Reporting Factor Analysis in SPSS
From Pattern Matrix to Loadings Table
Reporting factor analysis in SPSS means stating how a solution was built and then showing it: the extraction method, the retention rule, the rotation, the sampling checks and a pattern loadings table. Each decision earns a sentence of justification.
IN SHORT
The Decisions and Figures a Factor Report States
A factor analysis report lets a reader judge whether the solution is trustworthy and repeat it. It states the extraction method, how the number of factors was decided, the rotation and the reason for it, the sampling checks, the loadings that define each factor and the variance the solution accounts for. Reporting KMO and Bartlett’s test, the rotated matrices and the variance explained all draw on values from the Factor procedure under Analyze, Dimension Reduction, and labels differ between releases. Each figure comes from a table SPSS has already printed, so the difficulty lies in picking the right table and giving its figures the right name. OnlineSPSS is a specialist statistical consultancy for academic, clinical and professional research, and a factor analysis report is among the documents its PhD-level statisticians write up or verify from saved output.
In This Guide
METHOD SUPPORT
Where Reporting Meets the Method Itself
A solution still being decided, or an approach built on components instead of factors, is handled on the two method pages below.
Exploratory Factor Analysis (EFA)
Item pools examined for their latent structure, with the retention and rotation decisions made and justified for your data.
Principal Component Analysis (PCA)
Component scores that summarize correlated items, where the aim is data reduction and not a model of latent constructs.
BEFORE YOU REPORT IT
Decisions the Report Must Defend
Each decision is stated in a sentence, because a reader cannot recover any of them from the loadings table alone.
Extraction Method State whether principal axis factoring, maximum likelihood or another method was used. The Factor dialog starts from principal components, a different technique, so the choice must be made on purpose.
Retention Rule Report the rule that fixed the number of factors. Eigenvalues above 1 are a weak basis alone, so the scree plot and a comparison with eigenvalues from random data are reported beside them.
Rotation and Its Reason An orthogonal rotation fits only when uncorrelated factors are plausible. Constructs that are likely related call for an oblique rotation, and the factor correlations are then reported.
Cutoffs Set in Advance State the minimum loading, the cross-loading rule and the sample size before items are dropped, and report which items were removed, in what order and for what reason.
IN SPSS
Choosing the Source Table for Each Statement
The decisions behind the solution are made in EFA in SPSS, and this guide starts from the printed output.
Take the KMO and Bartlett Values
From the KMO and Bartlett’s Test table, take the Kaiser-Meyer-Olkin measure, and Bartlett’s chi-square with its df and Sig.
Check: Report KMO as a screen, with values below about .50 a warning. A significant Bartlett result is expected in most samples and does not show that the data are suitable.
Take the Variance Explained
From Total Variance Explained, take the Extraction Sums of Squared Loadings for the retained factors, with the % of Variance and Cumulative %.
Watch: Initial Eigenvalues describe the correlation matrix before extraction. With principal axis factoring the extraction figures are lower, and they are the ones that describe the factor solution.
Record the Evidence for Retention
Record the eigenvalues from Total Variance Explained, the Scree Plot and the random-data comparison produced by the macro or program used for it.
Tip: The Factor dialog offers no random-data comparison in many releases, so name the macro or program that produced it, with its version, in the methods text.
Take the Pattern Matrix Loadings
After an oblique rotation, take the loadings from the Pattern Matrix, not the Structure Matrix, and read them with the Communalities table.
Check: Sorted by size and Suppress small coefficients, both under Options, make the table readable. The cutoff used for suppression is then stated, since it hides the cross-loadings.
Take the Factor Correlations
For an oblique rotation, take the Factor Correlation Matrix and report the correlation between each pair of factors.
Watch: A very high correlation between two factors suggests they may be one. Look at it before naming the factors, and consider whether a solution with fewer factors fits the items better.
Match Items and Signs to the Matrix
Enter the loadings and variance figures in the table below, then compare it with the rotated output for the same items, the same signs and the same order.
Check: Count the items in the final table against the items entered, and list those dropped with the reason. A reader can then see that the tabled solution is the one that was analyzed.
READING THE OUTPUT
Where Each Factor Analysis Value Belongs
KMO and Bartlett's Test, Total Variance Explained and the rotated matrices carry the report, and the type of rotation decides which matrix holds the loadings.
| Output value | Where SPSS prints it | Where it goes and what to check |
|---|---|---|
| KMO Measure of Sampling Adequacy | KMO and Bartlett’s Test table | The sampling screen; item-level values on the anti-image diagonal are inspected if the overall value is low |
| Bartlett’s Test of Sphericity | KMO and Bartlett’s Test table | Chi-square, df and Sig. in one clause; a significant result is expected and says little alone |
| Communalities, Extraction | Communalities table | The h² column of the loadings table; a low value flags an item that fits the solution poorly |
| Initial Eigenvalues | Total Variance Explained | Evidence for retention, not the variance the factors explain |
| Extraction Sums of Squared Loadings | Total Variance Explained | The variance explained by the retained factors, named as extraction sums |
| Rotation Sums of Squared Loadings | Total Variance Explained | After an oblique rotation these cannot be added, so no cumulative figure is taken from them |
| Pattern Matrix, or Rotated Factor Matrix after Varimax | Printed after rotation | The loadings table, one column per factor, with the cutoff stated |
| Structure Matrix | Printed after an oblique rotation | Correlations that include shared variance; supplementary, and not reported as loadings |
| Factor Correlation Matrix | Printed after an oblique rotation | The correlations in the table note or in a line below the table |
REPORTING
A Methods Paragraph, Loadings Table and Variance Labels
The paragraph names each decision, and the table shows pattern loadings with communalities, so a reader can follow the solution from item to factor.
The factor loadings table APA 7 readers expect gives each retained factor a column and each item a row, with loadings below the stated cutoff left blank or set in a lighter style. The communality of each item and the factor correlations sit in the table or its note. The frame below describes a two-factor solution with an oblique rotation, and a different number of factors adds or removes columns. Variance explained is labeled by what it is, the extraction sums of squared loadings, and the sample size and the number of items belong in the methods text.
A factor analysis with [principal axis factoring / maximum likelihood] extraction and [direct oblimin / promax] rotation was run on ___ items from N = ___ participants. Sampling adequacy was [adequate / limited], KMO = ___, and Bartlett’s test was [significant / not significant], χ²(___) = ___, p = ___. The number of factors, ___, was set by [the scree plot and the comparison with random data]. After extraction the factors explained ___% of the variance, and they correlated at r = ___.
| Table 1. Pattern Matrix Loadings and Communalities for the [Scale] Items | |||
|---|---|---|---|
| Item | Factor 1 | Factor 2 | h² |
| [Item 1] | ___ | ___ | ___ |
| [Item 2] | ___ | ___ | ___ |
| [Item 3] | ___ | ___ | ___ |
| % of variance (extraction sums) | ___ | ___ | |
| Note. Extraction: ___. Rotation: ___. Loadings below ___ are suppressed. Factor correlation: r = ___. h² = communality. | |||
Variance explained is quoted from the extraction sums of squared loadings, because the initial eigenvalues describe the correlation matrix before any factor is extracted. With principal axis factoring the extraction percentages are lower than the initial ones, and the report uses the lower figures. After an oblique rotation the percentages for rotated factors cannot be added, so the extraction total is given. Communalities show what each item shares with the retained factors, and the loading cutoff, such as .30 or .40, is stated with its source.
COMMON MISTAKES
A Solution Can Be Right and Still Misreported
Reading the final table against the printed matrices catches most of these, and the methods text catches the rest.
Structure Matrix Reported as Loadings After an oblique rotation the Structure Matrix mixes an item's own loading with variance shared through correlated factors. The loadings come from the Pattern Matrix.
Eigenvalues as Variance Explained The Initial Eigenvalues columns describe the matrix before extraction. The variance explained by the solution comes from the extraction sums of squared loadings.
Rotated Sums Added Into a Total After an oblique rotation SPSS prints no cumulative column for the rotation sums, because the factors overlap. Adding them by hand counts shared variance more than once, so the extraction total is quoted instead.
Cross-Loadings Left Unmentioned An item loading above the cutoff on two factors needs a sentence. Dropping it silently, or keeping it on the stronger factor, changes the scale analyzed later.
Negative Loadings Left Unexplained A reverse-worded item usually loads with the opposite sign to the other items on its factor until it is recoded. The table shows the sign with a note on the reverse keying, or reports the recoded item, and the text says which was done.
Items Dropped Without a Trail Removing items one pass at a time changes the solution on each pass. The report lists the items removed, in order, with the reason, and tables the final solution.
Not sure the loadings and communalities came from the right matrices? Independent Statistical Review sets every number in the table against the output table it came from.
FREQUENTLY ASKED QUESTIONS
Reporting Factor Analysis in SPSS Beyond the Loadings Table
A loadings table cannot name its factors, settle an item that loads twice or show that the sample was large enough, so the text has to.
How are factor names justified in the write-up?
From the items with the highest pattern loadings on each factor, listed so that a reader can judge the label. A name is an interpretation and not a result of the analysis, so a factor defined by only two items is named cautiously. Whether the names hold is tested later, in a new sample, with a confirmatory model.
How do I report an item that loads on two factors?
State the cutoff and the cross-loading rule, then say what was done with the item and why. An item may stay on its stronger factor when the gap between its loadings is clear, or be dropped when it is not. If the decision changes the factors, a note reports the solution before and after the change.
Does a ratio of cases to items justify my sample size?
No single ratio does, so the report gives N, the number of items, the communalities and the clarity of the factors together. Solutions with high communalities and well-defined factors hold up in smaller samples than those with low communalities, which is why rules based on a fixed number of cases per item are treated as rough guidance.
What happens to a loadings table sent to OnlineSPSS?
Either a table exists already or only the saved output does. In the first case an in-house PhD-level statistician at OnlineSPSS compares the table with the rotated output, matrix and variance labels included. In the second, the report is built from the output. Both fall under Results Reporting & Interpretation, and naming the factors is your interpretive step.
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You share your research question, design and relevant materials, such as the dataset or the output you have, and a statistically trained Research Coordinator reviews the request and responds the same day. The analysis, deliverables, timeline and one fixed project fee are then confirmed before any work begins, so the timeline follows from the agreed scope.
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NEXT STEP
Would the Table Survive
a Reviewer?
A factor report is judged on whether its matrix, its variance labels and its retention evidence can be traced. A table taken from the Structure Matrix fails the first of these.
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