Factorial Experiments in Incomplete Block Designs

Statistical Design. Factorial Experiments in. Incomplete Block Designs. Recent work has shown how to use in complete block de signs for industrial...
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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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