Download CHAPTER 8. RANDOMIZED COMPLETE BLOCK DESIGN ... and more Study notes Design in PDF only on Docsity!
CHAPTER 8. RANDOMIZED COMPLETE BLOCK DESIGN WITH AND WITHOUT SUBSAMPLES
The randomized complete block design (RCBD) is perhaps the most commonly encountered design that can be analyzed as a two-way AOV. In this design, a set of experimental units is grouped (blocked) in a way that minimizes the variability among the units within groups (blocks). The objective is to keep the experimental error within each block as well as possible. Each block contains a complete set of treatments, therefore differences among blocks are not due to treatments, and this variability can be estimated as a separate source of variation. The removal of an appreciable amount of this source of variation reduces experimental error and improves the ability of the experiment to detect smaller treatment differences. The greater the variability among blocks the more efficient the design becomes. In the absence of appreciable block differences the design is not as efficient as a completely randomized design (CRD). The CRD has more degrees of freedom for error and a smaller F value is required for significant difference among treatments. The paired sample experiment discussed in Chapter 6 is the simplest case of using the concept of blocking, where pairs are blocks.
8.1 Randomized Complete Block Design Without Subsamples
In animal studies, to achieve the uniformity within blocks, animals may be classified on the basis of age, weight, litter size, or other characteristics that will provide a basis for grouping for more uniformity within blocks. For plants in field trials, land is normally laid out in equal- sized blocks, each block being subdivided into as many equal-sized plots as there are treatments to be studied. In general, it is most efficient to have a single replicate of each treatment per block. There may be situations, however, when it is desirable to have more than one replicate per blocks.
Randomization
After experimental units have been grouped into blocks, treatments are assigned randomly within a block, and separate randomizations are made for each block.
To illustrate the randomization and the AOV for a RCBD, consider the layout of the field plots in Figure 8-1. The sugar beet root yield data shown in Figure 8-1 are the same as in Table 7-1 and Figure 7-1. This experiment was actually performed as a RCBD but was analyzed as a CRD in Chapter 7 to provide a basis for comparing the two designs.
The six treatments in each block were randomly assigned to the six plots by drawing random numbers from Appendix Table A-1 in the manner described in Chapter 7. Note in this case that there are only six random numbers (1 - 6) to be drawn for each block, e.g., for block 1 the random sequence was 3, 6, 5, 2, 1, and 4. Assigning treatments A-F to numbers 1-6 results in the block 1 treatment sequence.
Block 1 Block 2 Block 3 Block 4 Block 5 C 1 (40.9)
A 7 (33.4)
B 13 (37.4)
D 19 (40.1)
C 25 (39.8) F 2 (40.6)
D 8 (41.7)
C 14 (39.5)
C 20 (38.6)
D 26 (40.0) E 3 (39.7)
B 9 (37.5)
D 15 (39.4)
E 21 (38.7)
A 27 (33.9) B 4 (38.8)
F 10 (41.0)
E 16 (39.2)
A 22 (32.2)
B 28 (38.4) A 5 (31.3)
E 11 (40.6)
F 17 (41.5)
F 23 (41.1)
E 29 (41.9) D 6 (40.9)
C 12 (39.2)
A 18 (29.2)
B 24 (35.8)
F 30 (39.8)
Figure 8-1. Field plots layout in a RCBD. Plots are numbered in the lower left. Treatments A-F are levels of nitrogen fertilizer from 0 - 250 lbs/acre in 50 lb increments. The number in parenthesis is the root yield per plot in tons/acre.
Analysis of Variance
To proceed with the AOV, the results shown in Figure 8-1 are organized by blocks and treatments in Table 8-1.
Table 8-1. Sugar beet root yield data (tons/acre).
A (0) 31.3 33.4 29.2 32.2 33.9 160.0 32. B (50) 38.8 37.5 37.4 35.8 38.4 187.9 37. C (100) 40.9 39.2 39.5 38.6 39.8 198.0 39. D (150) 40.9 41.7 39.4 40.1 40.0 202.1 40. E (200) 39.7 40.6 39.2 38.7 41.9 200.1 40. F (250) 40.6 41.0 41.5 41.1 39.8 204.0 40. Block (Y.j) Total
232.2 233.4 226.2 226.5 233.8 Y..=1152.
Block ( (^) Y. )j mean
38.70 38.90 37.70 37.75 38.
Y.. = 38 40.
A generalized outline of the AOV for a RCBD is shown in Table 8-2. Our main concern in this design is still to test the equality of treatment means. However, now we can also test for a significant block effect.
SST r Y Y
Y i r C C
= (^) i−
= − = − =
Σ
Σ
(. ..)
./..
2
(^2) 44522 17 277 69
MST = SST/(k-1) = 277.69/5 = 55.
Step 6. Error sum of squares (SEE) and mean square (MSE).
SST = TSS - SSB - SST = 311.13 - 9.44 - 277.69 = 24. MSE = SSE/(k-1) (r-1) = 24/(5) (4) = 1.
MSE represents the variability among experimental units that is not accounted for by any known source of variation. Thus the sum of squares for error is most easily obtained by subtracting the known sources of variation, i.e., blocks and treatments, from the total variation. To gain some insight into the nature of "experimental error" for this design, each observation can be expressed in the following form:
Yij = μ + (μi. - μ) + (μ.j - μ) + εij
where μ is the overall mean (estimated by Y.. = 38 4. ), μI. - μ represents the ith treatment effect (estimated by Yi .− Y..), and μ.j - μ represents the jth block effect (estimated by Y .j − Y..). Thus
the experimental error, εij, is the difference between the observation, Yij and the effects of known sources of variation,
εij = Yij - μ - (μi. - μ) - (μ.j - μ)
= Yij - (μi. + μj - μ)
To illustrate, we will calculate the estimated error component for plot 1 (which is treatment C in block 1),
ε^ $^31 = Y 31 − ( Y 3. + Y. 1 −Y..) = 40.9 - (39.6 + 38.7 - 38.4) = 1.
Performing this calculation for each plot of the experiment will yield the estimated errors. Squaring and summing these errors will result in the sum of squares for experimental error, i.e.,
SSE = ΣΣ( Yij − Yi .− Y .j +Y. .)^2
Step 7. Calculate F values
For blocks: F = MSB/MSE
= 2.36/1.20 = 1.97 with 4 and 20 df.
For treatments: F = MST/MSE
= 55.54/1.
= 46.28 with 5 and 20 df.
The F value for blocks is not significant at the 5% level (Appendix Table A-7), but the F value for treatment is highly significant (P < 0.01) and is considerably larger than the F value obtained when block effects are ignored in the AOV in Chapter 7.
8.2 Design Efficiency
In testing treatment differences, several alternative experimental designs may be used. However, the several designs that may be equally valid for testing treatment effects are rarely equally efficient. Efficiency may be defined in terms of the cost of experimentation, time to collect data, precision of the data obtained, etc. A commonly used index for comparing the efficiency of two different designs is the inverse ratio of the variance pr unit, i.e., the MSE's. Since different designs may have different degrees of freedom for error, a correction factor, suggested by Fisher, which multiplies the inverse ratio of variances will give a better measure of the relative efficiency (RE).
RE (design A to design B) =
( ) ( ) ( ) ( )
df df df df
MSE MSE
A B B A
B A
1 3 1 3
where MSEA is the mean square error obtained from design-A with degrees of freedom dfA, and MSEB is the mean square of design-B with degrees of freedom dfB. If RE>1, design A is more efficient. If RE<1, the converse is true.
If a randomized complete block design (say, design-A) is used, one may want to estimate the relative efficiency compared with a completely randomized design (say, design-B).
This is possible by using the following equation to estimate the MSE of CRD (MSEB) from the information obtained in the AOV of RCBD,
MSEB = [SSBA + SSEA + (k-1)MSEA] / (kr - 1)
where k is the treatment number and r is the number of blocks.
To illustrate, we will use the experiment shown in Figure 8-1 and the results presented in Table 8-3.
MSEB = [9.44 + 24.00 + 5(1.20)] / 29 = 1.
and
RE (RCBD to CRD) ( 100 )
20
36 ( 24 1 )( 20 3 )
( 20 1 )( 24 3 )
=
= (0.986) (1.133) (100) = 111.8%
Thus the estimated RE, 111.8%, implies the RCBD is slightly more efficient than the CRD for this experiment. It should be pointed out that although the F test for block effects is not
Table 8-4. % sucrose for two beet samples per plot from an RCBD.
Blocks Treatment (lb N/acre) Total (Yi ..) Mean 2 2 2 2 2 2 Block (Y.j.) total 182.30 172.3 185.3 185.1 182.8 907.8 = Y... Block (Y.j.) total Block ( (^) Y .j.) 15.19 14.36 15.44 15.43 15.23 (^) 15.13 ~ = ~ Y
- sample Sub-
- A(0) ( Yi ..) - 16. Yij. - 16. - 32. - 16. - 15. - 32. - 15. - 15. - 31. - 16. - 16. - 32. - 16. - 16. - 32. - 161.60 16.
- B(50) - 16. Y2j. - 16. - 32. - 14. - 13. - 28. - 15. - 16. - 32. - 15. - 16. - 31. - 16. - 16. - 32. - 157.40 15.
- C(100) - 15. Y3j. - 15. - 30. - 15. - 14. - 29. - 15. - 16. - 32. - 16. - 15. - 31. - 15. - 14. - 29. - 152.90 15.
- D(150) - 15. Y4j. - 15. - 31. - 14. - 15. - 29. - 15. - 15. - 31. - 15. - 14. - 30. - 15. - 15. - 30. - 152.90 15.
- E(200) - 13. Y5j. - 14. - 27. - 14. - 13. - 27. - 14. - 15. - 29. - 15. - 15. - 30. - 14. - 13. - 28. - 143.60 14.
- F(250) - 14. Y6j. - 13. - 27. - 12. - 12. - 25. - 15. - 14. - 29. - 14. - 14. - 28. - 14. - 14. - 28. - 139.40 13.
Table 8-5. AOV of RCBD with n subsamples.
Source df
Sum of squares
Mean square
Observed F Total (samples) krn-1 TSS Exp. units kr-1 SSU Blocks r-1 SSB MSB MSB/MSE Treatments k-1 SST MST MST/MSE Exp. error (r-1)(k-1) SSE MSE MSE/MSS Sampling error kr(n-1) SSS MSS
The AOV for Table 8-4 is shown in Table 8-6 and the steps for completing this table are given below.
Table 8-6. AOV for the data of Table 8-4.
Source df SS MS F Total (samples) 59 62. Plots (exp. units) 29 55. Blocks 4 9.53 2.38 4. Treatments 5 34.94 6.99 12. Exp. error 20 11.24 0.56 2. Sampling error 30 6.94 0.
Step 1: Degrees of freedom
Degrees of freedom are determined as shown in Table 8-5. For this experiment, treatments (k) = 6, blocks (r) = 5, and samples (n) = 2
Step 2: Correction factor
C = Y^2 .../krn = 907.8^2 /6(5) (2) = 13,735.
Step 3: Total (samples) sum of squares.
TSS Y Y
Yijh C
= (^) ijh−
= −
ΣΣΣ
ΣΣΣ
( ...) 2 2
= 16.5^2 + 16.4^2 + ... + 14.6^2 - 13,735.
= 62.
Step 4: Sum of squares for plots (exp. units).
MSS = SSS/kr (n-1)
= 6.94/6(5) (2-1)
= 0.
Step 9: Calculate F values.
For testing the equality of block effects; F = MSB/MSE = 2.38/0.56 = 4.25, which is nearly significant at the 1% level (F0.01,4,20 = 4.43), indicating that not all block means are equal. Note that the block effect for sucrose content is considerably higher than the block effect for the root yield in Table 8-3, where F = 1.97. Thus, the relative efficiency of the RCBD versus CRD is higher for % sucrose than it was for root yield. As an exercise the reader may wish to calculate the RE.
For testing the equality of treatment effects; F = MST/MSE = 6.99/0.56 = 12.48, which exceeds the 1% tabular value (F0.01,5,20 = 4.10). Thus there are some significant treatment differences. For testing the significance of variability among experimental units treated alike over and above the variability due to sampling units, F = MSE/MSS = 0.56/0.23 = 2.43, which is greater than the 5% tabular value (F0.05,20,30 = 1.93). This indicates the existence of variability from plot to plot within treatments apart from sampling variability.
Note that MSE is used as the denominator in the F tests for blocks and treatments. This is because MSE contains all the random variation due to samples and experimental units. In fact
MSE estimates is the variance component due to samples and the
component due to the experimental units. Both MSB and MST contain these random variations plus additional variation due to block or treatment effects, i.e
σ (^2) s^ + nσ (^2) e where σ^2 s
)
)
2
σ (^2) e
MST MSE (^) s n (^) e s n (^) e
MSB MSE (^) s n (^) e s n (^) e
where rn k and kn r
t
B
g B j
/ ~ ( _ ) / (
/ ~ ( ) / (
( ) / ( ) (. ) / ( )
σ σ δ σ σ
σ σ δ σ σ
δ μ δ μ μ
2 2 2 2
2 2 2 2
1 1
2
2
= − = − −
Σ Σ
Since MSS only estimates is a part of the random variation in the experiment, it should not
be used as the divisor for testing block or treatment effects. For example:
σ (^2) e
MST / MSS ~ ( σ (^2) s + n σ (^) e^2 +δ (^) t) /σs
Now a significant would result in a significant F test for treatment whether or not
there is any real treatment effect. Therefore, replications of experimental units are essential to
σ (^2) e
provide a valid estimate of experimental error for comparisons among treatments and should not be replaced by taking multiple samples from a single experimental unit per treatment.
8.4 The Relationship Between t and F Tests
In section 6.2, we discussed the use of paired samples to compare two treatments. Actually, this is the simplest RCBD and can be analyzed by AOV to yield the same statistical conclusion. To illustrate, the data in Table 6-2 are repeated in Table 8-7. Note that pairs of plots are now called blocks.
Table 8-7. Sugar beet root yield (tons/acre) paired plots (blocks).
Blocks Treatment Treatment (lb N/acre) 1 2 3 4 5
Total Yi.
Mean Yi. 50 38.8 37.6 37.4 35.8 38.4 188.0 37. 100 40.9 39.2 39.5 38.6 39.8 198.0 39. Block total Y.j 79.7 76.8 7.69 74.4 78.2 Y.. = 386. Block total Y.j Block mean Y .j
39.85 38.40 38.45 37.2 39.
Y .. = 38.
Table 8-8. AOV of data in Table 8-7.
Source of variation df
Sum of squares
Mean square F Total 9 18.
Block 4 7.67 1.92 12. Nitrogen 1 10.00 10.00 66. Exp. error 4 0.59 0.
C = 386.0^2 /10 = 14899.
TSS = 38.82^2 + ... + 39.8^2 - C = 18.
SSB = (79.7^2 + ... + 78.2^2 )/2 - C = 7.
SST = (188.0^2 + 198.02^2 )/5 - C = 10.
SSE = TSS - SSB - SST = 0.
curve of Figure 8-3 which indicates a large quantity of nitrogen is needed for maximal yield. A factorial experiment with several levels of nitrogen and phosphorous will reveal the fact that less nitrogen is required in the presence of a higher level of phosphorous.
Even when there is no interaction, factorial experiments have the advantage of enlarging inferences about the main effects of each factor as each is tested over a wider range of conditions. Also, time and materials are saved in comparison to conducting single factor experiments. Consider testing two methods of cultivation on two wheat cultivars; the single factor approach would require twice the total number of experimental units for the same precision in evaluating the main effects as would be provided by a 2x2 factorial experiment.
Figure 8-3. The interaction of nitrogen and phosphorus on crop yield.
2x2 factorial experiment Two single factor experiments
Cultivar Cultivar Method Method A B Ave rep A B rep 1 2 1 Y 11 Y 12 1 1 C 11 C 12 1 M 11 M 12 2 Y 21 Y 22 2 2 C 21 C 22 2 M 21 M 22 Ave 1 2 Ave 1 2 Ave (^1 )
Figure 8-4. A hypothetical 2x2 factorial experiment without replication gives the same precision with respect to the averages of cultivars and methods as obtained from two single factor experiments, each with two replications. Eight experimental units are used in the single factor experiments compared to four in the 2x factorial experiment.
To illustrate the analysis of a factorial experiment, we consider an experiment designed to study the effect of 5 nitrogen levels on winter wheat irrigated one or two times. The
treatments of the experiment are the 10 combinations of 0, 80, 160, 240, and 320 lb N/acre times 1 or 2 irrigations. With 20 plots of a suitable size available for the experiment, the 10 treatments could be completely randomized with 2 replications each, or, as was actually done, the plots can be divided into 2 blocks and the treatments can be assigned randomly within each block. Thus we have an RCBD with 10 treatments and 2 blocks. The field plots along with wheat grain yield are shown in Figure 8-5.
Block I Block II
I 1 N 0 I 2 N 0 I 2 N 240 I 1 N 80 31.7 28.9 72.9 48. I 2 N 160 I 1 N 240 I 2 N 160 I 2 N 320 68.5 73.5 66.7 72. I 1 N 160 I 2 N 80 I 1 N 240 I 1 N 320 56.8 61.4 59.4 54. I 1 N 80 I 2 N 320 I 2 N 0 I 2 N 80 50.0 70.6 40.0 53. I 1 N 240 I 1 N 320 I 1 N 160 N 1 N 0 57.3 53.1 60.6 39. Block Totals Y.. 1 = 561.8 Y.. 2 = 566. Figure 8-5. 10 treatments (5 levels of nitrogen X 2 levels of irrigation) arranged in an RCBD with grain yield data, 100 lbs/acre.
The treatment totals and means are organized in Table 8-9 to facilitate the AOV given in Table 8-10. Calculations are shown below.
C = 1128.6^2 /20 = 63686. TSS = 31.7^2 + ... + 39.1^2 - C = 66624.56 - C = 2937. SBB = (561.8^2 + 566.8^2 )/10 - C = 63688.15 - C = 1. STT = (70.8^2 + ... + 142.9^2 )/2 - C = 66547.98 - C = 2861. SSI = (510.7^2 + 617.9^2 )/10 - C = 64261.49 - C = 574. SSN = (149.7^2 + ... + 250.1^2 )/4 - C = 65850.02 - C = 2163. `SSIxN = SST - SSI - SSN = 2861.08 - 574.59 - 2163.12 = 123. SSE = TSS - SSB - SST = 2937.66 - 1.25 - 2861.08 = 75.
It can be calculated directly by the formula: r is the number of blocks (or the number of replications per treatment combination in a CRD).
SSIxN = r ΣΣ( Yij .− Yi ..− Y. .j+ Y...) 2
- The observed F for interaction (F = 3.68) is significant at the 5% level (F0.05,4,9 = 3.63). This indicates that the trend of the yield response for nitrogen-levels depends on the number of irrigations. It is equivalent to conclude that the difference of yields to irrigations depends on the applied nitrogen-level.
Figure 8-6 illustrates the nature of the interaction. It is obvious that 2-irrigations produced higher yields than 1-irrigation for all levels of N applications. However, the rates of yield increase or yield response curves over different levels of N application are different between the two irrigation schemes. For instance, the required N-level for maximum yield is greater for 2 than 1 irrigations.
SUMMARY
- Blocking is a design technique that is used to remove the known variation among experimental units from the unexplainable random variation. Thus the error mean square can be reduced or the power of detecting treatment differences can be increased.
- A randomized complete block design is a design which controls one criterion of heterogeneity besides the treatments, e.g., given kn experimental units these can be grouped into n blocks of k units each in such a way that the units are as uniform as possible within a block (experimental units may be classified on the basis of age, weight, general vigor, soil fertility, soil moisture, etc. Any one of these characteristics can be used as a criterion for blocking). The treatments are randomized within each block, the randomization being carried out separately for each block. With k treatments in blocks of size k, each treatment occurs once. This constitutes completeness.
- The relative efficiency of design-A to design-B is estimated by:
RE A to B
df df df df
MSE MSE
A B B A
B A
( )
( )( ) ( )( )
=
1 3 1 3
If RE > 1, design-A is more efficient, If RE < 1, the converse is true.
- If the effect of one factor depends on the effect of another factor, they are said to interact. Factorial experiments are usually designed to discover interactions between factors which are different types of treatments.
If there is more than one replication of all treatments in each block, the treatment by block interaction can also be separated from the experimental error term. The term treatment by block interaction means that the differences among treatments change between blocks. Sometimes, blocks may represent soil fertility or soil moisture; other times, they may be the animal's initial body weight, or age, etc. Thus, if a significant treatment by block interaction is found, the interpretation of differences among treatments must be refined within each block. A simple graphic presentation of treatment means for each block will usually clarify the type of interaction.
EXERCISES
- Five judges score 4 products on a 10-point scale. The results are shown in the following table. Analyze the data and make an inference whether there is a significant difference between products or judges. (F=2.59 and 12.97)
Product (or treatment)
Judge (or block) A B C D Total 1 7 10 7 8 32 2 9 10 5 6 30 3 8 8 5 7 28 4 7 8 4 4 23 5 8 9 6 4 27 Total 39 45 27 29 140 Mean 7.8 9.0 5.4 5.
- A plant breeder conducted an experiment to study the inheritance of some agronomic characters in a cross of safflower. Five generations (parents, F 1 , F 2 and backcross) and 3 locations were used in the study. The average number of seeds per head from 5 plants are shown in the following table.
Number of seeds/head of 5 generations of safflower.
Generation/location 1 2 3 Total parent 33.63 30.25 26.32 90. parent 32.39 29.57 28.11 90. F 1 35.86 31.32 29.15 96. F 2 33.92 31.09 28.86 93. Backcross 38.03 35.13 31.52 104. Total 173.83 157.36 1243.96 475.
Can we conclude that there is a difference between generations or locations? (F=22.42 and 83.60)
- In a randomized complete block experiment, four equally spaced levels of nitrogen were applied to a variety of barley. Blocking was based on the level of soil moisture. The yields (kg/plot) are,
N-fertilizer
Block 1 2 3 4 1 4.37 4.50 4.41 4. 2 6.72 8.80 7.82 8. 3 8.32 8.73 8.91 9. 4 8.03 8.31 9.62 9.
Construct the analysis of variance table. Estimate the means and their standard errors for treatments. Estimate the relative efficiency compared with a CRD. (F=65.76 and 3.32) (RE=32.42 RCBD to CRD)
- In a beef-feeding experiment, animals were randomly assigned to each of 4 diets with different percent of protein in 3 feedlots. The average body weight gains (kg/day) for each diet and feedlot are:
Feedlot
Diets 1 2 3 1 0.85 0.93 0. 2 1.03 0.97 1. 3 0.95 0.99 1. 4 1.15 1.23 0.
What is your conclusion on diet effects and feedlot differences? What is the 95% confidence interval of the difference between means of diet 1 and diet 2? (F=1.889 and <1)
- Suppose that a drug company wishes to test the effects of five new compounds on the growth rate of white rats. It is possible that rats within the same litter may have similar response. Hence blocks are defined as litters. Twenty-five rats within each of 4 litters are chosen at random from a large group of rats, and 5 rats of each litter are placed in one pen to be given one of the five treatments. Body weight gains (g/day/animal) on the pen basis are given in the following table after 3 months of feeding the compound.
Compound
Litter 1 2 3 4 5 1 1.45 1.08 1.72 1.04 0. 2 1.39 1.21 1.45 0.79 1. 3 0.86 0.99 1.42 1.01 1. 4 1.04 0.76 0.97 1.05 0.
