Design and Analysis of MultiFactored Experiments. Twolevel Factorial Designs. The 2 k Factorial Design. Special case of the general factorial design; k factors, all at two levels The two levels are usually called low and high (they could be either quantitative or qualitative)
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
DOE Course
“” and “+” denote the low and high levels of a factor, respectively
Low and high are arbitrary terms
Geometrically, the four runs form the corners of a square
Factors can be quantitative or qualitative, although their treatment in the final model will be different
DOE Course
Estimate of the effect of A
a1b1  a0b1 estimate of effect of A at high B
a1b0  a0b0 estimate of effect of A at low B
sum/2 estimate of effect of A over all B
Or average of high As – average of low As.
Estimate of the effect of B
a1b1  a1b0 estimate of effect of B at high A
a0b1  a0b0 estimate of effect of B at high A
sum/2 estimate of effect of B over all A
Or average of high Bs – average of low Bs
DOE Course
Estimate the interaction of A and B
a1b1  a0b1 estimate of effect of A at high B
a1b0  a0b0 estimate of effect of A at low B
difference/2 estimate of effect of B on the effect of A
called as the interaction of A and B
a1b1  a1b0 estimate of effect of B at high A
a0b1  a0b0 estimate of effect of B at low A
difference/2 estimate of the effect of A on the effect of B
Called the interaction of B and A
Or average of like signs – average of unlike signs
DOE Course
Note that the two differences in the interaction estimate are
identical; by definition, the interaction of A and B is the
same as the interaction of B and A. In a given experiment one
of the two literary statements of interaction may be preferred
by the experimenter to the other; but both have the same
numerical value.
DOE Course
A, the effect of B, and the effect of the interaction of A and
B; all four yields are needed and are used in each estimates.
interaction can be and are assessed separately, this in an
experiment in which both factors vary simultaneously.
themselves together with their interaction are, logically, all
that can be studied. These are among the merits of these
factorial designs.
DOE Course
Many scientists feel the need for experiments which will
reveal the effect, on the variable under study, of factors
acting jointly. This is what we have called interaction. The
simple experimental design discussed here evidently
provides a way of estimating such interaction, with the latter
defined in a way which corresponds to what many scientists
have in mind when they think of interaction.
It is useful to note that interaction was not invented by
statisticians. It is a joint effect existing, often prominently, in
the real world. Statisticians have merely provided ways and
means to measure it.
DOE Course
A is called a main effect. Our estimate of A is often simply written A.
B is called a main effect. Our estimate of B is often simply written B.
AB is called an interaction effect. Our estimate of AB is often simply written AB.
So the same letter is used, generally without confusion, to describe the factor, to describe its effect, and to describe our estimate of its effect. Keep in mind that it is only for economy in writing that we sometimes speak of an effect rather than an estimate of the effect. We should always remember that all quantities formed from the yields are merely estimates.
DOE Course
The following table is useful:
Notice that in estimating A, the two treatments with A at high level are compared to the two treatments with A at low level. Similarly B. This is, of course, logical. Note that the signs of treatments in the estimate of AB are the products of the signs of A and B. Note that in each estimate, plus and minus signs are equal in number
DOE Course
B
A
Example 2
Example 1
B
B
Example 2
A+
Low
High
B+
Low
High
A=2.5
B=2
B
A
Low
Low
A
A
B
A
High
High
B+
A=3
B
B
B
Example 4
Example 4
Example 3
Low
High
Low
High
A
B, B+
Low
Example 3
Low
A
15
A
14
13
A
12
High
High
11
Y
10
9
2
1
0
1
Discussion of examples:
Notice that in examples 2 & 3 interaction is as large as or larger than main effects.
*A = [(1)  b + a + ab]/2
= [10  12 + 13 + 15]/2
= 3
DOE Course
constant, multiplies each estimate by the constant but does not affect the relationship of estimates to each other.
DOE Course
Modern notation:
a0b0 = 1 a0b1 = b a1b0 = a a1b1 = ab
We also introduce Yates’ (standard) order of treatments and yields;
each letter in turn followed by all combinations of that letter and
letters already introduced. This will be the preferred order for the
purpose of analysis of the yields. It is not necessarily the order in
which the experiment is conducted; that will be discussed later.
For a twofactor twolevel factorial design, Yates’ order is
1 a b ab
Using modern notation and Yates’ order, the estimates of effects
become:
A = (1 + a  b + ab)/2
B = (1  a + b +ab)/2
AB = (1 a  b + ab)/2
DOE Course
Example: The variable is the yield of a nitration process. The yield forms the base material for certain dye stuffs and medicines.
Lowhigh
A time of addition of nitric acid 2 hours 7 hours
B stirring time 1/2 hour 4 hours
C heel absent present
Treatments (also yields) (i) old notation (ii) new notation.
(i) a0b0c0 a0b0c1 a0b1c0 a0b1c1 a1b0c0 a1b0c1 a1b1c0 a1b1c1
(ii) 1 c b bc a ac ab abc
Yates’ order:
1 a b ab c ac bc abc
DOE Course
DOE Course
Estimate of A
(1) a  1 estimate of A, with B low and C low
(2) ab  b estimate of A, with B high and C low
(3) ac  c estimate of A, with B low and C high
(4) abc  bc estimate of A, with B high and C high
= (a+ab+ac+abc  1bcbc)/4,
= (1+ab+abc+acbc+abc)/4
(in Yates’ order)
DOE Course
Effect of A with B high  effect of A with B low, all at C high
plus
effect of A with B high  effect of A with B low, all at C low
Note that interactions are averages. Just as our estimate of A is an average of response to A over all B and all C, so our estimate of AB is an average response to AB over all C.
AB = {[(4)(3)] + [(2)  (1)]}/4
= {1ab+ab+cacbc+abc)/4, in Yates’ order
or, = [(abc+ab+c+1)  (a+b+ac+bc)]/4
DOE Course
interaction of A and B, at C high
minus
interaction of A and B at C low
ABC = {[(4)  (3)]  [(2)  (1)]}/4
=(1+a+bab+cacbc+abc)/4, in Yates’ order
or, =[abc+a+b+c  (1+ab+ac+bc)]/4
DOE Course
measures the impact, on the yield of the nitration process, of
interaction AB when C (heel) goes from C absent to C
present. Or it measures the impact on yield of interaction AC
when B (stirring time) goes from 1/2 hour to 4 hours. Or
finally, it measures the impact on yield of interaction BC
when A (time of addition of nitric acid) goes from 2 hours to
7 hours.
As with twofactor twolevel factorial designs, the formation
of estimates in threefactor twolevel factorial designs can be
summarized in a table.
DOE Course
DOE Course
Yield of nitration process discussed earlier:
1 a b ab c ac bc abc Y = 7.2 8.4 2.0 3.0 6.7 9.2 3.4 3.7
A = main effect of nitric acid time = 1.25
B = main effect of stirring time = 4.85
AB = interaction of A and B = 0.60
C = main effect of heel = 0.60
AC = interaction of A and C = 0.15
BC = interaction of B and C = 0.45
ABC = interaction of A, B, and C = 0.50
NOTE: ac = largest yield; AC = smallest effect
DOE Course
analysis of this example, taking into account the unreliability
of estimates based on a small number (eight) of yields, some
estimates may turn out to be so small in magnitude as not to
contradict the conjecture that the corresponding true effect is
zero. The largest estimate is 4.85, the estimate of B; an
increase in stirring time, from 1/2 to 4 hours, is associated
with a decline in yield. The interaction AB = 0.6; an increase
in stirring time from 1/2 to 4 hours reduces the effect of A,
whatever it is (A = 1.25), on yield. Or equivalently
DOE Course
(makes more negative) the already negative effect (B = 485)
of stirring time on yield. Finally, ABC = 0.5. Going from no
heel to heel, the negative interaction effect AB on yield
becomes even more negative. Or going from low to high
stirring time, the positive interaction effect AC is reduced.
Or going from low to high nitric acid time, the positive
interaction effect BC is reduced. All three descriptions of
ABC have the same numerical value; but the chemist would
select one of them, then say it better.
DOE Course
We introduce the notation 2k. This means a factor design with each factor at two levels. The number of treatments in an unreplicated 2k design is 2k.
The following table shows the number of each kind of effect for each of the six twolevel designs shown across the top.
DOE Course
2 factor interaction
3 factor interaction
4 factor interaction
5 factor interaction
6 factor interaction
7 factor interaction
3
7
15
31
63
127
In a 2k design, the number of rfactor effects is Ckr = k!/[r!(kr)!]
DOE Course
In a 22 design, there are 22=4 treatments; we estimate 221= 3 effects. In a 23 design, there are 23=8 treatments; we estimate 231= 7 effects
One need not repeat the earlier logic to determine the forms of estimates in 2k designs for higher values of k.
A table going up to 25 follows.
DOE Course
1. Applied to Complete Factorials (Yates, 1937)
A systematic method of calculating estimates of effects.
For complete factorials first arrange the yields in Yates’
(standard) order. Addition, then subtraction of adjacent
yields. The addition and subtraction operations are
repeated until 2k terms appear in each line: for a 2k there
will be k columns of calculations
DOE Course
Example:
Yield of a nitration process
Tr.
Yield
1stCol
2ndCol
3rdCol
1 7.2 15.6 20.6 43.6 Contrast of µ
a 8.4 5.0 23.0 5.0 Contrast of A
b 2.0 15.9 2.2 19.4 Contrast of B
ab 3.0 7.1 2.8 2.4 Contrast of AB
c 6.7 1.2 10.6 2.4 Contrast of C
ac 9.2 1.0 8.8 0.6 Contrast of AC
bc 3.4 2.5 0.2 1.8 Contrast of BC
abc 3.7 0.3 2.2 2.0 Contrast of ABC
Again, note the linebyline correspondence between treatments
and estimates; both are in Yates’ order.
DOE Course
Several writers have cautioned against making statements
about main effects when the corresponding interactions
are large; interactions describe the dependence of the
impact of one factor on the level of another; in the
presence of large interaction, main effects may not be
meaningful.
DOE Course
Example (Adapted from Kempthorne)
Yields are in bushels of potatoes per plot. The two factors are
nitrate (N) and phosphate (P) fertilizers.
low level (1)high level (+1)
N (A) blood sulphate of ammonia
P (B) superphosphate steamed bone flower;
The yields are
1 = 746.75 n = 625.75 p = 611.00 np = 656.00
the estimates are
N = 38.00 P = 52.75 NP = 83.00
In the face of such high interaction we now specialize the main
effect of each factor to particular levels of the other factor.
Effect of N at high level P = npp = 656.00611.00 = 45.0
Effect of N at low level P = n1 = 625.71746.75 = 121.0, which
appear to be more valuable for fertilizer policy than the mean (38.00)
of such disparate numbers
746.75
P+
Keep both
low is best
121
Y
656
38
611.0
P
625.75
N
DOE Course
In general we accept high interactions wherever found and seek to explain them; in the process of explanation, main effects (and lowerorder interactions) may have to be replaced in our interest by more meaningful specialized or conditional effects.
DOE Course
DOE Course
In any experiment, factors other than those studied may be influential. Their presence is sometimes acknowledged under the dubious title “experimental error”. They may be neglected, but the usual cost of neglect is high. For they often have uneven impact, systematically affecting some treatments more than others, and thereby seriously confounding inferences on the studied factors. It is important to deal explicitly with them; even more, it is important to measure their impact. How?
DOE Course
2. Randomize their effects.
3. Estimate their magnitude by replicating the experiment.
4. Estimate their magnitude via side or earlier experiments.
5. Argue (convincingly) that the effects of some of these nonstudied factors are zero, either in advance of the experiment or in the light of the yields.
6. Confound certain nonstudied factors.
DOE Course
DOE Course
Factor Name Units Low Level (1) High Level (+1)
A Distance m 2 5
B Direction Deg. 0 90
C Shot type Set Jump
DOE Course
Order A B C Combination Y
1 1 1 1 (1) 9
2 +1 1 1 a 5
3 1 +1 1 b 7
4 +1 +1 1 ab 3
5 1 1 +1 c 6
6 +1 1 +1 ac 5
7 1 +1 +1 bc 4
8 +1 +1 +1 abc 2
DOE Course
Order A B AB C AC BC ABC Comb Y
1 1 1 +1 1 +1 +1 1 (1) 9
2 +1 1 1 1 1 +1 +1 a 5
3 1 +1 1 1 +1 1 +1 b 7
4 +1 +1 +1 1 1 1 1 ab 3
5 1 1 +1 +1 1 1 +1 c 6
6 +1 1 1 +1 +1 1 1 ac 5
7 1 +1 1 +1 1 +1 1 bc 4
8 +1 +1 +1 +1 +1 +1 +1 abc 2
Effect A = (a + ab + ac + abc)/4  (1 + b + c + bc)/4
= (5 + 3 + 5 + 2)/4  (9 + 7 + 6 + 4)/4 = 2.75
DOE Course
Using the sign table, all 7 effects can be calculated:
Effect A = 2.75
Effect B = 2.25
Effect C = 1.75
Effect AC = 1.25
Effect AB = 0.25
Effect BC = 0.25
Effect ABC = 0.25
The overall average value = (9 + 5 + 7 + 3 + 6 + 5 + 4 + 2)/8
= 5.13
DOE Course
The most important effects are: A, B, C, and AC
Model: Y = b0 + b1 X1 + b2 X2 + b3 X3 + b13 X1X3
b0 = overall average = 5.13
b1 = Effect [A]/2 = 2.75/2 = 1.375
b2 = Effect [B]/2 = 2.25/2 = 1.125
b3 = Effect [C]/2 = 1.75/2 =  0.875
b13 = Effect [AC]/2 = 1.25/2 = 0.625
Model in coded units:
Y = 5.13 1.375 X1  1.125 X2  0.875 X3 + 0.625 X1 X3
DOE Course
ActualPredictedValueValue
9.00 9.13 5.00 5.13 7.00 6.88 3.00 2.87 6.00 6.13 5.00 4.63 4.00 3.88 2.00 2.37
Amazing fit!!
DOE Course
10
Effect of B=46.25= 2.25
(9+5+6+5)/4=6.25
# out
of 10
8
6
(7+3+4+2)/4=4
4
2
0
90
B
C: Shot type
10
Interaction of A and C = 1.25
8
# out
of 10
C(1)
6
At 5m, Jump or set shot about the same BUT at 2m, set shot gave higher values compared to jump shots
4
2
C (+1)
2m
5m
A
DOE Course
Analysis of 2k Experiments
Statistical Details
DOE Course
1.Meaning of 2
Assume that each treatment has variance 2. This has the following meaning: consider any one treatment and imagine many replicates of it. As all factors under study are constant throughout these repetitions, the only sources of any variability in yield are the factors not under study. Any variability in yield is due to them and is measured by 2.
DOE Course
2. Effect of the number of factors on the error of an estimate
What is the variance of an estimate of an effect? In a 2k design, 2k treatments go into each estimate; the signs of the treatments are + or , depending on the effect being estimated.
So, any estimate = 1/2k1[generalized (+ or ) sum of 2k treatments]
2(any estimate) = 1/22k2 [2k 2] = 2/2k2;
The larger the number of factors, the smaller the error of each estimate.
Note: 2(kx) = k2 2(x)
DOE Course
3. Effect of replication on the error of an estimate
What is the effect of replication on the error of an estimate? Consider a 2k design with each treatment replicated n times.
1
a
b
abc
d






















DOE Course
Any estimate = 1/2k1 [sums of 2k terms, all of them means based on samples of size n]
2(any estimate) = 1/22k2 [2k 2/n] = 2/(n2k2);
The larger the replication per treatment, the smaller the error of each estimate.
DOE Course
DOE Course
a) p values from ANOVA
Compute pvalue of calculated F. IF p < , then effect is significant.
b) Comparing std. error of effect to size of effect
DOE Course
If effect ± 2 (se), contains zero, then that effect is not significant. These intervals are approximately the 95% CI.
e.g. 3.375 ± 1.56 (significant)
1.125 ± 1.56 (not significant)
DOE Course
Significant effects are those that do not fit on normal probability plot. i. e. nonsignificant effects will lie along the line of a normal probability plot of the effects.
Good visual tool  available in DesignExpert software.
DOE Course
DOE Course
A = (a + ab  1  b)/2n
= (100 + 90  60  80)/(2 x 3)
= 8.33
B = (b + ab  1  a)/2n
= 5.00
C = (ab + 1  a  b)/2n
= 1.67
The effect estimates are: A = 8.33, B = 5.00, AB = 1.67
DesignExpertanalysis
DOE Course
Term Effect SumSqr % Contribution Model Intercept Model A 8.33333 208.333 64.4995 Model B 5 75 23.2198 Model AB 1.66667 8.33333 2.57998 Error Lack Of Fit 0 0 Error P Error 31.3333 9.70072
Lenth's ME 6.15809 Lenth's SME 7.95671
DOE Course
Response:Conversion ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 291.67 3 97.22 24.82 0.0002A208.331208.3353.19< 0.0001B75.00175.0019.150.0024AB8.3318.332.130.1828 Pure Error 31.33 8 3.92 Cor Total 323.00 11
Std. Dev. 1.98 RSquared 0.9030 Mean 27.50 Adj RSquared 0.8666 C.V. 7.20 Pred RSquared 0.7817
PRESS 70.50 Adeq Precision 11.669
The Ftest for the “model” source is testing the significance of the overall model; that is, is either A, B, or AB or some combination of these effects important?
DOE Course
CoefficientStandard95% CI95% CI
FactorEstimateDFErrorLowHighVIF Intercept 27.50 1 0.57 26.18 28.82 AConcent 4.17 1 0.57 2.85 5.48 1.00 BCatalyst 2.50 1 0.57 3.82 1.18 1.00 AB 0.83 1 0.57 0.48 2.15 1.00
General formulas for the standard errors of the model coefficients and the confidence intervals are available. They will be given later.
DOE Course
Response:Conversion ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 283.33 2 141.67 32.14 < 0.0001A208.331208.3347.27< 0.0001B75.00175.0017.020.0026 Residual 39.67 9 4.41Lack of Fit8.3318.332.130.1828Pure Error31.3383.92 Cor Total 323.00 11
Std. Dev. 2.10 RSquared 0.8772 Mean 27.50 Adj RSquared 0.8499 C.V. 7.63 Pred RSquared 0.7817
PRESS 70.52 Adeq Precision 12.702
There is now a residual sum of squares, partitioned into a “lack of fit” component (the AB interaction) and a “pure error” component
DOE Course
Regression Model for the Process
DOE Course
DOE Course
DOE Course
A = carbonation, B = pressure, C = speed, y = fill deviation
DOE Course
Term Effect SumSqr % Contribution Model Intercept Error A 3 36 46.1538 Error B 2.25 20.25 25.9615 Error C 1.75 12.25 15.7051 Error AB 0.75 2.25 2.88462 Error AC 0.25 0.25 0.320513 Error BC 0.5 1 1.28205 Error ABC 0.5 1 1.28205 Error LOF 0 Error P Error 5 6.41026
Lenth's ME 1.25382 Lenth's SME 1.88156
DOE Course
Response:Filldeviation ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 73.00 7 10.43 16.69 0.0003A36.00136.0057.60< 0.0001B20.25120.2532.400.0005C12.25112.2519.600.0022AB2.2512.253.600.0943AC0.2510.250.400.5447BC1.0011.001.600.2415ABC1.0011.001.600.2415 Pure Error 5.00 8 0.63 Cor Total 78.00 15
Std. Dev. 0.79 RSquared 0.9359 Mean 1.00 Adj RSquared 0.8798 C.V. 79.06 Pred RSquared 0.7436
PRESS 20.00 Adeq Precision 13.416
DOE Course
CoefficientStandard95% CI 95% CI
FactorEstimateDFErrorLowHighVIF
Intercept 1.00 1 0.20 0.54 1.46
ACarbonation 1.50 1 0.20 1.04 1.96 1.00 BPressure 1.13 1 0.20 0.67 1.58 1.00 CSpeed 0.88 1 0.20 0.42 1.33 1.00 AB 0.38 1 0.20 0.081 0.83 1.00 AC 0.13 1 0.20 0.33 0.58 1.00 BC 0.25 1 0.20 0.21 0.71 1.00 ABC 0.25 1 0.20 0.21 0.71 1.00
DOE Course
Response:Filldeviation ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 70.75 4 17.69 26.84 < 0.0001A36.00136.0054.62< 0.0001B20.25120.2530.720.0002C12.25112.2518.590.0012AB2.2512.253.410.0917 Residual 7.25 11 0.66LOF2.2530.751.200.3700Pure E5.0080.63 C Total 78.00 15
Std. Dev. 0.81 RSquared 0.9071 Mean 1.00 Adj RSquared 0.8733 C.V. 81.18 Pred RSquared 0.8033
PRESS 15.34 Adeq Precision 15.424
DOE Course
CoefficientStandard95% CI 95% CIFactorEstimateDFErrorLowHigh Intercept 1.00 1 0.20 0.55 1.45 ACarbonation 1.50 1 0.20 1.05 1.95 BPressure 1.13 1 0.20 0.68 1.57 CSpeed 0.88 1 0.20 0.43 1.32 AB 0.38 1 0.20 0.072 0.82
DOE Course
DOE Course
Final Equation in Terms of Coded Factors:
Filldeviation = +1.00 +1.50 * A +1.13 * B +0.88 * C +0.38 * A * B Final Equation in Terms of Actual Factors: Filldeviation = +9.62500 2.62500 * Carbonation 1.20000 * Pressure +0.035000 * Speed +0.15000 * Carbonation * Pressure
DOE Course
DOE Course
DOE Course
If the factors are spaced too closely, it increases the chances that the noise will overwhelm the signal in the data
More aggressive spacing is usually best
DOE Course
DOE Course
DOE Course
DOE Course
DOE Course
Term Effect SumSqr % Contribution Model Intercept Error A 21.625 1870.56 32.6397 Error B 3.125 39.0625 0.681608 Error C 9.875 390.062 6.80626 Error D 14.625 855.563 14.9288 Error AB 0.125 0.0625 0.00109057 Error AC 18.125 1314.06 22.9293 Error AD 16.625 1105.56 19.2911 Error BC 2.375 22.5625 0.393696 Error BD 0.375 0.5625 0.00981515 Error CD 1.125 5.0625 0.0883363 Error ABC 1.875 14.0625 0.245379 Error ABD 4.125 68.0625 1.18763 Error ACD 1.625 10.5625 0.184307 Error BCD 2.625 27.5625 0.480942 Error ABCD 1.375 7.5625 0.131959
Lenth's ME 6.74778 Lenth's SME 13.699
DOE Course
DOE Course
DOE Course
Response:Filtration Rate ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum ofMeanFSourceSquaresDFSquareValueProb >F Model 5535.81 5 1107.16 56.74 < 0.0001A1870.5611870.5695.86< 0.0001C390.061390.0619.990.0012D855.561855.5643.85< 0.0001AC1314.0611314.0667.34< 0.0001AD1105.5611105.5656.66< 0.0001 Residual 195.12 10 19.51 Cor Total 5730.94 15
Std. Dev. 4.42 RSquared 0.9660 Mean 70.06 Adj RSquared 0.9489 C.V. 6.30 Pred RSquared 0.9128
PRESS 499.52 Adeq Precision 20.841
DOE Course
Final Equation in Terms of Coded Factors:
Filtration Rate = +70.06250 +10.81250 * Temperature +4.93750 * Concentration +7.31250 * Stirring Rate 9.06250 * Temperature * Concentration +8.31250 * Temperature * Stirring Rate
DOE Course
DOE Course
DOE Course
If one factor is dropped, the unreplicated 24 design will project into two replicates of a 23
Design projection is an extremely useful property, carrying over into fractional factorials
DOE Course
With concentration at either the low or high level, high temperature and high stirring rate results in high filtration rates
DOE Course
A = drill load, B = flow, C = speed, D = type of mud, y = advance rate of the drill
DOE Course
Term Effect SumSqr % Contribution Model Intercept Error A 0.9175 3.36722 1.28072 Error B 6.4375 165.766 63.0489 Error C 3.2925 43.3622 16.4928 Error D 2.29 20.9764 7.97837 Error AB 0.59 1.3924 0.529599 Error AC 0.155 0.0961 0.0365516 Error AD 0.8375 2.80563 1.06712 Error BC 1.51 9.1204 3.46894 Error BD 1.5925 10.1442 3.85835 Error CD 0.4475 0.801025 0.30467 Error ABC 0.1625 0.105625 0.0401744 Error ABD 0.76 2.3104 0.87876 Error ACD 0.585 1.3689 0.520661 Error BCD 0.175 0.1225 0.0465928 Error ABCD 0.5425 1.17722 0.447757
Lenth's ME 2.27496 Lenth's SME 4.61851
DOE Course
DOE Course
DOE Course
DOE Course
DOE Course
A log transformation is recommended
The procedure provides a confidenceinterval on the transformation parameter lambda
If unity is included in the confidence interval, no transformation would be needed
DOE Course
Three main effects are large
No indication of large interaction effects
What happened to the interactions?
DOE Course
Response:adv._rateTransform:Natural logConstant: 0.000 ANOVA for Selected Factorial Model
Analysis of variance table [Partial sum of squares]Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 7.11 3 2.37 164.82 < 0.0001B5.3515.35371.49< 0.0001C1.3411.3493.05< 0.0001D0.4310.4329.920.0001 Residual 0.17 12 0.014 Cor Total 7.29 15
Std. Dev. 0.12 RSquared 0.9763 Mean 1.60 Adj RSquared 0.9704 C.V. 7.51 Pred RSquared 0.9579
PRESS 0.31 Adeq Precision 34.391
DOE Course
Final Equation in Terms of Coded Factors:
Ln(adv._rate) = +1.60 +0.58 * B +0.29 * C +0.16 * D
DOE Course
DOE Course
DOE Course
DOE Course
DOE Course
Usually between 3 and 6 center points will work well
DesignExpert provides the analysis, including the Ftest for pure quadratic curvature
DOE Course
Response:yield ANOVA for Selected Factorial ModelAnalysis of variance table [Partial sum of squares]
Sum ofMeanFSourceSquaresDFSquareValueProb > F Model 2.83 3 0.94 21.92 0.0060A2.4012.4055.870.0017B0.4210.429.830.0350AB2.500E00312.500E0030.0580.8213 Curvature 2.722E003 1 2.722E003 0.063 0.8137 Pure Error 0.17 4 0.043 Cor Total 3.00 8
Std. Dev. 0.21 RSquared 0.9427 Mean 40.44 Adj RSquared 0.8996
C.V. 0.51 Pred RSquared N/A
PRESS N/A Adeq Precision 14.234
DOE Course
If curvature is significant, augment the design with axial runs to create a central composite design. The CCD is a very effective design for fitting a secondorder response surface model
DOE Course
DOE Course