What Is a Rotated Matrix and How to Fix a Messy EFA in SPSS
When to use EFA and read the rotated matrix
You use Exploratory Factor Analysis, usually abbreviated as EFA, when you want to check which factors the items in your questionnaire actually group into. For example, your model may include Service Quality, Convenience, and Continuance Intention. After entering the data into SPSS, you need to check whether the items load onto the expected groups or cross-load onto other groups.
The rotated matrix is the Rotated Component Matrix table in the SPSS output. Each row is an item, each column is a factor, and each cell contains a factor loading. This table gives you the basis for deciding which items represent a factor well and which items are adding noise.
EFA is appropriate when your research question asks how many factors these items can be reduced to, or whether the scale structure in your research context resembles the theoretical model. You can also read the overviews of EFA and Principal Component Analysis to distinguish the goal of exploring factors from the extraction method used in SPSS.
EFA is usually performed after checking reliability with Cronbach's Alpha and before running regression or a structural model. If the questionnaire already has a clearly confirmed structure from previous research, CFA may be more appropriate. In that situation, read the guide to CFA instead of forcing EFA into every case.
Preparing the data before running EFA
A messy rotated matrix can sometimes come from the data rather than from the EFA settings. Before running the analysis, open Data View and Variable View and check each group of items.
First, items must be entered as numbers. With a 5-point Likert scale, responses need to be coded consistently from 1 to 5. The Measure column is usually set to Scale. Identification variables such as respondent ID, gender, or occupational group should not be entered into EFA together with the scale items.
Second, check missing values. A row with too many blank cells can reduce the number of observations used in the analysis. If you coded missing responses with a number such as 99, declare it as a missing value or handle it before running EFA. Leaving 99 in the dataset can seriously distort the correlations between items.
Third, check reverse-coded items. If an item is negatively worded, so that a high level of agreement represents a low evaluation, you must reverse-code it before running Cronbach's Alpha and EFA. In SPSS, you can use Transform > Recode into Different Variables, create a new variable, and then check the coding again. Do not delete a negatively worded item just because its factor loading is low before confirming that the reverse coding was done correctly.
Fourth, review the variable codes. Names such as CLDV1, CLDV2, TT1, and TT2 help you identify item groups in the output. Avoid very long names or spaces because tracking the results across multiple runs becomes harder.
You also need to decide which items to include in the same EFA run. If the model has several independent scales and one dependent scale, many theses run EFA on all suitable items together. Some designs run EFA separately for the independent and dependent item groups. The important point is that your choice must fit the model and be reported consistently, rather than being selected simply because it produces a cleaner table.
If you are unsure whether the data meet the conditions, check communality after a trial run. Low communality shows that an item is explained poorly by the extracted factors, but you should not delete an item immediately based on one number alone.
Steps for running EFA in SPSS
Open the Factor Analysis dialog box
Go to Analyze > Dimension Reduction > Factor. From the list on the left, select the items you want to analyze and move them to the Variables box. Do not include a composite dependent variable, respondent ID, or demographic variable unless it is an item belonging to the scale.
Select the data checks in Descriptives
Click Descriptives. Select KMO and Bartlett's test of sphericity so that SPSS creates the KMO and Bartlett's Test table. You can also select Anti-image if you need to inspect the MSA for each item. Select Initial solution to obtain initial information about variance and eigenvalues.
KMO and Bartlett help you assess whether the data are suitable for factor analysis. This is a conditions check, not a conclusion that the rotated matrix has passed.
Select the extraction method in Extraction
Click Extraction. Under Method, choose the method that fits your research design. Principal components is often used in theses to reduce data, while actual factor-analysis methods such as Principal axis factoring are appropriate when the goal is to identify latent factors. State your choice clearly in the methods chapter.
Select Scree plot to view the scree plot. Under Extract, you can leave Eigenvalues over 1 if you are using the rule that the eigenvalue must be greater than 1, or specify the number of factors based on the theoretical model. The eigenvalue-greater-than-1 rule is associated with (Kaiser, 1960), but you should still examine the scree plot and theoretical meaning rather than mechanically retaining every factor SPSS extracts.
Select the rotation method in Rotation
Click Rotation. Choose Varimax when you assume that the factors are uncorrelated, or an oblique rotation such as Direct Oblimin when the concepts may be related. In behavioral research, factors are often related, so the rotation method needs to be justified by the model rather than selected only because it produces an easier table to read.
If you choose Varimax, SPSS usually creates a Rotated Component Matrix table. With an oblique rotation, you may see a Pattern Matrix and a Structure Matrix. Do not call every post-rotation table a Rotated Component Matrix when the actual output shows a different table.
Select the display settings in Options
Click Options, select Sorted by size so that SPSS sorts the factor loadings in each row by size. Select Suppress small coefficients and enter a value such as 0.50 if you want to hide small coefficients in the table. This option only makes the table easier to read. It does not remove data from the analysis.
After clicking Continue and then OK, save the output from each run. Name the files in sequence, such as EFA_vong_1 and EFA_vong_2. If you need to remove an item, you must know which item was removed and how the results changed.
Reading the output
Read the output in order instead of jumping straight to the Rotated Component Matrix.
The KMO and Bartlett's Test table reports KMO, Bartlett's test, and the significance level. The Total Variance Explained table reports the number of extracted factors, eigenvalues, and the percentage of explained variance. The Communalities table reports how much of each item is explained by the factors. Finally, read the Rotated Component Matrix or Pattern Matrix to examine the factor-loading structure.
The following is illustrative output, not the result of a real study:
| Item | Component 1 | Component 2 | Component 3 | Illustrative decision |
|---|---|---|---|---|
| CLDV1 | 0.812 | 0.184 | 0.102 | Retain, loads well on factor 1 |
| CLDV2 | 0.764 | 0.221 | 0.116 | Retain, loads well on factor 1 |
| TT1 | 0.193 | 0.781 | 0.205 | Retain, loads well on factor 2 |
| TT2 | 0.276 | 0.694 | 0.248 | Retain, loads well on factor 2 |
| YD1 | 0.401 | 0.438 | 0.512 | Examine cross-loading, do not remove yet |
| YD2 | 0.126 | 0.208 | 0.823 | Retain, loads well on factor 3 |
In this example, CLDV1 has its highest loading on Component 1. YD1 has 0.401 on Component 1 and 0.438 on Component 2, while its highest loading is only 0.512 on Component 3. This indicates a cross-loading. Before deciding, examine the item wording, the difference between the loadings, and the theoretical structure.
A messy rotated matrix often shows several warning signs: many items do not load clearly onto their expected group, one item has two similar loadings, items from the same scale split across several columns, or the rotated matrix does not appear at all. EFA rarely looks clean on the first run. Removing items, rerunning the analysis, and recording each round is common, provided every decision has a reason.
Evaluation criteria
The thresholds below are reference points that need to be considered alongside the research context, sample size, and scale content. Each row includes a source you can trace when writing your thesis.
| Criterion | Common benchmark | Source |
|---|---|---|
| KMO | 0.5 or above | (Kaiser, 1974) |
| Bartlett's Test | Sig. below 0.05 | (Kaiser, 1974) |
| Eigenvalue | Greater than 1 | (Kaiser, 1960) |
| Factor loading | 0.5 or above | (Hair et al., 2010) |
| Total variance explained | 50% or above | (Hair et al., 2010) |
| Corrected Item-Total Correlation | 0.3 or above at the reliability stage | (Nunnally and Bernstein, 1994) |
| Observations per item | Approximately 5 to 10 observations | (Hair et al., 2010) |
These benchmarks do not turn a poor table into an acceptable one simply by deleting items repeatedly. A factor loading of 0.5 or above is a commonly used benchmark (Hair et al., 2010), but you also need to examine cross-loadings. If an item loads 0.62 on its expected factor and 0.58 on another factor, retaining it only because 0.62 is slightly higher may weaken the discriminant validity of the scale.
When evaluating a cross-loading item, examine four points: the size of the highest loading, the difference from the second loading, whether the item wording could belong to both concepts, and whether deleting it would remove important content from the scale. No decision should be based on one isolated number.
If KMO is low, review the correlations between items, the MSA values in the Anti-image table, and items with very low communality. If Bartlett's test is not significant, the data do not show that the correlation matrix is suitable for factor extraction. Running several more rounds will usually not solve the root problem.
How to write this in your thesis
You can report the extraction method, rotation method, number of factors, KMO, Bartlett's test, total variance explained, and the factor-loading results. Do not copy the entire SPSS output into the main text when the table is too long. Put a formatted results table in Chapter 4 and keep the original output for comparison if you are asked about it.
How to write this in your thesis: “The EFA results using the extraction method [Principal components/Principal axis factoring] and the rotation method [Varimax/Direct Oblimin] showed that the KMO value was [KMO], while Bartlett's test had Sig. = [Sig.], indicating that the data were [suitable/not suitable] for factor analysis. [Number of factors] factors were extracted, explaining [x]% of the total variance. The factor loadings ranged from [min] to [max], except for item [item code], which was [removed/retained] because [reason].”
If you removed an item, record the order and reason: cross-loading, low loading, low communality, or content mismatch. Avoid writing that “SPSS automatically removed the item” because the software only provides the output. The decision belongs to the researcher.
Common errors when running EFA
The rotated matrix is messy and items jump across groups
Check reverse-coded items first, then examine the content of the cross-loading items. Rerun EFA after removing one clearly problematic item, changing one controlled element at a time. Do not remove many items simultaneously because you will not know which change caused the structure to shift.
If the factors are theoretically correlated, consider an oblique rotation instead of Varimax. Changing the rotation is not a way to hide a poorly designed scale. The new result still needs to fit the concepts and research hypotheses.
The Rotated Component Matrix does not appear
Check whether you selected Rotation in the Factor dialog box. If only one factor was extracted, SPSS may not create a rotated table because there are not multiple factors to rotate. You should also check the Total Variance Explained table to see how many factors SPSS actually extracted.
If the items have very low correlations or the data contain missing-value errors, EFA may not run correctly. Read the message in the Output window, check the number of valid observations, and review codes such as 99, 999, or accidentally entered text symbols.
One item loads highly on two factors
Do not remove an item just because it has two large numbers. Read the item wording and check whether it is measuring two concepts at the same time. If the item clearly belongs to one concept but cross-loads in the data, check data-entry errors, reverse coding, and response quality before making a decision.
Items from the same scale split across several factors
Review the specified number of factors, extraction method, rotation method, and the quality of the questionnaire translation. If several items continue to split across rounds, the theoretical structure or the wording in your context may differ from what you expected. Report the actual result and discuss it with your supervisor instead of forcing the items into their original group.
Cronbach's Alpha decreases after an item is removed
EFA and Cronbach's Alpha answer different questions. An item can have a reasonably good loading but reduce internal reliability, or the reverse can happen. Rerun the reliability analysis for the final structure after EFA. Do not use the Alpha for the pre-removal item set to draw conclusions about the new item set.
Frequently asked questions
What is a rotated matrix in SPSS?
The rotated matrix is a table showing the factor loading of each item on the factors after SPSS performs the rotation. With Varimax, the table is usually called Rotated Component Matrix. Read across each row to see which factor column has the strongest loading for the item.
What factor loading is acceptable in a rotated matrix?
A factor loading of 0.5 or above is commonly used as a reference point (Hair et al., 2010). In addition to the highest value, you need to examine cross-loadings, the differences between loadings, KMO, Bartlett's test, total variance explained, and the substantive meaning of the item.
How should you handle a messy EFA?
Check missing values, reverse-coded items, incorrect coding, and the item groups first. Then examine the loading, communality, and content of each item, remove problematic items one at a time with a reason, and rerun the analysis. Changing the rotation may be appropriate when factors are correlated, but it should not be used simply to make the output look better.
Why does SPSS not show an EFA rotated matrix?
A common reason is that you did not select a rotation under Analyze > Dimension Reduction > Factor > Rotation, or that the result extracted only one factor. Check Total Variance Explained, review the data, and make sure that at least two factors were extracted before looking for the rotated table.
Should you delete a cross-loading item immediately?
Review the item content, highest loading, second loading, difference between them, and the effect on the scale's content coverage. If you delete the item, record the reason and rerun EFA and Cronbach's Alpha for the remaining items. Removing items in bulk just to meet a threshold can make the scale lose its meaning.
Open your .sav file, save the first EFA output, mark each item with a low loading or cross-loading, and compare the results with the thresholds and sources in this article. If you need to run the analysis on your own SPSS or CSV data, M4 analysis performs this step with SPSS and SmartPLS without generating data.