A Multifactor Experiment CUTHBERT DANIEL 116 Pinehurst Ave., New York, N. Y .
EARL W. RIBLETT Petroleum und>ChemicalResearch Laborutory, T h e M . ,W. Kellogg Co., Jersey C i t y , N . J . Effect of factor A is measured by comparing the average of the first two runs with the average of the second two. The fact that B is present a t its high level in run 1 is of no consequence, since factor B produces the same effect in run 3. Similarly, levels of C do not influence the effect of A. The effect of factor B is estimated by the difference between the average of runs 1 and 3, and the average of runs 2 and 4. The effect of factor C is similarly estimated by the difference between the average of runs 2 and 3 and the average of runs 1 and 4. All four runs are used to measure each main effect with as much precision as if the whole set of four runs had been devoted solely to measuring a single effect. The principle used in this example is capable of wide extension.
GEKERAL method of solution has been developed for a fairly broad class of multifactor problems. Data from an experiment on catalyst manufacture are used here for illustration. Preparation of a catalyst may involve a hundred steps, each of which can be varied independently. Precipitations may be made slowly or rapidly, cold or hot, from dilute or concentrated solution, in agitated or unagitated me’dia, a t high or low pH. Other operations, such as filtration, washing, drying, calcining, sizing, activation, pretreatment, and induction similarly involve steps which must be specified before a catalyst can be made. Each detailed step is called a factor and the various states or conditions of that factor are called its levels-for example, rate of precipitation may be a factor, and actual rates used are levels of that factor. Factors may be capable of continuous variation (pH, concentration, temperature, etc.) or of discontinuous variation (source of raw material, type of catalyst support, etc.). Level is defined as either a value of a continuous factor, or the name of one condition of a discontinuous factor. Measurement of all possible combinations of 8 factors, each a t two levels, would require 28 runs. The present experiment was limited to 32 runs, permitting evaluation of effects of 8 factors and judgment of additivity of effects for 13 pairs of factors. All 32 results can be used to assess the effect of changing each factor from one level to another and to judge additivity of effect of a factor at two levels of each of the other factors. Each conclusion will be as precise as if the whole series of 32 runs had been devoted to testing that conclusion alone. The method used in choosing the 32 catalysts was that given by Brownlee, Kelly, and Loraine ( 4 ) . Finney (6, 7), and Davies and Hay (6) have discussed the technique of “fractional replications.”
TABLE I. UNBALASCED EXPERIXENTAL DESIGN Run NO.
B
C
Litera! Abbreuiation
TABLE 11. B.kL.kNC>:D EXPERIMENTAL DESIGX (Three factors)
Run
Factor Level
KO.
h
B
1
1 1 0 0
1 0 1 0
2 3 4
C 0 1
1
0
Literal Abbreviation ab ac
bc
(1)
It is often assumed that effects of A, B, and C are consistent. or additive, in the sense that if A has a given effect when B and C are a t zero levels, it will have the same effect a t any other combination of levels of B and C. However, it is possible to judge additivity of effects of factors A and B by sacrificing knowledge about factor C. Dropping column C leaves an experiment that is still balanced; the average effects of A and B can be appraised as before and additivity of A and B can be measured. The effect of A a t level 0 of B is estimated by the difference between results of runs 2 and 4. The effect of A a t level 1 of B is estimated by the difference between runs 1 and 3. If these two effects are not the same, hypothesis of additivity is denied and, in statistical language, factors A and B interact-i.e., the effect of A is dependent on the level of B. As in the previous three-factor balanced experiment, all four runs are used in reaching each of three conelusions: A-effect, B-effect, and AB interaction.
MULTIPLE BALANCE IN EXPERIMENTAL DESIGN
The concept of multiple balance in experimental design, due to Fisher (S), is first discussed by reference to an example in which effects of three factors are to be estimated. The three factors are designated as A, B, and C; levels of these factors are indicated by numbers, 0 symbolizing a particular level and 1 another level. The four runs usually made when principles of balanced experimental design are ignored are shown in Table I. A run in which A is a t level 1and B and C are a t 0 levels is conveniently symbolized by 100 or, more concisely, by a, the presence of the lower case letter indicating that corresponding factor is a t level 1 and absence of letter that factor is a t level 0. The symbol (1) is used to indicate that all factors are a t level 0. The effect of changing each factor is measured by comparing the corresponding run with control run 4. Two runs are used in judging each effect in context of a single set of levels for all other factors. This experiment is unbalanced in two ways; only three out of twelve factor-levels are 1’s and control run 4 is used in all three comparisons while each of the other runs is used only once. An experiment incorporating principle of multiple balance is given in Table 11.
Factor Level
A
EFFECTS OF EIGHT FACTORS
A group of 32 runs, each specifying a set of levels for 8 factors A to H, is given in Table 111. I t may be seen that the over-all effect of factor A is measured by the difference between average of results for runs in columns I and 11, and average of results for runs in columns I11 and IV. This estimate is not disturbed by the over-all effect of any of the seven other factors. 1465
INDUSTRIAL AND ENGINEERING CHEMISTRY
1466
TABLE111. BALAXCED EXPERIMENTAL DESIGN (Eight factors)
I1
I Run NO. 1 2 3 4 5
6 7 8
NO.
acegh ace
abejgh abej abg abh
13 14 15 16
adeg adeh adjoh
No. Aot.
5 6
7
8
4.99
5,OO
5.61 4.76
5.23 4.77 4.99 5.17
Run Select. No. 92.2 9 9 3 . 9 10 9 4 . 6 11 95.1 12 91.8 94.1 95.4 93.4
13 14 15 16
Run NO.
NO.
9 10 11 12
TABLE
1 2 3 4
Run
abcdefg abcdejh abcdgh abcd
Run
IV
I11
Run
WfQ acjh
ad/
17 18 19 20
bceg bceh bcfgh bcj
25 26 27 28
cdejgh cdej cdg cdh
21 22 23 24
bdegh bde bdfg bdjh
29 30 31 32
ejg ejh gh (1)
II-. ADJESTEDDATA
Act. 4.90 4.90 5.24 4.95
Select. 94.1 93.2 92.8 93.8
4.96 5.03 514 5.05
91.6 92.3 90.6 93.4
Run No. Act. 17 4 . 9 7 18 4 . 8 3 19 5 , 2 7 20 5 . 2 0 21 22 23 24
5.34 5.00 5.28 4.93
Select. 93.1 93.3 92.0 92.6 91.9 92.1 91.9, 93 7
Run
No. Act. Select. 25 26 27 28
4.91 4.71 4.99 4.91
91.0 92.9 94.8 94.1
29 30 31 32
4.86 4.65 5.24 5.05
91.7 89.4 92.8 93.7
Vol. 46, No. 7
fluctuations in the 32 runs. All 23 numbers were used in estimating the precision of single runs. Standard deviations derived were: selectivity 0.96, act,ivity 0.16. Confidence intervals having a predetermined probability of covering the true values of the several effects were obtained byuse of Student-Fisher Gvalues as multipliers of standard deviations. There were 16 judgments to be made: effects of eight factors on two dependent variables. The probability of making a single erroneous statement of significance in all 16 judgments was set a t 0.1. (For this public report, a high “lgvel of significance” has been used, equivalent to a low over-all likelihood of’ erroneously reporting a factor to have a real effect. The level of significance, of course, may be set a t will, cost of erroneously calling an effect real being balanced against cost of missing a real effect.) The t-value for a probability of (1/16)(0.1) with 23 degrees of freedom is 3.0. Multiplying the two standard deviations given above by i 1.06 [3.0 (1/16 1/16)1’2 = i 1.061 gives confidence ranges of i 0.17 and 5 1.02 for activity and selectivity, respectively. Addition of these ranges to the corresponding observed average values gives the desired confidence intervals.
+
B
Two-factor interaction AB is measured undistorted by any main effect. The measure of AB interaction is the discrepancy between difference (11-IV) and difference (I-111)-Le., (I-11-III+IV). Since a appears on the plus side as many times as on the minus side, AB interaction is measured unconfounded with A-effect. It is not possible in 32 runs to get all 28 two-factor interactions unconfounded with each other. The 13 two-factor interactions of G and H are all unconfounded. The remaining 15 interactions are confounded in 7 groups:
+ CD + E F + BD iiD + BC AE + BF AB -4C
AF + B E CE DF CF + D E
+
The tn-o factors suspected of having the largest interactions were assigned letters G and H. The symmetry of the “confounding pattern” is shown in Figure 1, where each point in the 28-point grid represents a two-factor interaction. Tie lines connect those interactions xhich are confounded-i.e., whose sums are measured.
C D
E
F
G
0
0
0
0
0
0
H
0
0
0
0
0
0
0
A
B
C
D
E
F
G
Figure 1.
TABLE
\‘. EFFECT01’ FACTORS Activity Effect,
DATA AND ANALYSIS
Catalysts were prepared and tested in an intentionally random order, to minimize any possible chronological bias. Tests Twre made a t constant space velocity, 7%-hichresulted in change in product quality and yield with change in catalyst activity. Activit,y was defined as space velocity required to give product, of a certain quality; selectivity was defined as yielJ of product of same quality. Observed yields and space velocities w r e adjusted, by standard correlation techniques, to yields and space velocities a t the desired level of product quality Table IV gives the adjusted data for activity and selectivity. The effect of each factor on each dependent variable, expressed as an average difference, is given in Table V. I l x h interaction was calculated in a similar way, using all 32 results in two groups of 16. For example, the AC BD interaction is one sixteenth of the difference between sum of runs 1 to 4, $1 to 12, 21 to 24, and 29 to 32 and sum of remaining runs. Interactions were small, indicating that each of factors A to H probably operated to produce the same c h m p a t all levels of other factors. Variations among the 23 numlxrs within each column of interactions reflect operation of rand sm. uncontrolled
+
Confounding Pattern
Di
A B C D E F G H AB AC , AD f AE AF AG AH BG BH CE CF CG CH DG DH EG EH FG FIT GH ABG ABH SCE
i CD f BD
BC ++ BBFE
DF ++ DE
0 0 -0 0 -0 -0 0
EF
0344 1156 0344 0244 1706 0294 1881
n n m
-0 .oi.iS
0.03% 0.0244 0 0294 0.03R6 - 0.0094 0.0844 0.0644 0,0906 -0 0444 0 0794 0 0144 - 0 0094 0 0406 0 0,544 -0 0294 0 0044 0 0194 -0 0894 0 0831 0 03R9 0 1131 0 OD69
O N rTARI.4BLES
Selectivity Effect,
Dz
0.713 0.550 0.850 -0.082 -1.000 -. 1 . 1 1 3 -0.538 -0.488 0,538 0.038 -0.550 0.263 0.225 -0.225 0.075 -0.113 0.263 0.250 -0.288
-0.013
0.513 -0.575 0.250 0.063 0.088 -0.675 -0.178
-0.100 0.250 -0.826
-0.238
INDUSTRIAL AND ENGINEERING CHEMISTRY
July 1954
1467
The error sum of squares and products consists of the sum of 23 matrices for two- and three-factor interactions. This sum, called ( J ) ,is:
1210-
(J)
0.5778 (-1.003
- 1.003
21.37)
0 8 -
Here 0.5778 = nsl', 21.37 = ns2, and -1.003 = ~ S I S Z T ~ where Z, SI is observed standard deviation for activity, sp is observed standard deviation for selectivity, n is number of degrees of freedom for error (here 23), and 712 is observed correlation coefficient bet\\-een errors in the measurements. Determinant IJI of this matrix is equal to 11.341. The significance of simultaneous effect of factor E on activity and selectivity was tested by comparison of calculated with critical F values. As above, ( E ) was calculated and added to ( J ) , and determinant IJ El was found:
0 6 -
O"
04.-
k
!
02-
w >.
0 -.
t ?
I-
-0 2
-'
+
W -I
I - 0 4 . .O 6
-.
-0 8 -
The test criterion, U,is the ratio of IJI to IJ
+ El:
-1 0--
-.
-1
2
-1
4 - 0 2 4 -020 -016
-012
-008 -004
0
004 0 0 8
012
016
020
I
When the null hypothesis is true-i.e., effect of E on activity or selectivity
when there is no real
ACTIVITY EFFECT D l
Figure 2.
Confidence Areas on Activity-Selectivity Effects
Comparison of these values with those given in Table V shows that only effects of E and G on activity can be judged significantly large. The only factor significantly affecting selectivity was F. These results are summarized in Figure 2, where the datum points are plotted and a "confidence rectangle" is drawn. The ellipse is discussed in the next section.
I n other words, the right side of the equation is distributed as the F ratio, with 2m and 2(n - 1) degrees of freedom, where n is the number of degrees of freedom for error and m is the number of degrees of freedom for effect tested:
CONFIDENCE INTERVALS FOR ACTIVITY AND SELECTIVITY
Since the significant value of F a t the 0.10/8 level, for 2 and 44 degrees of freedom, is 4.85, the measured value is clearly significant and there is little doubt that factor E has a real effect on the activity and selectivity point. Testing the effects of factors F and G in the same way, Fvalues of 5.50 and 5.03, respectively, were found, showing that these factors were also significantly effective on both activity and selectivity. The boundary between significant and nonsignificant regions forms an ellipse, the equation for which is:
Rectangular confidence regions constructed by the above method may be improved by use of a multivariate method which allows for any observed correlation between experimental errors in measuring activity and selectivity. The area covered by the inclined ellipse which results from the multivariate method is always smaller than that of the corresponding rectangle. In the present example, the reduction in size is not great, because the observed error correlation is small (r12 = -0.2855). I n cases where the correlation is higher, the reduction in area will be greater. The form used here, due to Anderson ( I ) , was first presented by Wilks (IO). It is equivalent to Hotelling's generalized Student's ratio (9) and'to a test by Bartlett ( 2 ) . If D1 is the effect of a treatment on activity, D2 the effect on selectivity, and N the number of runs (32 in the present example), then the sum of squares for effect on activity is NDl2/4, and on selectivity is ND2'/4. The sum of products, ND1D2/4, is used to measure correlation between two measurements. These numbers can be conveniently presented as a two-by-two matrix:
22 = 9.79 m2x 1
F# = 0.308
(1
SP 712
8( -0.171)' ( E ) = (8(-0.171)(--1.000)
8( -0.171)( -1.000) 8(-1.000)'
)= 0.2339 1.368 (1.368 8.000)
Matrices are added by adding values in corresponding cells.
F a 4n
+ 2) Tl
where F a is the critical F-value a t significance level C Y ; in this instance CY is 0.0125. This equation, on a graph of activity effects versus selectivity effects, defines the region outside which a datum point must fall to be considered significant. I n the present example : SI
For example, the sums of squares and products for the effect of E is written:
- T122)
= 0.1585 = 0.9640 = - 0.2855
n = 23 N = 32 F = 4.85
and the equation for the ellipse becomes:
(&)lf
0.571
(oG)(&6) + (&)'
= 1.298
The graph of this ellipse is given in Figure 2, with observed D1 and DZpoints for eight main effects. A confidence region for any particular effect is obtained by moving this ellipse until its center corresponds to its observed DI, DZpoint.
1468
INDUSTRIAL AND ENGINEERING CHEMISTRY
When activity and selectivity are simultaneously tested for significance, the same factors are found effective but effect of factor E is now least doubtful, while effects of factors F and G are on the boundary between significance and nonsignificance. The power of multiple balance in experimental design has been illustrated by an experiment of 32 runs on effects of eight preparation variables on performance of catalysts. Each conclusion m-aa as precise as if all 32 runs had been devoted t o testing that conclusion alone. ACKN0WLEDGB.I ENT
The authors are indebted to A. W. McIGnnej- for technical assistance in the preparation of this paper, and to the referees for several valuable suggestions. NOMENCLATURE
A , B. C---H a, b, c---h
Di D2
= factors, independent variables = upper levels of factors
= effect of changing factor level on activity
effect of changing factor level on selectivity ( E ) = matrix of effects of factor E F = observed value of ratio of two variances FZm, z ( n - l ) = observed value of ratio of two variances Piith degrees of freedom indicated F E = observed value of ratio of two variances for factor E Pa = upper critical value a t significance level 01 =
iJJ l )
.m
n TIS
sp
Vol. 46, No. 7
= sum of matrices used in evaluation of error = determinant of ( J ) = number of degrees of freedom for effect
= number of runs = number of degrees of freedom for error = = = =
observed correlation coefficient Observed standard deviation for observed standard deviation for selectivity ratio of ~ J toI I J + REFERENCES
(1) Anderson, T. W., unpublished lecture notes. (2) Bartlett, hf. S., J . Roy. Statistical SOC.(Supplement), B9, 176
(1947).
(3) Brownlee, K. A,, “Industrial Experimentation,” 3rd American ed., Chap. 11, New York, Chemical Publishing Co., 1950. (4) Brownlee, K. A., Kelly, B. K., and Loraine, P. K., Biometnka, 35. 268 (1948). (5) Davies, O.’L., and Hay, W. A., Biometrics, 6 , No. 3 , 233 (1950). ~ c 291 s , (1945). (6) Finney, D. J., Ann. E u Q ~ ~ 12, (7) Finney, D. J., J . Agr. Sci., 36, 184 (1946). (8) Fisher, R. A., “Design of Experiments,” 4th ed., London,
Oliver and Boyd, 1947. (9) Hotelling, H., Ann. Math. Statistics. 2 , 360 (1931). (10) milks, S. S., Biometrika, 24, 471 (193%).
RECEIVED for review November 5 , 1953. ACCEPTED X a r c h 9, 1954. Presented before the Division of Industrial and Engineering Chemistry, Symposium on Statistics in the Design of Experiments, a t t h e 124th Meeting of the A h r E R I C A h . C H E M I C A L SOCIETY, Chicago, 111.
ixation of Nitro en in a Crossed Sy’ILLIA11 S . PARTRIDGE, RANSQ3I B. PARLIN, AND BRUNO J. ZKOLISSKI1 Department of Chemistry, University of Utah, Salt Lake City, U t a h
1118 study arose in connection with an extensive investigation on the induction of chemical changes in a high frequency arc established betu-een a pair of metal electrodes. Through the specific advantages of various kinds of electrical discharges in bringing about a chemical change in systems characterized by high energies of activation and/or instability of the final products-e.g., endothermic processes-are well k n o m , little can be said as to whether ions, molecular or atomic fragments, or some combination of these are involved in the detailed mechanism. An investigation of the formation of nitric oxide when air is subjected to the action of such an arc in the frequency range of from 1 to 10 megacycles was initiated, and the present discussion forme a part of this general study. I n a recent series of patents, Cotton ( 2 ) disclosed the use of a new kind of electrical discharge for the fixation of nitrogen in air. Employing a discharge tube in a.hich four metal electrodes are symmetrically arranged in a plane, one pair of opposing electrodes was connected to a high voltage (ea. 2000 volts), 60-cycle power source; the other pair of electrodes was supplied with a radio-frequency energy source in the range of 1 to 10 me. This arrangement permitted him simultaneously to establish a high frequency and a low frequency discharge in the same spatial region. Essentially, Cotton’s claim has been that this superposition of a high and a low frequency discharge permits operation of a discharge in a relatively high pressure range (100 to 700 mm. of mercury), which a t the same time exhibits all the visible characteristics of a true glow discharge with none of the usual disadvantages of an arc-namely] high currents and high temperat ures. Under these favorable conditions of operation] conversion 1
Present address, Stanford Research Institute, Stanford, Calif.
yields tvere obtained for the formation of nitric oxide greatly in excess of those normally encountered in the Birkeland-Eyde process, Of especial interest is the appearance of critical frcquencies for the high frequency power which increased the yields by factors of from 2 to 10. Some of these frequencies appeared to be characteristic of the reaction mixture only, while others \yere characteristics of the electrode material. The former were designated as “critical reaction frequencies” and the latter as “critical electrode frequencies.” I n addition, a best or critical pressure of operation was found a t a value of 335 mm. of mercury. APPARATUS AND PROCEDURE
For the preparation and handling of the air csed in the study, a conventional flow system was employed.
L2
Figure 1. Apparatus
