Deck 19: Factor Analysis

Question 1

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

A) The total variance in the data is considered.
B) 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.
C) The factors are estimated based only on the common variance.
D) The diagonal of the correlation matrix consist of unities.

Answer 1

C

Question 2

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

Answer 2

B

Question 3

are simple correlations between the variables and the factors.

A) Factor scores
B) Correlation loadings
C) Factor loadings
D) Both A and B are correct

Answer 3

C

Question 4

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) the variables are not correlated
C) small values of the KMO statistic are found
D) the variables are correlated

Question 5

Which is not a method of factor analysis?

A) common factor analysis
B) omega method
C) unweighted least squares
D) principal components analysis

Question 6

If the variables are standardized, the factor model may be presented as .

A) X_i = A_i1 F₁ + A_i2 F₂ + A_i3 F₃ + ... + A_im F_m + V_iU_i
B) F_i = W_i1 X₁ + W_i2 X₂ + W_i3 X₃ + ... + W_im X_m + V_iU_i
C) X_i = W_i1 F₁ + W_i2 F₂ + W_i3 F₃ + ... + W_im F_m + V_iU_i
D) ^Xi ^{= A}i1 ^F1 ^{+ A}i2 ^F2 ^{+ A}i3 ^F3 + ... + ^Aim ^Fm

Question 7

Factor analysis can be used in which of the following circumstances?

A) to identify a new, smaller set of uncorrelated variables to replace the original set of correlated variables in subsequent multivariate analysis
B) to identify underlying dimensions, or factors, that explain the correlations among a set of variables
C) to identify a smaller set of salient variables from a larger set for use in subsequent multivariate analysis
D) All are correct circumstances.

Question 8

If the goal of factor analysis is to reduce the original set of variables to a smaller set of composite variables for use in subsequent multivariate analysis, it is useful to _.

A) compute factor scores for each respondent
B) compute discriminant scores for each respondent
C) select surrogate variables
D) both A and B are correct

Question 9

Which method of analysis does not classify variables as dependent or independent?

A) discriminant analysis
B) regression analysis
C) analysis of variance
D) factor analysis

Question 10

The amount of variance a variable shares with all other variables included in the factor analysis is referred to as .

A) percentage of variance
B) total variance
C) shared variance
D) communality

Question 11

The factor scores for the ith factor may be estimated as follows: .

A) X_i = W_i1 F₁ + W_i2 F₂ + W_i3 F₃ + ... + W_im F_m
B) F_i = W_i1 X₁ + W_i2 X₂ + W_i3 Y₃ + ... + W_ik Y_k
C) F_i = W_i1 X₁ + W_i2 X₂ + W_i3 X₃ + ... + W_ik X_k
D) X_i = A_i1 F₁ + A_i2 F₂ + A_i3 F₃ + ... + A_im F_m

Question 12

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

A) Wilks' lambda
B) KMO measure of sampling adequacy
C) Bartlett's test of sphericity
D) Mahalanobis ratio

Question 13

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

Question 14

is a class of procedures primarily used for data reduction and summarization.

A) Factor analysis
B) Discriminant analysis
C) Conjoint analysis
D) Regression analysis

Question 15

F 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 common factor
B) the number of common factors
C) a unique factor variable
D) the number of variables

Question 16

The first step in conducting factor analysis is .

A) determine the method of factor analysis
B) construct the correlation matrix
C) formulate the problem
D) determine the number of factors

Question 17

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

A) 50 %
B) 60%
C) 65%
D) 70%

Question 18

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

A) scree plot
B) scattergram
C) factor loading plot
D) territorial map

Question 19

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

A) Different methods of rotation may result in the identification of different factors.
B) Through rotation, the factor matrix is transformed into a simpler one that is easier to interpret.
C) Rotation affects the communalities and the percentage of total variance explained.
D) Preferably, each factor should have a nonzero, or significant, loadings or coefficients for only some of the variables.

Question 20

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

Question 21

The differences between the observed correlations (as given in the input correlation matrix) and the reproduced correlations (as estimated from the factor matrix) can be examined to determine model fit.

Question 22

The represents the total variance explained by each factor.

A) residual
B) eigenvalue
C) communality
D) percentage of variance

Question 23

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

Question 24

Factor loadings represent the correlations between sets of variables.

Question 25

The factor matrix is also called the factor pattern matrix.

Question 26

When using eigenvalues to determine the number of factors, only factors with eigenvalues greater than .05 are retained.

Question 27

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 common factors
B) the mth standardized variable
C) the common factors
D) the number of variables

Question 28

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

A) Common factor analysis
B) Unweighted least squares
C) Omega method
D) Principal components analysis

Question 29

Factor analysis is a(n) in that the entire set of interdependent relationships is examined.

A) varimax procedure
B) orthogonal procedure
C) KMO measure of sampling adequacy
D) interdependence technique

Question 30

When selecting surrogate variables from variables with similarly high loadings, the choice should be based on _ .

A) theoretical considerations
B) measurement considerations
C) the variable with the highest loading on a factor
D) Both A and B are correct

Question 31

The percentage of the total variance attributed to each factor analysis model is called the percentage of variance.

Question 32

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

A) Principal components analysis
B) Unweighted least squares
C) Omega method
D) Common factor analysis

Question 33

is the last step in factor analysis.

A) Rotate the factors
B) Interpret the factors
C) Determine the number of factors
D) Determine the model fit

Question 34

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

A) Orthogonal rotation
B) Oblique rotation
C) The varimax procedure
D) None of the above

Question 35

Which statement is not correct concerning factor scores?

A) In common factor analysis, estimates of factor scores are obtained, and there is no guarantee that factors will be uncorrelated with each other.
B) In principal components analysis it is possible to compute exact factor scores that are uncorrelated.
C) Factor scores can be used instead of the original variables in subsequent multivariate analysis.
D) All statements are correct.

Question 36

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) determination based on significance tests
B) determination based on split- half reliability
C) a priori determination
D) determination based on scree plot

Question 37

It is possible to compute as many principal components as there are variables; in doing so, parsimony is gained.

Question 38

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) total correlation matrix
D) correlation matrix

Question 39

Generally, the number of factors determined by a scree plot will be one or a few less than that determined by the eigenvalue criterion.

Question 40

Which of the following applications is appropriate for using factor analysis?

A) to understand the media consumption habits of the target market
B) to determine if variation in market share can be accounted for by the size of the sales force and advertising expenditures
C) to identify the characteristics of price- sensitive consumers
D) Both A and C are correct.

Question 41

The variables to be included in the factor analysis should be specified based on past research, theory, and the judgment of the researcher.

Question 42

A factor model that provides a good fit to the data has many large residuals.

Question 43

Bartlett's test of sphericity is one test of the appropriateness of the factor model.

Question 44

Factor analysis is somewhat similar to discriminant analysis in that each variable is expressed as a linear combination of underlying factors.

Question 45

Interpretation is facilitated by identifying the variables that have small loadings on the same factor.

Question 46

The unrotated factor matrix seldom results in factors that can be interpreted because the factors are correlated with many variables.

Question 47

The test statistic for sphericity is based on a chi- square transformation of the determinant of the correlation matrix. A large value of the test statistic will favor the acceptance of the null hypothesis.

Question 48

Rotation does not affect the communalities and the percentage of total variance explained.

Question 49

A factor is an underlying dimension that explains the correlations among a set of variables.

Question 50

Only in the case of principal components analysis is it possible to compute exact factor scores.

Question 51

For the factor analysis to be appropriate, the variables must be correlated.

Question 52

Factor scores should be computed if the goal of factor analysis is to use the results in subsequent multivariate analysis.

Question 53

The equation X_i = A_i1 F₁ + A_i2 F_{2 +} A_i3 F₃ + ... + A_im F_{m +} V_iU_i , represents the common factors expressed as linear combinations of the observed variables.

Question 54

The various methods of factor analysis are differentiated by the approach used to derive the weights or factor score coefficients.

Question 55

The factors identified in factor analysis are overtly observed in the population.

Question 56

Factor analysis does not classify variables as dependent or independent.

Question 57

In order to use factor analysis, it is important that the variables be appropriately measured on an ordinal or nominal scale.

Question 58

Residuals are the differences between the observed correlations, as given in the input correlation matrix, and the reproduced correlations, as estimated from the factor matrix.

Question 59

Selecting surrogate variables works well if one factor loading for a variable is clearly higher than all other factor loadings.

Question 60

If interpreting factors using the rotated factor matrix, a negative coefficient for a negative variable (prevention of tooth decay is not important) would lead to a positive interpretation (prevention of tooth decay is important).

Question 61

Percentage of variance accounted for, scree plot, and a priori determination are all procedures for determining the number of factors.

Question 62

When selecting variables to serve as surrogate variables, you should look for the variable with the highest loading on a factor.

Question 63

Discuss the process of selecting surrogate variables. Also discuss how the researcher should decide on which variable to choose in complex situations.

Question 64

Under what general circumstances is factor analysis used?

Question 65

Factor analysis examines the whole set of interdependent relationships among variables.

Question 66

Factors can be estimated so that their factor scores are not correlated and the first factor accounts for the highest variance in the data, the second factor the second highest and so on.

Question 67

List at least four of the procedures used for determining the number of factors.

Question 68

Describe principal components analysis and common factor analysis and the differences between the two methods of factor analysis.

Question 69

What are the steps involved in conducting factor analysis (Figure 19.3 in the text)?

Question 70

Principal components analysis is appropriate when the primary concern is to identify the underlying dimensions and the common variance is of interest.