I/EC
Statistical
Design
Factorial Experiments in Incomplete Block Designs EXPERIMENTATION consists
Recent work has shown how to use in complete block de signs for industrial experimentation to study the effects of several factors on response variables
of
ap
plying selected treatments to some material and observing the responses. It sometimes happens that there is not enough material of one kind for all of the treatments to be applied, so that several different materials are used. If a material will accommodate a block of k treatments, then consideration needs to be given to the selection of the k treatments. The treatments may be distributed at random to the mate rials, or they may be assigned sys tematically. Systematic assignments have been developed by statisticians under the name of "block" designs. Block designs have the advantage that they facilitate the elimination of variation among materials from comparisons among treatments. Re cent work has shown how to arrange factorial treatments into blocks and how to analyze the resulting re sponses. An example of eight treat ments arranged in eight blocks is discussed here. Test of M i l k Filter Disks
by W. S. Connor, The Research Triangle Institute
— Circle No. 59 on Readers' Service Card
The experiment dealt with milk filter disks, which are used after milking to strain out dirt and debris, but not butter fat. Purpose of the experiment was to assess changes in the construction of the disks on the speed of flow of milk through them. The disks were constructed by bond ing a fiber web on the top and bot tom of the body of the disk. The factors studied were the solution used for bonding on the top surface, the solution used for bonding on the bottom surface, and the loft, or thickness, of the disk. Each factor was considered at two levels, which are denoted by L and H, light and heavy, for the solutions, and Ν and L, normal and light, for the loft. Descriptions of the treatments and
Numbering of Factorial Table 1. Treatments and Estimated Responses Treat ment Number
Descrip tion
Estimated Response
1 2 3 4 5 6 7 8
LLN LLL LHN LHL HLN HLL HHN HHL
269 365 321 364 251 262 313 285
estimated responses are given in Table I. Disks were tested on dairy farms, by pouring a quantity of milk through the disks and noting the times required for the milk to flow through them. It was found that no more than three disks could be handled successfully at one time and that there were large differences in milk between farms and between different milkings at the same farm. Accordingly, it was decided to use a block design to eliminate the effects of these differences from compari sons among the disks. The design used is R5 from Bose, Clatworthy, and Shrikhande (7). The design and observed responses are given in Table II.
T a b l e II. Assignment o f Treatments to Farm Tests a n d O b s e r v e d Responses in Seconds" Farm
Pouring Position First Third Second 1 (246) 2 (321) 4 (320) 2 (409) 3 (344) 5 (276) 3 (368) 4(413) 6 (303) 4 (357) 5 (250) 7 (290) 5 (249) 6 (279) 8 (275) 6 (237) 7 (297) 1 (262) 7 (290) 8 (244) 2 (322) 8 (322) 1 (266) 3 (348) parentheses.
Test
Number 1 2 3 4 5 6 7 8
» In
VOL. 52, NO. 12
·
DECEMBER 1960
83 A
STATISTICAL DESIGN Conclusions Table Farm Test
III.
Estimation of Treatment 1
Effect
of
The principal conclusions, drawn from statis tical analysis, are set forth in this section.
Multiplier and Observed Response" 14(1) 1(2) 0(3) 1(4) -2(5) -7(6) 0(7) -7(8)
-7(2) 1(3) 0(4) -2(5) 1(6) -7(7) 0(8) 14(1)
• Disks with the heavy solution on the top surface are faster t h a n disks with the light solution on the top surface
-7(4) -2(5) 0(6)
1(7) 1(8) 14(1) 0(2) -7(3)
" E.g., (1) is 246 for the data considered.
• Disks with the light solution on the bottom surface are faster t h a n disks with the heavy solu tion on the bottom surface • Disks with n o r m a l loft a r e faster t h a n disks with low loft
Estimation of Responses
T h e effect of a t r e a t m e n t is the deviation of the response to the treat m e n t from the average response to all of the treatments in the experiment. It is estimated from the observed responses by a calculation for treat m e n t 1 (Table I I I ) . T r e a t m e n t numbers, enclosed in parentheses, denote observed responses, a n d the numbers in front of the parentheses are coefficients which multiply the observed responses. A d d i n g a n d subtracting the products, as indi cated, a n d dividing by 48 yields — 35 as the estimated effect of treat ment 1. Formulas for other effects are given in (7). T h e average of the 24 observed responses is 304, so that the estimated response for treat m e n t 1 is 304 - 35 = 269. Esti mated responses arc given in T a b l e I. T h e calculation (Table I I I ) shows how estimates of the t r e a t m e n t effects are unaffected by differences a m o n g farm tests. For each farm test, the sum of the coefficients is zero. For example, in farm test 1, the coeffi cients are 14, —7, a n d — 7. Ac cordingly, if the farm test imparts some additive constant (possibly
Table
IV. Estimated Responses Levels of the Factors Factor
Solution on top surface
to
• For disks with the light solution on the top surface, n o r m a l loft is faster t h a n low loft; but, for disks with the heavy solution on the top surface, low loft is faster t h a n normal loft • As noted above, disks with n o r m a l loft are, on the average, faster t h a n disks with low loft. H o w
negative) to each observed response, its effect is cancelled out in the cal culation of the estimated t r e a t m e n t effect. I n a similar way, responses of each farm test m a y be estimated, unaf fected by the particular treatments tested : Estimated Response
1 2 3 4 5 6 7 8
267 335 350 294 305 288 268 324
Light
Heavy
287
321
Low
Normal
Loft
319
289
84 A
INDUSTRIAL AND ENGINEERING CHEMISTRY
Pouring Position 1 2 3
Estimated Response 310 302 300
References
Level Light Heavy 278
sons to be m a d e a m o n g the pouring positions without being affected by treatments. Estimated responses for pouring positions a r e :
Estimated responses to levels of the factors are shown in T a b l e s I V a n d V. T h e s e averages s u m m a r i z e the statistically significant effects of the factors.
It was t h o u g h t t h a t the outcome for a disk might depend on whether it was first, second, or third in a farm test. T o gain information about this point, treatments were arranged so that every one of t h e m c a m e once in each position. This kept comparisons a m o n g treatments from being affected by the order of pouring a n d also permitted compari-
330
Solution on bottom surface
Farm Test
ever, this difference is markedly greater w h e n the light solution is on the bottom surface t h a n when the heavy solution is on the bottom surface • T h e r e is no indication of inter action between the solution on the top surface a n d the solution on the bottom surface, or ot interaction a m o n g the t h r e e factors simultaneously • F a r m tests differ in average speed • For three disks per farm test, there is no evidence that pouring posi tion matters »The estimated standard deviation of a single response is 18 seconds. Because the general average was 304, the estimated coefficient of variation is 6 %
(1) Bose, R. C , Clatworthy, W. H., Shrikhande, S. S., "Tables of Partially Balanced Designs with Two Associate Classes," Institute of Statistics of the Consolidated University of North Caro lina, Reprint Series 50 (1954). (2) Connor, W. S., "Experiences with In complete Block Designs," in "Experi mental Designs in Industry," Wiley, New York, New York, 1958. (3) Kramer, C. Y., Bradley, R. Α., Biometrics 13, No. 2 (1957). (4) Zelen, Marvin, Ann. Math. Statistics 29, No. 1 (1958).
Table V. Estimated Responses to Combinations of Levels of the Factors
Loft Low Normal
Soin . on Soin . on Top Surface Bottom Surface Light Heavy Light Heavy 324 364 274 314 295
282
260
317
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