5tati!itiedl Design Gun problem illustrates diverse classes of statistical designs available in research work by
W.J. Youden
mental error appropriate for the comparison of the averages. There may be some reluctance to perimental ammunitions for large 16-inch guns. Two important ex- assume that there is a linear fall off perimental restrictions confronted in velocity as a result of barrel wear. the individual in charge of the firing Fortunately it is not necessary to tests. First, only four rounds of postulate any particular function for each ammunition mere available. the effect of wear on velocity in ordes Second, the wear on the gun barrel to arrive a t a satisfactory firing in the course of firing 20 rounds could order. A random firing order, that is, one determined by drawing 20 not be ignored. lettered slips of paper out of a hat, will distribute the letters so they Systematic firing order sample the unknown function in a versus random arrangement manner that is fair for each letter. The point was made that a sys- Indeed, considering the whole setematic order such as AAAABBBB quence, the average separation beCCCCDDDDEEEE for firing the 20 tween like letters tends to attain the rounds of ammunition designated A, same value as the separation between B, C,D,and E was manifestly unfair unlike letters. This leads to the imto the ammunition tested last. If portant consequence that the variavelocity was the performance re- tion shown among the four results corded it seems reasonable to expect for each ammunition is a valid yarda worn barrel to penalize the last stick for appraising differences exrounds fired. Differences in per- hibited between the five averages. formance between the amniunitions The idea of complete randomizamight be due solely to barrel wear, tion appears t o raise a t least two not to any real differences between questions. One question has to do the ammunitions tested. Further- with the theoretical possibility that more, the relatively good agreement the letters, though drawn from a between successive rounds of t h e hat, may turn up in the systematic same ammunition would lead to a order that was discredited above. misplaced confidence in the pre- The probability of this happening is cision of the comparisons. small but “bunching” to a greater It was further seen that a sys- or lesser degree will often happen. tematic firing order of the type The experimenter may feel that randomization has let him down and ABCDECADEB IBEDACEDCBA that the various letters are not, in fact, fairly spaced. Should he rewould exactly compensate for barrel ject the drawing and try again? wear if the wear caused a linear fall This leads to the second question. off in velocity. The arrangement is Is it possible to devise arrangements symmetrical about the center point that are preferable to purely chance so that the advantage that accrues arrangements? The two questions can be answered to any round in the first half of the trials is counterbalanced by the together. When the experimenter equivalent disadvantage of the backs away from a chance arrangematching position in the second half ment of the letters in bunches, his of the trials. The repeat rounds of a reaction comes from his knowledge particular ammunition will not show of the experimental situation (as as good agreement as they mould if in the gun problem). The experifired in succession, but i t is easy to menter’s knowledge may be incomobtain an estimate of the experi- plete but it warns him that bunchHE
August column discussed a test
Tprogram to compare five new ex-
October 1955
ing of the letters is undesirable. This warning is accompanied by a conviction that he could make a better balanced arrangement by making sure that the letters are evenly spread throughout the sequence. The answer can now be stated. If the experimenter does have some knowledge of the experimental situation, and he usually does, this knowledge can provide the basis for a more even distribution of the letters in the sequence. But a t some point the knowledge gives out and a t this point randomization must again be used. The reason for the presence of restricted randomination is presented in the next example of a firing sequence. Many arrangements are based on restricted randomization
The whole sequence may be regarded as made up of four sections of five letters. Certainly experimental conditions are more uniform within each fourth than over the whole sequence. What is more natural than to make sure that a complete set of the five letters, A,B, C,D,and E,appears in each quarter? This is a proper and generally desirable restriction to impose. But what about the order of the letters in each quarter? Whatever the order used in the first quarter, if it is repeated identically in the other three quarters, it may leave the work vulnerable to some systematic bias. Certainly in the gun problem the first position in each quarter carries an advantage over the fifth position. To assign a particular letter to the first position in each fourth of the sequence would be unfair. One way to be fair is to randomize the five letters anew for each of the four sections. Randomization provides protection against any favoritism. The firing scheme just described is an example of a randomized com-
INDUSTRIAL AND ENGINEERING CHEMISTRY
99 A
Statistical Design
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.- 7 plete block design. The block refers to the choice of a unit large enough to accommodate a complete set of the items under test. The word randomized emphasizes t h a t any unknown persistant trend repeated in several blocks is allowed for by the chance assignment of letters within the blocks. The use of systematic repetition involves the risk that the repetition may coincide with some unsuspected cycle in the test environment. There is a n often overlooked dividend that results from the use of these blocks. Each block has a complete set of the ammunitions under test. If there were no gun barrel wear then the average for the five rounds in any one block should have the same expected composite value as the averages for the other blocks. I n other words, the use of different ammunitions will not introduce differences among the block averages provided each block contains a complete set of the five ammunitions. The actual block averages will, however, reflect the trend introduced by wear. By plotting these averages opposite 3, 8, 13, and 18 on the z-axis (these are the center points of the blocks) a very fair picture of the effect of gun barrel wear will be obtained. The information available in the block averages should not be ignored. It is feasible t o combine two studies that normally mould require separate programs and consequently save half the work. A study of the resistance to wear of different barrel linings may be contemplated. Clearly the ammunition experiment in complete blocks has made available a trend line for wear for the particular gun barrel used. Another ammunition experiment using another gun barrel mill provide another trend line for wear. These lines are as suitable for comparing gun barrels as if a separate project had been set up. Even the single study using 20 rounds can, with a somewhat more firing using two guns, furnish five points for the wear line for each gun. The size of the block is determined by the number of ammunitions, five in this case. Should it be impractical t o fire all 20 rounds in a single session then every effort should be made t o fire the five rounds of a com-
plete block in a single session or day. Advantage will then be taken of the relatively uniform experimental conditions associated with a compact working period. Even if conditions change for subsequent blocks the comparison of the ammunitions will not suffer because the design ensures that each ammunition is present in each block. If there is reason to believe that the wear on the gun barrel induces a similar trend within each block the experimenter may also consider trying to equalize the ammunitions in regard t o the positions within the blocks rather than t o depend on a random arrangement for each block. Given five rounds of each ammunition, it would be easy to arrange the firing order so that any letter, say A, will be found in all five within block positions. The sequence of five blocks
ABCDE 1 BDEAC I CABED DEACB [ ECDBA when written one block another Position within block
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above
Block
1
2
3
4
5
1
A B C D E
B D A E C
C E B A D
D A E C B
E C D B A
2 3 4
5
shows a complete set of letters a t each position within the block. Such a two-way equalization of the exposure of the test items to sources of experimental disturbance is known as a Latin square design. This double restriction on the randomization procedure does not fix the position of every letter in the firing sequence. There are 161,280 different 5 X 5 Latin squares so t h a t considerable latitude exists in filling out a square with randomly drawn letters. Of course any letter already present in either the row or column determined by the position in the square must be rejected and another letter drawn. The last row of a Latin square may be omitted and adjustments made by formulas. Correspondence concerning this column will be forwarded if addressed to the author, YoEditor, INDUSTRIALA N D ENGINEERIXQ CHEMISTRY, 1155--16th St., X.W., Washington 6, D. C.
I N D U S T R I A L A N D E N G I N E E R I N G CHE MISTRY
Vol. 47, No. 10
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