doe in excel

8/12/2019 DOE in Excel

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Experimental Design

A Factorial Design Example

Factorial designs are a type of experimental design for screening experiments.

The theory of factorial designs is explained in the document 'Factorial Designs' availalble fo

Software suitable for analysis of factorial designs includes well-known programs such as Mi

However these packages are quite expensive and, as most experimenters have access to

has been set up to illustrate using Excel to analyse a factorial design.

The data used for this design is from the article 'Screening and Sequential Experimentation:

Flame Atomic Absorption Spectrometry Experiments', J. Chem. Ed., 74, 216 (Feb 1997)


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r download as a pdf file.

nitab and Statistica.

xcel, this spreadsheet

: Simulations and


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The Design

The experiment in this case is the analysis of silver by flame AAS

To set up the experimental design the following steps need to be carried out:

1. Define the Variables

What variables affect the outcome of the experiment?

In this case 6 variables are considered to affect the result:

A Flame Height above Base (mm)

B Flame Stoichiometry

C Acetic Acid (%)

D Lamp Current (mA)

E Wavelength (mm)

F Slit Width (nm)

2. Define the Response Variable(s)

In this case the resonse variable is the AA signal (mAbs)

3. Define the Experimental Domain

We need to specify an appropriate range for each variable I.e. a low and high va

This range needs to be wide enough to include the optimal conditions but be wit

settings for the instrument. The following limits have been defined:

Variable Low High

A 6 12

B lean rich

C 0 5

D 4 8

E 328.1 338.1

F 0.2 0.7

4. Choice of Design

The aim of the experiment is to carry out a screening I.e. determine which variabresponse. If a variable doesn't significantly affect the result then it can be 'scree

This means the variable is set at its mid-point value and not varied in subsequen

This is often a necessary step before a full optimization study, to reduce the nu

a manageable numer (preferrably 2-4).

Factorial designs are commonly used for screening. In this case, with 6 variable

design - I.e all combinations of each variable at the two levels- would require 2^6

The full factorial design, in coded form , is shown on the next sheet.


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In coded form the low settings for each variable are shown as -1 and the high se


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lue

in achievable

les significantly affect theed out'.

t experiments.

ber of variables to

, to carry out a full factorial

= 64 experiments.


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ttings as +1


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Full Factorial Design for 6 variables

A B C D E F

-1 -1 -1 -1 -1 -1

1 -1 -1 -1 -1 -1

-1 1 -1 -1 -1 -11 1 -1 -1 -1 -1

-1 -1 1 -1 -1 -1

1 -1 1 -1 -1 -1

-1 1 1 -1 -1 -1

1 1 1 -1 -1 -1

-1 -1 -1 1 -1 -1

1 -1 -1 1 -1 -1

-1 1 -1 1 -1 -1

1 1 -1 1 -1 -1

-1 -1 1 1 -1 -1

1 -1 1 1 -1 -1

-1 1 1 1 -1 -11 1 1 1 -1 -1

-1 -1 -1 -1 1 -1

1 -1 -1 -1 1 -1

-1 1 -1 -1 1 -1

1 1 -1 -1 1 -1

-1 -1 1 -1 1 -1

1 -1 1 -1 1 -1

-1 1 1 -1 1 -1

1 1 1 -1 1 -1

-1 -1 -1 1 1 -1

1 -1 -1 1 1 -1

-1 1 -1 1 1 -1

1 1 -1 1 1 -1

-1 -1 1 1 1 -1

1 -1 1 1 1 -1

-1 1 1 1 1 -1

1 1 1 1 1 -1

-1 -1 -1 -1 -1 1

1 -1 -1 -1 -1 1

-1 1 -1 -1 -1 1

1 1 -1 -1 -1 1

-1 -1 1 -1 -1 1

1 -1 1 -1 -1 1

-1 1 1 -1 -1 1

1 1 1 -1 -1 1-1 -1 -1 1 -1 1

1 -1 -1 1 -1 1

-1 1 -1 1 -1 1

1 1 -1 1 -1 1

-1 -1 1 1 -1 1

1 -1 1 1 -1 1

-1 1 1 1 -1 1

1 1 1 1 -1 1


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-1 -1 -1 -1 1 1

1 -1 -1 -1 1 1

-1 1 -1 -1 1 1

1 1 -1 -1 1 1

-1 -1 1 -1 1 1

1 -1 1 -1 1 1

-1 1 1 -1 1 11 1 1 -1 1 1

-1 -1 -1 1 1 1

1 -1 -1 1 1 1

-1 1 -1 1 1 1

1 1 -1 1 1 1

-1 -1 1 1 1 1

1 -1 1 1 1 1

-1 1 1 1 1 1

1 1 1 1 1 1


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Fractional Factorial Design

It might be decided that the previous design contains too may experiments

The following is a reduced Fractional Factorial Design containing 16 experime

A B C D E F

-1 -1 -1 -1 -1 -1

1 -1 -1 -1 1 -1

-1 1 -1 -1 1 1

1 1 -1 -1 -1 1

-1 -1 1 -1 1 1

1 -1 1 -1 -1 1

-1 1 1 -1 -1 -1

1 1 1 -1 1 -1

-1 -1 -1 1 -1 1

1 -1 -1 1 1 1-1 1 -1 1 1 -1

1 1 -1 1 -1 -1

-1 -1 1 1 1 -1

1 -1 1 1 -1 -1

-1 1 1 1 -1 1

1 1 1 1 1 1

How was this design arrived at?

The columns A-D contain a full factorial design in these 4 variables I.e. all combinations of t

Column E was created by multiplying the coefficients in columns A, B and C row-wise

I.e E = ABC e.g. for row 10 -1 (cell F10) = -1(B10) * -1(C10) * -1(D10)Similalry column F was created by B*C*D I.e F = BCD

This creates a resolution 4 design since the defining word is I = ABCE or I = BCDF

A fuller explanation is contained in the document 'Factorial Designs'

On the next sheet the above design is displayed in actual levels. The responses were meas

16 experiments and the results displayed in the results column.


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nts

e two levels

ured for the


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Response

A B C D E F Signal

6 lean 0 4 0.2 328.1 95

12 lean 0 4 0.7 328.1 416 rich 0 4 0.7 338.1 63

12 rich 0 4 0.2 338.1 83

6 lean 5 4 0.7 338.1 59

12 lean 5 4 0.2 338.1 114

6 rich 5 4 0.2 328.1 121

12 rich 5 4 0.7 328.1 59

6 lean 0 8 0.2 338.1 107

12 lean 0 8 0.7 338.1 38

6 rich 0 8 0.7 328.1 44

12 rich 0 8 0.2 328.1 73

6 lean 5 8 0.7 328.1 60

12 lean 5 8 0.2 328.1 976 rich 5 8 0.2 338.1 105

12 rich 5 8 0.7 338.1 53


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Calculation of Main Effects

Response

A B C D E F Signal A

-1 -1 -1 -1 -1 -1 95 -95

1 -1 -1 -1 1 -1 41 41

-1 1 -1 -1 1 1 63 -63

1 1 -1 -1 -1 1 83 83

-1 -1 1 -1 1 1 59 -59

1 -1 1 -1 -1 1 114 114

-1 1 1 -1 -1 -1 121 -121

1 1 1 -1 1 -1 59 59

-1 -1 -1 1 -1 1 107 -107

1 -1 -1 1 1 1 38 38

-1 1 -1 1 1 -1 44 -44

1 1 -1 1 -1 -1 73 73

-1 -1 1 1 1 -1 60 -60

1 -1 1 1 -1 -1 97 97-1 1 1 1 -1 1 105 -105

1 1 1 1 1 1 53 53

-12

How do we determine if the variable has a significant effect on the response?

To determine this the main effectsfor each variable are calculated.

To do this we average the responses for the variable at the highlevel and subtract from it t

at the low level

This is equivalent to multiplying the response column (H) by the column of coefficients for th

(e.g. column B for variable A) and dividing by half the number of experiments (8)

How do we interpret the results?

The main effects give the relative importance of each variable. The (numerically) largest eff

wavelength (variable E), followed by Flame height and % Acetic Acid

The signof the effect also gives information. A negative effect means that the response is h

In this case, for example, the absorbance is higher at the low wavelength setting of 328.1n

From these experiments we could definitely 'screen out' flame stoichiometry and lamp curre

experiments I.e set them at mid point values (stoichiometry between lean and rich and curre


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B C D E F

-95 -95 -95 -95 -95

-41 -41 -41 41 -41

63 -63 -63 63 63

83 -83 -83 -83 83

-59 59 -59 59 59

-114 114 -114 -114 114

121 121 -121 -121 -121

59 59 -59 59 -59

-107 -107 107 -107 107

-38 -38 38 38 38

44 -44 44 44 -44

73 -73 73 -73 -73

-60 60 60 60 -60

-97 97 97 -97 -97105 105 105 -105 105

53 53 53 53 53

-1.25 15.5 -7.25 -47.25 4

Main effects

e average response

e variable

ct is for

igher at the low setting.

t from further

nt of 6 mA)


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Main Effects Plots

These plots give us another way to compare the effects of the variables

This is the data used to calculate the main effects

A B C D E F

-95 -95 -95 -95 -95 -95

41 -41 -41 -41 41 -41

-63 63 -63 -63 63 63

83 83 -83 -83 -83 83

-59 -59 59 -59 59 59

114 -114 114 -114 -114 114

-121 121 121 -121 -121 -121

59 59 59 -59 59 -59

-107 -107 -107 107 -107 107

38 -38 -38 38 38 38

-44 44 -44 44 44 -44

73 73 -73 73 -73 -73

-60 -60 60 60 60 -6097 -97 97 97 -97 -97

-105 105 105 105 -105 105

53 53 53 53 53 53

Step 1 Order each column from lowest to highest

A B C D E F

-121 -114 -107 -121 -121 -121

-107 -107 -95 -114 -114 -97

-105 -97 -83 -95 -107 -95

-95 -95 -73 -83 -105 -73

-63 -60 -63 -63 -97 -60-60 -59 -44 -59 -95 -59

-59 -41 -41 -59 -83 -44

-44 -38 -38 -41 -73 -41

38 44 53 38 38 38

41 53 59 44 41 53

53 59 59 53 44 59

59 63 60 60 53 63

73 73 97 73 59 83

83 83 105 97 59 105

97 105 114 105 60 107

114 121 121 107 63 114

Step 2:In the above table responses at the low (-1 ) settings have a negative sign and responses at the high (+We need to get the average of the absolute values at each setting. A main effects plot compares these

A B C D E F

low 81.75 76.375 68 79.375 99.375 73.75

high 69.75 75.125 83.5 72.125 52.125 77.75

Step 3: Plot the data


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A variable with the biggest difference between the 'high' and 'low' values will be the most sig

I.e E followed by A, C. These are shown by the steepest slopes in the above graphs

0

20

40

60

80

100

120

low high


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1) settings are positive.

two averages graphically


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nificant

A

B

C

D

E

F


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Interactions

The interactions between variables can also be calculated. The column of coded coefficient

is calculated by multiplying the columns of coefficients of the corresponding variables

The interaction effect is then found by multiplying the response column by this column of co

summing the column and dividing by 8

Response

A B C D E F Signal A*B A*C

-1 -1 -1 -1 -1 -1 95 1 1

1 -1 -1 -1 1 -1 41 -1 -1

-1 1 -1 -1 1 1 63 -1 1

1 1 -1 -1 -1 1 83 1 -1

-1 -1 1 -1 1 1 59 1 -1

1 -1 1 -1 -1 1 114 -1 1

-1 1 1 -1 -1 -1 121 -1 -1

1 1 1 -1 1 -1 59 1 1

-1 -1 -1 1 -1 1 107 1 1

1 -1 -1 1 1 1 38 -1 -1

-1 1 -1 1 1 -1 44 -1 11 1 -1 1 -1 -1 73 1 -1

-1 -1 1 1 1 -1 60 1 -1

1 -1 1 1 -1 -1 97 -1 1

-1 1 1 1 -1 1 105 -1 -1

1 1 1 1 1 1 53 1 1

Coefficients for main effects

95 95

-41 -41

The two largest interaction effects are A*C , B*E, B*D and C*F -63 63

83 -83

CAUTION! 59 -59

-114 114It is no coincidence that A*B and B*E are the same value. In experimental -121 -121

design language the two interaction effects are confounded. Confoundings 59 59

occur because of the reduced nature of the fractional factorial design. 107 107

A discussion of confoundings (aliases) can be found in -38 -38

the Factorial Designs document -44 44

73 -73

60 -60

Alias Table -97 97

-105 -105

A BCE DEF ABCDF 53 53

B ACE CDF ABDEF

C ABE BDF ACDEF -4.25 6.5

D AEF BCF ABCDE

E ABC ADF BCDEFF ADE BCD ABCEF

AB CE ACDF BDEF

AC BE ABDF CDEF

AD EF ABCF BCDEF

AE BCE DF ABCDEF

AF DE ABCD BCEF

BD CF ABEF ACDEF


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BF CD ABDE ACEF

ABD ACF BEF CDEF

ABF ACD BDE CEF


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s for each interaction

efficients,

A*D A*E A*F B*C B*D B*E B*F C*D C*E

1 1 1 1 1 1 1 1 1

-1 1 -1 1 1 -1 1 1 -1

1 -1 -1 -1 -1 1 1 1 -1

-1 -1 1 -1 -1 -1 1 1 1

1 -1 -1 -1 1 -1 -1 -1 1

-1 -1 1 -1 1 1 -1 -1 -1

1 1 1 1 -1 -1 -1 -1 -1

-1 1 -1 1 -1 1 -1 -1 1

-1 1 -1 1 -1 1 -1 -1 1

1 1 1 1 -1 -1 -1 -1 -1

-1 -1 1 -1 1 1 -1 -1 -11 -1 -1 -1 1 -1 -1 -1 1

-1 -1 1 -1 -1 -1 1 1 1

1 -1 -1 -1 -1 1 1 1 -1

-1 1 -1 1 1 -1 1 1 -1

1 1 1 1 1 1 1 1 1

Coefficients for interactions

95 95 95 95 95 95 95 95 95

-41 41 -41 41 41 -41 41 41 -41

63 -63 -63 -63 -63 63 63 63 -63

-83 -83 83 -83 -83 -83 83 83 83

59 -59 -59 -59 59 -59 -59 -59 59

-114 -114 114 -114 114 114 -114 -114 -114121 121 121 121 -121 -121 -121 -121 -121

-59 59 -59 59 -59 59 -59 -59 59

-107 107 -107 107 -107 107 -107 -107 107

38 38 38 38 -38 -38 -38 -38 -38

-44 -44 44 -44 44 44 -44 -44 -44

73 -73 -73 -73 73 -73 -73 -73 73

-60 -60 60 -60 -60 -60 60 60 60

97 -97 -97 -97 -97 97 97 97 -97

-105 105 -105 105 105 -105 105 105 -105

53 53 53 53 53 53 53 53 53

-1.75 3.25 0.5 3.25 -5.5 6.5 -2.25 -2.25 -4.25


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C*F D*E D*F E*F

1 1 1 1

1 -1 1 -1

-1 -1 -1 1

-1 1 -1 -1

1 -1 -1 1

1 1 -1 -1

-1 1 1 1

-1 -1 1 -1

-1 -1 1 -1

-1 1 1 1

1 1 -1 -11 -1 -1 1

-1 1 -1 -1

-1 -1 -1 1

1 -1 1 -1

1 1 1 1

95 95 95 95

41 -41 41 -41

-63 -63 -63 63

-83 83 -83 -83

59 -59 -59 59

114 114 -114 -114-121 121 121 121

-59 -59 59 -59

-107 -107 107 -107

-38 38 38 38

44 44 -44 -44

73 -73 -73 73

-60 60 -60 -60

-97 -97 -97 97

105 -105 105 -105

53 53 53 53

-5.5 0.5 3.25 -1.75


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What don't these experiments tell us?

(1) What are the best settings for each variable? To determine this we would n

optimization design such as the Central Composite Design. Optimization design

for each variable - this is why we carry out screening first

(2) Is there curvature in the design? Consider variable B - although the main eff

missing something - perhaps the resonse is significantly higher (or lower) in the

This means there is curvature in the design.

(3) The above analysis tells us the relativeeffect of each variable but it does n

whether the variable has a significant effect.

(2) and (3) can be tested for by modifying the design to include centre points. Texperiments with variables set at their mid-points, and given codes of 0.

The mid-point values are:- 9 mm(A), lean/rich(B), 2.5% (C), 6mA (D), 333.2 (E)

The extended design, in coded form, is shown below, with the responses.

Response

A B C D E F Signal

-1 -1 -1 -1 -1 -1 95

1 -1 -1 -1 1 -1 41

-1 1 -1 -1 1 1 63

1 1 -1 -1 -1 1 83

-1 -1 1 -1 1 1 59

1 -1 1 -1 -1 1 114

-1 1 1 -1 -1 -1 121

1 1 1 -1 1 -1 59

-1 -1 -1 1 -1 1 107

1 -1 -1 1 1 1 38

-1 1 -1 1 1 -1 44

1 1 -1 1 -1 -1 73

-1 -1 1 1 1 -1 60

1 -1 1 1 -1 -1 97

-1 1 1 1 -1 1 105

1 1 1 1 1 1 530 0 0 0 0 0 79

0 0 0 0 0 0 74

0 0 0 0 0 0 77

We will illustrate an alterantive analysis on the next sheet. The data will be fitted to a polynomial, with lin

interaction terms, as follows:-

constant y = b0


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first order terms +b1*A + b2*B +b3*C +b4*D +b5*E +b6*F

two way interactions +b12*A*B +b13*A*C +b14*A*D +b15*A*E +b16*A*F +b24*B*D +b26

three way interactions *b134*A*B*D +b126*A*B*F

Note: not all possible terms can be included due to confounding (see alias table on previous

The coefficients are then determined using least squares regression . This can be carried out in Excelusing the array function LINEST (consult the Excel help files for use of this function and using array func

However before performing regression columns corresponfing to the interaction coefficients need to be

as previously shown.


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ed to carry out a full

need at least three settings

ct is small perhaps we are

ange between 4 - 8?

t tell us absolutely

hese are

and 0.45nm(F)

ar and


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B*F

sheet )

tions)

onstructed


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Multivariate Regression

A B C D E F A*B A*C A*D

-1 -1 -1 -1 -1 -1 1 1 1

1 -1 -1 -1 1 -1 -1 -1 -1

-1 1 -1 -1 1 1 -1 1 1

1 1 -1 -1 -1 1 1 -1 -1

-1 -1 1 -1 1 1 1 -1 1

1 -1 1 -1 -1 1 -1 1 -1

-1 1 1 -1 -1 -1 -1 -1 1

1 1 1 -1 1 -1 1 1 -1

-1 -1 -1 1 -1 1 1 1 -1

1 -1 -1 1 1 1 -1 -1 1

-1 1 -1 1 1 -1 -1 1 -1

1 1 -1 1 -1 -1 1 -1 1

-1 -1 1 1 1 -1 1 -1 -11 -1 1 1 -1 -1 -1 1 1

-1 1 1 1 -1 1 -1 -1 -1

1 1 1 1 1 1 1 1 1

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

b126 b134 b26 b24 b16 b15 b14 b13 b12

-0.125 3.25 -1.125 -2.75 0.25 1.625 -0.875 3.25 -2.125

0.55508 0.55508 0.55508 0.55508 0.55508 0.55508 0.55508 0.55508 0.55508

0.998699 2.220321 #N/A #N/A #N/A #N/A #N/A #N/A #N/A

153.5552 3 #N/A #N/A #N/A #N/A #N/A #N/A #N/A

11355 14.78947 #N/A #N/A #N/A #N/A #N/A #N/A #N/A

the first line above contains the parameters. The second line is the standard errors, third line is R 2 an

the fourth line is the F statistic and degrees of freedom and the fifth line regression and residual sum of

critical t value 4.3

df =2

(since the error is based on the 3 replicates of the centre point)

the confidence interval for each coefficient is b +/-t*se where se is the standard error in the second line

2.386845 2.386845 2.386845 2.386845 2.386845 2.386845 2.386845 2.386845 2.386845

1015

Regression coefficients


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Viewing this graph now gives us an answer to which effects are significant

An effect is significant (at the 95% confidence level ) if its' regression coefficient is significan

I.e. its' confidnce interval does not include zero. This applies to b5, b4, b1, b3,b13,b24,b134

This means the variables E (wavelength), A( Flame Height), C(% acetic acid) and D(lamp c

are significant. There are also significant interactions but due to confounding we cannot defiwhich are significant. The only way to remove the confounding is to do more experiments. H

screen out variables B and F we could do a full factorial on 4 variables = 16 experiments an

interactions.

Note that it is still possible B and F may have significant interactions even though their main

Note You may see a connection between the regression coefficients and the main effhalf the size of the main effects so both give the same information.

Curvature? Compare the average of the response for the factorial points (first 16

average response for factorial points 75.75

average response for centre points 76.66667

Since these averages are very similar there is little curvature in the model.

Note that if significant curvature is indicated these experiments cannot tell which variable ca

curvature; a design such as a central composite design is needed to determine this.

-30-25-20-15-10

-505

b126 b134 b26 b24 b16 b15 b14 b13 b12 b6 b5


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Response

A*E A*F B*D B*F A*B*D A*B*F Signal

1 1 1 1 -1 -1 95

1 -1 1 1 1 1 41

-1 -1 -1 1 1 -1 63

-1 1 -1 1 -1 1 83

-1 -1 1 -1 -1 1 59

-1 1 1 -1 1 -1 114

1 1 -1 -1 1 1 121

1 -1 -1 -1 -1 -1 59

1 -1 -1 -1 1 1 107

1 1 -1 -1 -1 -1 38

-1 1 1 -1 -1 1 44

-1 -1 1 -1 1 -1 73

-1 1 -1 1 1 -1 60-1 -1 -1 1 -1 1 97 75.75

1 -1 1 1 -1 -1 105 76.66667

1 1 1 1 1 1 53

0 0 0 0 0 0 79

0 0 0 0 0 0 74

0 0 0 0 0 0 77

b6 b5 b4 b3 b2 b1 b0

2 -23.625 -3.625 7.75 -0.625 -6 75.89474

0.55508 0.55508 0.55508 0.55508 0.55508 0.55508 0.509377

#N/A #N/A #N/A #N/A #N/A #N/A #N/A



standard error of y

squares

of the output

2.386845 2.386845 2.386845 2.386845 2.386845 2.386845


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tly non-zero

urrent)

nitely say owever since we can

determine all

effects are not significant.

cts. The coefficeints are

) and the centre points (last 3)

uses the

b4 b3 b2 b1

doe in excel

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