THE ESTIMATION OF MISSING VALUES IN INCOMPLETE RANDOMIZED BLOCK EXPERIMENTS

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1 THE ESTIMATION OF MISSING VALUES IN INCOMPLETE RANDOMIZED BLOCK EXPERIMENTS BY E. A. CORNISH Waite Agricultural Research Institute, Xouth Australia D u RIN a recent years Yates has developed the series of experimental arrangements known as quasi-factorial designs. These designs are likely to be of great utility in all experimental work in which large numbers of treatments have to be compared and in which the experimental material contains only limited numbers of units in naturally occurring, homqgeneous groups. A prominent member of this series is the design called incomplete randomized blocks (Yates, 936). The balanced structure of this experimental design confers on it the great practical advantage of a simple analysis, but if, through some unforeseen circumstance, one or more of the observations is rendered unreliable or non-existent, the symmetry is destroyed. The introduction of asymmetry into a design in which confounding is already extensively employed will tend, in some circumstances, to increase the complexity of a general analysis for non-orthogonal data made by the methods given in Yates (933~). In the case of the ordinary randomized block or Latin square, it was found (Allan & Wishart, 930; Yates, 9333) that the most satisfactory method of dealing with the problem presented by the occurrence of missing values was to estimate these values and analyse the completed set of observations. The equations of estimation were obtained by minimizing the residual variance after substituting unknowns for the missing values (Yates, 933b). The same method is applicable in the case of balanced incomplete blocks. It will be found that the formulae are very little more complicated than those appropriate to the ordinary randomized block or Latin square. Throughout this paper the notation first employed in Yates (936) will be used. It is convenient, however, to introduce slight modifications in some of the symbols: Value of a missing observation Total of existing values in the block containing x Total of existing replicates of the treatment containing x Total of blocks containing this treatment other than the block containing x Total of existing values Then &, = k(tx+x)-(bz+x)-aqt. If a replicate of treatment i occurs in the same block as x, then Qi = kti-(r,+x)-si.

2 (Qi E. A. CORNISH 3 SINQLE MISSINQ VALUE (a) Incomplete block When only one value is missing the function to be minimized is - ~ k re x2 - (B, +x)~ + Q4 + Qg +...), where Qi, Qj,... are the Q s of the remaining (k- ) treatments in the same block as the missing value. This expression is a minimum for variations in x, when t- X = N-b-t+l p x + T j {k(k - ) T, - (k - ) x, - k (q +!q +... ) + xi + 8, +...} f (b) Youden square In the Youden (937) square, b = t and k = r. If R, stands for the total of existing values in the row containing x, the formula for a single missing value is X= (b-l)(r-2) SEVERAL MISSING VALUES When more than one value is missing there will be two or more unknowns and the process of minimizing the residual sum of squares will yield as many equations as there are unknowns, analogous to the simultaneous equations of partial regression. The form assumed by these equations, however, depends on the structural relationships between the treatments and blocks from which the values are missing. Consider the case of the ordinary incomplete block and denote the missing observations by x, y,... If the first two values, x and y, belong to Werent blocks but are both replicates of the same treatment, and assuming these classes involve no other missing value, the quadratic function, in so far as it involves x and y, is of the form x2+yy2+...-~{(b,+x)~+(b~+~)~+...}-kare{q~,+&q+q4+...+q~+q~+...}, where Qi, Q,,... are the Q functions of the (k - ) remaining treatments in the block containing x and Q,, Qm,... are similar functions for the treatments in the block containing y. The first two equations in this case are Ax-By+... = bb,+---- (t- ) k(k - ) x {k(k - ) Tz, - (k - ) s,, - (k - ) B, - k( q +!q +... ) + si + 8, +...}, -Bx+Ay+... =bb,+---- (t-) k(k- ) x{k(k-)tx,-(k-)s,,-(k-)b,-k(t,+t,,+...) +X,+X,+...}, EUGENICS 0, I 8

3 4 RANDOMIZED BLOCK EXPERIMENTS where and A = (N-b-t+l) B =--(k-l)(t-). k The labour of deriving these equations and solving them can be avoided by repeated applicstion of the formula for a single missing value, substituting approximations for the remaining missing values (Yates, 933b). If care is taken in selecting the approximations the solutions converge rapidly. A special case is worth noting. If the second missing value is neither a member of the same block, nor occurs in a block containing a replicate of any of the k treatments in the block containing the first missing value, then each equation involves only its leading unknown. Each equation thus degenerates to the formula for a single missing value, which can be solved directly. If a third value is missing and account is taken of the first two values, each equation will involve only its appropriate unknown; similarly for the fourth and so on. As an example, consider the arrangement given in Table. I a@) a a a b f d e C h 9 i ' I $ b I0 I 2 _ - ~ ~ _ ~ ~ _ ~ _ b C C c d(y) 9(hz) d d e f e i I h fy h 9 i f i Suppose the missing values are those corresponding to treatments a, d and g in blocks, and 2 respectively (they might equally have belonged to the remaining treatments of these blocks). Clearly, Q, does not involve the totals of blocks and 2, and therefore the equation with x as leading term involves neither y nor z; similarly for Qd and Qg. Hence, in this particular case, as many as three missing values can be estimated directly. Example. Fisher & Yates (938) have given an example showing the analysis of an incomplete block experiment involving nine treatments in eighteen blocks of four units each. Table 2, quoted from their book, gives the scores; x, y and z stand for the observations assumed missing and a, 6,..., i for the treatments. Table 3 gives the additional data required for estimating the missing values. Using the direct method of solution, the quadratic function to be minimized is x2+ y2 + z2 - i((2.8 + x )~ + ( Y ) + ~ (4.8 + z )~} -- [{4(36-9 +x) - (2.8 +x) - ( y) } {4(3.9+~)-(4.8+~)- (2*8+~)- 5*2}2].

4 ~ 5 f 2.6 a x i 2.4 i 5'0 d 0. 3'9 h 4.0 b 2.8 d 9'7 f 4.6 d 4.0 h 7'4 f 6. f 2.6 5' '0 e 0.3 f Y e '4 e 2.8 e 6.9 c 3'3 f '4 h 7.5 i 6.3 c 3'3 h 3' z '5 ; 5'7 b 4'7 a 3.0 c 7'5 c 3'7 i 3.0 g 2.6 b 7'3 9'3 g 6.6 h.4 g 2'2 a 5.2 g 2.6 e 4'9 e 5'4 5'4 a 5'5 z 4-2 e 2.6 d 2.4 e 4'7 d 6.0 f 5'7 i 6. h 5'3 d 2.8 a 4'4 b 2.4 a 2.4 h 4.6 a 4' ' ' '7 I bl ' f 36.;+Z IG- 9., lp<.r 29'9+y :*8 99' I x 27.3 ~ + 8 [2 x ~ 25*3-4(43.6+4*2+42-8) 46 4x3~46 a b C d e f 9 h i Total ~~ ~ Q o

5 ~ ' ~ G RANDOMIZED BLOCK EXPERIMENTS Example 2. Table 4 gives a set of estimates of the percentage of digestible fibre in a series of dietary treatments fed to sheep (A. E. Scott, unpublished data). The design consists of nine treatments denoted by a, b,..., i, in twelve blocks of three units each; x, y and z stand for the values assumed missing. 9 2 h 63. i z b 65'4 e 64.8 h d Y e 63.9 f 67.8 I3I'7+Y c 69.2 e 64.3 g '7 a x b 66.2 c 7. ~. 37'3 +g b 66.6 f 67.2 g 6.0 I948 a 6.0 f 67.0 h 62.9 b 66.0 d 6. i c 69. f 67'3 i ~ 202'2 c 69.9 d 62.0 h '5 a 58-8 e 63'4 i Table 5 gives the additional data required for the estimation of the missing values. Table 5. Treatment a b C d e f 9 h i T 76.7+z o+y '4 567'2 575' 587' '5 58'5 X = Applying the formula for a single missing value, 2 x '3x2~6 similarly y = 62.2 and z = 6.3. [6 x x ( ) = 59.9; TESTS OF SIGNIFICANCE The steps in fahe completion of the analysis and the test of significance of the treatment mean square proceed in exactly the same manner as described by Pates (9336). The calculations needed to complete the analysis of the data of example, above, are given in Tables 6 and 7. I Table 6. Analysis of completed values (test for treatment) Total Residual Block + treatment constants Blocks Treatment Degrees of freedom I Sum squares 324'9787 I I ~062 2'307 Mean squares

6 E. A. CORNISH 7 Table 7. Analysis of existing values (test for treatment) Total Residual Block + treatment, constants Blocks Remainder Degrees of freedom. Sum squares I Mean squares Note the reduction (0.797) in the sum of squares for testing treatments. It can be shown that the correct mean square for testing treatments is always less than the treatment mean square in the analysis of the completed values. Yates ( 933 b) has given aformula for evaluating the corresponding reduction in the case of the ordinary randomized block. A similar formula is applicable in the present case. Denote the estimated yield of a missing value in a block containing only one such value by a and the block total including this value by V,. If the corresponding quantities in a block containing two missing values are b,, b, and V,, and so on, the formula is &(v, - kay+ 2k(k - 2) &{2&- k(h,+ b2)}2 + +S,(b, - b,)2+ k(k-.... ) S,, S,,... denote summation over all blocks containing one, two or more missing values. In the majority of cases a test of significance for blocks is not necessary. If required, it is made in a similar manner to the test for treatment. In this case, also, the correct mean square is always less than the.block mean square in the analysis of the completed set of observations. The necessary reduction is calculable by a formula similar to that above. V will now stand for a treatment total and r is substituted for k. In the example given, this reduction is , giving a corrected mean square of In the Youden square with a single missing value the reduction in the sum of squares of treatments is given by the expression (hb, + rr, - G, - bra),, hr(h - ) (r- ) where a is the estimate of the missing value, B, and R, are the totals of the corresponding block and row including the estimate, and G, is the grand total including the estimate. The exact analysis of the Youden square with more than one missing value is made in a similar manner to the analysis given by Yates (933 h) for the ordinary Latin square. ERRORS OF TREATMENT DIFFERENCES The variance of the difference between treatments whose Q's do not involve estimates of the missing values is 2s2/rE. If' exact values of the variances of other treatment comparisons are required they can be obtained by the method given in Fisher (936, a 49.).

7 8 RANDOMIZED BLOCK EXPERIMENTS Slight modifications have to be introduced owing to the presence of redundant constants (Yates & Hale, 939). This procedure involves lengthy computations which will, in general, not be worth while. If only a few values are missing and the exact test of significance is made in the analysis of variance, the use of the formula 2s2/rE will not result in serious error. REFERENCES F. E. ALLAN & J. WISHART (930). A method of estimating the yield of a missing plot in field experimental work. J. Agric. Sci. 20, R. A. FISHER (936). Statlstical MeUlods for Research Workers, 6th ed. Edinburgh: Oliver and Boyd. R. A. FISHER & F. YATES (938). Statistid Tables for Biological, Agricultural and Medical Research. Edinburgh: Oliver and Boyd. F. YATES (933~). The principles of orthogonality and confounding in replicated experiments. J. Agric. Sci. 23, (933b). The analysis of replicated experiments when the field results are incomplete. Emp. J. E~J. Agric., (936). Incomplete randomized blocks. Ann. Eugen., Lond., 7, F. YATES & R. W. HALE (939). The analysis of Latin squares when two or more rows, columns or treatments are missing. Supp. J.R. Slat. SOC. 6, W. J. YOUDEN (937). TJse of incomplete block replicaticins in estimating tobacco moseic vim. Contr. Boyce Thompson Inat. 9,4-8.

Construction of Partially Balanced Incomplete Block Designs

International Journal of Statistics and Systems ISS 0973-675 Volume, umber (06), pp. 67-76 Research India Publications http://www.ripublication.com Construction of Partially Balanced Incomplete Block Designs