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(solution) Has to deal with experimental design and fractional factorials


Has to deal with experimental design and fractional factorials


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 e?ects. Write out the factor-level combinations that go into each block.

 

2. Suppose that we divide a 28 factorial into 16 blocks of size 16 using ABCF,

 

ABDE, ACDE, and BCDH as de?ning contrasts. Find all the confounded

 

e?ects.

 

3. Individuals perceive odors at di?erent intensities. We have a procedure that

 

allows us to determine the concentration of a solution at which an individual

 

?rst senses the odor (the threshold concentration). We would like to determine

 

how the threshold concentrations vary over sixteen solutions. The thresholddetermining procedure is time consuming and any individual judge can only

 

be used to ?nd threshold concentrations for four solutions.

 

Each solution is a combination of ?ve compounds in various ratios. The sixteen

 

solutions are formed by manipulating four factors, each at two levels. Factor

 

A is the ratio of the concentration of compound 1 to the concentration of

 

compound 5. Factor B is the ratio of the concentration of compound 2 to the

 

concentration of compound 5, and so on.

 

We have eight judges. Two judges are assigned at random to each of the solution sets [(1), bc, abd, acd], [a, abc, bd, cd], [ab, ac, d, bcd], and [b, c, ad, abcd]. The

 

threshold concentrations are normalized by dividing by a reference concentration. The results are given below. 1 Judge

 

1

 

(1)

 

bc

 

abd

 

acd 2

 

8389

 

816

 

4

 

46 3 a

 

abc

 

bd

 

cd 4351

 

78

 

5941

 

27138

 

5

 

6

 

(1) 56034 a

 

2346

 

bc

 

25046 abc

 

35

 

abd

 

109 bd

 

228

 

acd

 

490 cd

 

6842 ab

 

ac

 

d

 

bcd 4

 

6

 

262

 

1230

 

98 b

 

c

 

ad

 

abcd 67

 

3081

 

50991

 

784 b

 

c

 

ad

 

abcd 7

 

ab

 

ac

 

d

 

bcd 375

 

33551

 

246

 

10

 

8

 

40581

 

90293

 

19103

 

61 (a) Analyze the data to determine how the factors a?ect the threshold concentration.

 

(b) Re-analyze them after taking logs. Does it matter whether you take common logs (base 10) or natural logs (base e)?

 

(c) What de?ciencies can you ?nd in the design? 2

 


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