Statistical Design
I/EC
Fractional Factorial Experiment Designs of Mixed 2m3n Series - Part II Analysis and interpretation of a statistical design problem are illustrated with data to show how the factors at two levels are used to determine probable result by W. S. Connor, The Research Triangle Institute
I HE last column dealt with frac tional factorial designs in which m factors are studied at two levels and η factors are studied at three levels. The method of constructing the de signs was described, and illustrated for a one-half replicate of the 2332 complete factorial. This column discusses how to estimate the effects and how to interpret the estimates. Data from the last column (Table III, p. 94A, I / E C , June 1960) are used for illustration.
sponses a r e replaced by t h e asso ciated y ' s . T h e n , for a n y effect a n d each equation, t h e coefficient of the effect is multiplied by Y, a n d these products a r e a d d e d . T h e resulting sum is t h e g for t h a t effect. For example, g(Ai) is the sum of t h e re sponses for which factor Aχ is a t level 1 minus t h e s u m of t h e responses for which it is at level 0. T h e estimates of Ai a n d Α>Α3 a r e At = iig(At)
-
g(A,A,)\/96
and Estimation of Effects The random response to the treat ment combination (*ι*2·*3ΖιΖ2) is denoted by Y{x\x2.Ϊ3Ζ1Ζ2), and a par ticular observed response by y (χ 1*2*3 Z1Z2). For example, for treatment combination (00000), ^(00000) = 85.9. The estimates of the effects obtained by the method of least squares can be conveniently ex pressed in terms of certain linear functions of the Y's, which are de noted by g's. Because there is a g which corresponds to each effect, it is convenient to identify each g by putting the effect in parentheses. For example, the g which corre sponds to the main effect of factor A\ is^(^i). The g's are readily determined by reference to the set of equations which expresses the expected re sponses as linear functions of the effects. These equations were dis played in Table VII of the June column. First, the expected re-
mates of Αχ, A2, A3, A1A2, ΑχΑ3, a n d A2A3. Also, a m o n g t h e A's, t h e only correlation is between m e m b e r s of the pairs Ai, A2A3; A2, A1A3; a n d A3, A1A2. Accordingly, of 351 pairs of estimates, only 3 contain correlated estimates. For t h e m t h e coefficient of correlation is —V3. T h e variances of t h e estimates a r e σ2/32 for t h e A effects, a n d σ2 over t h e associated divisor from T a b l e I for t h e r e m a i n ing effects. An Example
Λ2Α, = {-giA,)
+ 3 5 (/M 3 )]/96.
Estimates of A-i a n d AiAz a r e o b tained by i n t e r c h a n g i n g A ι a n d A 2 in t h e above formulas. Similarly, for A3 a n d ΑχΑι, Αχ a n d A3 a r e inter c h a n g e d . T h e estimates for t h e re m a i n i n g effects a r e obtained by dividing their g's by a p p r o p r i a t e divisors, as is indicated in T a b l e I. U n d e r t h e assumptions t h a t t h e Y's have c o m m o n variance σ2 a n d a r e uncorrelated, t h e estimates of t h e effects in T a b l e I a r e uncorrelated with each other a n d with t h e esti-
Table 1.
Divisors o f g's for Estimating Effects
Form of Effect
Divisor
AB, Β AB', B"B1B1 BiBl, B\B~_ B\B%
36 24 72 16 48 144
β
T h e d a t a from T a b l e I I I (p. 94A, J u n e 1960) a r e used to calculate numerical estimates of t h e effects a n d to explain additional analysis a n d interpretation. T h e d a t a were obtained from a n experiment carried out by W . J. Y o u d e n a n d P. W . Zimmerman ["Field Trials with Fiber Pots," Contr. Boyce Thompson Inst. 8, 317-31 (1936)] to c o m p a r e various methods of producing t o m a t o plant seedlings prior to t r a n s p l a n t i n g in the field. Actually, Y o u d e n a n d Zimmerman had treatment combina tions in addition to those in t h e table. T h e factors a n d their levels are given in T a b l e I I . T h e object of t h e experiment was to evaluate t h e effects of these factors on t h e yield of tomatoes, measured in units of weight. A factor is q u a n t i t a t i v e if it m a y be set a t a n y value o n some scale of m e a s u r e m e n t , or qualitative if no scale exists. I n t h e present experi ment, factors Αχ a n d A% a r e q u a n VOL. 52, NO. 8
·
AUGUST 1960
59 A
STATISTICAL
Table II.
DESIGN
Factors and Levels for Tomato Experiment
Factor
Size of pot, Ai Variety of tomato, A% Method of production, H,
Field soil Plus fertilizer Three-inch Four-inch Bonny-Best Mar globe Flat Fiber Fiber + N O j 0, 1
0 1
Effect μ A, Ai .43 A1A2 AuU ΑΪΑΖ
Βι(1, #i(2, #i(2, #2(1, B2(2, lh(2,
60 A
0) 0) 1) 0) 0) 1)
Estimei t e s o f Effects Confidence Limits Estimate 134.3 11.6 7.5 14.1 6.0 7.3 -2.2 17.7 31.1 13.5 7.8 37.1 29.3
± ± i ± ± i ± ± =fc ± ± ± =t
9.5 10.1 10.1 10.1 10.1 10.1 10.1 23.2 23.2 23.2 23.2 23.2 23.2
Sum 0/ Squares
Mean Square
F
Αι, AiAs A2, A1A1 Az, Αι Α2
2 2 2
4,421 5,308 10,568
2210 2654 5284
8.3
A1B1, AiB\ AiB,, AiB\ AiBi, AtB\ A.2B2, A2B2 Α,Βι, AtBÎ Λ,Βι, AzBl
2 2 2 2 2 2
4,341 138 32 435 1,282 2,551
2171 69 16 218 641 1275
Bu B\ B2, By BiBt, BiBl \ ΒχΒί, BiBi]
2 2
5,851 9,193
2925 4597
4
6,284
1571 634
0 1 0 1 0 1 2 2
titative a n d factors A-6 a n d B2 are qualitative. Factor B\ is mixed, for there is a quantitative change from level 1 to level 2, but not from level 0 to levels 1 a n d 2. For a quantitative or qualitative factor at two levels, the main effect is half the average difference in re sponse at the two levels, and hence is a meaningful quantity. Also, for a quantitative factor at three equally spaced levels, the effects Β and B2 are meaningful because they esti m a t e the linear and q u a d r a t i c com ponents in the response. However, if the levels are not equally spaced, or if the factor is qualitative or mixed, Β and B2 are interpreted differently. T h e effect Β is one half the average difference in response between levels 2 and 0, and the effect Br is one third the average response at levels 0 and 2 minus the average response at level 1. T h e former comparison usually will be of interest, but the latter often will not be. In this case, it m a y be of more interest to estimate the average
Table III.
Analysis of Variance
Degrees of Freedom
Source
Levels
Soil condition, At
Location on field, !-!••
Table IV.
α
and
Error
9
5,708
Total
35
56,112
differences between levels. Defining the expected difference between level 1 and 0 by Β(1,0), and the other ex pected differences by 5(2,0) and B(2,\), these new parameters may be expressed as functions of Β and B2 as follows: 5(1,0) = Β -
3Λ2
#(2,0) = 2 « £(2,1) = Β + 3B 2
Estimates are obtained by substitut ing (he estimates of Β and B2. T h e variances of the estimates are σ 2 /β· T a b i c I I I contains estimates of effects which contribute to inter preting the d a t a . These are the effects discussed above, plus inter action effects a m o n g A\, /I2, and A3. Confidence limits for the effects are constructed by use of the i-statistic, which applies exactly if the Y's are normally distributed and approxi mately if they obey any of a wide class of distributions. T h e limits in T a b l e I I I have confidence coeffi cient 0.95. T h e estimate of σ2 used in calculating the limits is the residual mean square, 634, given in T a b l e I V . If a confidence interval does not contain zero, then the hypothesis that the effect is zero is discredited, and the effect is described as signifi cant. T h e main effects of factors A\ and A3 are significant, and so are the differences 5 i ( 2 , 0 ) , B2(2,0), and B-,{2,\). T h u s , it is concluded that the addition of fertilizer increases the yield, that Marglobe tomatoes yield more than Bonny Best, that fiber
INDUSTRIAL AND ENGINEERING CHEMISTRY
4.6 7.2
pots plus NO3 are superior to flats, and that location 2 is better t h a n lo cations 0 a n d 1. T a b l e I V contains an analysis of variance by sources of variation. Values of the F ratio which are significant at the 0.05 level have been shown. Information from this table m a y be used to supplement the analysis already given. In partic ular, none of the interactions of A factors with Β factors and B\ with B2 are significant as judged by the F ratio. T h e sum of squares for a source is the sum of the products of the esti mates of the effects associated with the source with their corresponding g's. For example, for the source A1, AiA-i, the sum of squares is Aig{Ai) -\- A2A3g(AzA3). The num ber of degrees of freedom is the n u m ber of effects associated with the source. T h e sources have been chosen so that effects associated with a source are uncorrclated with all other effects. Because of this, the component sums of squares add up to the total sum of squares.
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