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Randomized Complete Block Designs (RCBD)
Defn : A Randomized Complete Block Design is a variant of the completely randomized design that we recently learned. In this design, blocks of experimental units are chosen where the units within are block are more similar to each other (homogeneous) than to units in other blocks. In a complete block design, there are at least t experimental units in each block.
Examples of blocks:
a litter of animals could be considered a block since they all have similar genetic structure, similar prenatal/parental care, etc.
a field or pasture that can be divided into quadrants since soil properties, environmental conditions, etc are similar within a field
a greenhouse with multiple benches since environmental conditions are usually more similar within a greenhouse than between greenhouses
a year in which the experiment is performed since environmental conditions are similar within a year
Example of a CRBD: A nutritionist is interested in comparing the effect of three diets on weight gain in piglets. In order to perform the experiment, the researcher chooses 10 litters, each with at least three healthy and similarly sized piglets that have just been weaned. In each litter, three piglets are selected and one treatment is randomly assigned to each piglet. Diets are labeled A, B or C.
Litter Piglet 1 2 3 1 A C B 2 B C A … 10 C B A
In a design without blocking, the researcher would pick 30 piglets from different litters and randomly assign treatments to them. This is known as unrestricted randomization. Blocking designs have restricted randomization since the treatments are randomly assigned WITHIN each block.
An RCBD has two factors: the factor of interest that includes the treatments to be studied and the “Blocking Factor” that identifies the blocks used in the experiment.
There are several forms of Blocking Designs:
the RCBD that we will study
incomplete block designs in which not every block has t experimental units
block designs in which the blocks have more than t experimental units that are used in the experiment
Latin square designs which have very specific forms of randomization of treatments within blocks (example is usually relates to time ordering of treatments)
Notation
t the number of treatments of interest in the “research” factor
b the number of blocks containing t experimental units
N = t × b , the total sample size
yij observed value for the experimental unit in the jth^ block assigned to the ith^ treatment, j = 1,2,…,b and i = 1,2,…,t
y (^) i •
b
y
b
j
∑ ij
=
1 , the sample mean of the ith^ treatment
y • j
t
y
t
i
∑ ij
= =^1 , the sample mean of the jth^ block
y (^) ••
tb
y
t
i
b
j
∑ ∑ ij
= =
1 1 , the overall sample mean of the combined treatments
Example: piglet diet experiment with three litters
Diet Block Litter A B C Mean 1 yA1 = 54.3 yB1 = 53.1 yC1 = 59.7 (^) y • 1 = 55. 7 2 yA2 = 53.6 yB2 = 52.4 yC2 = 59.7 (^) y • 2 = 55. 2 3 yA3 = 55.2 yB3 = 57.1 yC3 = 67.2 y • 3 = 62. 2
Treatment Mean
y (^) A • = 54. 4 yB (^) • = 55. 2 yC (^) • = 59. 8 Grand Mean y • •= 56. 9
Model:
Yij = μ + α i + β j + ε ij
where
- μ is the overall (grand) mean,
- α i is the effect due to the ith^ treatment,
- β j is the effect due to the jth^ block, and,
- ε ij is the error term where the error terms, are independent observations from an approximately Normal distribution with mean = 0 and constant variance (^) σ ε^2
ANOVA Table for a Randomized Complete Block Design Source Sum of Squares
Degrees of Freedom
Mean Square
F-stat
Treatment SST t – 1 MST F*=MST/MSE Block SSB b – 1 MSB Error SSE (t – 1)(b – 1) MSE Total TSS tb – 1
Again, the test of a treatment effect
H 0 : μ 1 = μ 2 =…= μ t
HA: at least one mean differs
uses the statistic
F*=MST/MSE.
If the null hypothesis is true then F* has an F-Distribution on numerator degrees of freedom t – 1 and denominator degrees of freedom (t – 1)(b – 1).
In addition to the similarity of the F-test of equality of treatment means, the tests and comparisons of treatment means are done exactly the same as before as well.
Example: piglet experiment.
data pigsblocked; input litter diet$ gain @@; datalines; 1 I 54.3 2 I 53. 3 I 55.2 1 II 53. 2 II 52.4 3 II 57. 1 III 59.7 2 III 59. 3 III 67. run; proc glm data=pigsblocked; class diet litter; model gain = diet litter; quit ;
Dependent Variable: gain
Sum of
Source DF Squares Mean Square F Value Pr > F
diet 2 125.39 62.69 19.02 0. litter 2 38.46 19.23 5.83 0.
Error 4 13.18 3. CTotal 8 177.
Same experiment ignoring the litter effect:
proc glm data=pigsblocked; class diet litter; model gain = diet; quit ;
Sum of
Source DF Squares Mean Square F Value Pr > F
Model 2 125.39 62.69 7.28 0. Error 6 51.65 8.
CTotal 8 177.
Least Squares Means Table Level Least Sq Mean Std Error I 54.366667 1. II 54.200000 1. III 62.200000 1.
LSMeans Differences Tukey HSD Alpha= 0.050 Q= 3.
Mean[i]-Mean[j] Std Err Dif
I II III
I 0 0
-7.
II -0.
0 0
III 7.
8
0 0
Level Least Sq Mean III A 62. I B 54. II B 54. Levels not connected by same letter are significantly different
Advantages of the RCBD as compared to the CRD:
- reduce the error variance by “explaining” or identifying one source of some of the variability in the observations a. book refers to this as “filtering” out some of the variation
- the design is easy to construct, i.e. when there are natural or obvious blocks with at least t experimental units, the restricted randomization is easy to achieve
Disadvantages
- need homogeneous blocks in order for the blocking factor to be effective
- the effect of the treatments in the Factor under study must be the same in every block, i.e. the effect of a treatment cannot depend on which block it is being applied to.
e.g. experiment to compare the unused red light time for five different traffic light signal sequences during morning rush hour. Traffic engineer chose several intersections and performed the different sequences at each intersection in random order. Suppose the effect of a particular sequence depends on which intersection you are studying, e.g. in intersections with heavy traffic, the average unused red light time is greater than the average time at intersections with lighter traffic maybe. This is known as interaction of factors.
Choosing Variables On Which To Block:
We want experimental units within each block to be as similar as possible to each other with respect to any characteristic which can effect or influence the response variable (Y). So, if a study relates to weight gain, we want each block to have similar characteristics with respect to growth such as starting weight, metabolic rates, etc.
Example: in the piglet experiment, SSBRCBD = 38.46, SSERCBD = 13.18, t = 3, b = 3, MSERCBD = 3.
RE RCBDCRD =
This implies that it would have taken more than 2.5 times as many experimental units/treatment to get the same MSE as we got using the litters as blocks. I.e. we would have needed approximately 8 (≈ 2.61*3) piglets per treatment in a CRD experiment testing the three diets.
