What Is an Eigenvalue in EFA and SPSS
What Is an Eigenvalue
You open the Total Variance Explained table after running EFA and see values such as 4.862, 1.734, and 0.928. These numbers are eigenvalues, also called characteristic roots. In EFA, an eigenvalue shows how much variation in the set of observed variables is explained by a factor. When the observed variables have been standardized, each original variable has a variance of 1. An eigenvalue can therefore be understood approximately as the number of equivalent standardized observed variables explained by a factor.
For example, an eigenvalue of 4.862 means that the factor explains an amount of information roughly equivalent to 4.862 standardized observed variables. To convert this value into a percentage of variance, SPSS divides the eigenvalue by the total number of observed variables and multiplies the result by 100%:
% of variance explained by a factor = eigenvalue / number of observed variables × 100%
If you have 20 observed variables and the first factor has an eigenvalue of 4.862, its individual percentage of variance is 24.31%. This is illustrative output used to explain the calculation, not the result of a specific study.
Eigenvalues appear in analytical methods such as Principal Component Analysis and Common Factor Analysis. In your thesis, check the correct output table, extraction method, and factor-retention criterion instead of seeing one large number and drawing a conclusion immediately. If you are distinguishing EFA from other procedures, you can also read What Is EFA and Principal Component Analysis.
The Meaning of Eigenvalue in Quantitative Research
An eigenvalue helps you decide whether a group of observed variables can be retained as a factor. Each factor represents part of the latent structure in your data. For example, AT1, AT2, AT3, and AT4 may converge on a factor called “Attitude”, while NT1, NT2, and NT3 form a factor called “Perception”. The eigenvalue shows how much information from the original variables is retained by that factor.
In EFA, you usually interpret eigenvalue together with three other sources of information: the Scree Plot, the % of Variance, and the theoretical meaning of the result. The eigenvalue criterion helps prevent you from retaining too many factors that contain very little information. It is a supporting decision criterion, not the only evidence for naming a factor or confirming a measurement model.
Eigenvalue is also connected to the number of extracted factors. If you ask SPSS to extract factors using Eigenvalues over 1, the software retains factors with eigenvalues greater than 1. If you select Fixed number of factors, SPSS extracts exactly the number of factors you enter, even when some of those factors have eigenvalues below 1. These two choices can produce different numbers of factors.
Place eigenvalue within the full scale-validation process. Before EFA, check reliability with Cronbach's Alpha and review Corrected Item-Total Correlation. After EFA, read the factor loading, total variance explained, and the structure of the rotated matrix. Eigenvalue does not replace the rotated matrix, because the rotated matrix shows which factor each observed variable loads on most strongly.
What Eigenvalue Is Acceptable
The rule commonly used in EFA is to retain factors with eigenvalues greater than 1. This rule is called the Kaiser criterion and is associated with (Kaiser, 1960). The reasoning is that a factor with an eigenvalue below 1 explains less variance than one single standardized observed variable.
You should not write that every eigenvalue greater than 1 must automatically be retained. Eigenvalue depends on the number of variables, sample size, correlations among the variables, and extraction method. A factor with an eigenvalue of 1.08 may need further review using the Scree Plot and theoretical meaning, while a factor with an eigenvalue of 0.92 may still require further consideration when there is a clear theoretical basis.
| Criterion | How to read it in EFA | Permitted source to cite |
|---|---|---|
| Eigenvalue | Retain factors with eigenvalues greater than 1 under the Kaiser criterion | (Kaiser, 1960) |
| KMO | A KMO of 0.5 or above indicates the minimum level of suitability for factor analysis | (Kaiser, 1974) |
| Bartlett's Test | Sig. below 0.05 indicates that the correlation matrix differs from an identity matrix | (Kaiser, 1974) |
| Total Variance Explained | Total variance explained of 50% or above is commonly used as a reference point | (Hair et al., 2010) |
| Factor loading | A factor loading of 0.5 or above is commonly considered acceptable | (Hair et al., 2010) |
The table above is a set of reference criteria, not an automatic decision rule. For example, if the fourth factor has an eigenvalue of 1.12 but only one variable loads strongly on it, review the scale structure. Conversely, if the third factor has an eigenvalue of 0.96, three observed variables load clearly on it, and your theoretical model predicts this factor, discuss the decision with your supervisor before removing it.
When reporting the result, state the extraction method and factor-retention criterion clearly. The sentence “EFA retained four factors because their eigenvalues were greater than 1” is more informative than “the EFA results showed that the model was appropriate”, because the reader can see exactly what supported the decision.
How to Read Eigenvalue in the Output
In SPSS, go to Analyze > Dimension Reduction > Factor, select the observed variables, click Extraction, and choose the extraction method appropriate for your research design. If you select Scree plot and Unrotated factor solution, the output will include the Total Variance Explained table. This is the first table to open when checking eigenvalues.
This table usually contains three groups of columns: Initial Eigenvalues, Extraction Sums of Squared Loadings, and Rotation Sums of Squared Loadings. Each group contains Total, % of Variance, and Cumulative %. For PCA, the Initial Eigenvalues group lists the eigenvalues of all components before the software decides how many components to retain. After extraction, SPSS displays only the portion corresponding to the retained factors. After rotation, the cumulative variance explained by the retained factors remains the same, but the individual percentage for each factor may change.
Illustrative output
The table below simulates part of the SPSS Total Variance Explained output. The numbers are intended only to show you how to read the table, not to report the findings of an actual project. This is illustrative output.
| Component | Initial Eigenvalues Total | % of Variance | Cumulative % | Extraction Sums of Squared Loadings Total |
|---|---|---|---|---|
| 1 | 4.862 | 24.310 | 24.310 | 4.862 |
| 2 | 2.137 | 10.685 | 34.995 | 2.137 |
| 3 | 1.428 | 7.140 | 42.135 | 1.428 |
| 4 | 1.106 | 5.530 | 47.665 | 1.106 |
| 5 | 0.928 | 4.640 | 52.305 | Not retained |
| 6 | 0.771 | 3.855 | 56.160 | Not retained |
In this example, the first four components have eigenvalues greater than 1, so the Kaiser criterion suggests retaining four factors. Component 1 explains 24.310% of the variance, component 2 explains an additional 10.685%, and the first four factors together explain 47.665% before you consider how the software displays the extraction or rotation section. If the total variance explained by the retained factors reaches 50% or above in your actual output, you can compare it with the reference point in (Hair et al., 2010).
You also need to inspect the Scree Plot. When the line drops sharply and then begins to flatten, the bend is often used as a reference for the number of factors. If the Scree Plot suggests three factors but eigenvalues greater than 1 suggest four, compare both results with the theoretical model, the number of observed variables under each factor, and the rotated matrix. Do not choose a number simply because it produces a cleaner-looking table.
What to Do When the Eigenvalue Does Not Meet the Criterion
If a factor has an eigenvalue below 1, first check that you are reading the correct column and using the correct extraction method. A common mistake is confusing Initial Eigenvalues with Rotation Sums of Squared Loadings, or reading a component that was not retained as if it were the final factor in the model.
Recheck the Extraction Selection
Open the Extraction dialog again and check whether Analyze is set to Correlation matrix or another option. Check whether Extract is set to Eigenvalues over 1 or Fixed number of factors. If you entered a fixed number of factors, an eigenvalue below 1 may appear because SPSS is following the instruction you entered.
Check the Quality of the Observed Variables
If the EFA result is unclear, review Cronbach's Alpha, Corrected Item-Total Correlation, communality, and factor loading. An item with low communality or a high cross-loading can make the factor structure difficult to interpret. You can also read about communality to understand how much of each variable's variance is explained by the common factors.
Removing an item must be based on both the statistics and the content of the scale. Do not keep deleting several items simply to push the eigenvalue above 1. Record every removal round, including the item code, the reason for removal, the change in eigenvalue, and the factor structure after rerunning the analysis. The committee may ask why an item was removed, not only whether the final table looks good.
Consider the Number of Factors Based on Theory
If the eigenvalue is close to 1, review the Scree Plot and the conceptual model. In some cases, you can run EFA with a fixed number of factors based on the model you developed and then inspect the rotated matrix. Explain this approach clearly in the methodology and results chapters. Do not quietly change the criterion simply to obtain the result you want.
Distinguishing Eigenvalue from Eigenvector and Factor Loading
Eigenvalue and eigenvector often appear together in linear algebra, but they answer different questions. An eigenvalue is a scalar showing the amount of variation or information associated with a direction or factor. An eigenvector is a vector showing the direction, which you can think of as a weighted combination of the variables that forms that direction.
In EFA, you do not read the eigenvector as a coefficient for deciding which factor an observed variable belongs to. When reading SPSS output, you usually focus on factor loading in the Rotated Component Matrix or Pattern Matrix, depending on the rotation and extraction methods. Factor loading shows the relationship between an observed variable and a factor. A factor loading of 0.5 or above is commonly cited under (Hair et al., 2010).
Eigenvalue and factor loading also operate at different levels of interpretation. Eigenvalue describes the amount of information in the factor as a whole, while factor loading describes the relationship between one observed variable and one factor. A factor with a high eigenvalue does not necessarily mean that every variable under that factor has a good loading.
You should also distinguish eigenvalue from communality. Communality shows the proportion of an observed variable's variance explained by the common factors. Eigenvalue is aggregated at the factor level. The two measures can therefore appear in the same EFA process while answering different questions.
Common Mistakes
The first mistake is using the sentence “the higher the eigenvalue, the better” as a complete conclusion. A high eigenvalue shows that a factor explains more variance, but you still need to check whether the factor is formed by observed variables that make sense together.
The second mistake is looking only at eigenvalues greater than 1 while ignoring the Scree Plot, total variance explained, and rotated matrix. The Kaiser criterion is useful for initial screening, but it can suggest a different number of factors from the theoretical structure of the scale.
The third mistake is citing a threshold without giving its source. When you write “the eigenvalue must be greater than 1”, cite (Kaiser, 1960). When you write “total variance explained should be 50% or above”, cite (Hair et al., 2010). Do not assign the factor-loading threshold to eigenvalue, because these are different measures.
The fourth mistake is calling every result in the table a “factor” without checking the extraction method. If you use PCA, SPSS displays components. If you use a common factor method, the interpretation may refer to factors. Use terminology that matches the selection in the dialog and the method reported in your methodology chapter.
The final mistake is repeatedly changing the data or deleting items without saving versions. Keep the original data file, create a separate analysis copy, and name the versions according to each EFA round. This allows you to explain why the eigenvalue, number of factors, and total variance explained changed.
Frequently asked questions
Must a factor with an eigenvalue greater than 1 be retained?
This is a commonly used criterion for suggesting the number of factors and is associated with (Kaiser, 1960). You should also compare it with the Scree Plot, total variance explained, factor loading, and theoretical model. A factor with an eigenvalue only slightly above 1 requires careful interpretation.
What eigenvalue is acceptable in EFA?
The commonly used reference point is an eigenvalue greater than 1. This is a factor-retention rule under the Kaiser criterion, not a criterion for evaluating the entire scale. The scale also needs to be assessed through KMO, Bartlett's Test, factor loading, total variance explained, and the structure after rotation.
Why did I select Fixed number of factors but obtain an eigenvalue below 1?
When you select Fixed number of factors, you instruct SPSS to extract the exact number of factors you entered. The software can therefore retain a factor with an eigenvalue below 1. Check whether that number of factors fits the theoretical model and rotated matrix, then explain the selection criterion clearly.
Are eigenvalue and eigenvector the same?
These concepts are mathematically related but are not the same. An eigenvalue is a value showing the amount of information or variance associated with a direction, while an eigenvector describes that direction. When reading EFA output, you usually focus more on eigenvalue, factor loading, and the rotated matrix than on the eigenvector.
Can eigenvalue replace factor loading?
No. Eigenvalue describes the amount of variance explained by a factor at the overall factor level. Factor loading shows which factor each observed variable is associated with. You need to read both measures to evaluate the number of factors and the quality of the scale structure.
Open the Total Variance Explained table in your SPSS file, record the eigenvalue for each component, check the extraction criterion in Extraction, and compare the result with the Scree Plot and rotated matrix before writing the findings. If you need to run EFA on your own .sav or .csv file and turn the output into Chapter 4 writing, you can use M4 data analysis.