Design of Experiments (DOE)

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

Important Design of Experiments Quiz 2

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The post contains MCQs on the Design of Experiments Quiz (DOE). Most of the MCQs on the Design of Experiments Quiz are from factorial Experiments. Let us start with Online MCQs on the Design of Experiments Quiz.

Multiple Choice Questions about the Design of Experiments for preparation of examinations related to PPSC, FPSC, NTS, and Statistics job- and education-related examinations

1. Let ADE and BCE be two effects confounded in blocks. Then generalized interaction is

 
 
 
 

2. When a number of confounded arrangements for factorial designs are made in Latin Squares, the designs are called

 
 
 
 

3. When all pairs of treatments are compared with approximately the same precision, even though the differences among blocks may be large, called

 
 
 
 

4. An experiment was designed to investigate the effect of the amount of water and seed variety on the subsequent growth of plants. Each plant was potted in a clay plot, and a measured amount of water was given weekly. The height of the plant at the end of the experiment was measured. Which of the following is not correct?

 
 
 
 

5. The parameter $\lambda$ for a balanced incomplete block design with $a=4, b=4, k=3, r=3$ as usual notation is

 
 
 
 

6. An experiment was conducted where you analyzed the results of the plant growth experiment after you manipulated the amount of water and seed variety. Which of the following is correct?

 
 
 
 

7. In $3^k$ factorial design with $n$ replicates in the experiment, the $df$ of error are

 
 
 
 

8. The designs in which the number of treatments must be an exact square, the size of a block is the square root of this form separate replications are called

 
 
 
 

9. Which of the following is NOT CORRECT about a randomized complete block experiment?

 
 
 
 

10. In a $2^3$ factorial experiment with partial confounding in three replications of 6 blocks, the error degrees of freedom would be

 
 
 
 

11. For a Latin Square design

 
 
 
 

12. In a randomized complete block design, the block should be constructed so that

 
 
 
 

13. When a factorial experiment is performed in fractional replication, the two factorial effects that are represented by the same comparisons are called

 
 
 
 

14. If ABC is confounded in replicate I,
AB is confounded in replicated II,
BC is confounded in replicate III,
then the design technique is called

 
 
 
 

15. The number of aliases of two-factor interactions in a $2^6$-factorial experiment (1/4 replicate) would be

 
 
 
 

16. An important relationship between the coefficient of determination $R^2$ and the F-ratio used in ANOVA is

 
 
 
 

17. The models in which the levels of treatment factors are specifically chosen are known as

 
 
 
 

18. For a two-factor factorial design, if there are ‘$a$’ levels of Factor-A and ‘$b$’ levels of Factor-B, then $df$ of interaction are:

 
 
 
 

19. A design with $v$ treatment labels, each occurring $r$ times, and with $bk$ experimental units grouped into $b=v$ blocks of size $k<v$ in such a way that the units within a block are alike and units in different blocks are substantially different is

 
 
 
 

20. In a factorial experiment, if $r$ is the number of replicates then each factorial effect has the same variance, that is

 
 
 
 

Design of experiments (DOE) is a systematic method used to plan, conduct, analyze, and interpret controlled tests to study the relationship between factors and outcomes. Design of Experiment is a powerful tool used in various fields, including science, engineering, and business, to gain insights and optimize processes.

Design of Experiments Quiz

Design of Experiments Quiz

  • The parameter $\lambda$ for a balanced incomplete block design with $a=4, b=4, k=3, r=3$ as usual notation is
  • For a two-factor factorial design, if there are ‘$a$’ levels of Factor-A and ‘$b$’ levels of Factor-B, then $df$ of interaction are:
  • In $3^k$ factorial design with $n$ replicates in the experiment, the $df$ of error are
  • Let ADE and BCE be two effects confounded in blocks. Then generalized interaction is
  • If ABC is confounded in replicate I, AB is confounded in replicated II, BC is confounded in replicate III, then the design technique is called
  • An important relationship between the coefficient of determination $R^2$ and the F-ratio used in ANOVA is
  • In a randomized complete block design, the block should be constructed so that
  • For a Latin Square design
  • In a factorial experiment, if $r$ is the number of replicates then each factorial effect has the same variance, that is
  • When all pairs of treatments are compared with approximately the same precision, even though the differences among blocks may be large, called
  • In a $2^3$ factorial experiment with partial confounding in three replications of 6 blocks, the error degrees of freedom would be
  • When a factorial experiment is performed in fractional replication, the two factorial effects that are represented by the same comparisons are called
  • When a number of confounded arrangements for factorial designs are made in Latin Squares, the designs are called
  • The number of aliases of two-factor interactions in a $2^6$-factorial experiment (1/4 replicate) would be
  • The designs in which the number of treatments must be an exact square, the size of a block is the square root of this form, and separate replications are called
  • A design with $v$ treatment labels, each occurring $r$ times, and with $bk$ experimental units grouped into $b=v$ blocks of size $k<v$ in such a way that the units within a block are alike and units in different blocks are substantially different is
  • An experiment was designed to investigate the effect of the amount of water and seed variety on the subsequent growth of plants. Each plant was potted in a clay plot, and a measured amount of water was given weekly. The height of the plant at the end of the experiment was measured. Which of the following is not correct?
  • The models in which the levels of treatment factors are specifically chosen are known as
  • Which of the following is NOT CORRECT about a randomized complete block experiment?
  • An experiment was conducted where you analyzed the results of the plant growth experiment after you manipulated the amount of water and seed variety. Which of the following is correct?
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Block Design, Incidence, and Concurrence Matrix (2018)

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Block Design Properties

The necessary conditions that the parameters of a Balanced Incomplete Block Design (BIB design) must satisfy are

  • $bk = vr$, where $r=\frac{bk}{v}$ each treatment has $r$ replications
  • no treatment appears more than once in any block
  • all unordered pairs of treatments appear exactly in $\lambda$ blocks (equi-concurrence)
    where $\lambda=\frac{r(k-1)}{v-1}=\frac{bk(k-1}{v(v-1)}$ is often referred to as the concurrence parameter of a BIB design.

A design say $d$ with parameters $(v, b, r, k, \lambda)$ can be represented as a $v \times b$ treatment block incidence matrix (having $v$ rows and $b$ columns). Let denote it by $N=n_{ij}$ whose elements $n_{ij}$ signify the number of units in block $j$ allocated to treatment $i$. The rows of the incidence matrix are labeled with varieties (treatments) of the design and the columns with the blocks.

We have to put 1 in the ($i$, $j$)th cell of the matrix if variety $i$ is contained in block $j$ and 0 otherwise. Each row of the incidence matrix has $r$ 1’s, each column has $k$ 1’s, and each pair of distinct rows has $\lambda$ column 1’s, leading to a useful identity matrix.
The matrix $NN’$ has $v$ rows and $v$ columns, referred to as the concurrence matrix of design $d$, and its entries, the concurrence parameters are denoted by $\lambda_{dij}$. For a BIBD, $n_{ij}$ is either one or zero, and $n_{ij}^2= n_{ij}$.

Theorem: If $N$ is the incidence matrix of a $(v, b, r, k, \lambda)$-design then $NN’=(r-\lambda)I+\lambda J$ where $I$ is $v\times v$ identity matrix and $J$ is the $v\times v$ matrix of all 1’s.

Example: For Block Design {1,2,3}, {2,3,4}, {3,4,1}, {4,1,2} construct incidence matrix

Block Design: incidence matrix
Incidence and Concurrence matrix


Denoting the elements of $NN’$ by $q_{ih}$, we see that $q_{ii}=\sum_j n_{ij}^2$ and $q_{ih}=\sum_j n_{ij} n_{hj}, (i \ne h)$. For any block design $NN’$, the treatment concurrence with diagonal elements equal to $q_{ii}=r$ and off-diagonal elements are $q_{ih}=\lambda, (i\ne h)$ equal to the number of times any pairs of treatment occur together within the block. In a balanced design, the off-diagonal entries in $NN’$ are all equal to a constant $\lambda$ i.e., the common replication for a BIBD is $r$, and the common pairwise treatment concurrence is $\lambda$.

$N$ is a matrix of $v$ rows and $b$ columns that $r(N)\le min(b, c)$. Hence, $t\le min(b, v)$. If design is symmetric $b=v$ and $N$ is square the $|NN’|=|N|^2$, so $(r-\lambda)^{v-1}r^2$ is a perfect square.

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Data Collection Methods

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There are many methods to collect data. These Data Collection Methods can be classified into four main methods (sources) of collecting data: used in statistical inference.

Data Collection Methods

The Data Collection Methods are (i) Survey Method (ii) Simulation (iii) Controlled Experiments (iv) Observational Study. Let us discuss Data Collection Methods one by one in detail.

(i) Survey Method

A very popular and widely used method is the survey, where people with special training go out and record observations of, the number of vehicles, traveling along a road, the acres of fields that farmers are using to grow a particular food crop; the number of households that own more than one motor vehicle, the number of passengers using Metro transport and so on. Here the person making the study has no direct control over generating the data that can be recorded, although the recording methods need care and control.

(ii) Simulation

Simulation is also one of the most important data collection methods. In Simulation, a computer model for the operation of an (industrial)  system is set up in which an important measurement is the percentage purity of a (chemical) product. A very large number of realizations of the model can be run to look for any pattern in the results. Here the success of the approach depends on how well the model can explain that measurement and this has to be tested by carrying out at least a small amount of work on the actual system in operation.

(iii) Controlled Experiments

An experiment is possible when the background conditions can be controlled, at least to some extent. For example, we may be interested in choosing the best type of grass seed to use in the sports field.

The first stage of work is to grow all the competing varieties of seed at the same place and make suitable records of their growth and development. The competing varieties should be grown in quite small units close together in the field as in the figure below

Data Collection Methods: Controlled Experiments

This is a controlled experiment as it has certain constraints such as;

i) River on the right side
ii) Shadow of trees on the left side
iii) There are 3 different varieties (say, $v_1, v_2, v_3$) and are distributed in 12 units.

In the diagram below, much more control of local environmental conditions than there would have been if one variety had been replaced in the strip in the shelter of the trees, another close by the river while the third one is more exposed in the center of the field;

Data Collection Methods: Controlled Experiments 2

There are 3 experimental units. One is close to the stream and the other is to trees while the third one is between them which is more beneficial than the others. It is now our choice where to place any one of them on any of the sides.

(iv) Observational Study

Like experiments, observational studies try to understand cause-and-effect relationships. However, unlike experiments, the researcher is not able to control (1) how subjects are assigned to groups and/or (2) which treatments each group receives.

Note that small units of land or plots are called experimental units or simply units.

There is no “right” side for a unit, it depends on the type of crop, the work that is to be done on it, and the measurements that are to be taken. Similarly, the measurements upon which inferences are eventually going to be based are to be taken as accurately as possible. The unit must, therefore, need not be so large as to make recording very tedious because that leads to errors and inaccuracy. On the other hand, if a unit is very small there is the danger that relatively minor physical errors in recording can lead to large percentage errors.

Experimenters and statisticians who collaborate with them, need to gain a good knowledge of their experimental material or units as a research program proceeds.

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