Question 1: You are conducting a factor analysis on a dataset with 10 variables. After performing the analysis, you obtain the following eigenvalues: 3.2, 2.8, 2.5, 2.1, 1.9, 1.6, 1.4, 1.2, 0.9, and 0.7. Determine the number of factors to retain based on the eigenvalues and explain your decision.
Answer: To determine the number of factors to retain, you can use the Kaiser's criterion, which suggests retaining factors with eigenvalues greater than 1. According to this criterion, eigenvalues greater than 1 indicate that the corresponding factor explains more variance than an individual variable.
In this case, the eigenvalues are as follows: 3.2, 2.8, 2.5, 2.1, 1.9, 1.6, 1.4, 1.2, 0.9, and 0.7. Based on Kaiser's criterion, you should retain factors with eigenvalues greater than 1.
From the given eigenvalues, the number of factors that should be retained is 4. The eigenvalues 3.2, 2.8, 2.5, and 2.1 are all greater than 1, indicating that these factors explain a significant amount of variance in the dataset. The remaining eigenvalues (1.9, 1.6, 1.4, 1.2, 0.9, and 0.7) are less than 1, suggesting that they explain relatively less variance and can be considered insignificant. Therefore, based on the eigenvalues, you should retain 4 factors in the factor analysis.
Question 2:
You are conducting a factor analysis on a dataset with 8 variables. After performing the analysis, you obtain the following communalities: 0.75, 0.60, 0.85, 0.70, 0.80, 0.65, 0.55, and 0.90. Determine the total variance explained by the retained factors based on these communalities.
Answer: To determine the total variance explained by the retained factors, you need to sum up the communalities of the variables.
In this case, the communalities for the 8 variables are as follows: 0.75, 0.60, 0.85, 0.70, 0.80, 0.65, 0.55, and 0.90.
To calculate the total variance explained, you sum up these communalities:
Total Variance Explained = Sum of Communalities
Total Variance Explained = 0.75 + 0.60 + 0.85 + 0.70 + 0.80 + 0.65 + 0.55 + 0.90
Total Variance Explained = 5.80
Therefore, the retained factors in the factor analysis explain a total variance of 5.80.
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Question 3:
You are conducting a factor analysis on a dataset with 12 variables. After performing the analysis, you obtain the following factor loadings for the retained factors:
Factor 1: 0.75, 0.85, 0.70, 0.80, 0.65, 0.55 Factor 2: 0.90, 0.80, 0.70, 0.75, 0.60, 0.50
Determine the total variance explained by these two factors based on the factor loadings.
Answer: To determine the total variance explained by the retained factors, you need to calculate the sum of the squared factor loadings for each factor.
In this case, the factor loadings for Factor 1 are: 0.75, 0.85, 0.70, 0.80, 0.65, 0.55. The factor loadings for Factor 2 are: 0.90, 0.80, 0.70, 0.75, 0.60, 0.50.
To calculate the total variance explained by each factor, you square each factor loading, sum them up, and then multiply by the number of variables: Total Variance
+ 0.55^2) * 6
* 6 = 14.445
Therefore, the two retained factors in the factor analysis explain a total variance of 12.225 and 14.445, respectively.
Question 4:
You are conducting a factor analysis on a dataset with 6 variables. After performing the analysis, you obtain the following factor loadings for the retained factors:
Determine the cumulative proportion of variance explained by these two factors based on the factor loadings. For
Answer: To determine the cumulative proportion of variance explained by the retained factors, you need to calculate the proportion of variance explained by each factor and then calculate the cumulative sum. In this case, the factor loadings for Factor 1 are: 0.85, 0.75, 0.60, 0.80, 0.70, 0.65.The factor loadings for Factor 2 are: 0.90, 0.80, 0.70, 0.75, 0.60, 0.55.To calculate the proportion of variance explained by each factor, you square each factor loading and divide by the total sum of squared loadings:
the values:
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0.491Now, to calculate the cumulative proportion of variance explained, we sum up the proportions of variance explained by each factor:
Cumulative Proportion of Variance Explained = Proportion of Variance Explained by Factor 1 + Proportion of Variance Explained by Factor 2Cumulative Proportion of Variance Explained = 0.509 + 0.491Cumulative Proportion of Variance Explained = 1.000Therefore, the two retained factors in the factor analysis explain a cumulative proportion of variance of 1.000, indicating that they capture all the variance in the dataset.
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Question 5:
You are conducting a factor analysis on a dataset with 10 variables. After performing the analysis, you obtain the following factor loadings for the retained factors:
0.80, 0.75, 0.60, 0.75, 0.65, 0.70, 0.75, 0.80 Determine the cumulative variance explained by these two factors based on the factor loadings.
Answer: To determine the cumulative variance explained by the retained factors, you need to calculate the variance explained by each factor and then calculate the cumulative sum. In this case, the factor loadings for Factor 1 are: 0.80,
calculate the variance explained by each factor, you square each factor loading, sum them up, and then divide by the total number of variables:
0.524Now, to calculate the cumulative variance explained, we sum up the variances explained by each factor:
/
Cumulative Variance Explained = Variance Explained by Factor 1 + Variance Explained by Factor 2Cumulative Variance Explained = 0.535 + 0.524Cumulative Variance Explained = 1.059Therefore, the two retained factors in the factor analysis explain a cumulative variance of 1.059, which exceeds 100%. This indicates an error in the calculation. Variance explained should not exceed 100% since it represents the proportion of total variance accounted for by the factors.To correct this issue, it's important to review the factor loadings and re-evaluate the calculations to ensure accuracy. Please double-check the factor loadings provided and confirm their correctness.
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