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Switchback Trials for More than Two Treatments1
SWITCHBACK
T R I A L S F O R MORE' T H A N T W O T R E A T M E N T S ~
H. L. LUCAS Depart~e~t of Experimental Statistics, North Carolina State College, Raleigh I t is often desirable to take advantage of the sensitivity of switch-back or double-reversal trials when doing research with d a i r y cattle. Basically, however, these designs p e r m i t the comparison of only two treatments, a n d this has imposed some limitation on their usefulness. The present p a p e r is devoted to switch-back designs t h a t permit comparisons of three or more treatments. The basic switch-back design is as follows :
Experimental period
T r e a t m e n t sequence 1
2
1
1~
2
2 3
2 1
1 2
"~The n u m b e r s in the h e a r t of t he t a b l e r e f e r to the t r e a t m e n t s b e i n g c o m p a r e d .
The t r e a t m e n t p a t t e r n results in p a r t i c u l a r l y sensitive comparisons, because it permits the elimination f r o m error of (a) the period effects due to changes in environment, (b) the between-cow variation in production level, and (c) most of the between-cow variation in slope of the lactation curve. B r a n d t (1) has elucidated an efficient statistical analysis for switch-back trials. This analysis is also described by Sncdecor (.-7). B r a n d t first computes for each cow a q u a n t i t y D = Y1 -- 2Y~ + Y;~ in which Y~, Y~, and Y:~ represent the p e r f o r m a n c e s during periods 1, 2, and 3, respectively. To outline B r a n d t ' s analysis when the n u m b e r of cows on the two sequences is equal, let D~ = p e r f o r m a n c e of the jth cow on the i th sequence (i = 1, 2; j = 1, 2, •
.
.
r)
G~ = sum of the D ' s for the itt~ sequence The estimate of the error variance is given by 1
~
s~ =
G~ + G~o ~D-~ii
6(2) i ; ' - - 1) }
-
r
I
t~eceived for publication July 21, 1955. 1Published with the approval of the Director of Research of the North Carolina Agricul-rural Experiment Station as Puper No. 662 in the Journal Series. 146
SWITCHBACK
TRIALS
FOR MORE THAN TWO TREATMENTS
147
in which 2 ( r -- 1) = degrees of freedom for error. The divisor 6 is introduced to put the mean square on a per cow per period basis. The a d v a n t a g e for t r e a t m e n t 1 over t r e a t m e n t 2 and its s t a n d a r d error are estimated b y G~ - Gz
4r
and
/ 3s 2 ~f 4r
respectively. The divisor 4 is introduced to place the contrasts on a per cow per period basis. An extension of B r a n d t ' s analysis is used in the present paper. Seath (2) devised a four-sequence switch-back design to s t u d y 2 × 2 factorial p a t t e r n s of treatments. The design provides a sensitive test of the two m a i n effects but yields a poor test of the interaction. The designs in the present p a p e r provide equal sensitivity for all t r e a t m e n t contrasts. T a y l o r and A r m s t r o n g (4) have pointed out that the switch-back designs can be extended to a n y n u m b e r of t r e a t m e n t s by setting out the basic switch-back p a t t e r n for each possible p a i r of t r e a t m e n t s and have briefly indicated the method of analysis. The present p a p e r is devoted to designs of the extended type. The analysis is given in detail, and designs of reduced size are discussed along with the introduction of a blocking feature. Missing-value formulas also are given. THE
DESIGNS
The designs combine the switch-back and the balanced incomplete blocks principles. I n general, the comparison of p t r e a t m e n t s requires p ( p -- 1) treatment sequences (complete designs). I f p is odd and 5 or greater, however, designs t h a t require only p (p -- 1 ) / 2 sequences can be used (reduced designs). T r e a t m e n t p a t t e r n s for a selection of complete and reduced designs are given in the following section. Complete designs are shown for 3, 4, and 6 treatments, but only the reduced designs are given for 5, 7, and 9 treatments. The p a t t e r n s for l a r g e r numbers of t r e a t m e n t s are easily derived but p r o b a b l y are of limited usefulness because they require so m a n y cows. F o r each reduced design a conlplementary design is easily obtained b y writing the second row of the reduced design as the first and t h i r d rows of the complement, and the first row of the reduced design as the second row of the complement. The reduced and the c o m p l e m e n t a r y designs together f o r m the complete design. The complete designs are subdivided into p -- 1 blocks of p sequences each, and the reduced designs into (p - 1 ) / 2 blocks of p sequences each. The rows represent experimental periods, and the colunms t r e a t m e n t sequences. TREATMENT PATTERNS 3 T r e a t m e n t s (complete design):
Block I 1 2 3 2 3 1
Block 2 1 2 3 3 1 2
1
1
2
3
2
3
148
H.n. LUCAS
4 Treatments (complete design) : Block 1 1 2 1
2 3 2
Block 3
Block 2
3 4 3
4 1 4
1 3 1
2 4 2
3 1 3
1 4 1
4 2 4
2 1 2
3 2 3
4 3 4
5 Treatments (redfaced design): Block 2
Block 1 1 2 1
2 3 2
3 4 3
4 5 4
1 3 1
5 1 5
2 4 2
3 5 3
4 1 4
5 2 5
6 Treatments (complete design) : Block 1 1 2 1
2 3 2
3 4 3
4 5 4
Block 3
Block 2 5 6 5
6 1 6
1 3 1
2 4 2
3 5 3
4 6 4
5 1 5
6 2 6
2 6 2
3 1 3
4 2 4
2 5 2
5 4 5
6 5 6
3 6 3
4 1 4
5 2 5
6 3 6
Block 5
Block 4 1 5 1
1 4 1
5 3 5
1 6 1
6 4 6
2 1 2
3 2 3
4 3 4
7 Treatments (redfaced desig~) : Block 1 1 2 1
2 3 2
3 4 3
4 5 4
5 6 5
Block 3
Block 2 6 7 6
7 1 7
1 3 1
2 4 2
3 5 3
4 6 4
5 7 5
6 1 6
7 2 7
1 3 1
2 4 2
1 4 1
2 5 2
3 4 67 3 4
5 1 5
6 7 23 6 7
7 9 7
8 1 8
9 2 9
7 2 7
8 3 8
9 4 9
9 Treatments (redfaced design): Block 2
Block 1 1 2 1
2 3 2
3 4 3
4 5 4
5 6 5
6 7 6
7 8 7
8 9 8
9 1 9
3 5 3
2 5 2
3 6 3
4 7 4
5 8 5
6 9 6
5 7 5
6 8 6
Block 4
Block 3 1 4 1
4 6 4
7 1 7
8 2 8
9 3 9
1 5 1
2 6 2
3 7 3
4 8 4
5 9 5
6 1 6
USE OF DESIGNS The first step in using these designs is to allot r a n d o m l y the treatments to the numbers used in the pattern. Next, the cows are allotted to sequences. I f there are enough cows available to start all sequences at the same time, the allotmerit to the sequences should be completely random. No attempts should be made to group the cows for similarity prior to allotment nor to balance cow differences across treatments. As was noted, the designs and the method of analysis automatically remove from experimental error the between-cow variation in production level and most of the variation in persistency. Thus, balancing on these
SWITCHBACK TRIALS FOR MORE THAN TWO TREATMENTS
149
factors or on factors associated with them results in no f u r t h e r reduction of experimental error, and a question can be raised as to whether restrictions on allotment reduce the degrees of freedom for error. F o r m i n i m u m experimental errors all cows should have passed the lactation peak before being placed on experiment, a n d no cows should have passed midgestation at the close of the experiment. Often, the n u m b e r of available cows satisfying these conditions is not sufficient to p e r m i t s t a r t i n g all sequences of a design at one time. I n such an event, a blocking procedure can be used. One blocking procedure, usable with either the complete or the reduced designs, is to s t a r t the blocks shown in the p a t t e r n one at the time. The order of s t a r t i n g the blocks and the allotment of the cows to t r e a t m e n t sequences within each block should be at random. A n alternative blocking procedure, usable only with the complete designs, is as follows: D i s r e g a r d the blocks showu in the p a t t e r n s and p a i r off the sequences by treatments. F o r example, the sequence pairs of the 4-treatment design are : Pair I Pair 2 Pair 3 Pair 4 Pair 5 Pair 6 1 2 2 3 3 4 4 1 1 3 2 4 2 1 3 2 4 3 1 4 3 1 4 2 1 2 2 3 3 4 4 1 1 3 2 4 I f necessary, each p a i r m a y serve as a block, but such is not advisable for experiments with few animals because the degrees of freedom for error will be small. Blocks also m a y be formed by combining two or more pairs. The n u m b e r of pairs m a y v a r y f r o m block to block, but p a i r m a t e s should not he split, a n d the allocation of pairs to blocks should be random. O r d i n a r i l y an e x p e r i m e n t e r will kuow the n u m b e r of cows that will be available at a certain time a n d can f o r m the blocks before a n y cows s t a r t on test. F o r example, two cows m i g h t be available on a certain date, six niore cows 3 weeks later, and four more 2 weeks a f t e r that. I t thus would be known that the first block would consist of one sequence pair, the second of three pairs, and the last block of two pairs. The sequence pairs would be randonIly allotted to blocks accordingly, and the cows to sequences within blocks. E v e n with one cow per sequence, a n d a m a x i n m m subdivision into blocks, all of the design p a t t e r n s given in this p a p e r provide degrees of freedom for error. I n the smaller designs, however, the precision of comparison will be relatively low if only one cow is used p e r sequence. This is because there will be few replications and few degrees of freedom for error. I t seems advisable o r d i n a r i l y to use enough cows on each sequence to provide a total of at least 8 or 10 degrees of freedom f o r error. The method of analysis to be outlined permits one or more cows p e r sequence but requires t h a t the m m l b e r of cows be the same on all sequences. THE
STATISTICAL ANALYSIS
Preli+~inary con~putations: The first step of analysis is to compute for each cow, as did B r a n d t , the q u a n t i t y D = Y 1 - 2Y., + Ya
150
H.L. LUC.~S
in which Y~, Y~, and Y3 represent the performances in periods 1, 2, and 3, respectively. The next step is to compute M = sum of the D ' s for all cows on all sequences, and, if the design is blocked, to compute for each block B = stun of the D ' s in the block. The third step is to compute for each treatment Q = stun of the D ' s for the cows receiving the treatment in the first and third periods minus the stun of the D ' s for the coivs receiving the treatment in the second period. TABLE 1 C o m p u t a t i o n s f o r t h e analysis o f ,~'ariance o f the D val~ws Sunl
of
squnres
Variance source
Degrees of freedom
Abbreviation
Correction factor
1
CF
M: 3 n p ( p - 1)
TSS
~/~ X,Z~D~j -°- C F
¼ "2~ B~-".m~
Total ~,cuH v c t e u )
~tp ( p - 1) 2
Blocks (if present)
v- 1
SSB
Treatments
p-
SST
Error Blocks present No blocks
up(p-
1
1
1) - 2 ( p + v) + 2 2
np"- - (u + 2 ) p 2
Computational formulas ~
_
] _ 6np
- CF
"~-
--kQk
2
SSE
TSS - SSB - SST
SSE
TSS - SST
* T h e d i v i s o r 6 is" u s e d to p l a e e t h e s u m s o f s q u a r e s o n a p e r cow p e r p e r i o d b a s i s .
Analysis of variance: The degrees of freedom and the computational formulas for the sums of squares required in the analysis of variance are shown in Table 1. The parameters of the design and the quantities used in the computational formulas are defined as follows: p
= the number of treatments
n
----
v
= the number of blocks in the design
r in a reduced design } where r = the number of cows per 2r in a complete design treatment sequence
m~ = the number of cows in the itth block D~s = the D value for the
jth cow on the i TM treatment sequence
B,, = tile B value for the ~d~' block Qk = the Q value for the k TM treatment.
SWITCHBACK TRIALS FOR MORE THAN TWO TREATMENTS
151
Mean squares m a y be computed a n d the F test f o r t r e a t m e n t s made in the usual fashion. T r e a t m e n t n~eans: The mean for the k '~ t r e a t m e n t is obtained as Y + Q~/2np
in which Y is the g r a n d mean p e r f o r m a n c e in the experiment, i.e., the mean of the original data, not the m e a n of the D ' s . The s t a n d a r d e r r o r of a t r e a t m e n t difference is V3S~'/np, where s ~ : e r r o r m e a n square. A n a l y s i s w h e n there are missb~g or ab~wrmal p e r f o r m a ~ w e s : Occasionally, a cow started on experiment m a y not finish, or her production d u r i n g one or more experimental periods m a y be abnormal owing to accountable factors. W h e n this occurs, it will be necessary to provide a D value for t h a t cow in order to a p p l y the computational f o r m u l a s previously given for the analysis. I f the design is blocked, the missing D value is obtained b y the formula, n p B ' ~ + m~( Q'.~t - Q ' ~ ) n p ( n ~ - 1) -- 2m~
i n which n and p are defined as previously and ?n. s
the n u m b e r of cows intended to be in the block in which the missing value occurs
BP8
the sum of the D ' s for the non-missing cows in the block the Q value for the t r e a t m e n t occurring in the first and t h i r d periods on the missing cow, computed b y letting the missing D be zero
QFSC the Q value for the t r e a t m e n t occurring in the second period on the missing cow, computed by letting the missing D be zero. :If the design is not blocked the f o r m u l a becomes 2M' + (p -- 1) (Q'~I - Q ' ~ ) n p ~ -- ( n + 2 ) p in which M ' = the g r a n d sum of the non-missing D's, a n d the other quantities are defined as before. A COMPUTATIONAL EXAMPLE
Because no data f r o m feeding trials employing the present designs are yet available, u n i f o r m i t y d a t a will be used to illustrate the calculations. The data simulate an experiment c o m p a r i n g three t r e a t m e n t s with 12 cows. To illustrate the blocking feature it is assumed t h a t six of the cows were s t a r t e d at one time with the complete design, and that the r e m a i n i n g six cows were started at later times in two groups of three, each g r o u p being assigned to one block of the
H. L. LUCAS
152
TABLE 2 The data and preliminary computations
Treatment and pounds :FCMproduction per day Cow Block 1: 1 2 3 4 5 6 Sum Block 2 : 7 8 9
Period 1
Period 2
Period 3
(1) (2) (3) (1) (2) (3)
(2) (3) (1) (3) (1) (2)
(1) (2) (3) (1) (2) (3)
34.6 22.8 32.9 48.9 21.8 25.4
28.5 18.6 27.5 42.0 21.7 23.9
-1.5 ~ -0.6 -5.8 -2.9 -4.3 -2.7
186.4
183.2
162.2
-17.8
(1) 30.4 (2) 35.2 (3) 30.8
(3) 29.5 (1) 33.5 (2) 29.3
(1) 26.7 (2) 28.4 (3) 26.4
-1.9 -3.4 -1.4
96.4
92.3
81.5
-6.7
(1) 38.7 (2) 25.7 (3) 21.4
(2) 37.4 (3) 26.1 (1) 22.0
(1) 34.4 (2) 23.4 (3) 19.4
-1.7 -3.1 -3.2
Sum Block 3 : 10 11 12
32.3 21.0 33.1 46.9 23.9 26.0
D Value
Sum
85.8
85.5
77.2
-8.0
Grand sum
368.6
361.0
320.9
-32.5
Treatment 1 2 3
Q Values 8.7 b -4.1 -4.6
Sum
0.0
"-1.5 = 34.6 - 2(32.3) + 28.5. 8.7 = (-1.5 - 2.9 - ~.9 - 1.7) -
(-5.8
- 4.3 - 3.4 - 3.2).
s e q u e n c e p a t t e r n s . The d a t a are s h o w n p u t e d p r i o r to the a n a l y s i s of v a r i a n c e . t r e a t m e n t s , a n d the d a t a are a v e r a g e t h r e e p e r i o d s of 5 weeks each. N o t i n g t h a t p = 3, n = 2r = 2 ( 2 ) , for blocks 1, 2, a n d 3, respectively, the
i n T a b l e 2 a l o n g w i t h the q u a n t i t i e s comThe n m n b e r s i n p a r e n t h e s e s i n d i c a t e the d a i l y p o u n d s of F C S f p r o d u c e d d u r i n g v = 3, a n d m~, ~ , a n d m:~ = 6, 3, a n d 3 a n a l y s i s of v a r i a n c e proceeds as follows:
Correction factor : ( - 3 2 . 5 ) -" 3(2)(2)(3)(2)
= 14.67
T o t a l s u m of s q u a r e s : 1~ [ ( - - 1 . 5 ) 2 + (--0.6)2 + . . .
+ (--3.2)~] -- 14.67 = 3.72
Block s u m of s q u a r e s : J/~
(--17"8)2 (-6"7)2 + 6 ~ 3
- 14.67 = 0.18 3
SWITCHBACK TRIALS FOR MORE THAN TWO TREATMENTS
153
T r e a t m e n t s u m of s q u a r e s : 1 [(8.7)-' + ( - 4 . 1 ) : 6 ( 2 ) (2) (3)
+ ( - 4 . 6 ) '2] = 1.58
E r r o r sunl of s q u a r e s : 3.72 -- 0.18 -- 1.58 = 1.96 These r e s u l t s a r e s m n m a r i z e d in T a b l e 3. TABLE
3
The a~mlysis of variance summary Variance source
D e g r e e s of freedom
Sum of squares
Total Blocks Treatments Error
11 2 2 7
3.72 0.18 1.58 1.96
Mean square 0.34 a 0.09 0.79 0.28
]~
2.82
a O r d i n a r i l y a m e a n s q u a r e is n o t c o m p u t e d f o r the t o t a l v a r i a t i o n , b u t t he c o m p u t a t i o n w a s done h ere b e c a u s e the d a t a a r e u n i f o r m i t y d a t a .
I f all sequences h a d been a s s u m e d to h a v e s t a r t e d t o g e t h e r , t h e b l o c k s s u m of s q u a r e s w o u l d n o t h a v e been c o m p u t e d . The e r r o r s u m of s q u a r e s w o u l d t h e n h a v e b e e n 2.14 w i t h 9 d e g r e e s of f r e e d o m , y i e l d i n g a n e r r o r m e a n s q u a r e of 0.24. T h e f a i l u r e to o b t a i n s i g n i f i c a n t b l o c k or t r e a t m e n t effects is e x p e c t e d , of course, in uniformity data. T h e g r a n d m e a n is 368.6 -t- 361.0 + 320.9 = 29.18 36 The m e a n f o r t r e a t m e n t 1 t h u s is 29.18 +
8.7
2(2)(2)(3)
-- 29.54
a n d t h e m e a n s f o r t r e a t m e n t s 2 a n d 3 t u r n o u t to be 29.01 a n d 28.99, r e s p e c t i v e l y . The s t a n d a r d e r r o r of a t r e a t m e n t d i f f e r e n c e is /r 3(0.28) ~i 2 ( 2 ) ( 3 )
O.27
T h e h i g h s e n s i t i v i t y of t h e s w i t c h - b a c k t r i a l is a t t e s t e d b y t h e coefficient of v a r i a t i o n i n these d a t a . I f t h e d a t a w e r e f r o n i a n a c t u a l e x p e r i m e n t t h e coefficient o f v a r i a t i o n w o u l d be o b t a i n e d as 100 V 0.28 29.18 -- 1.8%
154
H.L. LUCAS
o n t h e p e r cow p e r p e r i o d basis. B e c a u s e n o block a n d t r e a t m e n t effects w e r e r e a l l y p r e s e n t here, t h e coefficient of v a r i a t i o n m a y be c o m p u t e d as 100 ~ / 0 . 3 4 29.18
--
2.07(
S o m e w h a t h i g h e r v a l u e s m i g h t be e x p e c t e d i n a c t u a l e x p e r i m e n t s . SUMMARY
To t a k e b e t t e r a d v a n t a g e of the h i g h s e n s i t i v i t y of s w i t c h - b a c k o r d o u b l e r e v e r s a l t r i a l s c o m m o n l y u s e d to c o m p a r e t w o t r e a t m e n t s a n d w h i c h h a v e been e x t e n d e d to p e r m i t t h e c o m p a r i s o n of t h r e e or m o r e t r e a t m e n t s , c e r t a i n conv e n i e n t a n d u s e f u l f e a t u r e s h a v e been a d d e d . D e s i g n s a r e g i v e n f o r 3, 4, 5, 6, 7, a n d 9 t r e a t m e n t s . The s t a t i s t i c a l a n a l y s i s is o u t l i n e d s y m b o l i c a l l y a n d is i l l u s trated numerically with uniformity data. Missing value formulas are given. ACKNOWLEDGMENT This work was supported in part by Bankhead-Jones funds administered through the U. S~ Department of Agriculture, Bureau of Agricultural Economics. REFERENCES (1) Bm~DT, A. E. Tests of Significance in Reversal or Switch-Back Trials. Iowa Agr. Expt. Sta., Researcl~ B~dt. 234. 1938. (2) SEATIt, D. ~[. A 2 X 2 Factorial Design for Double-Reversal Feeding Experiments. J. Dairy Sol., 27: ]59. 1944. (3) SNEDECOR,G. W. Statistical Metllods. 4th ed. Iowa State College Press, Ames. 1946. (4) TAYLOg~W. B., AND At~MSTRONG,P. J. The Efficiency of Some Experimental Designs Used in Dairy Husbandry Experiments. J. Agr. Sci., 43: 407. 1953.
T R I A L S F O R MORE' T H A N T W O T R E A T M E N T S ~
H. L. LUCAS Depart~e~t of Experimental Statistics, North Carolina State College, Raleigh I t is often desirable to take advantage of the sensitivity of switch-back or double-reversal trials when doing research with d a i r y cattle. Basically, however, these designs p e r m i t the comparison of only two treatments, a n d this has imposed some limitation on their usefulness. The present p a p e r is devoted to switch-back designs t h a t permit comparisons of three or more treatments. The basic switch-back design is as follows :
Experimental period
T r e a t m e n t sequence 1
2
1
1~
2
2 3
2 1
1 2
"~The n u m b e r s in the h e a r t of t he t a b l e r e f e r to the t r e a t m e n t s b e i n g c o m p a r e d .
The t r e a t m e n t p a t t e r n results in p a r t i c u l a r l y sensitive comparisons, because it permits the elimination f r o m error of (a) the period effects due to changes in environment, (b) the between-cow variation in production level, and (c) most of the between-cow variation in slope of the lactation curve. B r a n d t (1) has elucidated an efficient statistical analysis for switch-back trials. This analysis is also described by Sncdecor (.-7). B r a n d t first computes for each cow a q u a n t i t y D = Y1 -- 2Y~ + Y;~ in which Y~, Y~, and Y:~ represent the p e r f o r m a n c e s during periods 1, 2, and 3, respectively. To outline B r a n d t ' s analysis when the n u m b e r of cows on the two sequences is equal, let D~ = p e r f o r m a n c e of the jth cow on the i th sequence (i = 1, 2; j = 1, 2, •
.
.
r)
G~ = sum of the D ' s for the itt~ sequence The estimate of the error variance is given by 1
~
s~ =
G~ + G~o ~D-~ii
6(2) i ; ' - - 1) }
-
r
I
t~eceived for publication July 21, 1955. 1Published with the approval of the Director of Research of the North Carolina Agricul-rural Experiment Station as Puper No. 662 in the Journal Series. 146
SWITCHBACK
TRIALS
FOR MORE THAN TWO TREATMENTS
147
in which 2 ( r -- 1) = degrees of freedom for error. The divisor 6 is introduced to put the mean square on a per cow per period basis. The a d v a n t a g e for t r e a t m e n t 1 over t r e a t m e n t 2 and its s t a n d a r d error are estimated b y G~ - Gz
4r
and
/ 3s 2 ~f 4r
respectively. The divisor 4 is introduced to place the contrasts on a per cow per period basis. An extension of B r a n d t ' s analysis is used in the present paper. Seath (2) devised a four-sequence switch-back design to s t u d y 2 × 2 factorial p a t t e r n s of treatments. The design provides a sensitive test of the two m a i n effects but yields a poor test of the interaction. The designs in the present p a p e r provide equal sensitivity for all t r e a t m e n t contrasts. T a y l o r and A r m s t r o n g (4) have pointed out that the switch-back designs can be extended to a n y n u m b e r of t r e a t m e n t s by setting out the basic switch-back p a t t e r n for each possible p a i r of t r e a t m e n t s and have briefly indicated the method of analysis. The present p a p e r is devoted to designs of the extended type. The analysis is given in detail, and designs of reduced size are discussed along with the introduction of a blocking feature. Missing-value formulas also are given. THE
DESIGNS
The designs combine the switch-back and the balanced incomplete blocks principles. I n general, the comparison of p t r e a t m e n t s requires p ( p -- 1) treatment sequences (complete designs). I f p is odd and 5 or greater, however, designs t h a t require only p (p -- 1 ) / 2 sequences can be used (reduced designs). T r e a t m e n t p a t t e r n s for a selection of complete and reduced designs are given in the following section. Complete designs are shown for 3, 4, and 6 treatments, but only the reduced designs are given for 5, 7, and 9 treatments. The p a t t e r n s for l a r g e r numbers of t r e a t m e n t s are easily derived but p r o b a b l y are of limited usefulness because they require so m a n y cows. F o r each reduced design a conlplementary design is easily obtained b y writing the second row of the reduced design as the first and t h i r d rows of the complement, and the first row of the reduced design as the second row of the complement. The reduced and the c o m p l e m e n t a r y designs together f o r m the complete design. The complete designs are subdivided into p -- 1 blocks of p sequences each, and the reduced designs into (p - 1 ) / 2 blocks of p sequences each. The rows represent experimental periods, and the colunms t r e a t m e n t sequences. TREATMENT PATTERNS 3 T r e a t m e n t s (complete design):
Block I 1 2 3 2 3 1
Block 2 1 2 3 3 1 2
1
1
2
3
2
3
148
H.n. LUCAS
4 Treatments (complete design) : Block 1 1 2 1
2 3 2
Block 3
Block 2
3 4 3
4 1 4
1 3 1
2 4 2
3 1 3
1 4 1
4 2 4
2 1 2
3 2 3
4 3 4
5 Treatments (redfaced design): Block 2
Block 1 1 2 1
2 3 2
3 4 3
4 5 4
1 3 1
5 1 5
2 4 2
3 5 3
4 1 4
5 2 5
6 Treatments (complete design) : Block 1 1 2 1
2 3 2
3 4 3
4 5 4
Block 3
Block 2 5 6 5
6 1 6
1 3 1
2 4 2
3 5 3
4 6 4
5 1 5
6 2 6
2 6 2
3 1 3
4 2 4
2 5 2
5 4 5
6 5 6
3 6 3
4 1 4
5 2 5
6 3 6
Block 5
Block 4 1 5 1
1 4 1
5 3 5
1 6 1
6 4 6
2 1 2
3 2 3
4 3 4
7 Treatments (redfaced desig~) : Block 1 1 2 1
2 3 2
3 4 3
4 5 4
5 6 5
Block 3
Block 2 6 7 6
7 1 7
1 3 1
2 4 2
3 5 3
4 6 4
5 7 5
6 1 6
7 2 7
1 3 1
2 4 2
1 4 1
2 5 2
3 4 67 3 4
5 1 5
6 7 23 6 7
7 9 7
8 1 8
9 2 9
7 2 7
8 3 8
9 4 9
9 Treatments (redfaced design): Block 2
Block 1 1 2 1
2 3 2
3 4 3
4 5 4
5 6 5
6 7 6
7 8 7
8 9 8
9 1 9
3 5 3
2 5 2
3 6 3
4 7 4
5 8 5
6 9 6
5 7 5
6 8 6
Block 4
Block 3 1 4 1
4 6 4
7 1 7
8 2 8
9 3 9
1 5 1
2 6 2
3 7 3
4 8 4
5 9 5
6 1 6
USE OF DESIGNS The first step in using these designs is to allot r a n d o m l y the treatments to the numbers used in the pattern. Next, the cows are allotted to sequences. I f there are enough cows available to start all sequences at the same time, the allotmerit to the sequences should be completely random. No attempts should be made to group the cows for similarity prior to allotment nor to balance cow differences across treatments. As was noted, the designs and the method of analysis automatically remove from experimental error the between-cow variation in production level and most of the variation in persistency. Thus, balancing on these
SWITCHBACK TRIALS FOR MORE THAN TWO TREATMENTS
149
factors or on factors associated with them results in no f u r t h e r reduction of experimental error, and a question can be raised as to whether restrictions on allotment reduce the degrees of freedom for error. F o r m i n i m u m experimental errors all cows should have passed the lactation peak before being placed on experiment, a n d no cows should have passed midgestation at the close of the experiment. Often, the n u m b e r of available cows satisfying these conditions is not sufficient to p e r m i t s t a r t i n g all sequences of a design at one time. I n such an event, a blocking procedure can be used. One blocking procedure, usable with either the complete or the reduced designs, is to s t a r t the blocks shown in the p a t t e r n one at the time. The order of s t a r t i n g the blocks and the allotment of the cows to t r e a t m e n t sequences within each block should be at random. A n alternative blocking procedure, usable only with the complete designs, is as follows: D i s r e g a r d the blocks showu in the p a t t e r n s and p a i r off the sequences by treatments. F o r example, the sequence pairs of the 4-treatment design are : Pair I Pair 2 Pair 3 Pair 4 Pair 5 Pair 6 1 2 2 3 3 4 4 1 1 3 2 4 2 1 3 2 4 3 1 4 3 1 4 2 1 2 2 3 3 4 4 1 1 3 2 4 I f necessary, each p a i r m a y serve as a block, but such is not advisable for experiments with few animals because the degrees of freedom for error will be small. Blocks also m a y be formed by combining two or more pairs. The n u m b e r of pairs m a y v a r y f r o m block to block, but p a i r m a t e s should not he split, a n d the allocation of pairs to blocks should be random. O r d i n a r i l y an e x p e r i m e n t e r will kuow the n u m b e r of cows that will be available at a certain time a n d can f o r m the blocks before a n y cows s t a r t on test. F o r example, two cows m i g h t be available on a certain date, six niore cows 3 weeks later, and four more 2 weeks a f t e r that. I t thus would be known that the first block would consist of one sequence pair, the second of three pairs, and the last block of two pairs. The sequence pairs would be randonIly allotted to blocks accordingly, and the cows to sequences within blocks. E v e n with one cow per sequence, a n d a m a x i n m m subdivision into blocks, all of the design p a t t e r n s given in this p a p e r provide degrees of freedom for error. I n the smaller designs, however, the precision of comparison will be relatively low if only one cow is used p e r sequence. This is because there will be few replications and few degrees of freedom for error. I t seems advisable o r d i n a r i l y to use enough cows on each sequence to provide a total of at least 8 or 10 degrees of freedom f o r error. The method of analysis to be outlined permits one or more cows p e r sequence but requires t h a t the m m l b e r of cows be the same on all sequences. THE
STATISTICAL ANALYSIS
Preli+~inary con~putations: The first step of analysis is to compute for each cow, as did B r a n d t , the q u a n t i t y D = Y 1 - 2Y., + Ya
150
H.L. LUC.~S
in which Y~, Y~, and Y3 represent the performances in periods 1, 2, and 3, respectively. The next step is to compute M = sum of the D ' s for all cows on all sequences, and, if the design is blocked, to compute for each block B = stun of the D ' s in the block. The third step is to compute for each treatment Q = stun of the D ' s for the cows receiving the treatment in the first and third periods minus the stun of the D ' s for the coivs receiving the treatment in the second period. TABLE 1 C o m p u t a t i o n s f o r t h e analysis o f ,~'ariance o f the D val~ws Sunl
of
squnres
Variance source
Degrees of freedom
Abbreviation
Correction factor
1
CF
M: 3 n p ( p - 1)
TSS
~/~ X,Z~D~j -°- C F
¼ "2~ B~-".m~
Total ~,cuH v c t e u )
~tp ( p - 1) 2
Blocks (if present)
v- 1
SSB
Treatments
p-
SST
Error Blocks present No blocks
up(p-
1
1
1) - 2 ( p + v) + 2 2
np"- - (u + 2 ) p 2
Computational formulas ~
_
] _ 6np
- CF
"~-
--kQk
2
SSE
TSS - SSB - SST
SSE
TSS - SST
* T h e d i v i s o r 6 is" u s e d to p l a e e t h e s u m s o f s q u a r e s o n a p e r cow p e r p e r i o d b a s i s .
Analysis of variance: The degrees of freedom and the computational formulas for the sums of squares required in the analysis of variance are shown in Table 1. The parameters of the design and the quantities used in the computational formulas are defined as follows: p
= the number of treatments
n
----
v
= the number of blocks in the design
r in a reduced design } where r = the number of cows per 2r in a complete design treatment sequence
m~ = the number of cows in the itth block D~s = the D value for the
jth cow on the i TM treatment sequence
B,, = tile B value for the ~d~' block Qk = the Q value for the k TM treatment.
SWITCHBACK TRIALS FOR MORE THAN TWO TREATMENTS
151
Mean squares m a y be computed a n d the F test f o r t r e a t m e n t s made in the usual fashion. T r e a t m e n t n~eans: The mean for the k '~ t r e a t m e n t is obtained as Y + Q~/2np
in which Y is the g r a n d mean p e r f o r m a n c e in the experiment, i.e., the mean of the original data, not the m e a n of the D ' s . The s t a n d a r d e r r o r of a t r e a t m e n t difference is V3S~'/np, where s ~ : e r r o r m e a n square. A n a l y s i s w h e n there are missb~g or ab~wrmal p e r f o r m a ~ w e s : Occasionally, a cow started on experiment m a y not finish, or her production d u r i n g one or more experimental periods m a y be abnormal owing to accountable factors. W h e n this occurs, it will be necessary to provide a D value for t h a t cow in order to a p p l y the computational f o r m u l a s previously given for the analysis. I f the design is blocked, the missing D value is obtained b y the formula, n p B ' ~ + m~( Q'.~t - Q ' ~ ) n p ( n ~ - 1) -- 2m~
i n which n and p are defined as previously and ?n. s
the n u m b e r of cows intended to be in the block in which the missing value occurs
BP8
the sum of the D ' s for the non-missing cows in the block the Q value for the t r e a t m e n t occurring in the first and t h i r d periods on the missing cow, computed b y letting the missing D be zero
QFSC the Q value for the t r e a t m e n t occurring in the second period on the missing cow, computed by letting the missing D be zero. :If the design is not blocked the f o r m u l a becomes 2M' + (p -- 1) (Q'~I - Q ' ~ ) n p ~ -- ( n + 2 ) p in which M ' = the g r a n d sum of the non-missing D's, a n d the other quantities are defined as before. A COMPUTATIONAL EXAMPLE
Because no data f r o m feeding trials employing the present designs are yet available, u n i f o r m i t y d a t a will be used to illustrate the calculations. The data simulate an experiment c o m p a r i n g three t r e a t m e n t s with 12 cows. To illustrate the blocking feature it is assumed t h a t six of the cows were s t a r t e d at one time with the complete design, and that the r e m a i n i n g six cows were started at later times in two groups of three, each g r o u p being assigned to one block of the
H. L. LUCAS
152
TABLE 2 The data and preliminary computations
Treatment and pounds :FCMproduction per day Cow Block 1: 1 2 3 4 5 6 Sum Block 2 : 7 8 9
Period 1
Period 2
Period 3
(1) (2) (3) (1) (2) (3)
(2) (3) (1) (3) (1) (2)
(1) (2) (3) (1) (2) (3)
34.6 22.8 32.9 48.9 21.8 25.4
28.5 18.6 27.5 42.0 21.7 23.9
-1.5 ~ -0.6 -5.8 -2.9 -4.3 -2.7
186.4
183.2
162.2
-17.8
(1) 30.4 (2) 35.2 (3) 30.8
(3) 29.5 (1) 33.5 (2) 29.3
(1) 26.7 (2) 28.4 (3) 26.4
-1.9 -3.4 -1.4
96.4
92.3
81.5
-6.7
(1) 38.7 (2) 25.7 (3) 21.4
(2) 37.4 (3) 26.1 (1) 22.0
(1) 34.4 (2) 23.4 (3) 19.4
-1.7 -3.1 -3.2
Sum Block 3 : 10 11 12
32.3 21.0 33.1 46.9 23.9 26.0
D Value
Sum
85.8
85.5
77.2
-8.0
Grand sum
368.6
361.0
320.9
-32.5
Treatment 1 2 3
Q Values 8.7 b -4.1 -4.6
Sum
0.0
"-1.5 = 34.6 - 2(32.3) + 28.5. 8.7 = (-1.5 - 2.9 - ~.9 - 1.7) -
(-5.8
- 4.3 - 3.4 - 3.2).
s e q u e n c e p a t t e r n s . The d a t a are s h o w n p u t e d p r i o r to the a n a l y s i s of v a r i a n c e . t r e a t m e n t s , a n d the d a t a are a v e r a g e t h r e e p e r i o d s of 5 weeks each. N o t i n g t h a t p = 3, n = 2r = 2 ( 2 ) , for blocks 1, 2, a n d 3, respectively, the
i n T a b l e 2 a l o n g w i t h the q u a n t i t i e s comThe n m n b e r s i n p a r e n t h e s e s i n d i c a t e the d a i l y p o u n d s of F C S f p r o d u c e d d u r i n g v = 3, a n d m~, ~ , a n d m:~ = 6, 3, a n d 3 a n a l y s i s of v a r i a n c e proceeds as follows:
Correction factor : ( - 3 2 . 5 ) -" 3(2)(2)(3)(2)
= 14.67
T o t a l s u m of s q u a r e s : 1~ [ ( - - 1 . 5 ) 2 + (--0.6)2 + . . .
+ (--3.2)~] -- 14.67 = 3.72
Block s u m of s q u a r e s : J/~
(--17"8)2 (-6"7)2 + 6 ~ 3
- 14.67 = 0.18 3
SWITCHBACK TRIALS FOR MORE THAN TWO TREATMENTS
153
T r e a t m e n t s u m of s q u a r e s : 1 [(8.7)-' + ( - 4 . 1 ) : 6 ( 2 ) (2) (3)
+ ( - 4 . 6 ) '2] = 1.58
E r r o r sunl of s q u a r e s : 3.72 -- 0.18 -- 1.58 = 1.96 These r e s u l t s a r e s m n m a r i z e d in T a b l e 3. TABLE
3
The a~mlysis of variance summary Variance source
D e g r e e s of freedom
Sum of squares
Total Blocks Treatments Error
11 2 2 7
3.72 0.18 1.58 1.96
Mean square 0.34 a 0.09 0.79 0.28
]~
2.82
a O r d i n a r i l y a m e a n s q u a r e is n o t c o m p u t e d f o r the t o t a l v a r i a t i o n , b u t t he c o m p u t a t i o n w a s done h ere b e c a u s e the d a t a a r e u n i f o r m i t y d a t a .
I f all sequences h a d been a s s u m e d to h a v e s t a r t e d t o g e t h e r , t h e b l o c k s s u m of s q u a r e s w o u l d n o t h a v e been c o m p u t e d . The e r r o r s u m of s q u a r e s w o u l d t h e n h a v e b e e n 2.14 w i t h 9 d e g r e e s of f r e e d o m , y i e l d i n g a n e r r o r m e a n s q u a r e of 0.24. T h e f a i l u r e to o b t a i n s i g n i f i c a n t b l o c k or t r e a t m e n t effects is e x p e c t e d , of course, in uniformity data. T h e g r a n d m e a n is 368.6 -t- 361.0 + 320.9 = 29.18 36 The m e a n f o r t r e a t m e n t 1 t h u s is 29.18 +
8.7
2(2)(2)(3)
-- 29.54
a n d t h e m e a n s f o r t r e a t m e n t s 2 a n d 3 t u r n o u t to be 29.01 a n d 28.99, r e s p e c t i v e l y . The s t a n d a r d e r r o r of a t r e a t m e n t d i f f e r e n c e is /r 3(0.28) ~i 2 ( 2 ) ( 3 )
O.27
T h e h i g h s e n s i t i v i t y of t h e s w i t c h - b a c k t r i a l is a t t e s t e d b y t h e coefficient of v a r i a t i o n i n these d a t a . I f t h e d a t a w e r e f r o n i a n a c t u a l e x p e r i m e n t t h e coefficient o f v a r i a t i o n w o u l d be o b t a i n e d as 100 V 0.28 29.18 -- 1.8%
154
H.L. LUCAS
o n t h e p e r cow p e r p e r i o d basis. B e c a u s e n o block a n d t r e a t m e n t effects w e r e r e a l l y p r e s e n t here, t h e coefficient of v a r i a t i o n m a y be c o m p u t e d as 100 ~ / 0 . 3 4 29.18
--
2.07(
S o m e w h a t h i g h e r v a l u e s m i g h t be e x p e c t e d i n a c t u a l e x p e r i m e n t s . SUMMARY
To t a k e b e t t e r a d v a n t a g e of the h i g h s e n s i t i v i t y of s w i t c h - b a c k o r d o u b l e r e v e r s a l t r i a l s c o m m o n l y u s e d to c o m p a r e t w o t r e a t m e n t s a n d w h i c h h a v e been e x t e n d e d to p e r m i t t h e c o m p a r i s o n of t h r e e or m o r e t r e a t m e n t s , c e r t a i n conv e n i e n t a n d u s e f u l f e a t u r e s h a v e been a d d e d . D e s i g n s a r e g i v e n f o r 3, 4, 5, 6, 7, a n d 9 t r e a t m e n t s . The s t a t i s t i c a l a n a l y s i s is o u t l i n e d s y m b o l i c a l l y a n d is i l l u s trated numerically with uniformity data. Missing value formulas are given. ACKNOWLEDGMENT This work was supported in part by Bankhead-Jones funds administered through the U. S~ Department of Agriculture, Bureau of Agricultural Economics. REFERENCES (1) Bm~DT, A. E. Tests of Significance in Reversal or Switch-Back Trials. Iowa Agr. Expt. Sta., Researcl~ B~dt. 234. 1938. (2) SEATIt, D. ~[. A 2 X 2 Factorial Design for Double-Reversal Feeding Experiments. J. Dairy Sol., 27: ]59. 1944. (3) SNEDECOR,G. W. Statistical Metllods. 4th ed. Iowa State College Press, Ames. 1946. (4) TAYLOg~W. B., AND At~MSTRONG,P. J. The Efficiency of Some Experimental Designs Used in Dairy Husbandry Experiments. J. Agr. Sci., 43: 407. 1953.