Experimental units in a Completely Randomized Design (CRD) are always assumed to be homogeneous . In reality, this is not necessarily true, so other methods are needed that can describe this diversity. If we conduct an experiment on a plot of land that has different fertility levels, then the effect of the treatment that we ascribe to the treatment we are trying may not be correct, thus creating a Type I Error. this land in the RAL will be included in the JKG ( Within) so that the KTG will get bigger and F (KTP/KTG) will get smaller, as a result the experiment is no longer sensitive. Finally, if we repeat the treatment at locations that have different ( non-homogeneous ) variations, then the additional diversity needs to be removed from the analysis so that we focus more on the diversity caused by the treatment we are trying. If the group factor is included in the design, we can capture the variation it causes into the JK Block. This process will reduce SS Within (Error), compared to Completely Randomized Design.
A full discussion of Randomized Complete Block Design (RCBD) can be read in the following document.
Introduction
Randomized Block Design is a randomized design that is carried out by blocking experimental units into homogeneous groups called blocks and then determining the treatment randomly within each block. The Randomized Complete Block Design is a randomized block design with all treatments tried on each existing block. The purpose of blocking the experimental units is to make the variance of the experimental units within each block as small as possible while the differences between blocks are as large as possible. The level of accuracy usually decreases with the increase in units of experiments (sizes of experimental units) per block, so as much as possible make the size of the block as small as possible. The right blocking will produce with a higher degree of accuracy than a complete random design that is comparable in magnitude.
The advantages of Randomized Block Design are:
- More efficient and accurate than CRD
- An effective blocking will decrease the Number of Squared Errors, so it will increase the level of accuracy or can reduce the number of repeats.
- More Flexible.
- The multiplicity of treatments
- Number of tests/blocks
- not all blocks require the same unit of experimentation
- Drawing conclusions is broader, because we can also see differences between blocks
The disadvantages are:
- Requires additional assumptions for some hypothesis tests
- Interactions between Blocks*Treatment is very difficult
- The increase in blocking accuracy will decrease with the increasing number of experimental units in the block
- The block-degree of freedom will decrease the error-degree of freedom, so its sensitivity will decrease especially if the number of treatments is small or the variance in the experimental unit is small (homogeneous).
- Requires an additional understanding of the variance of experimental units for successful blocking.
- if any data is lost requires more complicated calculations.
As outlined above, successful blocking in the RCBD Environmental Design requires an additional understanding of the variance of experimental units. We must be able to identify the direction of the variance, so that the Nuisance factor (disturbing factor) can be minimized. Nuisance factor is any factor / variable outside the treatment that will affect the response. Table 11 presents several disruptive variables that can be used as a reference in blocking.
Table 11. Guidance in identifying factors that can be used as a reference in blocking.
Confounding Variables | Trial units |
Differences in the direction of fertility Differences in the direction of water/moisture content Slope differences Differences in soil composition | Experiment plot |
Direction to the angle of irradiation of the sun Water flow Heat/temperature spread | Greenhouse |
Age Density | Tree |
Gender Age IQ Income Education Attitude | People/Participants |
Observation time Location Experimental Materials Gauges |
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How to Randomize and Plan a Randomized Complete Block Design Experiment
The randomization steps in RCBD are the same as in CRD with blocks as replications. Take a look at the Image below. Blocking is carried out directly against the direction of variance so that the variance in each of the same block is relatively smaller. The experimental areas within each block were divided into numbers corresponding to the number of treatments to be attempted.
Figure 3. Example of blocking experimental plots
Before randomization, divide the experimental area or experiment unit into blocks according to the number of tests. Each block is then subdivided into several plots that correspond to the number of treatments to be tried. Randomization is carried out separately for each block, since in the RCBD the treatment must appear once in each test. For example, experiments with 6 treatments (A, B, C, D, E, F) and 4 blocks. A simpler way to do a draw. Make 6 rolls of paper, then on each paper write one treatment code to be tried from code A to F. Do a no-replacement draw for block I. After finishing the draw for block I, do the same for block II and so on.
Actually, the randomization process will be easier and more practical if we use computer assistance, for example by using Random Numbers (in Microsoft Excel for example by using the RAND() function). The following is given an example of randomization by using MS Excel. The detailed work steps are almost similar to the randomization process on CRD (see the randomization process on CRD using the help of MS Excel).
- Create a Table consisting of 4 columns, No; Treatment; Block; Random Numbers. The Number column is only for reference and is not randomized so it should not be highlighted (Block). The number of treatments and blocks in accordance with the Treatment Plan. For the example case above, the table shape is as in Figure... a. Next Highlight the Treatment, Block, and Random Number Columns, sort by the following hierarchy: First sorting by Block, and second by Random Number (Figure ... b).
Figure 4. How to randomize by using Microsoft Excel
- The result of the randomization looks like in the following Figure: Notice the Block Order is retained, what changes is the Random Order of the Treatment. Place the random Sequence according to the block (or place the Treatment Code based on the Number we have previously created on the Experiment Plan. Be careful, the numbering on the experimental plan should be sorted by block, Nos 1-6 are placed on Block I, 7-12 on block II etc.).
Figure 5. Randomized Complete Block Design Layout
The tabulation of data for a randomized design of the block from the results of randomization above is presented as follows :
Table 12. Tabulation of Data from Experimental Results Using a Randomized Complete Block Design
Treatment (t) | Block (r) | Total Treatment (Yi.) | |||
1 | 2 | 3 | 4 | ||
1 | Y11 | Y12 | Y13 | Y14 | Y1. |
2 | Y21 | Y22 | Y23 | Y24 | Y2. |
3 | Y31 | Y32 | Y33 | Y34 | Y3. |
4 | Y41 | Y42 | Y43 | Y44 | Y4. |
5 | Y51 | Y52 | Y53 | Y54 | Y5. |
6 | Y61 | Y62 | Y63 | Y64 | Y6. |
Total Block (Y.j) | Y.1 | Y.2 | Y.3 | Y.4 | Y.. |
Linear Model of Randomized Complete Block Design
The linear model of RCBD with the multiplicity of blocks (replications ) k and the multiplicity of treatments t is
$$Y_{ij}=\mu+\tau_i+\beta_j+\varepsilon_{ij} \\ \text{i =1, 2, ..., t and j = 1, 2, ..., r}$$
With:
Yij = observations on the i-th and j-th treatment
μ = population mean
τi = additive effect of the i-th treatment
βj = effect of additives of the j-th block
εij = random effect of the i-th and j-th treatment
Assumption:
The effect of fixed treatment | Effect of random treatment |
$E\left(\tau_i\right)=\tau_i\hspace{10mm} \sum_{i=1}^{t}\tau_i=0 \hspace{10mm}\varepsilon_{ij}\overset{bsi}{\sim}N\left(0,\sigma^2\right)$ | $\tau_i \overset{bsi}{\sim}N\left(0,{\sigma_\tau}^2\right)\ ; \beta_j\overset{bsi}{\sim}N\left(0,{\sigma_\beta}^2\right)\ ;\varepsilon_{ij}\overset{bsi}{\sim}N\left(0,\sigma^2\right)$ |
$E\left(\beta_i\right)=\beta_i \ \text{;} \hspace{10mm} \sum_{j=1}^{r}\beta_i=0$ |
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Hypothesis:
Hypotheses that To be Tested: | The effect of fixed treatment | Effect of random treatment |
H0 | All τi = 0 (i = 1, 2, ..., t) | στ2 = 0 (no variance in the treatment population) |
H1 | Not all τi = 0 (i = 1, 2, ..., t) | στ2 > 0 (there is variance in the treatment population) |
Analysis of variance:
Parameters | Estimators |
$\mu$ | $\hat{\mu}=\ Y..$ |
$\tau_i$ | ${\hat{\tau}}_{i\ }\ =\ Yi.-Y..$ |
$\beta_j$ | ${\hat{\beta}}_j\ =\ Y_{.j}-Y..$ |
$\varepsilon_{ij}$ | ${\hat{\varepsilon}}_{ij}=Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+{\overline{Y}}_{..}$ |
The data representation of the linear model Yij = μ + τi + βj + εij is as follows:
$$Y_{ij}={\overline{Y}}_{..}+\left({\overline{Y}}_{i.}-{\overline{Y}}_{..}\right)+\left({\overline{Y}}_{.j}-{\overline{Y}}_{..}\right)+\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+{\overline{Y}}_{..}\right)$$
Its total variance can be described as follows :
$$Y_{ij}={\overline{Y}}_{..}+\left({\overline{Y}}_{i.}-{\overline{Y}}_{..}\right)+\left({\overline{Y}}_{.j}-{\overline{Y}}_{..}\right)+\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+{\overline{Y}}_{..}\right)$$
$$\ Y_{ij}-{\overline{Y}}_{..}=\left({\overline{Y}}_{i.}-{\overline{Y}}_{..}\right)+\left({\overline{Y}}_{.j}-{\overline{Y}}_{..}\right)+\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+{\overline{Y}}_{..}\right)$$
So that the equation Sum Squares becomes:
$$\sum_{i=1}^{t}{\sum_{j=1}^{r}{(Y_{ij}-{\overline{Y}}_{..})^2}=r\sum_{i=1}^{t}{({\overline{Y}}_{i.}-{\overline{Y}}_{..})^2+t\sum_{j=1}^{r}{({\overline{Y}}_{.j}-{\overline{Y}}_{..})^2}+\sum_{i=1}^{t}\sum_{j=1}^{r}{(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+{\overline{Y}}_{..})^2}}}$$
Or: SSTOT = SSB + SST + SSE.
So
Sum of squares of total (SSTOT) = Sum of squares of blocks (SSB) + Sum of squares of treatment (SST) + Sum of squares of errors (SSE)
| Definition | Calculation by hand |
CF | $\frac{Y..^2}{tr}$ | $\frac{Y..^2}{tr}$ |
SSTOT | $\sum_{i=1}\sum_{j=1}{(Y_{ij}-\bar{Y}..)^2}=\sum_{i=1}\sum_{j=1}{Y_{ij}}^2-\frac{Y..^2}{tr}$ | $\sum_{i,j}{Y_{ij}}^2-CF$ |
SSB | $\sum_{i=1}\sum_{j=1}{({\bar{Y}}_{.j}-\bar{Y}..)^2}=\sum_{j}\frac{{Y_{.j}}^2}{t}-\frac{Y..^2}{tr}$ | $\sum_{j}\frac{{Y_{.j}}^2}{t}-CF$ |
SST | $\sum_{i=1} \sum_{j=1}(\bar{Y}i.-\bar{Y}..)^2=\sum_{i=1}\frac{Yi.^2}r-\frac{Y..^2}{tr}$ | $\sum_{i}\frac{{Y_{i.}}^2}{r}-CF$ |
SSE | $\sum_i\sum_j{(Y_{ij}-\bar{Y}_{i.}-\bar{Y}_{.j}+\bar{Y}_{..})}^2=\sum_i\sum_j{e_{ij}}$ | $ SSTOT-SSB-SST$ |
The analysis of variance table for a complete randomized block design with fixed block effects is as follows :
Table 13. Anova Table of Randomized Complete Block Designs with Fixed Block Effect
Sources of Variance (SK) | Sum of Squares (SS) | Degree of freedom (df) | Mean Square (MS) | E(MS) | |
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| Fixed treatment | Random treatment |
Block | SSB | r-1 | MSB | $\sigma^2+[\frac{t}{(r-1)}]\sum{\beta_j^2}$ | $\sigma^2+[\frac{t}{(r-1)}]\sum{\beta_j^2}$ |
Treatment | SST | t-1 | MST | $\sigma^2+[\frac{r}{(t-1)}]\sum{\tau_i^2}$ | $\sigma^2+r{{\sigma^2}_\tau}$ |
Error | SSE | (r-1) (t-1) | MSE | $\sigma^2$ | $\sigma^2$ |
Total | SSTOT | rt-1 |
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The test statistics used for the above tests are with decision rules at a significant level α as follows:
$$F_{stat}=\frac{MST}{MSE}$$
If accept H0: $F_{stat}\le F_{\alpha(db1,db2)}=F_{\alpha(t-1,(r-1)(t-1))}$ and vice versa reject H0. Fα is a broad F value to its right of α.
Sometimes we want to test the effect of the block, but it is usually the treatment that is the main concern , the blocking is carried out as a tool to reduce the variance of experimental errors.
Hypothesis to test the effect of the block :
- H0 : All βj = 0
- H1 : Not all βj = 0
The test statistics for the block's effect testing are with the decision to reject H0 $F_{stat}=\frac{MSB}{MSE}$ if and vice versa $F_{stat}>F_{\alpha(db1,db2)}=F_{\alpha(k-1,(r-1)(t-1))}$
Standard Error
The standard error for the difference among treatment averages is calculated by the following formula:
$$S_{\bar{Y}}=\sqrt{\frac{2MSE}{r}}$$
Blocking Efficiency Versus Complete Random Design
The relative efficiency of blocking compared to complete random plans is expressed as follows:
$$ E=\frac{(db2+1)(db1+3)}{(db2+3)(db1+1)}\frac{{S_a}^2}{MSE}$$
with E indicating how much greater replication is required on a Completely Randomized Design compared to a block design to obtain the sensitivity of a complete random design equal to that of a randomized block design. Whereas db1 states the test error-degree of freedom for the Completely Randomized Design and db2 states the test error-degree of freedom for the block draft, Sa2 states the test error variance estimator for the randomized draft block and the MSE states the error variance estimator for the randomized draft block.
Implementation Example 1
From the results of research on the effect of washing and removing excess moisture by layering or spraying air on the content of ascorbic acid in turnip green plants obtained data in milligrams per 100 gr of dry weight as follows:
Table 14. Turnip Green Data (mg/100gr Dry Weight)
Treatment | Block | Total Treatment | ||||
| 1 | 2 | 3 | 4 | 5 | (Yi.) |
control | 950 | 887 | 897 | 850 | 975 | 4559 |
Washed and wiped | 857 | 1189 | 918 | 968 | 909 | 4841 |
Washed and sprayed with air | 917 | 1072 | 975 | 930 | 954 | 4848 |
Total block (Y.j) | 2724 | 3148 | 2790 | 2748 | 2838 | Y.. = 14248 |
Analysis of variance calculation steps:
Step 1: Calculate the Correction Factor
$$ CF=\frac{Y..^2}{tr}=\frac{14248^2}{(3)(5)}=13533700$$
Step 2: Calculate the Sum of The Total Squares
$$ SSTOT=\sum_{i,j}{Y_{ij}}^2-CF=950^2+857^2+...+954^2-13533700=103216$$
Step 3: Calculate the Sum of Squares of Blocks
$$ SSB=\sum_{j}\frac{{Y_{.j}}^2}{t}-CF=\frac{2724^2+3148^2+...+2838^2}{3}-13533700=25148$$
Step 4: Calculate the Sum of Squares of Treatment
$$ SST=\sum_{i}\frac{{Y_{i.}}^2}{r}-CF=\frac{4559^2+4841^2+4848^2}{5}-13533700=10873$$
Step 5: Calculate the Sum of Squares of Errors
$$ SSE=SSTOT-SSB-SST=67194$$
Step 6: Create Anova Table
Table 15. Anova Table of Turnip Green
Sources of Variance (SK) | Degree of | Sum Squared (SS) | Mean Square (MS) | Fstat | F0.05 | F0.01 |
Block | 4 | 25148 | 6287 | 0.75 | 3.838 | 7.006 |
Treatment | 2 | 10873 | 5436 | 0.65 | 4.459 | 8.649 |
Error | 8 | 67194 | 8399 |
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Total | 14 | 103216 |
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F(0.05,4,8) = 3.838
F(0.0 1,4,8) = 7,006
F(0.05,2,8) = 4,459
F(0.0 1,2,8) = 8,649
Step 7: Make a Conclusion
Since Fstat (0.65) ≤ 4,459 then we fail to reject H0:μ1 = μ2 = μ3 at a confidence level of 95%. This means that at a confidence level of 95%, all treatment averages are no different from others. Or in other words, a decision can be made to accept Ho, meaning that there is no difference in the effect of the treatment on the observed response.
Information:
Usually, an not significant sign (ns) is given, when the F-stat value is less than F(0.05), an asterisk (*) is given, when the F-stat value is greater than F(0.05) and a two-asterisk (**) is given when the F-stat value is greater than F(0.01)
Step 8: Calculate the Coefficient of Variance (CV)
$$CV=\frac{\sqrt{MSE}}{\bar{Y}..}\times100\%=\frac{\sqrt{8399}}{949.867}\times100\%=9.65\%$$
Post-Hoc
Because based on the analysis of variance, the effect of treatment is not significant, it is not necessary to carry out further testing because the average between treatments is not different.
Example 2
The data in the following table are the results of rice (kg / plot) Genotif S-969 which was given 6 treatments. The factors studied were a combination of NPK fertilizers of 6 levels, namely Control, PK, N, NP, NK, NPK.
Table 16. Genotif S-969 Rice Yield Data (kg/plot)
Fertilization Combination | Block | Total Treatment | |||
1 | 2 | 3 | 4 | (Yi.) | |
Control | 27.7 | 33.0 | 26.3 | 37.7 | 124.7 |
PK | 36.6 | 33.8 | 27.0 | 39.0 | 136.4 |
N | 37.4 | 41.2 | 45.4 | 44.6 | 168.6 |
NP | 42.2 | 46.0 | 45.9 | 46.2 | 180.3 |
NK | 39.8 | 39.5 | 40.9 | 44.0 | 164.2 |
NPK | 42.9 | 45.9 | 43.9 | 45.6 | 178.3 |
Total block (Y.j) | 226.6 | 239.4 | 229.4 | 257.1 | 952.5 |
Analysis of variance calculation steps:
Step 1: Calculate the Correction Factor
$$ CF=\frac{Y..^2}{tr}=\frac{952.5^2}{(6)(4)}=37802.3438$$
Step 2: Calculate the Sum of The Total Squares
$$ MS=\sum_{i,j}{Y_{ij}}^2-CF=27.7^2+33.0^2+...+43.9^2+45.6^2-37802.3438=890.42625$$
Step 3: Calculate the Sum of Squares of Blocks
$$ SSB=\sum_{j}\frac{{Y_{.j}}^2}{t}-CF=\frac{226.6^2+239.4^2+229.4^2+257.1^2}{6}-37802.3438=95.1045833$$
Step 4: Calculate the Sum of Squares of Treatment
$$ SST=\sum_{i}\frac{{Y_{i.}}^2}{r}-CF=\frac{124.7^2+136.4^2+168.6^2+...+178.3^2}{4}-37802.3438$$
Step 5: Calculate the Sum of Squares of Errors
$$ SSE=SSTOT-SSB-SST=890.42625-95.1045833-658.06375=137.2579167$$
Step 6: Create an Analysis of Variance Table with its F-Values table
Table 17. ANOVA Table of Rice Yield
Sources of Variance (SK) | Degree of | Sum Squared (SS) | Mean Square (MS) | Fstat | F0.05 | F0.01 |
Block | 3 | 95.1045833 | 31.7015278 | 3.46 * | 3.287 | 5.417 |
Treatment | 5 | 658.06375 | 131.61275 | 14.38 ** | 2.901 | 4.556 |
Error | 15 | 137.257917 | 9.15052778 | - |
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Total | 23 | 890.42625 |
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F(0.05,3,15) = 3,287
F(0.0 1,3,15) = 2,901
F(0.05,5,15) = 5,417
F(0.0 1,5,15) = 4,556
Step 7: Make a Conclusion
Since Fstat (14.39) > 2,901 then we reject H0:μ1 = μ2 = μ3 at a confidence level of 95%. This means that at a confidence level of 95%, there is one or more of the average different treatments from others. Or in other words, a decision can be made to reject Ho, meaning that there is a difference in the effect of treatment on the observed response.
Information:
Usually, not significant sign (ns) is given, when the F-stat value is less than F(0.05), an asterisk (*) is given, when the F-stat value is greater than F(0.05) and a two-asterisk (**) is given when the F-stat value is greater than F(0.01)
Step 8: Calculate the Coefficient of Variance (CV)
$$\begin{matrix}CV=\frac{\sqrt{MSE}}{\bar{Y}..}\times100\%=\frac{\sqrt{9.1505}}{39.688}\times100\%\\=7.62\%\\\end{matrix}$$
Post-Hoc
The step of working on testing the average difference using the Tukey HSD test.
Calculate the value of Tukey HSD ():w
$$\begin{matrix}\omega=q_\alpha(p,\nu)\sqrt{\frac{MSE}{r}}\\=q_{0.05}(6,15)\sqrt{\frac{MSE}{r}}\\=4.595\times\sqrt{\frac{9.1505}{4}}\\=6.95\\\end{matrix}$$
Compare the average of the treatment with the Tukey HSD value ()w
- Sort treatment average (ascending/descending order)
- Create a Matrix Table of differences between treatment averages
- Compare the average difference with the HSD value
- $ if\ \ \left|\mu_i-\mu_j\right|\ \ \left\langle\ \ \begin{matrix}>6.95\ reject\ H_0,\ two\ means\ are\ significantly\ different\\\le 6.95\ accept\ H_0,\ two\ means\ are\ not\ significantly\ different.\\\end{matrix}\right.$
Control | PK | NK | N | NPK | NP | Notation | ||
average | 31.18 | 34.10 | 41.05 | 42.15 | 44.58 | 45.08 |
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Control | 31.18 | 0.00 | a | |||||
PK | 34.10 | 2.93 | 0.00 | a | ||||
NK | 41.05 | 9.88* | 6.95* | 0.00 | b | |||
N | 42.15 | 10.98* | 8.05* | 1.10 | 0.00 | b | ||
NPK | 44.58 | 13.40* | 10.48* | 3.53 | 2.43 | 0.00 | b | |
NP | 45.08 | 13.90* | 10.98* | 4.03 | 2.93 | 0.50 | 0.00 | b |
The result is as follows:
- The average table of treatments is returned in order according to the Order No. of the treatment)
Fertilizer (P) | Average |
Control | 31.18 a |
PK | 34.10 a |
N | 42.15 b |
NP | 45.08 b |
NK | 41.05 b |
NPK | 44.58 b |
Calculation by SmartstatXL Excel Add-In
