,
2
i=1 (xi
var(0 ) = 2
x
)2
!
1
STAT 424
Discussion 9
TA: Donggyu Kim
1. 2k factorial design
Normal Plot
Deduction Step: Null hypothesis H0 : all factorial effects = 0 . Under H0 , i N (0, 2 ) and the
resulting normal plot should follow a straight line.
Induction Step: By fitting a s
STAT 424
Discussion 7
TA: Donggyu Kim
1. Graeco-Latin Square Design
Two Latin squares are orthogonal if each pair of letters appears once in the two squares, when superimposed. The super-imposed square is called a Graeco-Latin square.
Model:
yijlm = + i
STAT 424
Discussion 10
TA: Donggyu Kim
1. Robust Parameter Design
Two types of factors in a system (product/process):
control factors: once chosen, values remain fixed.
noise factors: hard-to-control during normal process or usage
Robust Parameter des
STAT 424
Discussion 14
TA: Donggyu Kim
1. Fractional Factorial Experiments at Three Levels
The break-up of the degrees of freedom will be as follows (33 design):
Each main effect has two degrees of freedom because each factor has three levels.
Each two-f
Data Set
j
i
1 Low
2 Medium
3 High
1
1
6.88
2
8.13
3
9.20
grand mean
7.95
Residuals (from individual means)
i
1
1 Low
2 Medium
3 High
Residuals (from grand mean)
i
1 Low
2 Medium
3 High
ssE
2
7.6
6.7
8.5
3
8.2
8.1
9.7
2
0.72
-1.43
-0.70
1
3
1.32
-0.03
0.5
NTHU STAT 5510, 2014
Solution to Homework 1
Problem 1
(a)
typing experiment
model_1yijk = + i + j + k + ij
i manuscript effecti=1,2,.,6
j keyboard effectj=1,2 ( A,B)
k k learning effectk=1,2(k i,j
j keyboard i manuscript k=1 k=2)
yijk response
N(0,2I)
Stat/ME 424, Fall 2016
Practice Exam 1
This is a closed book exam.
You are allowed to use calculators.
There are five problems in this exam. In order to get full credit, please answer all problems.
Please put all your answers in a blue book with your
Stat/ME 424, Spring 2011
Solutions to Problems for Midterm II
(Total : 100 points)
This is a closed-book exam, except for several sides of notes.
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You are allowed to use calculators.
Problem 1 (20
Stat/ME 424, Spring 2008
Practice Exam for Second Midterm Examination
This is a closed book exam, except for four pages of notes.
You are allowed to use calculators.
There are four problems in this exam. In order to get full credit, please answer all p