Factorial One-Factor-at-a-Time Experiments

Factorial One-Factor-at-a-Time ExperimentsAuthor(s): Cuthbert DanielSource: The American Statistician, Vol. 48, No. 2 (May, 1994), pp. 132-135Published by: Taylor & Francis, Ltd. on behalf of the American Statistical AssociationStable URL: http://www.jstor.org/stable/2684266 .

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STATISTICAL PRACTICE This department publishes articles of interest to statistical practition-

ers. Innovative applications of known methodology may be suitable, but sizable case studies should be submitted to other journals. Brief

descriptions and illustrations of new developments that are potentially useful in statistical practice are appropriate. Acceptable articles should appeal to a substantial number of practitioners.

Factorial One-Factor-at-a-Time Experiments Cuthbert DANIEL Dedicated to the memory of W Edwards Deming

Factorial experiments are done to measure the average effects of varying each of several factors (experimental conditions, independent variables) over fixed ranges of the other factors. Factors at two levels only are consid- ered here. The system under study is assumed to permit the variation of only one factor per trial. Up to four factors may be of interest. The sequences proposed, of sizes 7-9, are easy to remember, simple to analyze, and straightforward to augment. Each result is compared with its immediate predecessor or successor only. Thus the three-factor one-at-a-time factorial follows the sequence. (1)1, a, ab, abc, bc, c, (1)2, where (1) means all factors at their low levels; a means 1 factor, A, at its high level; ab means A and B at high levels; and so forth.

KEY WORDS: Edge designs; One-at-a-time plans; Two- level sequential designs.

1. INTRODUCTION

An experimenter wishing to learn the effects of varying each of several factors may find it natural or even unavoid- able to vary only one factor at a time. He or she may wish to measure the first-order effects of three or four factors for some restricted range of each. Most factors are expected to operate additively, but one simple nonadditivity (defined in the following) should be detectable. Linear drift must be accommodated. Trials may be completed in short roughly equal time intervals.

What is desired, then, is a one-at-a-time n-factor, linear- drift-free experimental design for n = 3 or 4. This article gives the smallest designs with these properties.

A single experimental trial in three factors, A, B, and C, can be specified by the presence or absence of the cor- responding lowercase letter. Thus the symbol ac signals a trial with factors A and C at their high levels and factor B at its low level. Factor levels can be qualitative or quantitative. Two sources of a raw materials, two varieties of a plant, or two samples from different groups of sub- jects are levels of qualitative variables. Either alternative

may be called the high level. Two temperatures, two con- centrations of an ingredient, or two times of exposure are examples of quantitative levels.

2. RATIONALE

I have not been able to discover the name of the origina- tor of the following dictum: "When you wish to determine the effect on some property of varying each of several factors, you must hold constant all factors but one, and then vary that one." Your lab notebooks are probably full of records of short sequences of trials to see if your system is sensitive to small changes in experimental conditions. I am proposing a shorter and more informative way of doing this.

The wide success of R. A. Fisher and his followers in promulgating randomized factorial experiments has in- deed so dominated the field that many experimenters now take it as axiomatic that there is no case to be made for one-at-a-time experiments. But there must be situations that demand or at least favor the variation of one factor at a time. Primary among these is that stage in a series of trials at which only one piece of equipment, perhaps restricted to one experimenter or team, is available. A second restric- tion may be the long time required for a system to come to equilibrium after varying the levels of several factors.

3. DESIGN

To get results as quickly as possible, each effect difference may be started where the last left off. The simplest factorial trial starts under some standard condition, call it (1), and varies some factor, call it A, to some changed condition, call it a. The difference between the two measured responses may be indicated by the same symbols: Thus [a - (1)] is called the main effect of A at the low levels of B and C.

Changing the version of factor B from its current low level at a to its high level ab produces [ab - a], the main effect of B at high A, low C. Changing factor C to its alternative level c permits estimation of the main effect of C by [abc - ab] at high A and B.

We now have estimates of the effects of all three factors but no checks as to consistency or precision. By changing the level of factor A in the latest result, abc, one obtains bc and hence can now get a second estimate of the effect of A from [abc - bc], this time at high B and C. If this difference agrees well with the earlier difference, one has

Cuthbert Daniel is at 50 East 77th Street, New York, NY 10021- 1836. The author is grateful to J. M. Cameron for comments and en- couragement and for varification of calculations. He would like to thank J. W. Tukey for detailed criticism and for the method of randomization proposed. Brian Schlain made several very useful suggestions.

132 The American Statistician, May 1994, Vol. 48, No. 2 ? 1994 American Statistical Association



Table 1. Calculation of Confounding Patterns

Conditions Panel A Effects Panel B Effects

(1) 1 0 0 0 1 0 0 a -1 0 00 1 -1 0 0

b 0 -2 0 -1 0 0 ab 0 2 4 +AB -1 1 0 -4 -AB

c -2 00 -1 1 -2 0 ac 0 0 4 +AC -1 2 0

bc -1 0 2 0 1 1 -2 4 +BC abc 1 2 2 0 -1 -2 0

a crude indication that factor A operates additively, that is, consistently, over the chosen ranges of B and C. If not, A is said to interact with B and/or C, and the difference in the two A-effects measures AB + AC.

Passing from bc to c, by changing the level of B one obtains [bc - c], which gives a second measure of the effect of factor B, now at the low level of A but at high C. The average of the two simple B effects gives a more precise estimate of the effect of B over these ranges of A and C.

The difference of the two B effects, that is, (bc - c) - (ab - a), gives a measure of the difference of the two interactions, -AB + BC. This quantity is called for obvious reasons the difference of the interactions with B.

All effects thus far have been measured by differences between successive observations and so are minimally af- fected by time drift. If the trials can be made at nearly equal time intervals, then any linear time drift is elimi- nated from their average effects. In general, one should not take the results of [c - (1)] to estimate the C effect because these two results are too far apart in time. Rather, one should repeat the base trial with a subscript 2, thus (1)2, to measure the C effect. The average of the two simple C effects gives a duplicated and hence presumably more precise measure of the effect of C in this neighborhood. The difference between the two C effects gives a measure of the sum of two C interactions, that is, AC + BC.

(If these paragraphs are obvious, so much the better. If any part of them is confusing, I beg you to study them because their content is crucial for all that follows.)

The three-factor, two-level, one-at-a-time plan can then be put in a form easy to remember:

(1)1, a, ab, abc, bc, c, (1)2.

Similar plans can be constructed for four factors with the same properties. For four two-level factors A, B, C, D, the sequence of nine trials,

(l)1, a, ab, abc, abed, bed, cd, d, ()2,

permits estimation of four main effects and of three sets of three interaction sums each.

It should be clear that these arrangements automatically guarantee that the main effects are estimated clear of any linear drift. The elimination of linear trend from the interaction sums is explained in Section 5.

4. RANDOMIZATION

The letters A, B, and C have been assigned casually to the three factors under discussion, and it is assumed that they are roughly equally easy (or difficult) to vary. Two precautions have already been taken to minimize the effects of uncontrollable drift. The balanced raising and lowering of levels and the variation of only one factor at a time should both be stabilizing influences. In a steady system already known to vary little when undis- turbed, it may not be necessary to consider any further randomization of the order of application of factor levels. But when preliminary exploratory trials might mis- lead the experimenter through the operation of hidden but influential factors, some degree of objective random allocation of treatments to plan may be judged desir- able.

The three-factor plan may be viewed as a 2 x 2 x 2 or 23 from which one diagonal has been deleted. There are four diagonals, namely (1) and abc, a and bc, b and ac, and c and ab. Any one of these can be deleted and the remainder sequenced by varying along the edges of the remaining cube. There are six permutations of the letters A, B, and C, which can be assigned to each partial cube. There are then 6 x 4 or 24 plans, no two identical, but each meeting the one-at-a-time and the linear trend-free requirements. Any one can be chosen at random.

It is my guess that the proper sequencing and regular timing will nearly always suffice to give indication of the magnitudes and regularity of effects.

The common-sense requirements of typicality, good sampling, and uniform technique are skipped in this article, but of course will not be forgotten by the serious experimenter.

5. ANALYSIS OF RESULTS

The plans proposed permit unbiased estimates of all main effects, but the two-factor interactions are measured in pairs. The simplest method for determining which pairs are measured together is the use of Yates's addition and subtraction algorithm, placing ones in the positions of the performed trials. The two panels of Table 1 show the confounding patterns for the interactions with A and B for the three-factor one-at-a-time plan.

Thus in panel A of Table 1, the four ones in the first column represent the difference between the two A effects, (abc - bc) and (a - (1)), that is, abc - bc - a + (1). The

The American Statistician, May 1994, Vol. 48, No. 2 133



Table 2. Calculation of Effects and Interaction-pairs, With Three Main Effects But No Interactions Built In, and Unit Linear Drift

Specifications Results 2A 2B 2C 2AB + 2AC -2AB + 2BC 2AC + 2BC

(1)i 50 -1 +1 a 54 +1 -1 -1 +1

ab 61 +1 -1 -1 -1 abc 71 +1 +1 +1 +1 bc 69 -1 +1 -1 +1 c 64 -1 +1 -1 -1

(1)2 56 -1 +1

Totals 6 12 18 -2* -2* +2*

Main effects 3 6 9

*Unit linear drift.

Table 3. Calculation of Effects Plus Two Factor Interaction, With Same Main Effects Plus Interaction AB = 1.33 Built In, and Unit Linear Drift

Specifications Results 2A 2B 2C 2AB + 2AC -2AB + 2BC 2AC + 2BC

(1)i 51.33 -1 +1 a 54.00 +1 -1 -1 +1

ab 62.33 +1 -1 -1 -1 abc 72.33 +1 +1 +1 +1 bc 69.00 -1 +1 -1 +1 c 65.33 -1 +1 -1 -1

(1)2 57.33 -1 +1

Totals 6 12 18 .66 -4.66 2.00

Main effects 3 6 9 -2 x Drift +2.00 +2.00 -2.00 - 2 x Interaction ? AB +2.66 -2.66 .00 ? AB +1.33 -1.33 .00

addition and subtraction algorithm proceeds in the second column by first adding the successive pairs of entries in the first column to get the first four zeros, and then subtracting the members of each of the same pairs to get -2, 0, 0, + 2.

The third column repeats the same operations on the values in the preceding column. And so also for the last column. The labeling 4AB + 4AC shows that this combi- nation of the four trials listed in the first column measures only AB + AC.

Though not shown, the sum of the final columns of Panels A and B will give the pair of interactions AC + BC measured by the difference of the two C effects, that is, by (abc - ab) - (c - (1)).

These calculations need not be memorized. They are listed for the three-factor one-at-a-time plan in Tables 2 and 3.

The combinations of results given here are not the same as those given by exact least squares, but they are more intuitive, and their deficiencies are negligible.

It does not seem fair to me to compare the precision of these plans with that of a fully randomized seven-trial plan. The requirements here, varying one factor at a time with all two-factor interactions separated from main effects, cannot be met by any plan that varies two or more factors in each trial; however, for the reader who insists on comparing this three-factor plan with a seven-run randomized factorial plan, the efficiency factors for A, B, C,

and the linear trend are .6027, .5830, .6157, and .6786, respectively.

Table 2 summarizes the calculations of effects and interactions. Thus for the fictitious results given, the effect of factor A is found by

[(abc - bc) + (a -(1))

= (71 - 69) + (54 - 50)

=6=2A.

SoA = 3.

Table 3 is added as an error-free example with a two- factor interaction, AB, of 1.33. The linear trend, now deduced only from the sum in the last column, must be subtracted from the last three column totals before one can see the tell-tale agreement in the other two columns, as is explained. The pattern of ? I 's is identical in Tables 2 and 3.

As the last three columns in Table 2 indicate, the two- factor interactions are not separable but are confounded in pairs. The unit built-in linear drift also shows up as -2, -2, +2. If, then, the results in these three columns come out to be of the same magnitude but with signs -, -, + (or the reverse), one can be quite sure that there is no large interaction. But all is not invariably lost. There

134 The American Statistician, May 1994, Vol. 48, No. 2



are some patterns of responses in these columns that are safely interpretable. If none of these appears, further trials will be required. The informative patterns are shown as follows.

For compactness put

P =AB+AC from (1), -a-bc +abc

Q = -AB + BC from a - ab - c + bc

R = AC + BC from(1)2-ab-c + abc.

Next, then AB AC BC

if P 1 1 Q -1 1 R 1 1

For example, if P and Q come out large and nearly equal but of opposite signs, then it stands to reason that only a large AB interaction could produce this pattern.

The following shows the six interpretable outcomes for the three interaction-combinations from afour-factor one- at-a-time plan.

For compactness,

P = AB +AC+AD from (1)1 -a-bcd + abcd

Q = -AB+BC+BD from a-ab-cd+bcd

R = -AC-BC + CD from ab-abc-d + cd;

Next,

then AB AC AD BC BD CD

if P +1 +1 +1 Q -1 +1 +1 R -1 -1 +1

If none of these combinations appears, there may be ad- ditivity among the four factors. To decide more defi- nitely, a new sequence of trials, perhaps even several sequences, will be required. The magnitudes of the main effects, together with practical considerations, will tell the experimenter whether to repeat trials in the same range or to extend or shrink the range for one or more factors. The second set of trials should be analyzed by the same method. The two sets of effects can then be integrated (a micro-meta-analysis).

A similar plan for five two-level factors is conceivable but in my judgment will rarely be needed. The plan, by analogy with the three-and four-factor designs just de- scribed, can be written by starting with (1) and then changing the level of one factor in each of five new trials up to abcde. The use of five more trials, removing one letter at a time in the same order down to a final (1)2 for 11 trials, will yield five main effects and five strings of four two-factor interactions each.

6. NESTING

Some experimental situations cannot be accurately characterized by a number of equally flexible factors. Some factors may be difficult to vary, others may be easy. For example, suppose that one wished to measure the effects of varying some easily changed operating conditions in two different plants, or pilot plants, or other large operating systems. The easy-to-vary factors may be varied over the same ranges in each plant, but it will usually not be easy to change back and forth from plant to plant. The easy factors are then said to be nested inside the hard factor.

It will never be as easy to move from plant to plant as to change operating conditions within a plant. Random uncontrolled variation between plants cannot be expected to match random variation within a plant. The equal time spacing and linear time trend assumed earlier will not exist either.

Thus when there is nesting it will be necessary to carry out a one-factor-at-a-time sequence of the type discussed at each level of the hard-to-vary factor.

7. TESTING FOR LINEARITY

It is possible to test a system for linearity, even if only two levels of input are attainable at one time. At its crud- est, the same factor can be used at two levels called (1) and a, with a further equal increment of A, now called b, and finally at a third equally spaced level of the same factor. Curvature will be detected by the inequality of the three pseudoeffects.

8. CONCLUSIONS

It is sometimes necessary to study the effects of several factors, varying them one at a time. The three-factor sequence is (1), a, ab, abc, bc, c, (1); the four-factor sequence is (1), a, ab, abc, abcd, bcd, cd, d, (1).

Each plan gives two estimates of each main effect and hence a preliminary indication of precision (or of nonadditivity).

Each plan also gives three estimates of pairs of two- factor interactions. If only one interaction is large, it will usually be isolable.

If results can be obtained at uniform time intervals, such a plan gives all main effects free of bias because of linear trend.

Each effect is estimated by two pairs of successive observations. No matrix arithmetic or least squares manipu- lation is required.

REFERENCE

Daniel, C. (1973), "One-At-a-Time Plans; The Fisher Memorial Lecture for 1971 ," Journal of the American Statistical Association, 68, 353- 360.

Thle American Statistician, May 1994, Vol. 48, No. 2 135



Factorial One-Factor-at-a-Time Experiments

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