Week 7

Question 1

A 2 level full factorial design in f factors has how many treatments

Answer 1

t = 2^f

Question 2

Unit treatment model for 2 level full 3 factorial design

Answer 2

Question 3

Main effect for 2 level factor

Answer 3

Where first term is high level, second term is low level

Question 4

2 factor interaction between 2 level factors

Answer 4

Where the first and second indices correspond to factors A and B respectively

Question 5

Interaction between p 2-level factors

Answer 5

Question 6

Generic2 level factorial effect estimator

Answer 6

Question 7

Variance of generic factorial effect estimator

Answer 7

(Which can be calculated as such because they are unbiased)

Question 8

Required assumptions for Inference

Answer 8

r > 1 (must be replication)

Error terms must be normally distributed

Question 9

Why is replication required for inference

Answer 9

r = 1 would result in there being no residual DoF and hence no estimate for σ^2

r > 1 also ensure estimate of s^2 provided by MSE from full treatment model is independent of which factorial effects we choose to estimate

Question 10

Comparisons made in inference

Answer 10

Is compared to appropriate t dist with N - 2^f DoF

Question 11

how to determine r

Answer 11

(#of trials)/(#factors in full factorial)

Question 12

X fractional factorial design confounds Y factorial effects with the mean

Answer 12

X = 2^f-q
Y = 2^q - 1

q of these effects can be chosen independently
The others are formed as the set of all elementwide products of the first q

Question 13

General steps to choose a 2^f-q design with f factors

Answer 13

1) choose q factorial effects to confound with mean, E1, …, Eq

2) DEFINING RELATION: formed from set of 2^q - 1 effects consisting of E1, .., Eq and all of their products: I = E1 = … = Eq = E1E2 = … = E1…Eq

3) find aliasing scheme by multiplying defining relation by each combination of factors

4) find treatment combinations in design (in coded units) by finding treatments that satisfy q equations:
E1 = +/- 1 , E2 = +/- 1, … , Eq +/- 1

Question 14

Resolution of a 2^f-q design is

Answer 14

Length of shortest word in defining relation

Question 15

Word length pattern

Answer 15

For a 2^f-q design

A_i denotes number of factorial effects including i factors in the defining relation

Question 16

Minimum aberration

Answer 16

For two 2^f-q designs labelled δ1 and δ2
let l b the smallest value such that A_i(δ1) != A_i(δ2)

Then δ1 is said to have LESS aberration than δ2 if A_i(δ1) <= A_i(δ2)

If no design has less aberration than δ1 then δ1 has minimum aberration

Basically the design with higher resolution is preferred (according to ex sheet 6)

Question 17

How to find full aliasing scheme given generators

Answer 17

Multiply generators together, any squared characters become I