Design of Experiments (DOE)

Lecture notes and quizzes related to the Design of Experiments (DOE).

Incomplete Block Design: A Quick Guide

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When the block size is less than the number of treatments to be tested is known as an incomplete block design (IBD). Yates introduced incomplete block designs to eliminate the heterogeneity when the number of treatments becomes very large.

It is known that the precision of the estimate of a treatment effect depends on the number of replications of the treatment, that is, the larger the number of replications, the more the precision. A similar criterion holds for the precision of estimating the difference between two treatment effects. If two treatments occur together in a block, then we say that these are replicated once in that block.

Different patterns of values of the numbers of replications or different pairs of treatments in a design have given rise to different types of incomplete block designs.

The randomized block designs in which every treatment is not present in every block then these designs are known as randomized incomplete block designs. The choice of incomplete block designs depends on factors such as the number of treatments.

Examples of Incomplete Block Designs

Example 1: When the set of treatments is larger than the block size, we use incomplete block designs. Suppose, we want to test the quality of six tires on a given car only 4 tires can be tested, such a block would be incomplete, as it is not possible to test all 6 tires on a given car at once.

Example 2: Consider a study comparing the effectiveness of three fertilizers ($A$, $B$, and $C$) on crop yield. If there are 12 experimental plots, a BIBD with 4 blocks of 3 plots each could be used. Each fertilizer would appear in 4 blocks, and each pair of fertilizers would appear together in 2 blocks.

Example 3: A pharmaceutical company wants to compare the effectiveness of four new drugs for treating a disease. Due to ethical considerations, patients cannot receive all four drugs. An IBD can be used to assign the drugs to different groups of patients, ensuring that each drug is tested against a variety of patient characteristics.

Incomplete block design

Using an IBD, the experimenter can control for variability between plots while still comparing the effects of the fertilizers.

Types of Incomplete Block Designs

The following are types of incomplete block design:

  • Balanced Incomplete Block Designs (BIBDs):
    • Each treatment appears in an equal number of blocks.
    • Each block contains an equal number of experimental units.
    • Every pair of treatments appears together in an equal number of blocks.
  • Partially Balanced Incomplete Block Designs (PBIBDs):
    • Similar to BIBDs but with a more relaxed constraint on the number of times pairs of treatments appear together.
  • Cyclic Designs:
    • A special type of BIBD where the treatments are arranged in a cyclic order within each block.

    Advantages and Disadvantages of IBDs

    • Advantages:
      • Reduced Experiment Size: IBD can require fewer experimental units compared to complete block designs.
      • Feasibility: IBD can be more practical when it is difficult or impossible to apply all treatments to every experimental unit.
    • Disadvantages:
      • Increased Complexity: Analysis can be more complex compared to complete block designs.
      • Reduced Efficiency: May not be as efficient as complete block designs in terms of precision.

    Applications of IBD

    • Agricultural Experiments: Testing different crop varieties or fertilizer treatments.
    • Industrial Experiments: Evaluating different manufacturing processes or materials.
    • Medical Research: Comparing the effectiveness of different treatments for a disease.

    Analysis of IBD

    • Hypothesis Testing: Testing hypotheses about the effects of treatments.
    • Estimation of Treatment Effects: Estimating the differences between treatment effects.
    • Analysis of Variance (ANOVA): IBD Can be used to assess the effects of treatments and blocks.
    • Least Squares Estimation: IBD is used to estimate treatment effects and block effects.
    • Tukey’s HSD: IBD can be used for multiple comparisons to identify significant differences between treatments.

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    Important MCQs on Experimental Design 1

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    The post is about MCQs on Experimental Design with Answers. There are 20 multiple-choice questions. The quiz is related to the Basics of the Design of Experiments, Analysis of variation, assumptions of ANOVA, One-Way ANOVA, Single-factor designs, and Two-Way ANOVA. Let us start with the MCQs on Experimental Design Quiz.

    MCQs about Designs of Experiment

    1. If there are 6 treatments with 3 blocks in a RCBD then the degrees of freedom for error are

     
     
     
     

    2. If the treatments consist of all combinations that can be formed from the different factors then the experiment is

     
     
     
     

    3. In one-way ANOVA, given $SSB = 2580, SSE =1656, k = 4, n = 20$ then the value of F is

     
     
     
     

    4. In two-way ANOVA with $m$ rows and $n$ columns, the error degrees of freedom is

     
     
     
     

    5. In one-way ANOVA, with the usual notation, the error degree of freedom is

     
     
     
     

    6. Consider $k$ independent samples each containing $n_1, n_2, \cdots, n_k$ items such that $n_1+n_2+\cdots+ n_k=n$. In ANOVA we use F-distribution with a degree of freedom

     
     
     
     

    7. An experiment is performed in CRD with 10 replications to compare two treatments. The total experimental units will be

     
     
     
     

    8. In ANOVA we use

     
     
     
     

    9. In two-way ANOVA with $m=5$, $n=4$, then the total degrees of freedom is

     
     
     
     

    10. In one-way ANOVA, the calculated F value is less than the table F value then

     
     
     
     

    11. Consider an experiment to investigate the efficacy of different insecticides in controlling pests and their effects on subsequent yield. What is the best reason for randomly assigning treatment levels (spraying or not spraying) to the experimental units (farms)?

     
     
     
     

    12. A teacher uses different teaching ways for different groups in his class to see which yields the best results. In this example a treatment is

     
     
     
     

    13. For a single-factor ANOVA involving five populations, which of the following statements is true about the alternative hypothesis?

     
     
     
     

    14. In one-way ANOVA with the total number of observations is 15 with 5 treatments then the total degrees of freedom is

     
     
     
     

    15. Analysis of variance

     
     
     
     

    16. If the total degrees of freedom between treatments in a CRD are 15 and 4 respectively, the degrees of freedom for error will be

     
     
     
     

    17. The assumption used in ANOVA is

     
     
     
     

    18. Which of the following are important in designing an experiment?

     
     
     
     

    19. Analysis of variance is used to test

     
     
     
     

    20. A Mean Square is

     
     
     
     

    Online MCQs on Experimental Design

    MCQs on Experimental Design Quiz
    • Analysis of variance is used to test
    • The assumption used in ANOVA is
    • In ANOVA we use
    • Consider $k$ independent samples each containing $n_1, n_2, \cdots, n_k$ items such that $n_1+n_2+\cdots+ n_k=n$. In ANOVA we use F-distribution with a degree of freedom
    • In one-way ANOVA, with the usual notation, the error degree of freedom is
    • In one-way ANOVA, given $SSB = 2580, SSE =1656, k = 4, n = 20$ then the value of F is
    • In two-way ANOVA with $m$ rows and $n$ columns, the error degrees of freedom is
    • In one-way ANOVA, the calculated F value is less than the table F value then
    • In two-way ANOVA with $m=5$, $n=4$, then the total degrees of freedom is
    • In one-way ANOVA with the total number of observations is 15 with 5 treatments then the total degrees of freedom is
    • If the treatments consist of all combinations that can be formed from the different factors then the experiment is
    • Consider an experiment to investigate the efficacy of different insecticides in controlling pests and their effects on subsequent yield. What is the best reason for randomly assigning treatment levels (spraying or not spraying) to the experimental units (farms)?
    • Which of the following are important in designing an experiment?
    • Analysis of variance
    • A Mean Square is
    • For a single-factor ANOVA involving five populations, which of the following statements is true about the alternative hypothesis?
    • An experiment is performed in CRD with 10 replications to compare two treatments. The total experimental units will be
    • A teacher uses different teaching ways for different groups in his class to see which yields the best results. In this example a treatment is
    • If the total degrees of freedom between treatments in a CRD are 15 and 4 respectively, the degrees of freedom for error will be
    • If there are 6 treatments with 3 blocks in a RCBD then the degrees of freedom for error are
    Statistics Help MCQs on Experimental Design

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    Best Design of Experiments MCQS with Answers 5

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    Online Quiz about Design of Experiments MCQs with Answers. There are 20 MCQs in this test. Let us start with “Design of Experiments MCQs with Answer”.

    Please go to Best Design of Experiments MCQS with Answers 5 to view the test

    Design of Experiments MCQs with Answers

    Design of Experiments MCQs with Answers

    • Laboratory experiments are usually performed under:
    • Common applications of DOE in physical sciences include.
    • When do experimental factors include the proportions of ingredients we use?
    • Physical science is the systematic study of the inorganic world, consisting of astronomy, physics, chemistry, and:
    • Common applications of DOE in management sciences include.
    • An important application of DOE in management sciences is to?
    • DOE can be used in management sciences to organize:
    • What is the most common one-factor-at-a-time design in social sciences?
    • An important application of DOE in social sciences is to:
    • Changes in mean scores over three or more time points are compared under the:
    • Initial applications of DOE are in?
    • With the passage of time, Statisticians moved from?
    • Taguchi designs were presented ———- Plackett-Burman designs.
    • Which term is estimated through replication?
    • A single performance of an experiment is called?
    • The different states of a factor are called.
    • A phenomenon whose effect on the experimental unit is observed is called.
    • The process of choosing experimental units randomly is called
    • Accidental bias (where chance imbalances happen) is minimized through
    • Selection bias (where some groups are underrepresented) is eliminated

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    One Way Analysis of Variance: Made Easy

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    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.

    One way analysis of variance

    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 $\sigma^2$.

    Design of Experiments

    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.

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