Muhammad Imdad Ullah

Currently working as Assistant Professor of Statistics in Ghazi University, Dera Ghazi Khan. Completed my Ph.D. in Statistics from the Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan. l like Applied Statistics, Mathematics, and Statistical Computing. Statistical and Mathematical software used is SAS, STATA, Python, GRETL, EVIEWS, R, SPSS, VBA in MS-Excel. Like to use type-setting LaTeX for composing Articles, thesis, etc.

Designs of Experiment Terminology (2023)

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Planning an experiment to obtain appropriate data and drawing inferences from the data concerning any problem under investigation is known as the design and analysis of the experiment or simply the designs of experiment (DOE).

Important Designs of Experiment Terminology are:

EXPERIMENT: An experiment deliberately imposes a treatment on a group of objects or subjects in the interest of observing the response.

EXPERIMENTAL UNIT: The experimental unit is the basic entity or unit on which the experiment is performed. It is an object to which the treatment is applied and in which the variable under investigation is measured and analyzed. For example, the experimental unit may be a single person, animal, plant, manufactured item, or country that belongs to a larger collection of such entities being studied.

Designs of Experiment

Identify the Experimental Units

  • A teacher practices the different teaching methods on different groups in her class to see which yields the best results.
  • A doctor treats a patient with a skin condition with different creams to see which is most effective.

The experimental unit is the physical entity or subject exposed to the treatment independently of other units. In other words, it is the basic unit on which the experiment is performed (smallest division of experimental material).

TREATMENTS: In experiments, a treatment is something that researchers administer to experimental units. For example, a corn field is divided into four, each part is ‘treated’ with a different fertilizer to see which produces the most corn.

Treatment is an experimental condition whose effect is to be measured and compared. For example, animal diets, temperature, soil types, and brands of tires.

FACTOR: A factor of an experiment is a controlled independent variable; a variable whose levels are set by the experimenter. A factor is a general type or category of treatments. Different treatments constitute different levels of a factor.

Designs of Experiment

EXPERIMENTAL ERROR

It describes the variation among identically and independently treated experimental units. In the designs of experiments, various origins of experimental error include:

  • The natural variation among the experimental units.
  • Inability to reproduce the treatment conditions exactly from one unit to another.
  • Interaction of treatments and experimental units.
  • Any other extraneous factors that influence the measured characteristics.

There are two types of errors:

  1. Systematic Errors
    Systematic Errors are caused by a consistent bias in one direction, consistently pushing your results away from the true value. Systematic errors can be caused by a variety of factors, such as a faulty instrument, an incorrect calibration, or an error in the experimental design. Systematic errors will cause data points to shift all in the same direction, away from the true value.
  2. Random Error
    The random error is caused by small and unpredictable variations that occur in every experiment. Random errors can come from a variety of sources, such as slight differences in how a measurement is made, or fluctuations in environmental conditions. Random errors tend to cause data points to scatter randomly around the true value.

The experimental error can be controlled by

  • Blocking
  • Proper plot technique
  • Data Analysis

EXPERIMENTAL DESIGN

An experimental design is a plan to collect the data relevant to the problem under investigation. In such a way as to provide a basis for valid and objective inferences about the stated problem.

The plan usually consists of the selection of the treatments, specifications of the experimental layouts, allocation of the treatments, and collection of observations for analysis.

Hence designs of Experiments are simply a sequence of steps taken ahead of time to ensure that the appropriate data will be obtained in a way that permits an objective analysis leading to a valid analysis concerning the problem.

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Important Online Multivariate MCQ

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Multivariate Analysis term includes all statistics for more than two simultaneously analyzed variables. The post contains a Multivariate Quiz.

Online Multivariate MCQs

MCQs Cluster Analysis 6Multivariate Quiz 5Multivariate MCQs 4
Multivariate MCQs 3Multivariate MCQs 2Multivariate MCQs 1

Multivariate analysis is based upon an underlying probability model known as the Multivariate Normal Distribution (MND). The objective of scientific investigations to which multivariate methods most naturally lend themselves includes. Multivariate analysis is a powerful technique for analyzing data that goes beyond the limitations of simpler, single-variable methods.

Online Multivariate MCQ
  • Data reduction or structural simplification
    The phenomenon being studied is represented as simply as possible without sacrificing valuable information. It is hoped that this will make interpretation easier.
  • Sorting and Grouping
    Graphs of similar objects or variables are created, based on measured characteristics. Alternatively, rules for classifying objects into well-defined groups may be required.
  • Investigation of the dependence among variables
    The nature of the relationships among variables is of interest. Are all the variables mutually independent or are one or more variables depend on the observation of the other variables?
  • Prediction
    Relationships between variables must be determined for predicting the values of one or more variables based on observation of the other variables.
  • Hypothesis Construction and testing
    Specific statistical hypotheses, formulated in terms of the parameter of the multivariate population, are tested. This may be done to validate assumptions or to reinforce prior convictions.
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Multivariate analysis provides a comprehensive and robust way to analyze the data. It leads to better decision-making across various fields. Multivariate analysis is a vital tool for researchers and data scientists seeking to extract deeper insights from complex datasets.

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Easy Multivariate Analysis MCQs – 1

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Multivariate Analysis term includes all statistics for more than two simultaneously analyzed variables. The post contains Multivariate Analysis MCQs. Let us start with the Online Multivariate Analysis MCQs test.

Multiple Choice Questions about Multivariate and Multivariate Analysis

1. Eigenvalue is often introduced in the context of

 
 
 
 

2. Eigenvalues and Eigenvectors are only for the matrices

 
 
 
 

3. If $A$ is a square matrix, then $det(A – \lambda)=0$ is known as

 
 
 
 

4. Let $A$ be a $k\times k$ symmetric matrix and $X$ be a $k\times 1$ vector. Then

 
 
 
 

5. The pdf of multivariate normal distribution exists only when $\sigma$ is

 
 
 
 

6. Let $x_1, x_2, \cdots, x_n$ be a random sample of size $n$ from a p-variate normal distribution with mean $\mu$ and covariance matrix $\sigma$, then

 
 
 
 

7. Let $x$ be distributed as $N_p(\mu, \sigma)$ with $|\sigma | > 0$, then $(x-\mu)’ \sigma^{-1} (x-\mu)$ is distributed as:

 
 
 
 

8. A matrix $A_{m\times n}$ is defined to be orthogonal if

 
 
 
 

9. The rank of a matrix $\begin{bmatrix}1 & 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 1 & 2 \\ 1 & 1 & 0 & 0 & 2 \\ 0 & 1 & 1 & 1 & 3\end{bmatrix}$ is

 
 
 
 

10. How many Eigenvalues does a 2 by 2 matrix have?

 
 
 
 

11. If $A$ and $B$ are two $n \times n$ matrices, which of the following is not always true?

 
 
 
 

12. If $A$ is a square matrix of order ($m \times m$) then the sum of diagonal elements is called

 
 
 
 

13. A square matrix $A$ and its transpose have the Eigenvalues

 
 
 
 

14. The eigenvalue is the factor by which the Eigenvector is

 
 
 
 

15. What are Eigenvalues?

 
 
 
 

16. A matrix in which the number of rows and columns are equal is called

 
 
 
 

17. The set of all linear combination of $X_1, X_2, \cdots, X_k$ is called

 
 
 
 

18. A set of vectors $X_1, X_2, \cdots, X_n$ are linearly independent if

 
 
 
 

19. Length of vector $\underline{X}$ is calculated as

 
 
 
 

20. Let $x_1, x_2, \cdots, x_n$ be a random sample from a joint distribution with mean vector $\mu$ and covariance $\sigma$. Then $\overline{x}$ is an unbiased estimator of $\mu$ and its covariance matrix is:

 
 
 
 

Multivariate Analysis MCQs

Multivariate Analysis MCQs

  • If $A$ and $B$ are two $n \times n$ matrices, which of the following is not always true?
  • Let $x_1, x_2, \cdots, x_n$ be a random sample from a joint distribution with mean vector $\mu$ and covariance $\sigma$. Then $\overline{x}$ is an unbiased estimator of $\mu$ and its covariance matrix is:
  • Let $x$ be distributed as $N_p(\mu, \sigma)$ with $|\sigma | > 0$, then $(x-\mu)’ \sigma^{-1} (x-\mu)$ is distributed as:
  • Let $A$ be a $k\times k$ symmetric matrix and $X$ be a $k\times 1$ vector. Then
  • Let $x_1, x_2, \cdots, x_n$ be a random sample of size $n$ from a p-variate normal distribution with mean $\mu$ and covariance matrix $\sigma$, then
  • The set of all linear combination of $X_1, X_2, \cdots, X_k$ is called
  • A set of vectors $X_1, X_2, \cdots, X_n$ are linearly independent if
  • Length of vector $\underline{X}$ is calculated as
  • A matrix in which the number of rows and columns are equal is called
  • A matrix $A_{m\times n}$ is defined to be orthogonal if
  • If $A$ is a square matrix of order ($m \times m$) then the sum of diagonal elements is called
  • The rank of a matrix $\begin{bmatrix}1 & 0 & 1 & 0 & 2 \ 0 & 0 & 1 & 1 & 2 \ 1 & 1 & 0 & 0 & 2 \ 0 & 1 & 1 & 1 & 3\end{bmatrix}$ is
  • If $A$ is a square matrix, then $det(A – \lambda)=0$ is known as
  • The pdf of multivariate normal distribution exists only when $\sigma$ is
  • The eigenvalue is the factor by which the Eigenvector is
  • Eigenvalue is often introduced in the context of
  • How many Eigenvalues does a 2 by 2 matrix have?
  • What are Eigenvalues?
  • Eigenvalues and Eigenvectors are only for the matrices
  • A square matrix $A$ and its transpose have the Eigenvalues

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Inferential Statistics Tests List (2023)

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The following is the list of different parametric and non-parametric lists of the Inferential Statistics Tests List. A short description of each Inferential Statistics Test is also provided.

Inferential Statistics Tests: Parametric Statistics

Sr. No.Statistical TestShort Description of Inferential Test
1)Z testLarge sample test for one mean/average when sigma ($\sigma$) is known (or $n$ is large), population distribution is normal.
2)t testSmall sample test for one mean/average when sigma ($\sigma$) is unknown (and $n$ is small), population distribution is normal.
3)Z testLarge sample test for one proportion.
4)Z testSmall sample test for two means/averages when sigmas ($\sigma_1$ and $\sigma_2$) are unknown, samples are independent, and are from normal populations. The variances are NOT pooled.
5)t testSmall sample test for two means/averages when sigmas ($\sigma_1$ and $\sigma_2$) are unknown, samples are independent and are from normal populations. The variances are NOT pooled.
6)t testSmall sample test for two means/averages when sigmas ($\sigma_1$ and $\sigma_2$) are unknown, samples are independent and are from normal populations. The variances are NOT pooled.
7)t testA test for two means/averages for dependent (paired or related) samples where $d$ (The difference between samples) is normally distributed.
8)Z testLarge sample test for two proportions.
9)$\chi^2$Chi-square goodness of fit, or multinomial distribution., where each expected value is at least 5.
10)$\chi^2_{ii}$Chi-square for contingency tables (rows & columns) where each expected value is at least 5.
Either a test of independence, a test of homogeneity, or a test of association.
11)$\chi^2$Test for one variance or standard deviation.
12)F testTest for two variances or standard deviations for independent samples from the normal populations.
13)F (Anova)Test for three or more means for independent random samples from normal populations. The variances are assumed to be equal.
14)Tukey QA multiple comparison test for all pairs of means (usually for equal sample sizes).
15)Dunnett qA multiple comparison test for a control mean to other means.
16)Hartley H,
Bartlett,
Levene,
Brown-Forsythe,
O’Brien		Test for homoscedasticity, and homogeneity of variances.
17)Pearson $r$Pearson product-moment correlation coefficient.
18)SlopeTest on the slope of the linear regression line.
19)InterceptIntercept Test on the y-intercept of the linear regression line.

Inferential Statistics Tests: Non-Parametric Tests

The following is the list of non-parametric Tests with a short description of the tests.

Sr. No.Statistical TestsShort Description of Inferential Statistics Tests
1)Runs TestUsed to determine whether the sequence of data is random.
2)Mann-Whitney U TestAnalogous to Test # 5 from Parametric test list.
3)Sign TestAnalogous to Test # 2 from Parametric Test (Single sample Median Test). or Test # 7 from Parametric Test.
4)Wilcoxon Signed-Ranked TestSimilar to the sign test, but more efficient, analogous to Parametric Test # 7.
5)Kruskal-Wallis TestAnalogous to Parametric Test # 13.
6)Multiple Comparison TestAnalogous to Parametric Test # 14.
7)Spearmna $r_s$ Rank CorrelationAnalogous to Parametric Test # 15.

Advanced Inferential Statistics Tests

Following is the list of some advanced inferential Statistics tests with a short description of the test.

Sr. No. Statistical TestShort Description of Inferential Statistics Tests
1)Two-factor ANOVA With ReplicationInteraction is possible between two factors.
2)Two-factor ANOVA Without ReplicationOnly one observation per cell; no interaction effect is observed between the two factors.
3)One Datum StatUsed to compare one piece of data (datum) to a mean.
4)McNemar StAtUsed to test a 2 x 2 table of matched discordant pairs.
5)Two Poisson CountsUsed to compare two Poisson counts.
6)Two Regression SlopesUsed to compare two regression equation slopes.
7)Several Regression SlopesUsed to compare several regression equation slopes.
8)Multiple RegressionUsed to test a linear relationship with more than two variables.
9)Holgate StatisticUsed to determine spatial distribution.
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