Design of Experiments for Most Precise Slope Estimation Cuthbert Daniel 116 Pinehurst Awe., New York 33,
N. Y .
T h e well-known methods for calculating the variance of the slope of a least-square fitted straight line are given briefly, both for constant precision of y measurements and for variable though known precision. When the relation between two variables is known t o be linear, proper choice of r values at which y is to be measured can greatly increase the precision with which the slope is estimated. The gain, cornpared to the usual N equally spaced points, may be from one third to two thirds of N for a given precision of the slope estimate. This article illustrates how the the principles of statistics can be utilized by chemists and chemical engineers to obtain more precise results from their experimental w-orlr.
T
HE modulus of elmticity of many materials is iound by
measuring the amount of stretching of a standardized piece under various loadings. The data are plotted as a strain versus stress diagram and a straight line is drawn through the points. The slope of the line is equal to, or proportional to, the modulus of elasticity. Typically the points do not all lie c.xactfy on a straight line. Typically too, if the whole test is repeated even with the same test piece and a t the same loadings, the strain values do not exactly reproduce themselves. The true slope would be found if a very large (infinite) number of measurements could be made. The question should therefore be asked: How variable are the slopes estimated froni, say, N observations each? The rate of increase of concentration of a volatile component through a distillation column often is estimated by sampling the liquid or vapor phase in the column a t several equally spaced plates. Some function of the concentration--e.g., the abundance ratio-is often a linear function of the plate number. The rate of enrichment per plate will be estimated by the slope of the best line through the data points. Again it is useful to know by how much the random, uncontrollable unevenness of the column’s operation can influence the resultant slopc~ of the abundance ratio versus plate number line. The production rate of a pilot plant is often rather variable. Several runs, made for different lengths of time are sometimes pooled t o get a more reliable estimate of the true rate. Usually the best way t o weight the several runs is t o estimate the slope of the best line through the points when the amount of product produced is plotted against the duration of the run. Rut even the best way will not give the true value. Again, one statistical question t o be answered is: How do the results of many such ,sets of pilot plant runs spread around the true value? In other words, what, is the precision of the slope as estimated from N runs of specified durations? Many other examples of the measurement of rates-of-change of one engineering variable with another can be given. This paper will first outline very briefly some statistical methods already available for estimating rates or slopes, together with their precision. It will then show t h a t under certain conditions there are positions and distributions of measurements t h a t minimize the uncertainty of the slope.
expremed here in iiot,ntion consistent, with that used in i h two ~ preceding papers. 1. The relation between x and y is known to be lirieui,. The true relation is taken t o be 1- = a
+ 02
(1 1
and p are unknown constants. 2. The relation between 5 and u (where u is the struitlard deviat’ion of y values a t ). is known, except possibly for a fixed multiplier, K-Le., u = K J ( r ) ,wheref(r) is known and K is unknown. 3. It is possible to choose the z values a t which g mvasurements are to be made. (This rules out most time series.) 4. It is possible t o attain statistical independence in repeated me:surement’s of y a t fixed values of 5. a. The z measurements are essentially exact; oniy i/ ir subject to random error. where
CY
Over a century ago Legcndre &owed that the dopc1 b, and the height a t = O(a) of the least-squares straight line through 3 points (z~, ye) are given by the equations:
b =
(I
=
- bz
(3)
The “expected v a l u d of n and b, which would be hnuntl if a vary large number of observations were made are, m p w i ively, CY and 8. Calling the best estimate of Y obtainable from a set of data points Y’, we have then 1”
=
4-b s
(4) When the mean square deviation mectsured in the iiiiwtion of a very large (infinite) number of observations from the true straight line is constant for some range of z values, It is called the “variance of estimate,” here designated by uz(y) In terms of this quantity, which is a measure of the spread of observed 11 values at any fixed % due t o random, uncontrolled factors, it is possible to express the spread of b-values around b (I
THE STATISTICAL MODEL
The criteria given later for optimum z placement of data points are exact under the following five assumptions, which are
When the inean squaiu: deviation of a large set of g measurements is different for different values of r, it will be called e:,
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INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1951
corresponding to xi. The points are now weighted inversely by their variances. The equation corresponding to Equation 5 is: u2(bw) =
vhcrr
w2 = l/uf
and
1/2Wi(Z,
- Zw)Z
P
By the use of Equations 7 , 8 , and 9,
xz
=
k
2h
nl
=
722
u1 = u2 =
= Zwixi/Zw,.
For both optimum positions t o exist, the curve of u versus z must be concave upward. Equation 7 may be solved graphically for z1 and z~ when the relation between u and x is given graphically. Explicit solutions are given below for several simple relations between u and x. I n addition to specifying the x values a t which y should be measured, one must specify the distribution of the N available measurements. Calling nl and n2 the number of measurements to be made at z, and 5 2 , respectively, the minimum variance of slope will be obtained when
a
(8)
When Equation 8 is satisfied, the (minimum) variance of slope is given by
(13)
Thus in this case measurements are not made where the precision of y is best, but only a t the two points where u is twice its minimum. It is also possible t o calculate the variance of slope for Aequally spaced points when u varies parabolically with 5. The ratio of this variance to the optimum variance may be taken as u measure of the efficiency of the optimum arrangement. This ratio is 1.58 when N = 7, 1.69 when N = 21, and 2.00 when N is very large. When the standard deviation of y can be represented by one parabola on one side of its minimum, and by another on thp other side, a somewhat more general explicit solution can be given. Suppose that
+h u = c2(x - k)2 + h
u = cl(x - k ) *
3:
k
when
(14)
The substitution of these values in Equations 7 , 8, and 9 gives
Thus one set of measurements is made where u>2h, and one set where uC2h. It will sometimes happen that the standard deviation is a monotone increasing function of x, and t h a t for some reason y measurements cannot be made a t values of x smaller than, say, xl. If the relation between u and x is parabolic, with u = kx2, and x1>0, then the optimum conditions for a most precise slope are given by 52
=
2.41421
~2
i=
5.83k~f
nl/nP = 0.171 23 31
u*(b,) = A
N
h =do2
(11) (12)
u2(bw)= 4ch/N
I t is intuitively obvious that for fixed precision of measurements, U, and for a fixed number, N , of observations, the variance of b, d ( b ) , will be minimized when Z(s, - *)2 is maximized, and that this maximum is reached when the measurements are made a t two values of x, say 21 and XZ, as far apart as is physically possible, and in equal number. The variance of slope given by N points equally spaced over the same range, xl to 22, is 3(N-l)/ (iV+l) times as large as the variance of slope estimated by N points “equally bunched.” Similarly 3(N--I)/(N+l) times as many equally spacad observations will be required for a given uncertainty (variance) of slope as when the points are equally bunched. In this sense, equal bunching is about three times as efficient as equal spacing. Thus when the relation between x and y is known t o be linear, the maximum precision for a given number of observations will be attained by taking N / 2 measurements at eacth end of the measurable range. When, however, the precision of g measurement varies with x, Equation 6 holds. It is still true that the observations should be bunched at two values of x, but not necessarily in equal number, nor necessarily a s far apart as possible. It is perhaps intuitively clear t h a t when the precision of y measurement deteriorates very rapidly with x, i t may not improve the precision of the slope estimate if one extends XI or z~too far. The criterion for the limiting x values at which t o take y measurements is given by the following equation (1):
n
+ dvc
x1=k-- d h 7 (6)
MOST PRECISE ESTIMATION OF SLOPE
e
1299
(18)
(19) u2
x xf -3
As a final example, when u increases exponentially with xLe., when u = kern=,with m>0-then
xP = x1
+ 1.279/m
02
=i
3.59~1
(21 1
(9) An absolute minimum occurs when Equation 7 can also be satisfied. The Btandard error of the slope is of course immediately obtainable from Equation 9. When all measurements of y are made at ;h and x ~ then , the slope, bw, can be calculated from the and 71 are simple equation: b, = & - Td/(xs - XI), where the unweigheed means of the y measurements a t 2 2 and XI. SOME PARTICULAR CASES
An explicit solution can be given when the relation between and x i s parabolic, with a minimum rat u = h, x = k-Le., when u
4~ - k)* + h
u
(10)
RELATION TO PRECEDING PAPERS
The dependence of this paper on the concepts outlined in the papers of Scheff6 (3) and Mosteller ( 8 ) is immediate and perhaps obvious. The y values are random variables; statistical independence of y measurements is assumed; the normal distribution is not assumed above but in actual usage it would generally be safe t o assume normality for 91 and 92 by the central limit theorem with further conclusions outlined below. The straight line, Equation 1, gives the expected values of y when a! and B are known. The paper deals only with those cases in which the
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INDUSTRIAL AND ENGINEERING CHEMISTRY
parameter, u, is known. Optimum procedures when estimates of u must be based on small amounts of data are under study. As in almost all statistical reasoning, y is assumed t o be in a state of statistical control wherever measured. Mosteller has given a discussion of the mathematical model used in the present paper under the subtitle, “Regression in Two Dimensions.” The case handled here is the one in which the variability of the y distribution changes from one fixed value of x t o another. Mosteller did not discuss this case. A common null hypothesis in estimating slopes would be: I s the true slope of the line-of which there is usually only an estimate from a sample-equal to some standard value, say BI A statistical test made as indicated near the end of Mosteller’s paper will give a t value as large as 1.96 only one time in twenty nhen the true slope is B . This is the 0.05 significance level or the frequency of making an error of Type I. The power curve given in his paper can be used to decide how many 21 measurements should be made (A‘ = nl n ~ ) .It is only necessary to substitute ( B - b ) / u ( b ) for his abscissa. When measurements are made a t z1 and xz in the ratio U I / U Z ,
+
Vol. 43, No. 6
u ( b ) can be found directly from Equation 9 of this paper. After deciding how large a real difference in slope from the standard, B - b, must be detected with some definite probability (ordinate of Mosteller’s Figure I), the corresponding abscissa value can be read off and set equal t o
which can be solved for N. Conversely, if N is already fixed, the same graph can be used t o estimate the frequency of making a n error of Type I1 of any size B - b. LITERATURE CITED
(1) Daniel, C., and Heerema, N., J . Am. Statistical Assoc., 45, 546-56 (December 1950). ( 2 ) Mosteller, F., Ibid. ENG.CHEM.,43, 1292 (1951). (3) Scheffb, H., Ibid., 43, 1292 (1951). RECEWBD February 8, 1951. Presented before the Division of Industrial and Engineering Chemistry, Symposium on Statistics and Quality Contra in the Chemical Industry, a t the 117th Meeting of the AMERICANCHEMICAL SOCIETY, Detroit, Mich.
Factorial Experiments in Pilot Plant Studies J. R. Bainbridge Imperial Chemical Industries of Australia and New Zealand Ltd., 380 Collins Street, Melbourne, CI, Australia T h e factorial experiment reported in this paper was used to investigate unexpected performance in a small gaseous synthesis plant. This approach proved applicable and gave convincing and comprehensive information of a kind which had not been obtained in a much longer period of “noi+nial”plant operation. Of interest to chemical engineers is the detailed examination of the structure of the factorial experiment; the analysis of covariance; the information obtained and the types of error to which it is subject; and the superior effectiveness of the factorial . over other experimental approaches.
T
HE growing application of statistical procedures to Iaboratory and pilot scale investigations ( 2 , 3 ) is most commendable. Whenever a new kit of tools, such as statistical methods, becomes available the owner faces two problems: First, he must acquire technical proficiency in the use of these tools, and secondly, he must learn discrimination-when t o use and when not to use each tool and how t o modify his traditional procedures t o make best use of his new assets. This article is directed toward the second problem, though necessarily it must deal extensively with the first. It is the author’s opinion t h a t in the majority of problems requiring experimental investigation-whether laboratory, pilot, or commercial scale-the factorial experiment is the most effective and economical means of experimental approach. I n order to sustain this thesis a brief description is given of a factorial experiment performed on a small gaseous synthesis plant. This is followed by a detailed analysis of the results using analysis of variance and also analysis of covariance. Finally, the value of the information obtained is compared with that arising from Huhndorff’s repro-
ducibility study (3’) and from Gore’s approaches t o factorial designs ( 2 ) ; the shortcomings of nonstatistical approaches are indicated. TYPICAL EXPERIMENT
The factorial experiment described here was performed on a small plant carrying out a catalytic gaseous synthesis reaction and removing the product as a liquid solution. The plant consists of the following essential parts, which are indicated in Figure
1: 1. Gas preparation plant in which the concentration of the active constituent is controlled 2. Gas purification plant in which impurities causing undesirable side reactions are removed 3. Synthesis section consisting of a converter with catalyst and temperature control arrangements, means for removing the synthesized product in solution, means for recirculating exit gas through the converter and for purging inert gases Three important variables in most chemieal plants are converter reaction temperature, throughput rate through the converter, and the concentration of the active ingredient in the makeup gas. In general, and also in this experimental plant, these three factors are controllable. However, the degree of gas purification depended ultimately on cooling water temperature ; this was not readily controllable and constituted a potential cauee of nonreproducible results. The experimental investigation was required t o show the effect of running the plant a t several different levels of each of the three controllable variables-temperature, throughput and concentration. In addition, it was essential that there be reproducibility of results when running the plant on different occasions under allegedly the same conditions. Unless this reproducibility was determined, it would be impossible t o assess the reliability of
