Question 1: In a factor analysis, a researcher collected data on 10 variables and obtained a correlation matrix. How many factors should be extracted from this data?
Answer: The number of factors to be extracted from the data depends on various criteria, such as the eigenvalues, scree plot, and interpretability of the factors. A common approach is to use the Kaiser criterion, which suggests extracting factors with eigenvalues greater than 1. However, other criteria and theoretical considerations should also be taken into account.
Question 2: A researcher conducted a factor analysis on a dataset with 100 observations and 20 variables. What is the sample size-to-variable ratio?
Answer: The sample size-to-variable ratio is an important consideration in factor analysis to ensure reliable results. In this case, the sample size-to-variable ratio is 100/20 = 5, indicating a ratio of 5 observations per variable. Generally, a sample size-to-variable ratio of at least 5:1 is recommended, although a higher ratio is desirable for more accurate factor extraction.
Question 3: After performing an exploratory factor analysis on a dataset, the researcher obtained communalities ranging from 0.6 to 0.9. What do these values indicate?
Answer : Communalities represent the proportion of variance in each variable that can be explained by the extracted factors. In this case, communalities between 0.6 and 0.9 indicate that a substantial portion (60% to 90%) of the variance in each variable is accounted for by the factors extracted during the analysis. Higher communalities suggest a better fit between the variables and the extracted factors.
Question 4: In a confirmatory factor analysis, a researcher assessed the goodness of fit using the chi-square test. The obtained chi-square value was 150 with 80 degrees of freedom. Is the model a good fit?
Answer: In confirmatory factor analysis, the chi-square test assesses the discrepancy between the observed and expected covariance matrices. However, the chi-square test is sensitive to sample size, and with large samples, even trivial deviations from the hypothesized model can result in a significant chi-square value. It is advisable to consider additional fit indices, such as the Comparative Fit Index (CFI) or Root Mean Square Error of Approximation (RMSEA), to evaluate model fit more comprehensively.
Question 5: A researcher conducted a factor analysis and obtained factor loadings ranging from -0.2 to 0.8. How should these factor loadings be interpreted?
Answer: Factor loadings represent the strength and direction of the relationship between each variable and the factors. In this case, factor loadings ranging from -0.2 to 0.8 indicate the extent to which each variable is associated with the extracted factors. Positive loadings (e.g., 0.8) suggest a positive relationship, while negative loadings (e.g., -0.2) suggest a negative relationship. Higher absolute values indicate a stronger association between the variable and the factor.
Question 6: In a factor analysis, the researcher obtained eigenvalues for the first five factors as follows: 3.2, 2.1, 1.5, 1.2, and 0.9. How many factors should be retained based on the eigenvalue criterion?
Answer: The eigenvalue criterion suggests retaining factors with eigenvalues greater than 1. In this case, only the first two factors have eigenvalues above 1 (3.2 and 2.1). Therefore, two factors should be retained based on the eigenvalue criterion.
Question 7: A researcher performed a factor analysis on a dataset with 200 observations. The Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy was found to be 0.8. What does this value indicate about the suitability of the data for factor analysis?
Answer: The KMO measure of sampling adequacy assesses the suitability of data for factor analysis. It ranges from 0 to 1, with values closer to 1 indicating better suitability. In this case, a KMO value of 0.8 suggests that the dataset has good suitability for factor analysis, indicating that the variables share enough common variance to justify factor extraction.
Question 8: In a factor analysis, the researcher identified three factors. The factor loadings for a particular variable on these factors were 0.7, 0.3, and 0.1. How would you interpret these factor loadings?
Answer: Factor loadings indicate the strength and direction of the relationship between variables and factors. In this case, the variable has the highest loading of 0.7 on the first factor, indicating a strong positive association with that factor. The loadings of 0.3 and 0.1 on the second and third factors, respectively, suggest weaker associations. It is important to note that a loading of 0.1 might be considered relatively low and may indicate a less substantial relationship.