Question 1

Communality is ________.
A)the amount of variance a variable shares with all the other variables being considered
B)the percentage of the total variance attributed to each factor
C)the proportion of variance explained by the common factors
D)both A and C are correct

Accepted Answer

Communality represents the amount of variance a variable shares with all the other variables being considered, which is option A. It is also used to determine the proportion of variance explained by the common factors, which is option C. Therefore, both options A and C are correct, making option D the correct answer.

Question 2

m represents ________ in the factor model, X_i= A_i1 F₁+ A_i2 F₂+A_i3 F₃+ ...+ A_im F_m+V_iU_i,. A)the number of variables B)the number of common factors C)the mth standardized variable D)the common factors

Accepted Answer

In the factor model, "m" typically represents the number of common factors influencing the variables being analyzed.

Question 3

________ is an index that compares the magnitudes of the observed correlation coefficients to the magnitudes of the partial correlation coefficient.
A)Mahalanobis ratio
B)KMO measure of sampling adequacy
C)Bartlett's test of sphericity
D)Wilks' lambda

Accepted Answer

The KMO (Kaiser-Meyer-Olkin) measure of sampling adequacy compares the magnitudes of the observed correlation coefficients to the magnitudes of the partial correlation coefficients. It helps in determining the adequacy of the sample for conducting a factor analysis.

Question 4

Which is not a method of factor analysis?
A)principal components analysis
B)common factor analysis
C)omega method
D)unweighted least squares

Accepted Answer

Principal components analysis, common factor analysis, and unweighted least squares are all methods of factor analysis, while the omega method is a reliability estimate for a set of items.

Question 5

________ should be used when factors in the population are likely to be strongly correlated.
A)Orthogonal rotation
B)The varimax procedure
C)Oblique rotation
D)None of the above

Accepted Answer

Oblique rotation should be used when factors in the population are likely to be strongly correlated because it allows the factors to be correlated in the solution, whereas orthogonal rotation (such as varimax procedure) assumes that the factors are uncorrelated in the solution.

Question 6

As a rough guideline, there should be at least ________ times as many observations (sample size)as there are variables.
A)two or three
B)three or four
C)four or five
D)five or six

Accepted Answer

As a general rule of thumb, there should be at least four or five times as many observations as there are variables. This helps to ensure sufficient power and accuracy in statistical analyses.

Question 7

________ is an approach to factor analysis that considers the total variance in the data.
A)Principal components analysis
B)Common factor analysis
C)Omega method
D)Unweighted least squares

Accepted Answer

Principal components analysis is an approach to factor analysis that considers the total variance in the data. It identifies the underlying structure or patterns in the data by reducing the dimensionality of the data and creating a smaller set of variables, called principal components, that explain the majority of the variance in the original data set. Common factor analysis, on the other hand, assumes that each observed variable is influenced by a set of common factors and error terms, and seeks to estimate these common factors. Omega method and unweighted least squares are alternative statistical methods that are not directly related to factor analysis.

Question 8

Sometimes, because of prior knowledge, the researcher knows how many factors to expect and thus can specify the number of factors to be extracted beforehand. This is referred to as ________.
A)a priori determination
B)determination based on split-half reliability
C)determination based on significance tests
D)determination based on scree plot

Accepted Answer

A priori determination involves specifying the number of factors before conducting the factor analysis based on prior knowledge and theory. This is different from other methods such as split-half reliability or significance tests, which involve analyzing the data to determine the number of factors. The scree plot can provide a visual representation of the eigenvalues of the factors but does not specify the number of factors to be extracted.

Question 9

Factor analysis may not be appropriate in all of the following situations except ________.
A)a small value for Bartlett's test of sphericity is found
B)small values of the KMO statistic are found
C)the variables are not correlated
D)the variables are correlated

Accepted Answer

Factor analysis is appropriate when the variables under study are correlated, as it aims to identify underlying factors that explain the patterns of correlations among variables.

Question 10

________ is an approach to factor analysis that estimates the factors based only on the common variance.
A)Principal components analysis
B)Common factor analysis
C)Omega method
D)Unweighted least squares

Accepted Answer

Common factor analysis is an approach to factor analysis that estimates the factors based only on the common variance. This means that it only considers the variance shared between variables rather than the unique variance specific to each variable. Principal components analysis (A) is another approach to factor analysis, but it includes all of the variance in the data, rather than only the common variance. Omega method (C) and unweighted least squares (D) are not specific approaches to factor analysis.

Question 11

Which of the following statements is not true about factor rotation?
A)Through rotation, the factor matrix is transformed into a simpler one that is easier to interpret.
B)Rotation affects the communalities and the percentage of total variance explained.
C)Preferably, each factor should have a nonzero, or significant, loadings or coefficients for only some of the variables.
D)Different methods of rotation may result in the identification of different factors.

Accepted Answer

Rotation affects the shape and orientation of the factor loadings, but does not affect the communalities or the percentage of total variance explained.

Question 12

Which of the following is a way to interpret factors?
A)based on the variables that load high on a factor
B)by plotting the variables using the factor loadings as coordinates
C)both A and B
D)none of the above

Accepted Answer

Interpreting factors can be done by analyzing the variables that have high factor loadings and by plotting them using the factor loadings as coordinates. Therefore, both options A and B are correct ways to interpret factors.

Question 13

The first step in conducting factor analysis is ________.
A)construct the correlation matrix
B)determine the method of factor analysis
C)formulate the problem
D)determine the number of factors

Accepted Answer

Formulating the problem is the first step in conducting factor analysis, as it involves defining what you are trying to investigate or analyze before proceeding with any statistical methods or decisions on how many factors to consider.

Question 14

Which statement is not true about principal components analysis (PCA)?
A)The factors are estimated based only on the common variance.
B)The total variance in the data is considered.
C)PCA is recommended when the primary concern is to determine the minimum number of factors that will account for maximum variance in the data for use in subsequent multivariate analysis.
D)The diagonal of the correlation matrix consist of unities.

Accepted Answer

PCA considers both common and unique variances of the variables, not just the common variance. The other statements accurately describe aspects of PCA.

Question 15

A ________ is a lower triangle matrix showing the simple correlations, r, between all possible pairs of variables included in the analysis.
A)factor matrix
B)classification matrix
C)correlation matrix
D)total correlation matrix

Accepted Answer

A correlation matrix is a lower triangle matrix that displays the simple correlations between all possible pairs of variables included in the analysis. The other answer choices (factor matrix, classification matrix, and total correlation matrix) do not accurately describe this type of matrix.

Question 16

A principal components analysis was run and the following eigenvalue results were obtained: 2.731, 2.218, .442, .341, .183, and .085. How many factors would you retain using the eigenvalues to determine the number of factors?
A)one
B)two
C)four
D)six

Accepted Answer

According to the Kaiser rule, only factors with an eigenvalue greater than 1 should be retained. In this case, the first two eigenvalues are greater than 1, so two factors should be retained.

Question 17

A ________ is a plot of the original variables using the factor loadings as coordinates.
A)scree plot
B)territorial map
C)factor loading plot
D)scattergram

Accepted Answer

A factor loading plot is a graph that shows the relationship between the original variables and the factors. It uses the factor loadings as coordinates to plot the variables. The plot allows for visual identification of patterns in the data and can aid in interpreting the factors. A scree plot shows the eigenvalues of the factors, a territorial map is not a commonly used term in statistics, and a scattergram is a plot of two quantitative variables.

Question 18

The ________ represents the total variance explained by each factor.
A)eigenvalue
B)percentage of variance
C)residual
D)communality

Accepted Answer

The eigenvalue in factor analysis represents the total variance explained by each factor. It indicates the amount of the overall variance in the observed variables that can be attributed to that factor.

Question 19

It is recommended that the factors extracted should account for at least ________ of the variance.
A)50 %
B)60%
C)65%
D)70%

Accepted Answer

It is commonly recommended that the factors extracted in factor analysis should account for at least 60% of the variance to ensure that a significant portion of the data's variability is captured by the model.

Question 20

________ are simple correlations between the variables and the factors.
A)Factor scores
B)Factor loadings
C)Correlation loadings
D)Both A and B are correct

Accepted Answer

Factor loadings represent the relationship between observed variables and the underlying factors, whereas factor scores represent an individual's position on the factor. Correlation loadings are not a commonly used term. Therefore, only option B is correct.

Quiz 19: Factor Analysis

Communality is ________.

m represents ________ in the factor model, X_i= A_i1 F₁+ A_i2 F₂+A_i3 F₃+ ...+ A_im F_m+V_iU_i,.

________ is an index that compares the magnitudes of the observed correlation coefficients to the magnitudes of the partial correlation coefficient.

Which is not a method of factor analysis?

________ should be used when factors in the population are likely to be strongly correlated.

As a rough guideline, there should be at least ________ times as many observations (sample size) as there are variables.

________ is an approach to factor analysis that considers the total variance in the data.

Sometimes, because of prior knowledge, the researcher knows how many factors to expect and thus can specify the number of factors to be extracted beforehand. This is referred to as ________.

Factor analysis may not be appropriate in all of the following situations except ________.

________ is an approach to factor analysis that estimates the factors based only on the common variance.

Which of the following statements is not true about factor rotation?

Which of the following is a way to interpret factors?

The first step in conducting factor analysis is ________.

Which statement is not true about principal components analysis (PCA) ?

A ________ is a lower triangle matrix showing the simple correlations, r, between all possible pairs of variables included in the analysis.

A principal components analysis was run and the following eigenvalue results were obtained: 2.731, 2.218, .442, .341, .183, and .085. How many factors would you retain using the eigenvalues to determine the number of factors?

A ________ is a plot of the original variables using the factor loadings as coordinates.

The ________ represents the total variance explained by each factor.

It is recommended that the factors extracted should account for at least ________ of the variance.

________ are simple correlations between the variables and the factors.