Single Factor Experiments

Completely Randomized Block Designs (RCBD) is the design in which homogeneous experimental units are combined in a group called a Block. The experimental units are arranged in such a way that a block contains complete set of treatments. However, these designs are not as flexible as those of Completely Randomized Designs (CRD).

Introduction to Randomized Complete Block Designs

A Randomized Complete Block Design (RCBD or a completely randomized block design) is a statistical experimental design used to control variability in an experiment by grouping similar (homogeneous) experimental units into blocks. The main goal is to reduce the impact of known sources of variability (e.g., environmental factors, subject characteristics) that could otherwise obscure the effects of the treatments being tested.

The restriction in RCBD is that a single treatment occurs only once in a single block. These designs are the most frequently used. Mostly RCBD is applied in field experiments. Suppose, a field is distributed in block x treatment experimental units $(N = B \times T)$.

Suppose, there are four Treatments: (A, B, C, D), three Blocks: (Block 1, Block 2, Block 3), and randomization is performed, that is, treatments are randomly assigned within each block.

Key Features of RCBD

The key features of RCBD are:

• Control of Variability: By grouping/blocking similar units into blocks, RCBD isolates the variability due to the blocking factor, allowing for a more precise estimate of the treatment effects.
• Blocks: Experimental units are divided into homogeneous groups called blocks. Each block contains units that are similar to the blocking factor (e.g., soil type, age group, location).
• Randomization: Within each block, treatments are randomly assigned to the experimental units. This ensures that each treatment has an equal chance of being applied to any unit within a block. For example,

In agricultural research, if you are testing the effect of different fertilizers on crop yield, you might block the experimental field based on soil fertility. Each block represents a specific soil fertility level, and within each block, the fertilizers are randomly assigned to plots.

Advantages of Completely Randomized Block Designs

• Improved precision and accuracy in experiments.
• Efficient use of resources by reducing experimental error.
• Flexibility in handling heterogeneous experimental units.

When to Use Completely Randomized Block Designs

CRBD is useful in experiments where there is a known source of variability that can be controlled through grouping/ blocking. The following are some scenarios where CRBD is appropriate:

1. Heterogeneous Experimental Units: When the experimental units are not homogeneous (e.g., different soil types, varying patient health conditions), blocking helps control this variability.
2. Field Experiments: In agriculture, environmental factors like soil type, moisture, or sunlight can vary significantly across a field. Blocking helps account for these variations.
3. Clinical Trials: In medical research, patients may differ in age, gender, or health status. Blocking ensures that these factors do not confound the treatment effects.
4. Industrial Experiments: In manufacturing, machines or operators may introduce variability. Blocking by machine or operator can help isolate the treatment effects.
5. Small Sample Sizes: When the number of experimental units is limited, blocking can improve the precision of the experiment by reducing error variance.

When NOT to Use CRBD

The Completely Randomized Block Design should not be used in the following scenarios:

• If the experimental units are homogeneous, instead of RCBD a CRD may be more appropriate.
• If there are multiple sources of variability that cannot be controlled through blocking, more complex designs like Latin Square or Factorial Designs may be needed.

Common Mistakes to Avoid

• Incorrect blocking or failure to account for key sources of variability.
• Overcomplicating the design with too many blocks or treatments.
• Ignoring assumptions like normality and homogeneity of variance.

Assumptions of CRBD Analysis

1. Normality: The residuals (errors) should be normally distributed.
2. Homogeneity of Variance: The variance of residuals should be constant across treatments and blocks.
3. Additivity: The effects of treatments and blocks should be additive (no interaction between treatments and blocks).

Statistical Analysis of Design

The statistical analysis of a CRBD typically involves Analysis of Variance (ANOVA), which partitions the total variability in the data into components attributable to treatments, blocks, and random error.

Formulate Hypothesis:

$H_0$: All the treatments are equal
$S_1: At least two means are not equal

$H_0$: All the block means are equal
$H_1$: At least two block means are not equal

Partition of the Total Variability:

The total sum of squares (SST) is divided into:

• The sum of Squares due to Treatments (SSTr): Variability due to the treatments.
• The sum of Squares due to Blocks (SSB): Variability due to the blocks.
• The Sum of Squares due to Error (SSE): Unexplained variability (random error).

$$SST=SSTr+SSB+SSESST=SSTr+SSB+SSE$$

Degrees of Freedom

• df Treatments: Number of treatments minus one ($t-1$).
• df Blocks: Number of blocks minus one ($b-1$).
• df Error: $(t-1)(b-1)$.

Compute Mean Squares:

• Mean Square for Treatments (MSTr) = SSTr / df Treatments
• Mean Square for Blocks (MSB) = SSB / df Blocks
• Mean Square for Error (MSE) = SSE / df Error

Perform F-Tests:

• F-Test for Treatments: Compare MSTr to MSE.
  $F=\frac{MSTr}{MSE}$
  ​If the calculated F-value exceeds the critical F-value, reject the null hypothesis.
• F-Test for Blocks: Compare MSB to MSE (optional, depending on the research question).

ANOVA for RCBD and Computing Formulas

Suppose, for a certain problem, we have three blocks and 4 treatments, that is 12 experimental units are analyzed, and the ANOVA table is

SOV	df	SS	MS	F-value	P-value
Block	$b-1 = 2$	20.67	10.33	1.35	0.3285
Treatments	$t-1 = 3$	94.08	31.36	4.09	0.0617
Error	$(b-1)(t-1) = 6$	46.00	7.67
Total	$N-1 = 11$	348.92

\begin{align*}
CF &= \frac{(GT)^2}{N}\\
SS_{Total} &= \sum\limits_{j=1}^t \sum\limits_{i=1}^r y_{ij}^2 – CF\\
SS_{Treat} &= \frac{\sum\limits_{j=1}^t T_j^2}{r} – CF\\
SS_{Block} &= \frac{\sum\limits_{i=1}^b B_i^2}{t} – CF\\
SS_{Error} &= SS_{Total} – SS_{Treat} – SS_{Block}
\end{align*}

Summary

Randomized Complete Block Design is a powerful statistical tool for controlling variability and improving the precision of experiments. By understanding the principles, applications, and statistical analysis of RCBD, researchers, and statisticians can design more efficient and reliable experiments. Whether in agriculture, medicine, or industry, CRBD provides a robust framework for testing hypotheses and drawing meaningful conclusions.

FAQs on Completely Randomized Block Designs

• What is the main purpose of blocking in CRBD?
• Can CRBD be used for small sample sizes?
• How do I choose the right blocking factor?
• What are the assumptions of CRBD?

R Language Frequently Asked Questions

The Latin Square Designs is an effective tool that can simultaneously handle two sources of variation among the treatments, which are treated as two independent blocking criteria. These blocks are known as row-block and column-block, also called double-block. Both sources of variations (blocks) are perpendicular to each other. Latin Square Designs are used to simultaneously eliminate (or control) the two sources of nuisance variability (Rows and Columns).

Introduction

In a Latin square, treatments are arranged in a square matrix such that each treatment appears exactly once in each row and once in each column. This structure helps mitigate the influence of extraneous variables, allowing researchers to focus on the effects of the treatments themselves.

Latin square designs are widely used in agriculture (field experiments), psychology, and many fields where controlled experiments are necessary. The Latin Square Designs are applied in field trials, where

• the experimental area has two fertility gradients running perpendicular to each other
• in the greenhouse experiments in which the experimental pots are arranged in straight lines perpendicular to the sheets or walls of the greenhouse such that the difference between rows and the distance from the wall is expected to be two major extraneous sources of variation,
• in laboratory experiments where the trials are replicated over time such that the difference between the experimental units conducted at the same time and those conducted over different time period constitute the two known sources of variations

	Rows of Tree
Water Channel	A	B	C
	B	C	A
	C	A	B

Key Features of Latin Square Designs

The Latin square designs have the following key features:

• Control for Two Variables: The design simultaneously accounts for variability in two factors (e.g., time and location).
• Efficient Use of Resources: These designs allow for the evaluation of multiple treatments without requiring a full factorial design, which can be resource-intensive.
• Simple Analysis: The data collected can be analyzed using standard statistical techniques such as ANOVA.

Randomization and Layout Plan for Latin Square Designs

Suppose, there are five treatments (A, B, C, D, E) for this we need $5 \times 5$ LS-Designs, which means we should layout the experiment with five rows and five columns:

A	B	C	D	E
B	C	D	E	A
C	D	E	A	B
D	E	A	B	C
E	A	B	C	D

First of all, randomize the row arrangement by using random numbers then randomize the column arrangement by using random numbers. One can generate five random numbers on your calculator or computer. For example,

Random Numbers	Sequence	Rank
628	1	3
846	2	4
475	3	2
902	4	5
452	5	1

The first rank is 3, treatment c is allocated to cell-1 in column-1, then treatment D is allocated to cell-2 of column-1, and so on.

C	D	A	E	B
D	E	B	A	C
B	A	E	C	D
E	C	D	B	A
A	B	C	D	E

Now, generate random numbers for the columns

Random Numbers	Sequence	Rank
792	1	4
032	2	1
947	3	5
293	4	3
196	5	2

For the layout of LS-Designs, the 4th column from the first random generation is used as the 1st column of LS-Designs, then the 1st column as the 2nd of LS-Design, and so on. The complete Design is:

ANOVA Table for Latin Square Designs

For a statistical analysis, the ANOVA table for LS-Designs is used given as follows:

SOV	df	SS	MS	Fcal	F tab/P-value
Rows	$r-1 = 4$
Columns	$c-1 = 4$
Treatments	$t-1 = 4$
Error	$12$
Total	$rc-1 = 24$

Example: An experiment was conducted with three maize varieties and a check variety, the experiment was laid out under Latin Square Designs, Analyse the data given below

	$C$-1	$C$-2	$C$-3	$C$-4	$Total$
$R$-1	1640(B)	1210(D)	1425(C)	1345(A)
$R$-2	1475(C)	1185(A)	1400(D)	1290(B)
$R$-3	1670(A)	710(C)	1665(B)	1180(D)
$R$-4	1565(D)	1290(B)	1655(A)	660(C)
$Total$

Solution:

A	B	C	D
1670	1640	1475	1565
1185	1290	710	1210
1655	1665	1425	1400
1345	1290	660	1180

The following formulas may be used for the computation of Latin Square Design’s ANOVA Table.

\begin{align*}
CF &= \frac{GT^2}{N}\\
SS_{Total} &= \sum\limits_{j=1}^t \sum\limits_{i=1}^r y_{ij}^2 -CF\\
SS_{Treat} &= \frac{\sum\limits_{j=1}}{r} r_j^2 – CF\\
SS_{Rows} &= \frac{\sum\limits_{r=1}^r R_i^2}{t} – CF\\
SS_{Col} &= \frac{\sum\limits_{r=1}^b c_j^2}{t} – CF\\
SS_{Error} &=SS_{Total} – SS_{Treat} – SS_{Rows} – SS_{Col}
\end{align*}

SOV	df	SS	MS	Fcal	F tab (5%)	F tab (1%)
Rows	3	30154.69	10051.56	0.465NS	4.7571	9.7795
Columns	3	827342.19	275780.73	12.769**	4.7571	9.7795
Treatments	3	426842.19	142280.73	6.588*	4.7571	9.7795
Error	6	129584.38	21597.40
Total	15	1413923.44

In summary, the Latin square design is an effective tool for researchers looking to control for variability and conduct efficient, straightforward analyses in their experiments.

Learn about the Introduction of Design of Experiments

MCQs General Knowledge

The article is about one way Analysis of Variance. In the analysis of variance, the total variation in the data of the sample is split up into meaningful components that measure different sources of variation. Each component yields an estimate of the population variance, and these estimates are tested for homogeneity by using the F-distribution.

One Way Classification (Single Factor Experiments)

The classification of observations based on a single criterion or factor is called a one-way classification.

In single factor experiments, independent samples are selected from $k$ populations, each with $n$ observations. For samples, the word treatment is used and each treatment has $n$ repetitions or replications. By treatment, we mean the fertilizers applied to the fields, the varieties of a crop sown, or the temperature and humidity to which an item is subjected in a production process. The collected data consisting of $kn$ observations ($k$ samples of $n$ observations each) can be presented as.

where

$X_{ij}$ is the $i$th observation receiving the $j$th treatment

$X_{\cdot j}=\sum\limits_{i=1}^n X_{ij}$ is the total observations receiving the $j$th treatment

$\overline{X}_{\cdot j}=\frac{X_{\cdot j}}{n}$ is the mean of the observations receiving the $j$th treatment

$X_{\cdot \cdot}=\sum\limits_{i=j}^n X_{\cdot j} = \sum\limits_{j=1}^k \sum\limits_{i=1}^n X_{ij}$ is the total of all observations

$\overline{\overline{X}} = \frac{X_{\cdot \cdot}}{kn}$ is the mean of all observations.

The $k$ treatments are assumed to be homogeneous, and the random samples taken from the same parent population are approximately normal with mean $\mu$ and variance $\sigma^2$.

One Way Analysis of Variance Model

The linear model on which the one way analysis of variance is based is

$$X_{ij} = \mu + \alpha_j + e_{ij}, \quad\quad i=1,2,\cdots, n; \quad j=1,2,\cdots, k$$

Where $X_{ij}$ is the $i$th observation in the $j$th treatment, $\mu$ is the overall mean for all treatments, $\alpha_j$ is the effect of the $j$th treatment, and $e_{ij}$ is the random error associated with the $i$th observation in the $j$th treatment.

The One Way Analysis of Variance model is based on the following assumptions:

• The model assumes that each observation $X_{ij}$ is the sum of three linear components
  • The true mean effect $\mu$
  • The true effect of the $j$th treatment $\alpha_j$
  • The random error associated with the $j$th observation $e_{ij}$
• The observations to which the $k$ treatments are applied are homogeneous.
• Each of the $k$ samples is selected randomly and independently from a normal population with mean $\mu$ and variance $\sigma^2_e$.
• The random error $e_{ij}$ is a normally distributed random variable with $E(e_{ij})=0$ and $Var(e_{ij})=\sigma^2_{ij}$.
• The sum of all $k$ treatments effects must be zero $(\sum\limits_{j=1}^k \alpha_j =0)$.

Suppose you are comparing crop yields that were fertilized with different mixtures. The yield (numerical) is the dependent variable, and fertilizer type (categorical with 3 levels) is the independent variable. ANOVA helps you determine if the fertilizer mixtures have a statistically significant effect on the average yield.

https://rfaqs.com

https://gmstat.com