Latin Square Designs

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 ChannelABC
BCA
CAB

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:

ABCDE
BCDEA
CDEAB
DEABC
EABCD

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 NumbersSequenceRank
62813
84624
47532
90245
45251

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.

CDAEB
DEBAC
BAECD
ECDBA
ABCDE

Now, generate random numbers for the columns

Random NumbersSequenceRank
79214
03221
94735
29343
19652

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:

Latin Square Designs

ANOVA Table for Latin Square Designs

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

SOVdfSSMSFcalF 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$Total$
$R$-11640(B)1210(D)1425(C)1345(A) 
$R$-21475(C)1185(A)1400(D)1290(B) 
$R$-31670(A)710(C)1665(B)1180(D) 
$R$-41565(D)1290(B)1655(A)660(C) 
$Total$     

Solution:

ABCD
1670164014751565
118512907101210
1655166514251400
134512906601180
    

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*}

SOVdfSSMSFcalF tab (5%)F tab (1%)
Rows330154.6910051.560.465NS4.75719.7795
Columns3827342.19275780.7312.769**4.75719.7795
Treatments3426842.19142280.736.588*4.75719.7795
Error6129584.3821597.40   
Total151413923.44    

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.

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MCQs General Knowledge

Design of Experiments Quiz 8

Online Quiz about Design of Experiments Quiz Questions with Answers. There are 20 MCQs in this DOE Quiz cover the basics of the design of experiments, hypothesis testing, basic principles, and single-factor experiments, fixed effect models, random effect models. Let us start with “Design of Experiments MCQs with Answer”. Let us start with the Design of Experiments Quiz Questions with Answers now.

Design of Experiments Quiz with Answers

Online design of experiments quiz with Answers

1. One of the ANOVA assumptions is that treatments have:

 
 
 
 

2. If the experiment were to be repeated and the same set of treatments would be included, we choose:

 
 
 
 

3. In a random effects model ————- are randomly chosen from a large population.

 
 
 
 

4. Factorial experiments cannot be used to detect the presence of interaction.

 
 

5. One factor ANOVA means, there is only:

 
 
 
 

6. In a fixed effect model:

 
 
 
 

7. The experimenter is interested in treatment means only. The model used is called:

 
 
 
 

8. A fixed effect model is used when the effect of —————– is assumed to be fixed during the experiment.

 
 
 
 

9. If the treatments in a particular experiment are a random sample from a large population of similar treatments. we choose:

 
 
 
 

10. The treatment effect is associated with:

 
 
 
 

11. If the experimenter is interested in the variation among treatment means not the treatment means themselves. The model used is called:

 
 
 
 

12. For ANOVA we assume that treatments are applied to the experimental units:

 
 
 
 

13. For one factor ANOVA, the model contains:

 
 
 
 

14. To compare the IQ level of five students a series of tests is planned and IQ is computed based on their results. The model will be:

 
 
 
 

15. If an interaction effect in a factorial design is significant the main effects of the factors involved in that interaction may be difficult to interpret.

 
 

16. In a random effect model:

 
 
 
 

17. A researcher is interested in measuring the rate of production of five particular machines. The model will be a:

 
 
 
 

18. An interaction term in a factorial model with quantitative factors introduces curvature in the response surface representation of the results.

 
 

19. A factorial experiment can be run as an RCBD by assigning the runs from each replicate to separate blocks.

 
 

20. Single-factor ANOVA is also called:

 
 
 
 

Design of Experiments Quiz with Answers

  • If an interaction effect in a factorial design is significant the main effects of the factors involved in that interaction may be difficult to interpret.
  • Factorial experiments cannot be used to detect the presence of interaction.
  • An interaction term in a factorial model with quantitative factors introduces curvature in the response surface representation of the results.
  • A factorial experiment can be run as an RCBD by assigning the runs from each replicate to separate blocks.
  • One of the ANOVA assumptions is that treatments have:
  • For ANOVA we assume that treatments are applied to the experimental units:
  • One factor ANOVA means, there is only:
  • For one factor ANOVA, the model contains:
  • Single-factor ANOVA is also called:
  • In a fixed effect model:
  • In a random effect model:
  • The treatment effect is associated with:
  • If the experiment were to be repeated and the same set of treatments would be included, we choose:
  • The experimenter is interested in treatment means only. The model used is called:
  • A fixed effect model is used when the effect of —————– is assumed to be fixed during the experiment.
  • A researcher is interested in measuring the rate of production of five particular machines. The model will be a:
  • To compare the IQ level of five students a series of tests is planned and IQ is computed based on their results. The model will be:
  • If the treatments in a particular experiment are a random sample from a large population of similar treatments. we choose:
  • If the experimenter is interested in the variation among treatment means not the treatment means themselves. The model used is called:
  • In a random effects model ————- are randomly chosen from a large population.

MCQs General Knowledge

Design of Experiments Quiz Questions 7

Online Quiz about Design of Experiments Quiz Questions with Answers. There are 20 MCQs in this DOE Quiz covers the basics of the design of experiments, hypothesis testing, basic principles, and single-factor experiments. Let us start with “Design of Experiments MCQs with Answer”. Let us start with the Design of Experiments Quiz Questions with Answers now.

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Design of Experiments Quiz Questions with Answers

Design of Experiments Quiz Questions with Answers

  • Why is randomization an important aspect of conducting a designed experiment?
  • Why would an agricultural field trial require a different experimental strategy than a typical industrial experiment?
  • Sir Ronald A. Fisher is regarded as the modern pioneer of designed experiments because
  • The analysis of variance treats the factor as if it were qualitative even if it is a continuous variable such as temperature.
  • The Fisher LSD procedure used to compare pairs of treatment means following an ANOVA is extremely conservative.
  • If a single-factor experiment has a continuous factor with $a$ levels and a polynomial of degree $a – 1$ is fit to the data the error sum of squares for the polynomial model will be identical to the error sum of squares that resulted from the standard ANOVA.
  • In a single-factor random effects experiment we assume that the levels of the factor are selected at random from an infinitely large population of possible levels.
  • When comparing more than two population means at the same time we should not use:
  • In an independent samples t-test two samples:
  • When population variance is unknown and sample sizes are small we can estimate the variance by
  • To apply the t-test, two samples must be:
  • The t-test is used when:
  • Paired samples are:
  • A paired samples t-test is also called:
  • Paired samples t-test utilizes degree of freedom:
  • In case of pairing, samples are usually taken from:
  • Basic ANOVA measures ————— source/s of variation
  • ANOVA is suitable to compare —————- means
  • In ANOVA we use
  • For the validity of different inferential tools we assume that errors have:

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