Multivariate Statistics: A Comprehensive Guide for Psychology Students

Question 1

In multivariate analysis, which distribution is typically used to represent the distribution of correlated variables?

Accepted Answer

Multivariate normal distribution

Answer

Binomial distribution

Answer

Poisson distribution

Answer

Uniform distribution

Question 2

In discriminant analysis, which assumption is crucial for accurate classification?

Accepted Answer

Group covariance matrices are equal.

Answer

Predictor variables are normally distributed.

Answer

No multicollinearity among predictor variables.

Answer

Dependent variable is continuous.

Question 3

In multivariate analysis, which measure is specifically designed to compare the similarity between two vectors, considering correlations among variables?

Accepted Answer

Mahalanobis distance

Answer

Euclidean distance

Answer

Cosine similarity

Answer

Jaccard distance

Question 4

In discriminant analysis, which assumption implies that the predictor variables are not correlated with each other?

Accepted Answer

Independence of predictor variables

Answer

Multicollinearity among predictor variables

Answer

Normally distributed predictor variables

Answer

Equal covariance matrices between groups

Question 5

Which assumption of discriminant analysis is often more flexible and can be relaxed under certain conditions?

Accepted Answer

Multivariate normality

Answer

Linearity

Answer

Equal group covariance matrices

Answer

No outliers

Question 6

In multivariate statistics, Mahalanobis distance measures the:

Accepted Answer

Dissimilarity between two data points, considering their covariance.

Answer

Distance between a data point and the center of a data group.

Answer

Spread of values within a multivariate dataset.

Answer

Relationship strength between two multivariate datasets.

Question 7

Which assumption is essential for conducting discriminant analysis?

Accepted Answer

Covariance matrices of the groups are equal.

Answer

Predictor variables are normally distributed.

Answer

Predictor variables are linearly related.

Answer

Sample size is large enough.

Question 8

In multivariate discriminant analysis, which of the following is a crucial assumption that must be satisfied for the analysis to be valid?

Accepted Answer

The groups have equal variance-covariance matrices.

Answer

The predictor variables are normally distributed.

Answer

The number of observations is equal in all groups.

Question 9

In discriminant analysis, which assumption must be met regarding the predictor variables?

Accepted Answer

Predictor variables must have equal variance-covariance matrices.

Answer

Predictor variables must follow a normal distribution.

Answer

Predictor variables must have equal means.

Answer

Predictor variables must be independent of each other.

Question 10

In discriminant analysis, which assumption is crucial for the validity of the results?

Accepted Answer

The covariance matrices of the groups being discriminated are equal.

Answer

The dependent variable follows a normal distribution.

Answer

The independent variables are entirely independent.

Answer

The sample sizes of the groups being discriminated are identical.

Question 11

In discriminant analysis, the Mahalanobis distance is used to measure the similarity between an observation and the:

Accepted Answer

Centroid of its assigned group

Answer

Mean of each individual variable

Answer

Median of its assigned group

Question 12

In discriminant analysis, the Mahalanobis distance measures the:

Accepted Answer

Distance between an observation and its assigned group's centroid

Answer

Distance between an observation and the overall sample mean

Answer

Distance between an observation and the origin

Answer

Distance between two observations

Question 13

In factor analysis, which assumption is crucial?

Accepted Answer

Observed variables are linearly related.

Answer

Latent factors are known before analysis.

Answer

Data is normally distributed.

Answer

Data matrix has no missing values.

Question 14

In discriminant analysis, which assumption must be met?

Accepted Answer

Covariance matrices of the groups are equal.

Answer

Variables are independent.

Question 15

In discriminant analysis, which assumption is made about the distribution of the predictor variables?

Accepted Answer

They are normally distributed.

Answer

They are independent.

Answer

They have equal variances.

Answer

They are all continuous.

Question 16

In multivariate discriminant analysis, which assumption is critical for valid classification results?

Accepted Answer

Variance-covariance matrices of the groups are equal.

Answer

Independent variables follow a normal distribution.

Answer

Dependent variable is a quantitative measure.

Question 17

In multivariate analysis, which measure quantifies the distance between an observation and its cluster's center, taking into account the relationships among variables?

Accepted Answer

Mahalanobis distance

Answer

Euclidean distance

Answer

Cosine similarity

Answer

Jaccard index

Question 18

In factor analysis, the assumption of high correlation among observed variables implies that:

Accepted Answer

These variables share underlying common factors or constructs.

Answer

These variables are independent and have no influence on each other.

Answer

These variables are all normally distributed.

Question 19

In discriminant analysis, which assumption ensures that the groups being compared have similar levels of variability?

Accepted Answer

Equal variance-covariance matrices

Answer

Independence of predictor variables

Answer

Normality of the outcome variable

Answer

Sufficient sample size

Question 20

Which of the following is a unique assumption of discriminant analysis?

Accepted Answer

The groups have unequal covariance matrices.

Answer

The variables are independent.

Answer

The variables are normally distributed.

Question 21

In discriminant analysis, which assumption ensures that the groups being compared have similar levels of variability within each group?

Accepted Answer

Equal variance-covariance matrices assumption

Answer

Equal means assumption

Answer

Multivariate normality assumption

Answer

Linearity assumption

Question 22

Which of the following is a crucial assumption of principal component analysis (PCA)?

Accepted Answer

The variables must be measured on a consistent scale.

Answer

The data must be normally distributed.

Answer

The number of variables must significantly exceed the number of observations.

Question 23

In principal component analysis (PCA), what is the main reason for calculating eigenvalues and eigenvectors?

Accepted Answer

To determine the number of principal components to extract and the variance explained by each component.

Answer

To identify the variables that contribute most to each principal component.

Answer

To calculate the correlation between the original variables and the principal components.

Question 24

In which research scenario would discriminant analysis be the most appropriate statistical method to use?

Accepted Answer

Classifying individuals into distinct groups based on their scores on a set of variables

Answer

Examining the relationship between several independent variables and a single dependent variable

Answer

Determining the underlying factors that contribute to a set of observed variables

Answer

Reducing the dimensionality of a large dataset by identifying the most important variables

Question 25

The primary goal of factor analysis is to:

Accepted Answer

Identify underlying patterns and relationships in the data

Answer

Reduce the number of variables in a dataset

Answer

Predict the value of one variable based on the values of others

Answer

Classify individuals into different groups

Question 26

Discriminant analysis is primarily used for:

Accepted Answer

Identifying differences between groups of individuals

Answer

Reducing the dimensionality of a dataset

Answer

Predicting the value of a continuous variable

Answer

Identifying relationships between variables

Question 27

In factor analysis, a scree plot is used to:

Accepted Answer

Determine the number of factors to extract

Answer

Identify the communality of each variable

Answer

Determine the loadings of each variable on each factor

Answer

Calculate the eigenvalues of each factor

Question 28

In principal component analysis, the first principal component:

Accepted Answer

Accounts for the largest amount of variance in the dataset

Answer

Accounts for the smallest amount of variance in the dataset

Answer

Has an eigenvalue of 1

Answer

Is always positively correlated with all the variables in the dataset

Question 29

Which of the following measures the proportion of variance in a variable that is explained by the factors?

Accepted Answer

Communality

Answer

Eigenvalue

Answer

Loading

Answer

Factor score

Question 30

In discriminant analysis, what is the primary purpose of the Mahalanobis distance?

Accepted Answer

To measure the similarity or dissimilarity between observations and group centroids

Answer

To rank the importance of predictor variables

Answer

To determine the probability of an observation belonging to a group

Answer

To identify the optimal cutoff point for classifying observations

Question 31

What is the main purpose of discriminant analysis in multivariate statistics?

Accepted Answer

To predict group membership based on a set of predictor variables.

Answer

To identify the underlying structure of a dataset by grouping variables into factors.

Answer

To reduce the dimensionality of a dataset by identifying the most important variables.

Question 32

In discriminant analysis, which assumption is critical regarding the covariance matrices of the predictor variables?

Accepted Answer

They are equal across groups.

Answer

Their means are equal across groups.

Answer

They are highly correlated.

Answer

They are normally distributed.

Question 33

Which of the following is a key assumption of factor analysis?

Accepted Answer

Observed variables have linear relationships.

Answer

Observed variables are normally distributed.

Answer

The number of factors to extract is predetermined.

Answer

Sample size must meet a specific threshold for analysis validity.

Question 34

In multivariate statistics, what is the main objective of factor analysis?

Accepted Answer

To identify underlying variables that explain the relationships between multiple variables

Answer

To reduce the number of variables in a dataset by selecting the most significant ones

Answer

To determine the most suitable statistical test for a research question

Answer

To predict future outcomes using a set of predictor variables

Question 35

In multivariate analysis, what is the concept of effect size and why is it important for interpreting statistical results?

Accepted Answer

Effect size measures the strength of relationships between variables, helping researchers understand the practical significance of statistical findings.

Answer

Effect size is not applicable in multivariate analysis.

Answer

Effect size is always expressed as a percentage between 0 and 100%.

Question 36

In multivariate analysis, which measure quantifies the overall similarity between two observations, considering all variables and their interrelationships?

Accepted Answer

Mahalanobis distance

Answer

Kendall's tau

Answer

Pearson's correlation

Answer

Wilks' lambda

Question 37

Which of the following is the **primary objective** of factor analysis?

Accepted Answer

To identify the underlying structure of a dataset

Answer

To reduce the number of variables in a dataset

Answer

To predict the value of one variable based on other variables

Answer

To test the significance of relationships between variables

Question 38

What is the **key distinction** between principal component analysis (PCA) and factor analysis?

Accepted Answer

PCA focuses on maximizing explained variance, while factor analysis focuses on identifying underlying structure.

Answer

PCA is an exploratory technique, while factor analysis is a confirmatory technique.

Answer

PCA can only be used with continuous variables, while factor analysis can be used with any type of variable.

Answer

There is no difference between PCA and factor analysis.

Question 39

Which of the following assumptions **is not required** for factor analysis?

Accepted Answer

Independence of observations

Answer

Linearity

Answer

Multivariate normality

Answer

Equal variances and covariances

Question 40

How many components **should be extracted** in PCA?

Accepted Answer

As many as can be interpreted meaningfully

Answer

As many as possible to maximize explained variance

Answer

As many as there are variables in the dataset

Answer

A fixed number determined by the researcher

Question 41

What is the **main purpose** of discriminant function analysis?

Accepted Answer

To identify the variables that best predict group membership

Answer

To reduce the number of variables in a dataset

Answer

To test the significance of differences between groups

Answer

To find the underlying structure of a dataset

Question 42

Which technique is commonly employed in multivariate statistics for reducing the dimensionality of a dataset?

Accepted Answer

Factor analysis

Answer

Regression analysis

Answer

Cluster analysis

Answer

Discriminant analysis

Question 43

In factor analysis, the scree plot is utilized to determine the:

Accepted Answer

Optimal number of factors to extract

Answer

Communalities of variables

Answer

Eigenvalues of factors

Answer

Loadings of variables on factors

Question 44

Discriminant analysis is employed for the purpose of:

Accepted Answer

Classifying observations into distinct groups

Answer

Identifying relationships between variables

Answer

Predicting continuous outcomes

Answer

Reducing the dimensionality of a dataset

Question 45

In discriminant analysis, the discriminant function:

Accepted Answer

Is a linear combination of predictor variables that optimally discriminates between groups

Answer

Is a nonlinear combination of predictor variables

Answer

Is a function of the group membership variable

Answer

Is a random variable

Question 46

In factor analysis, the factor loadings represent:

Accepted Answer

The contribution of a variable to a factor

Answer

The correlation between a variable and a factor

Answer

The variance of a variable explained by a factor

Answer

The weight of a variable in a linear combination

Question 47

Which of the following is a core assumption of factor analysis?

Accepted Answer

The variables share a common underlying structure.

Answer

The variables are measured on a ratio scale.

Answer

The variables are normally distributed.

Answer

The variables are independent of each other.

Question 48

What is the primary goal of principal component analysis?

Accepted Answer

To reduce the dimensionality of a dataset.

Answer

To identify the latent factors underlying a dataset.

Answer

To classify observations into distinct groups.

Answer

To predict the value of one variable based on other variables.

Question 49

Which measure assesses the reliability of a discriminant analysis model?

Accepted Answer

Wilks' lambda

Answer

Cronbach's alpha

Answer

Pearson's correlation coefficient

Answer

Spearman's rank correlation coefficient

Question 50

In discriminant analysis, the classification matrix serves what purpose and how does it contribute to evaluating classification accuracy?

Accepted Answer

A table that summarizes predicted and actual group memberships, providing insights into classification accuracy.

Answer

A graphical representation of the discriminant function, displaying the separation between groups in multivariate space.

Answer

A matrix containing the coefficients of the discriminant function, indicating the relative importance of predictor variables.

Question 51

In multiple regression analysis, multicollinearity occurs when:

Accepted Answer

Two or more independent variables are highly correlated

Answer

There is a weak correlation between the dependent and independent variables

Answer

The sample size is insufficient

Answer

The dependent variable is not normally distributed

Question 52

In discriminant analysis, what is the primary reason for utilizing prior probabilities?

Accepted Answer

To improve the accuracy of group classification.

Answer

To reduce the computational complexity of the analysis.

Answer

To eliminate the need for cross-validation.

Answer

To guarantee equal representation of all groups in the analysis.

Question 53

In regression analysis, how does multicollinearity affect the interpretation of regression coefficients?

Accepted Answer

It increases the standard errors of the coefficients, making them less precise.

Answer

It improves the model's accuracy and reliability.

Answer

It has no impact on the interpretation of the regression coefficients.

Answer

It reduces the number of variables in the model, making it easier to interpret.

Question 54

Enhanced What is the primary purpose of factor analysis in psychological research?

Accepted Answer

To uncover the underlying structure of a dataset

Answer

To reduce the number of variables in a dataset

Answer

To predict the value of one variable based on the values of other variables

Answer

To compare the means of different groups on a set of variables

Question 55

Enhanced Which of the following is NOT a commonly used criterion for determining the number of factors to extract in factor analysis?

Accepted Answer

The number of variables in the dataset

Answer

The eigenvalue

Answer

The scree plot

Answer

The theoretical basis for the study

Question 56

Enhanced What is the primary purpose of cross-validation in discriminant analysis?

Accepted Answer

To evaluate the predictive accuracy of the discriminant function

Answer

To improve the predictive power of the discriminant function

Answer

To reduce the number of variables in the discriminant function

Answer

To test the statistical significance of the discriminant function

Question 57

Which of the following is NOT a type of multivariate statistical analysis?

Accepted Answer

T-test

Answer

Principal component analysis

Answer

Canonical correlation analysis

Answer

Discriminant analysis

Question 58

What is the main purpose of factor analysis?

Accepted Answer

To identify the underlying structure of a set of variables by grouping related variables into factors.

Answer

To predict future outcomes

Answer

To test for differences between means

Question 59

What is the primary goal of discriminant analysis?

Accepted Answer

To classify cases into different groups based on their characteristics

Answer

To identify the underlying structure of a set of variables

Answer

To test for differences between means

Question 60

Which of the following is a common application of multivariate statistics in practice?

Accepted Answer

Analyzing market research data to identify consumer segments

Answer

Calculating a correlation coefficient

Answer

Performing a t-test

Question 61

What is the purpose of examining a scree plot in factor analysis?

Accepted Answer

To determine the optimal number of factors to extract

Answer

To identify the underlying structure of the data

Answer

To test the statistical significance of the extracted factors

Question 62

Which of the following measures is commonly used to assess the reliability of a discriminant function?

Accepted Answer

Wilks' lambda

Answer

Cronbach's alpha

Answer

Pearson's correlation coefficient

Answer

R-squared

Question 63

Which of the following is NOT a key purpose of factor analysis?

Accepted Answer

To determine the causal relationships between variables

Answer

To identify relationships between variables

Answer

To reduce the number of variables in a dataset

Question 64

Which of the following techniques identifies the linear combinations of variables that account for the maximum variance in a dataset?

Accepted Answer

Principal component analysis

Answer

Cluster analysis

Answer

Factor analysis

Answer

Discriminant analysis