VI. Incomplete Block Designs A. Introduction What is the purpose of block designs? Blocks are formed by grouping EUs in what way? How are experimental units randomized to treatments? 550
What if we have more treatment levels than there are EUs in a block? Incomplete block designs retain the desired number of blocking factors, but do not allocate all treatments to every block. 551
If we allocate treatments to blocks cleverly, then we can still estimate all effects of interest. If we do not do this cleverly, then we can have confounded effects: the effects of one factor cannot be estimated separately from the effects of another factor. EX. In a two-way ANOVA with no replication, what two effects are confounded? 552
EXAMPLE: Four assays for determining HIV-RNA in blood samples are to be compared. Four blood samples are available and can be split into at most three parts. Each part is then analyzed with one assay. Blocks = Treatments = 553
Such a design might look like: 1 A C D Block 2 A B C 3 B C D 4 A B D t = b = # treatments per block = 554
EXAMPLE: Food science researchers are interested in the hardiness of the E. coli bacterium during various storage conditions. 2 refrigeration units are available, but the researchers would like to study 4 refrigeration termperatures: 5, 10, 15, and 20 C. Each day for 6 days the RA will prepare two samples of E. coli in growth medium. After 24 hours, the increase in E. coli will be measured. Blocks = Treatments = 555
Such a design might look like: 1 A D 2 B D Block 3 A B 4 C D 5 A C 6 B C t = b = # treatments per block = 556
B. Balanced Incomplete Block Designs: One Blocking Factor A BIBD allocates EUs such that all treatments are allocated to equally many EUs each treatment pair occurs in the same block an equal number of times across blocks. 557
Notation t = #treatments b = #blocks k = #EUs per block r = #replications of each trt N = total #EUs = = k < t r < b 558
In terms of t, b, k, and r, all treatments allocated to equally many EUs means what about r? 559
In terms of t, b, k, and r, each treatment pair occurs in the same block an equal number of times across blocks means what? Consider for example treatment A. Treatment A occurs in the same block as treatment B for λ of the b blocks, λ < b. Thus, λ(t 1) = 560
In terms of t, b, k, and r, We also know that treatment A occurs a total of r times, and that when it occurs there are k 1 other treatments in that same block. Thus, r(k 1) = 561
In terms of t, b, k, and r, λ(t 1) = r(k 1) λ = 562
What are t, b, r, k, and λ? Is λ an integer? EX. 1 B C D E 2 A B D E Block 3 A C D E 4 A B C D 5 A B C E 563
EX. 1 A B C Block 2 C D E 3 B E F 4 A B D 5 C D F 6 A E F 564
Randomization determine t, b, r, and k determine how the t treatment codes should be divided among the b blocks 565
e.g., for t = 4, b = 4, r = 3, and k = 3, use (A,B,C) (A,B,D) (A,C,D) (B,C,D) randomly assign each grouping to one block 566
within block, randomly assign treatment codes (e.g., A) to EUs randomly assign treatment codes to the actual treatment levels, e.g., A = Assay 4 B = Assay 1 C = Assay 3 D = Assay 2 567
Model Y ij = µ + τ i + ρ j + e ij e ij iid N(0, σ 2 e ) i = 1,..., t, j = 1,..., b but not every treatment appears in every block!! Thus treatments and blocks are no longer orthogonal. 568
Consequences: The two-way ANOVA table sums of squares are not correct: the model SS can no longer be partitioned into treatment SS and block SS. Observed treatment means Ȳ i are no longer unbiased estimates of µ i = µ + τ i. 569
Why the lack of orthogonality? Suppose we had: 1 A C D Block 2 A B C 3 B C D 4 A B D Comparing Block 1 to Block 4 is both a block-to-block comparison and a comparison of... 570
ANOVA table: No easy SS formulas. We use the regression approach. Tests: Use Type III sums of squares or General Linear F-tests. Estimation: Use least squares means for pairwise comparisons and other contrasts. Diagnostics: As before. 571
C. BIBD: Two Blocking Factors With more than one blocking factor, we could have incomplete blocks for either or both of the factors. If both blocking factors are incomplete, then we have a design similar to a Latin square but without the restrictions of a Latin square that t = b = r = k. 572
Row orthogonal designs These have complete row blocks and incomplete column blocks. They can be formed by taking a Latin square and omitting one or more rows. These are called Youden squares. This does not work for all Latin squares! It only works if you end up with an integer λ after the omission. 573
EXAMPLE: Certain fungi can disrupt proper growth of fingernails and toenails. Researchers wish to study a new topical cream with 7 levels of the active ingredient. They will recruit 7 participants and randomly allocate one treatment to each hand and to each foot of each participant in a Youden square design. Blocks = Treatments = 574
Block (participant) 1 2 3 4 5 6 7 1 A B C D E F G Block 2 B C D E F G A (hand or foot) 3 C D E F G A B 4 D E F G A B C 5 E F G A B C D 6 F G A B C D E 7 G A B C D E F How do we get from this Latin square to a Youden square? 575
EXAMPLE: Researchers are interested in collecting information on sexual practices. Since this is sensitive information, the survey questions must be designed carefully. They design five different surveys and will do a pilot study to determine which version has the highest response rate. They will block on age and on region of the country. One participant per block will answer the survey, and percent of questions answered will be recorded. 576
Block (age group) 1 2 3 4 5 1 A B C D E Block 2 B C D E A (region) 3 C D E A B 4 D E A B C 5 E A B C D 577
Model Y ijk = µ + τ i + ρ j + γ k + e ijk e ijk iid N(0, σ 2 e ) i = 1,..., t, j = 1,..., k, k = 1,..., b where ρ j are the row block effects and γ k are the column block effects. Treatments are orthogonal to but treatments are not orthogonal to 578
ANOVA table: No easy SS formulas. We use the regression approach. Tests: Use Type III sums of squares or General Linear F-tests. Estimation: Use least squares means for pairwise comparisons and other contrasts. Diagnostics: As before. 579
D. Other Incomplete Block Designs Other designs are possible: partially balanced incomplete block designs: when balance (λ = integer) is not possible for the desired t and/or b. Some treatment pairs will occur together more frequently than others. 580
resolvable block designs: blocking levels are grouped such that each group contains one replication of each treatment. These are useful for example when the grouping is done by time because a complete experiment cannot be conducted e.g. on one day. 581
factorial treatment structures within blocks: blocks are often not big enough to accommodate a full factorial structure so an incomplete block design is used. These designs often require some confounding, e.g., the block effect cannot be estimated separately from the highest order interaction among treatment factors. 582