4.5. Blocking

In order to remove the additional variability in the data due to other external sources, blocking is often desired. Likewise, if the external variability is present due to two independent sources or is present in two orthogonal directions, two-way blocking using the Latin square design is often used. If the data are available for all the cells in the Latin square, the orthogonality of the two blocking variables and the treatment is automatically accomplished. Hence the Type I, II, III, and IV analyses are identical. A problem, however, occurs if the data are not available for certain cells or the blocks are of unequal size. The question in that case is, which analysis is appropriate? As the treatments are to be compared only after eliminating the effects due to blocking variables, the Type I analysis is clearly the appropriate choice with the treatment variables listed after the blocking variables (in an appropriate sequence) in the MODEL statement of PROC GLM. As an illustration, see the following example, where data are collected under a Latin square design setup for all the cells except two.

EXAMPLE 5

Experiments in Blocks, Comparison of Corn Varieties We consider a part of the data from Srivastava and Carter (1983, p. 107), where a certain area of land was used for testing four varieties of corn represented by four levels of the variable VARIETY. Due to the slope of the land, differences from north to south (NS) and from east to west (EW) were possible. As a result, the experiment was conducted using a Latin square design with the corresponding layout given in Table 4.3.

Note that for our analysis the experiments corresponding to A₃B₂ and A₄B₄ are not included in Table 4.3, so the design is unbalanced. For each experiment, two characteristics, namely the height of the plant (HEIGHT) and the yield (YIELD), were measured. Thus, the additive bivariate model (p = 2) containing the variable VARIETY as well as the two blocking variables NS and EW will be fitted for the response variables HEIGHT and YIELD.

As indicated earlier, sequential MANOVA partitioning (Type I) is used to adjust the treatment SS&CP matrix for the block effects. The corresponding MODEL statement is

model height yield = ew ns variety/ss1;

An alternative is to use the SS3 option instead of SS1 to get all the SS&CP matrices adjusted for all the remaining terms in the right side of the MODEL statement. However, if in the MODEL statement given above, VARIETY was specified before NS and EW, then SS3 and not SS1 would be the correct option since under the SS1 option, the SS&CP matrix for variety would be adjusted only for the intercept.

Therefore, since the MODEL statement in Program 4.5 lists VARIETY first, we have chosen the Type III analysis and specified the SS3 option. The multivariate tests based on Type III analysis (that is, after adjusting for the two directions in the Latin squares, in this case) are presented as Output 4.5. Based on any of the four multivariate tests, there does not appear to be significant difference between the four varieties; the p value for Wilks' Λ is 0.2638. If there were indeed a significant difference, it would also have been interesting to perform the pairwise comparisons of the four varieties. This can be done using the CONTRAST statement. For instance, if we wanted to compare the varieties C₂ and C₄, then the corresponding SAS statement, which should follow the MODEL statement but must precede the MANOVA statement, is

contrast  'c2 vs. c4' variety 0 1 0 -1;

For the optional phrase C₂ versus C₄ enclosed by single quotation marks (' '), we could have used any other alternative identifier appropriately indicating the type of contrast.

/* Program 4.5 */

options ls=64 ps=45 nodate nonumber;
    data corn1;
    input ew  $ ns $ variety  $ height yield;
    lines ;
    a1 b1 c2 65 24
    a1 b2 c3 66 20
    a1 b3 c1 65 24
    a1 b4 c4 65 26
    a2 b1 c3 68 21
    a2 b2 c2 63 23
    a2 b3 c4 67 25
    a2 b4 c1 64 25

a3 b1 c4 67 26
    a3 b3 c2 64 19
    a3 b4 c3 64 25
    a4 b1 c1 68 27
    a4 b2 c4 67 24
    a4 b3 c3 63 20
    ;
    /* Source: Srivastava and Carter (1983, p. 109). */

    title1 'Output 4.5';
    title2 "Latin Square Design: Corn Yield and Plant Height";
    proc glm data = corn1;
    class ew ns variety;
    model height yield = variety ew ns/ss3 nouni;
    manova h =variety/printe printh;
    run;

Example 4.5. Output 4.5

Latin Square Design: Corn Yield and Plant Height



         Manova Test Criteria and F Approximations for
          the Hypothesis of no Overall VARIETY Effect
 H = Type III SS&CP Matrix for VARIETY   E = Error SS&CP Matrix

                      S=2    M=0    N=0.5

Statistic               Value        F    Num DF  Den DF  Pr > F

Wilks' Lambda          0.135447   1.7172       6       6  0.2638
Pillai's Trace         1.166323   1.8653       6       8  0.2033
Hotelling-Lawley Trace 4.154999    1.385       6       4  0.3925
Roy's Greatest Root    3.522509   4.6967       3       4  0.0846

  NOTE: F Statistic for Roy's Greatest Root is an upper bound.
         NOTE: F Statistic for Wilks' Lambda is exact.

Since the pairwise comparisons are not meaningful in the present context of nonsignificant effect on VARIETY, we do not pursue this analysis further for this example.