What Is a Factor in Data Analysis and EFA
What Is a Factor
You open your SPSS output and see columns such as Component, Eigenvalue, and Factor Loading, or a table called Rotated Component Matrix, but you are not sure what they mean. In data analysis, a factor is a latent concept inferred from the correlations among several observed variables. For example, ease of use, simple operation, and the ability to learn quickly may all reflect one latent factor, namely “ease of use.”
A factor is not a variable that you measure directly with one question. You measure it through a group of observed variables in the questionnaire, usually coded as SD1, SD2, SD3 or PU1, PU2, PU3. Exploratory factor analysis, usually called EFA, tries to group highly correlated variables together and separate them from other groups.
Technically, EFA analyzes the correlation matrix among observed variables. Each factor is represented by a linear combination of those variables. The results you usually need to examine include the number of extracted factors, Eigenvalue, the percentage of variance explained, and the factor loading, also called factor loading.
If your study uses concepts such as service quality, satisfaction, or usage intention, see what is a scale to distinguish the concept, scale, and observed variables before running EFA.
The Meaning of Factors in Quantitative Research
A factor helps you check the structure of a scale before putting the variables into regression, SEM, or PLS-SEM. In terms of the research question, you are checking whether the observed variables designed to measure one concept actually move together in the data you collected.
For example, your model has three concepts: information quality, trust, and continuance intention. Each concept has four observed variables. If EFA produces three groups and the variables for each concept load strongly onto the corresponding group, the scale structure is consistent with your original expectation.
If one observed variable loads strongly onto two factors, its content may overlap with both concepts. If two concepts that were expected to be distinct appear in the same group, review the questionnaire wording, survey context, or theoretical basis. EFA provides evidence from the data, but it does not decide the meaning of a factor for you. You still need to read the item content and compare it with the research model.
In a quantitative workflow, EFA usually comes after checking scale reliability and before regression or a structural model. You can see the Statistics topic to place EFA within the full data-processing workflow. If the data contain highly skewed values or unusual observations, check outliers before interpreting the factor results.
EFA can also reveal problems that Cronbach's Alpha does not show. A scale may have a reasonable Alpha while its items actually divide into two different groups. In that situation, looking only at Alpha is not enough to conclude that the scale is unidimensional.
How Many Factors Are Acceptable
There is no fixed number of factors that applies to every study. A reasonable number must fit the theoretical model, the content of the observed variables, and the indicators in the output at the same time. Do not choose a number of factors simply because the software produces a cleaner-looking table.
The thresholds below are commonly used when reading EFA. Each row includes a source that you can explain in Chapter 4.
| Indicator or criterion | Reference level | Source |
|---|---|---|
| KMO | 0.5 or above | (Kaiser, 1974) |
| Bartlett's Test of Sphericity | Sig. below 0.05 | (Kaiser, 1974) |
| Eigenvalue | Above 1 | (Kaiser, 1960) |
| Total Variance Explained | 50% or above | (Hair et al., 2010) |
| Factor loading | 0.5 or above | (Hair et al., 2010) |
| Corrected Item-Total Correlation | 0.3 or above | (Nunnally and Bernstein, 1994) |
| Observations per variable | About 5 to 10 observations per variable | (Hair et al., 2010) |
KMO indicates whether the data are suitable for factor analysis. Bartlett's Test checks whether the correlation matrix among the variables differs from an identity matrix. In SPSS output, you usually look at the Sig. column for Bartlett's Test. If Sig. is below 0.05, the variables have enough linear correlation to continue considering EFA.
Eigenvalue reflects the amount of variance explained by a factor. The rule of retaining an Eigenvalue above 1 is commonly used, but you should also examine the Scree Plot and the theoretical meaning of each group. A factor that only just exceeds 1 and has two difficult-to-interpret items is not necessarily a good factor.
Total Variance Explained shows what percentage of the variation in all observed variables is explained by the retained factors. A level of 50% or above is commonly used as a reference in quantitative research according to (Hair et al., 2010). This is a reference point, not a reason to retain a structure that has no meaningful content.
A factor loading of 0.5 or above is commonly considered acceptable according to (Hair et al., 2010). An item with a loading of 0.52 should not automatically be judged better than an item with 0.48 if the content, cross-loading, and scale reliability have not been considered together.
How to Read Factors in the Output
In SPSS, the common path for running EFA is Analyze > Dimension Reduction > Factor. Move the observed variables you want to check into the Variables box, select Descriptives to enable KMO and Bartlett's test, and select Initial solution. Under Extraction, you can choose Principal Axis Factoring when the goal is to explore the latent structure. If your department requires Principal Component Analysis, report the method you actually used in the thesis.
Under Rotation, choose Varimax when you assume the factors are uncorrelated. If the concepts in your model may be correlated, Promax is usually more suitable for interpretation. Under Options, select Sorted by size and set the display cutoff to a suitable level, such as 0.50, so the table is easier to read. Hiding small coefficients only makes the output shorter. It does not remove information from the calculation.
The key table for reading item groups is Rotated Component Matrix with Varimax, or Pattern Matrix and Structure Matrix when you use an oblique rotation such as Promax. Check which factor has the highest loading for each item, whether the item has a meaningful cross-loading on another factor, and then compare the item code with the questionnaire content.
Illustrative output
The table below is illustrative output, presented in the form of SPSS output. It is not the result of a specific study and must not be copied into a thesis as real data.
| Table in SPSS | Indicator or variable | Component 1 | Component 2 | Component 3 |
|---|---|---|---|---|
| KMO and Bartlett's Test | KMO Measure of Sampling Adequacy | 0.812 | ||
| KMO and Bartlett's Test | Bartlett's Test, Sig. | 0.000 | ||
| Total Variance Explained | Factor Eigenvalue | 4.218 | 2.106 | 1.347 |
| Total Variance Explained | Cumulative % | 35.150 | 52.700 | 63.920 |
| Rotated Component Matrix | PU1 | 0.812 | 0.174 | 0.092 |
| Rotated Component Matrix | PU2 | 0.779 | 0.213 | 0.118 |
| Rotated Component Matrix | TR1 | 0.146 | 0.824 | 0.201 |
| Rotated Component Matrix | TR2 | 0.205 | 0.756 | 0.244 |
| Rotated Component Matrix | YD1 | 0.101 | 0.226 | 0.801 |
| Rotated Component Matrix | YD2 | 0.187 | 0.281 | 0.748 |
With these illustrative figures, KMO reaches the reference level and Bartlett has Sig. equal to 0.000, so the data are suitable for considering EFA. Three factors have an Eigenvalue above 1, and the cumulative total variance explained is 63.920%. The variables PU1 and PU2 load highest on Component 1, TR1 and TR2 on Component 2, and YD1 and YD2 on Component 3.
When reading your real output, check every variable, not only two representative variables such as those in the example. An item with its highest loading at 0.60 on Component 1 and another loading of 0.55 on Component 2 needs attention because the gap between the two loadings is small. The decision to remove an item should be based on its content, theory, reliability, and the purpose of the model.
What to Do When the Factor Structure Does Not Meet the Criteria
It is rare for EFA to give a clean matrix on the first pass. Follow an order of checks so that you do not remove items arbitrarily.
Check the Data and Coding
Review the data type, missing values, reverse-coded items, and response codes outside the scale. A reverse-coded item that has not been recoded may correlate negatively with the other items. Also check unusual observations before running the analysis again, but do not delete a row simply because it makes the indicators look better.
Check Items With Low Loadings or Cross-Loadings
Read the Rotated Component Matrix or Pattern Matrix, and record items with low loadings, cross-loadings, or a single item standing alone on a factor. Compare the item wording with the concept. If an item does not measure the concept properly, you have a methodological basis for considering its removal.
Remove One Item at a Time and Rerun the Analysis
Remove only one item at a time, then rerun EFA to see how the structure changes. Save the output from every round, the code of the removed item, and the reason for removal. The committee may ask why PU3 was removed, so your answer should be based on its loading, cross-loading, or measurement content, rather than simply saying that “SPSS reported it.”
Review the Number of Factors
If the extracted number of factors does not match the model, examine the Scree Plot, Eigenvalue, and theoretical meaning. Two concepts may be too similar, or the questionnaire may not distinguish them well. Forcing the number of factors to match the hypotheses is appropriate only when you have a clear theoretical reason for doing so.
Return to Scale Design When Necessary
If the items still load onto mixed groups after several rounds, or a factor has only one item left, the problem may be in the scale or the sample data. The software cannot repair the wording of an item. Discuss whether to retain, remove, or redesign the scale with your supervisor instead of repeatedly running the same file.
Distinguishing Factors From Latent Variables and Moderators
A factor is a structure extracted from the relationships among observed variables in an analysis, usually EFA or CFA. A latent variable is a concept that is not directly observed and is measured through several observed variables in a measurement model. In many theses, the two terms are used in similar ways, but “factor” emphasizes the result of the analysis, while “latent variable” emphasizes the concept's role in the model.
A moderator changes the strength or direction of the relationship between an independent variable and a dependent variable. It is not a group of items formed by EFA. You can read more about moderator and moderating if your output involves interactions among variables.
A mediator explains the mechanism through which an independent variable affects a dependent variable. For example, service quality may affect loyalty through satisfaction. A mediator is also a role in a structural model, not a factor identified only through the Rotated Component Matrix. See mediator to keep these two concepts separate when writing hypotheses.
A concept may be a factor in the EFA results and a latent variable in the model at the same time. However, a factor does not automatically become a moderator or mediator. That role must be defined by the research model and hypotheses.
Common Errors
The first error is calling every column in Rotated Component Matrix a “factor” without checking the number of extracted factors. The Component column is only the way SPSS numbers groups in the output. You need to name each factor based on the items with high loadings and the theoretical basis.
The second error is retaining an item because its loading barely reaches 0.50 even though it has a high cross-loading or makes the group content difficult to interpret. Numerical thresholds need to be read together with the scale structure. Conversely, immediately removing every item below 0.70 is also mechanical, because the reference threshold for factor loading in this context is 0.50 according to (Hair et al., 2010).
The third error is using KMO to conclude that a scale is reliable. KMO concerns the suitability of the data for factor analysis. Internal reliability should be examined through Cronbach's Alpha, Corrected Item-Total Correlation, and other relevant indicators.
The fourth error is mixing EFA with PLS-SEM. In SmartPLS, you will see Outer Loadings, Composite Reliability, AVE, and HTMT, not SPSS's Rotated Component Matrix. If you run a PLS-SEM model, do not put CFI, TLI, or RMSEA from CB-SEM into the SmartPLS assessment.
The final error is reporting only that “factors were found” without stating the extraction method, rotation method, initial number of items, number of removed items, and total variance explained. The reader needs to know how you made decisions at each stage.
Frequently asked questions
What is a factor in data analysis?
A factor is a latent concept inferred from the correlations among several observed variables. In EFA, a factor represents a group of variables that tend to vary together and is expressed through factor loading.
When should you use exploratory factor analysis, EFA?
Use EFA when you want to explore or preliminarily check the structure of a scale, especially when the scale has been adapted to a new context, population, or language. EFA is usually conducted before regression or a structural model.
What Eigenvalue is acceptable for a factor?
The common reference rule is an Eigenvalue above 1, according to (Kaiser, 1960). You still need to examine the Scree Plot, total variance explained, and whether the factor can be interpreted. An Eigenvalue that just exceeds 1 is not enough to prove that a factor is meaningful.
What factor loading is acceptable?
A factor loading of 0.50 or above is commonly used as a reference threshold according to (Hair et al., 2010). If an item has a cross-loading, examine the gap between the loadings, the item content, and the effect of retaining or removing the item.
What should you do when the extracted number of factors is unsuitable?
First, check the coding, reverse-coded items, missing data, and unusual observations. Then examine the Scree Plot, loadings, and scale content. You can remove one item with a low loading or cross-loading at a time, rerun the analysis, and record the reason. If the structure remains unstable, discuss the scale and model design instead of forcing the output to match your preference.
Should you remove items to make KMO and loadings look better?
Remove an item only when you have a clear methodological and content-based reason. Record the indicators before and after removal, check the scale reliability, and explain the decision in the thesis. An attractive output without a defensible basis will be difficult to defend when someone asks about it.
Open your SPSS file and record the extraction method, rotation method, KMO, Bartlett, number of factors, total variance, and items with low loadings before deciding on the next run. If you need to run the analysis on your own .sav or .csv file, you can use M4 analysis by DoThesis.