Mathematics for Engineers—II Method of Graeco-Latin Squares

H. P O R I T S K Y

I N ENGINEERING in-vestigations and labora-

tory experiments it often hap-pens that the effect which is being investigated, or the per-formance of the device or of the particular material under consideration, depends upon a great many variables. It is of importance to be able to pick out the significant vari-ables, whose variation will affect the results of the experi-ment or the performance of the device or of the substance in question, from the insignificant ones whose variation has little if any effect, so as to minimize time and effort in devising experiments and building and testing samples. Added to the difficulty of too many variables is the fact that there are also accidental and random variations, due to quantities which are either unknown or which it is out of the question to investigate, and which thus are termed accidental.

A standard method of testing any one variable consists in varying it only while holding all the others constant. To cover a range of variation of many variables by this straight-forward method generally involves far too many tests, unless one is willing to make the assumption that if the variation of one variable Xi proved of little significance for one set of values of the other variables, #2, #3,. ., its variation also will be of little import for other sets of values—an obviously unwarranted assumption.

The method of Graeco-Latin squares enables one to cover a range of many variables in a somewhat systematic fashion with a minimum number of tests. It thus leads to results with the greatest economy of effort, and yields most infor-mation for a given number of tests.

To explain this method, suppose there are four variables: *i> *2> *3> *4· Suppose that three values are taken on by each variable, as shown in the array (1):

xi: I II III \

;; ί ? ? I » #4.* a b c /

If each value of each of these variables were employed and combined with every value of every other variable one would have to carry out altogether 34 = 81 samples or tests. By contrast the Graeco-Latin square method utilizes only a total of 9 combinations of values shown in Table I.

It is clear from this table just what nine combinations of values of each of the four variables are employed. Thus, if the values displayed in expression 1 proceed in increasing H. Poritsky is with the General Electric Company, Schenectady, N. Y.

numerical order, the upper left-hand corner of Table I corresponds to the smallest value of each of the four vari-ables, #i=I, x2=A, #3 *= 1, xA *= a»

It will be noted that each element of the first row of Table I contains the value A, each element of the second row the value B, each of the third row the quantity C. Like-

wise, each element of the first column contains the value I, each element of the second column the value II, each ele-

Table I. Combinations of Values of Four Variables

I II UI

A U 2b 3c B 2c 3a 16 C 3b ,1c 2a

ment of the third column III . In this way the samples picked are distributed evenly among the values of the first two variables xh #2 in the array (1), each value of χχ occur-ring once and only once with each value of x%. Likewise, they also can be shown to be uniformly distributed in all four variables xh #2, #3, #4 in that each of the 12 values in the array (1) occurs combined once and only once with each of the 9 values of the other variables occurring in the array.

Similar Graeco-Latin squares have been constructed for 5 and 6 variables provided that the number of values of each variable is one less than the number of variables (that is, 4 and 5). These squares are shown in Table II and Table III . They, too, have the same property that as many elements are assigned to each value of each variable as to any other value of the same variable and each value of each variable occurs once with each value of every other variable.

Table II. Combinations of Values of Five Variables

I II ΙΠ IV

A \aa 2bß 3cy 4dS

B 2ch Idy 4aß 3ba C 3dß 4ca \bb 2ay D 4by 3αδ 2da. \cß

Table III. Combinations of Values of Six Variables

I n III TV V _

A UaV 2bßW 3cyX 4dhY 5«Z B 2ciZ 3du>V 4ectW SaßX XbyY C 3eßY 4ayZ 5b8V UeW 2daX D 4dtX ScaY UßZ 2eyV 3d&W E 5dyW U&X 2aeT . . . . . .3daZ 4cßV

In this second part of a 3-part series, a method is described whereby time and effort in examin-ing problems involving a number of variables may be reduced greatly by selecting only those combinations of greatest significance. It has been applied for four, five, and six variables, provided the number of values of each variable

is one less than the number of variables.

NOVEMBER 1948 Poritsky—Mathematics for Engineers 1061

After the experiments have been designed, or the appara-tus built, and tests carried out in accordance with the squares just described, and measurements of the quanti ty of interest obtained, one proceeds to analyze them as follows. Con-sider, say the variable Xi whose values I, I I , I I I , correspond to the columns of the three tables. First the means Χτ, X1U

Xui- . .of the values of each column are found; then one obtains the over-all mean X, which is also the mean of the means of the individual columns. Next one calculates the variance σ2 of the difference of the column mean and the over-all mean, by means of

σ2=Γ Σ (*i-*)V(*-l)l (2) Ü - I , II. . . J

where n is the number of columns. In general, the variance σ2 of a population is the square of its standard deviation σ:

Γ N T/f

H Σ (*«-*>VAj Ö) However, for a sample of N members taken for a large population, under the normal law of distribution, the stand-ard deviation from the mean of the sample is somewhat higher than for the population as a whole, and is obtained by replacing N in equation 3 by (Λ"-1):

*sample= [ΣΑ-ί)ν(^-1)] V i (4)

It is now essential to find out whether the variance ob-tained is significant or whether it is merely accidental. For this purpose it is advisable to repeat the experiments and observe how closely individual entries repeat themselves. More precisely, say for the case of four variables, if the experiments are repeated once, the deviations of each set of values in Table I from their mean are obtained and the variance (of these deviations) is computed

<τα8=ίΣ(*-*)2/*α] (5)

where now na = 9 (due to the nine entries in Table I ) . To find whether the variable Χχ is significant or not, one

now compares the variance σ2 in the means of the columns with the accidental variance σα

2. Statisticians have worked out several methods by means of which one can decide (to within a certain degree of certainty) whether the first vari-ance σ2 is significant compared to σα

2 or not. If it is, then the variable Xi is an important variable; if it is not, then the variation of the function in question with xx is either too small to be of interest or will be covered up by a random variation due to the impossibility of repeating the results.

One convenient test for significance of variations is given by the "F-test" (see, for instance, reference 1). First, the ratio F of the two variances is found, say always in such a way as to lead to F> 1. I t will be noted that for Figure 1, σ2 was basod onw = 2 "degrees of freedom," while σα

2 was based on na = 9 degrees of freedom—by a "degree of free-d o m " is meant the number of independent values upon which the variance calculation is based; since the deviations Xf—X of a set of n values from their mean necessarily satis fies the condition of a vanishing sum, this set possesses n-l degrees of freedom. The table entry corresponding to these two values for say 5 per cent degree of significance gives an entry F = 8 . 0 2 . If the F value obtained from the

tests is larger than the table value F then the chances are 95 per cent that the variable Xi is a significant variable; if not, then within the foregoing certainty Χχ is not a significant variable.

The F test and the tables for it are based upon a Gaussian distribution of random variation.

After the significant variables have been singled by this method from the total number of variables considered, one must proceed with a more detailed exploration of their range so as to find the opt imum performance for the device or the substance under consideration.

Two instances may be mentioned where the method of Graeco-Latin squares has been applied in the General Electric Company, one where it is worked very successfully, the other one with rather little success. The first case was in design of a tachometer compensator, the other one in the study of an alloy.

In the tachometer compensator problem which was studied by P. E. Thompson of the works laboratory of the West Lynn Works, it was a question of obtaining the best control of temperature variations of the tachometer. This was accomplished by means of a compensator plate and the following four variables were considered to be the control-ling factors: location of the compensator plate, thickness of the plate, electrical conductivity of the drag disk, and ther-mal coefficient of electrical conductivity of the drag disk. Nine tachometers were built in accordance with Table I and calibration was obtained for them at four tempera-tures and was investigated by means of the F test. By analyzing the results it was found that the coefficient of thermal conductivity was not an important factor, but the three other factors were significant. Opt imum values for them then were determined to yield the least error.

The second instance was an application of Graeco-Latin squares to the study of a 6-component alloy by J . D. Nisbet and Ann Lindberg of the research laboratory at Schenec-tady. Here the percentages of five of the component ele-ments of the alloy were used as the variables X\. .x$ in ques-tion, and alloy samples were prepared in accordance with the square of Table I I . However, the analysis of the re-sults showed that it was impossible to pick out any one ele-ment as more significant than any other one. Except for indicating that all of the elements were significant, no useful result was thus obtained from application of the Graeco-Latin square method in this case.

Generally, in making up Graeco-Latin squares the values of any one variable are taken to be equally spaced. Thus in the array (1) the values denoted for I, I I , and I I I are equally spaced, and the same applies to the three values of each remaining variable. This need not be the case, how-ever, always.

Examples can be readily constructed where by replacing the original variables by a set of new variables which are functions of the original variables one will obtain significant and nonsignificant variables, when that is not the case with the original variables. However, this is a situation which the method of Graeco-Latin squares is not capable of detecting.

I t may be pointed out that if in case, say of four variables, the dependent quanti ty y is a linear function of the three

1062 Poritsky—Mathematics for Engineers ELECTRICAL ENGINEERING

variables, say *2, *s, *4, then it is possible not only to separate the significant variables from the insignificant ones, but also to obtain the functional variation of y with x\ from the column mean of the Graeco-Latin square. As will be noted from Table I, each value of each one of the variables *2, *3, and *4 occurs once and only once in the three ele-ments of each column of Table I. I t follows that the column mean will average out the values of those variables, and if the column mean is plotted against the values of the variable xh the points should lie on a straight line if there is linearity in #1 also, or on a curve if xx does not enter line-arly; discrepancies due to accidental variations also will

Standards of Very

Lack of standardization in capacity measuring equipment often has resulted in losses due to rejection by the purchaser of electron tubes whose interelectrode capacitance was not within the tolerance limits for acceptable performance. T h e result has been a demand for secondary reference standards of small capacitance. The National Bureau of Standards was requested to establish and maintain a group of primary capacitance standards ranging from 100 down to 0.001 micromicrofarads.

The capacitance values of several small capacitors have been determined by a process of stepping down or of sub-division from larger units, which ordinarily are measured by well-known bridge methods in terms of resistance and time. However, for capacitances from 5 micromicrofarads down it was considered desirable to check the accuracy of the subdivision by the use of absolute standards whose values could be computed from their mechanical dimensions.

In the range from 5 micromicrofarads to 0.1 micromicro-farad the Kelvin guard-ring type of capacitor was used as a primary standard. In this device the high-voltage elec-trode is supported at a fixed distance from a smaller meas-uring electrode, which is surrounded by a guard ring to eliminate fringing. The larger electrode is connected to the high-voltage terminal of a bridge and the measuring electrode and guard ring to the ground-potential terminals of the same instrument. Only that portion of the total flux from the high-voltage electrode which reaches the smaller electrode is measured by the bridge and is used in comparing the capacitor with a secondary standard. For precision work an improved design was developed at the bureau in which the guarded electrode and the guard ring, separated by a very small gap, as well as the high-voltage electrode, are flat and polished so that they can be tested readily by optical methods for parallelism, co-planarity, and symmetry. The guarded electrode, or "island," is rigidly and accurately centered in the guard ring by means of a Pyrex-glass collar held firmly in position.

For capacitance below 1 micromicrofarad, a new type of guarded-electrode capacitor was developed for the range from 0.1 micromicrofarad down to 0.001 micromicrofarad on the basis of a design suggested by Doctor F. B. Silsbee of the National Bureau of Standards. In the new capacitor the

show up , though with only three values keen judgement is required to decide whether the deviations from a straight line are due to a nonlinear dependence upon xx or to acciden-tal variations. Thus , under the assumption of partial linearity, it is possible not only to pick out a significant vari-able but also to obtain coefficients of variation and the opti-m u m value of the dependent variable. This naturally will occur at one end or the other of the range of each linear variable.

REFERENCE 1. H. A. Freeman. John Wiley and Sons, New York, N. Y., 1946. Table 8, pages

Small Capacitance

guarded electrode, instead of being coplanar with the guard ring as in the Kelvin type, is placed at the bottom of a cylindrical well of fixed depth below the surface of the guard. Fringing occurs, depending on the depth of the well, so that only a fraction of the electric flux from the high-potential electrode reaches the measuring electrode. By increasing the depth of the well, the capacitance can be made as small as desired; at the same time the capacitor is of such dimensions that it can be constructed and meas-ured accurately. The construction of the guarded plate and of the high-voltage electrode is identical with that of the guard-ring capacitor.

Formulas for computing the capacitance of both types of capacitors have been derived by Doctor Chester Snow of the National Bureau of Standards on the assumptions that the clearance between the island and the guard ring is infinitely small, that the edges of the hole in the guard ring are not rounded, and that the guard ring and voltage plates extend to infinity. However, measurements on an experi-mental model have shown that the clearance between the island and the guard ring can be as large as several thou-sandths of an inch without appreciably altering capacitance, and that the high-voltage plate need extend over the edge of the guard ring for a distance only three or four times the space between the high-voltage plate and the ring.

In addition to the primary standards, several secondary standards have been built. One of these is a decade of noval construction having two units of 0.1 micromicrofarad, two units of 0.2 micromicrofarad, and one unit of 0.4 micromicrofarad. Each unit of the decade consists of a pair of plates, insulated from the housing by mica insu-lation, together with a metallic blade which is connected to the housing. The individual units are switched out of the circuit by sliding the metallic blades between them. This completely isolates one terminal from the other, mak-ing the capacitance of the unit zero. The capacitance of each unit may be adjusted within very close limits by means of a vernier screw which controls the effective distance between plates. The units were adjusted at the bureau to be equal or exact multiples of each other within the limits of sensitivity of a bridge and have held this adjustment for over a year*

NOVEMBER 1948 Poritsky—Mathematics for Engineers 1063