Eigenvalues Greater Than 1: What They Mean and Limits in (2026)

You run a PCA on your dataset, and the software tells you to keep every component with an eigenvalue greater than 1. But what does that number actually represent? Why 1 and not 0.8 or 1.5? And is that rule always safe to follow?

Our team has spent years working through principal component analysis across datasets ranging from 5-variable psychological surveys to 500-feature genomic panels. In that time, we have seen the eigenvalue greater than one rule called the Kaiser criterion save researchers hours of subjective guessing. We have also seen it produce badly misleading component counts that led to wrong conclusions.

This article breaks down what eigenvalues greater than one in PCA actually measure, where the threshold comes from mathematically, and why it can fail in real-world analyses. By the end, you will know when to trust the Kaiser criterion and when to pair it with more reliable alternatives like parallel analysis.

Whether you are working in SPSS, R, or Python, the principles here apply directly to your component selection process. We will cover the math, the intuition, the failure modes, and the practical workflow that experienced statisticians use every day.

If you have ever stared at a Total Variance Explained table wondering whether to trust the eigenvalue greater than 1 default, this guide will give you the confidence to make that call yourself.

Core Concepts: What Eigenvalues Measure in PCA

Before tackling the specific threshold of 1, we need to define what eigenvalues represent in principal component analysis. An eigenvalue measures the amount of variance in the original data that a given principal component captures. It is the single number that summarizes how much informational value a component carries.

Think of it this way. Each principal component is a new axis built from linear combinations of your original variables. The eigenvalue attached to that component tells you how much information (variance) that single axis holds. A larger eigenvalue means the component is doing more work to summarize your data.

When you run PCA on a correlation matrix of p variables, the sum of all eigenvalues always equals p. This is because each standardized variable contributes exactly 1 unit of variance to the total. The eigenvalue decomposition redistributes that total variance across the new principal components.

So if you have 8 variables, the sum of all 8 eigenvalues equals 8. The first component might grab an eigenvalue of 3.2 (meaning it explains 3.2 units of variance), the second might get 1.8, and the remaining six split what is left. This redistribution is what makes eigenvalues so useful for deciding which components matter.

The eigenvalues are always sorted in descending order. The first eigenvalue is always the largest, representing the component that captures the most variance. The last eigenvalue is always the smallest. This ordering is built into the mathematics of eigenvalue decomposition.

Eigenvalues and Eigenvectors Work Together

The eigenvector defines the direction of the principal component in variable space. The eigenvalue defines how much variance sits along that direction. You need both pieces of information to interpret a PCA result fully.

Component loadings, which are the correlations between original variables and principal components, are derived from eigenvectors. The sum of squared loadings for a component equals its eigenvalue. This relationship connects the abstract eigenvalue back to something you can interpret practically: how strongly each original variable contributes to the component.

For example, if Variable A has a loading of 0.8 on Component 1, its squared loading is 0.64. That means Variable A contributes 0.64 units to the eigenvalue of Component 1. If you sum all squared loadings across all variables for that component, you get the eigenvalue.

This is why researchers care about component loadings. A high eigenvalue with many moderate loadings tells a different story than a high eigenvalue driven by one or two dominant variables. The eigenvalue alone does not reveal which variables are responsible for the variance.

Variance Explained: The Percentage View

Researchers often convert eigenvalues into percentages of variance explained. If you have 8 variables (total variance = 8) and the first eigenvalue is 3.2, that component explains 3.2 divided by 8, or 40% of the total variance.

This percentage framing makes eigenvalues intuitive. A component with an eigenvalue of 0.5 in the same 8-variable analysis explains only 6.25% of the variance. That is a small contribution, which is exactly why the eigenvalue greater than 1 threshold exists as a natural cutoff point.

Cumulative variance explained adds up the percentages across retained components. If Component 1 explains 40% and Component 2 explains 22%, your two-component solution explains 62% of the total variance. Researchers use this cumulative figure to decide whether they have retained enough components to adequately represent the data.

There is no universal threshold for adequate cumulative variance. Some fields accept 60% as sufficient, while others demand 90% or higher. The right threshold depends on how much information loss your downstream analysis can tolerate.

The Trace Property: Why the Sum Matters

The sum of all eigenvalues equals the trace of the correlation (or covariance) matrix. For a correlation matrix, the trace equals the number of variables because each diagonal element is 1 (each variable correlates perfectly with itself after standardization).

For a covariance matrix, the trace equals the sum of all variable variances. This distinction matters enormously for the Kaiser criterion, as we will explain in the limitations section. The entire logic of eigenvalue greater than 1 depends on which matrix you start with.

The trace property is not just a mathematical curiosity. It is the foundation of the Kaiser criterion. Without understanding that the trace equals p for correlation matrices, the threshold of 1 seems arbitrary. With it, the threshold becomes a logical consequence of how variance is distributed.

Standardized vs Unstandardized Variables

When you perform PCA on a correlation matrix, you are implicitly standardizing all variables to have a mean of 0 and a standard deviation of 1. This puts every variable on equal footing regardless of its original measurement scale.

Standardization is what makes the average eigenvalue equal to 1. Without it, variables with larger variances would dominate the first few components, and the eigenvalues would reflect measurement scale rather than underlying data structure.

This is why most PCA applications use the correlation matrix. If your variables are measured in the same units and you specifically want variance-based weighting, you might use the covariance matrix instead. But then the Kaiser criterion does not apply, as we will discuss at length.

The Kaiser Criterion: Where the Eigenvalue Greater Than 1 Rule Comes From

The eigenvalue greater than 1 rule comes from Henry Kaiser’s 1960 paper and builds on earlier work by Louis Guttman. Kaiser proposed that any principal component with an eigenvalue exceeding 1 should be retained. This became known as the Kaiser criterion or the Kaiser-Guttman rule.

The logic is straightforward and elegant. When you perform PCA on a correlation matrix of p variables, the average eigenvalue is exactly 1 (since the total variance equals p and there are p eigenvalues). A component with an eigenvalue above 1 captures more variance than a single original variable contributes. A component below 1 captures less.

Kaiser originally developed this rule for psychological research, where he frequently dealt with survey data containing 20 to 50 items. He needed a practical way to decide how many factors or components to extract without relying on subjective judgment alone. The eigenvalue greater than 1 threshold provided that objectivity.

Historical Context: Kaiser and Guttman

Louis Guttman laid the groundwork in 1954 when he proved that the weakest lower bound for the number of common factors equals the number of eigenvalues greater than 1 in certain matrices. Kaiser took this mathematical result and turned it into a practical decision rule.

The rule gained rapid acceptance because it was simple, objective, and easy to implement in the computing environments of the 1960s and 1970s. Before modern simulation methods like parallel analysis were computationally feasible, the Kaiser criterion was the best available option for many researchers.

SPSS adopted the Kaiser criterion as its default extraction method when it added factor analysis capabilities. This single software decision embedded the rule into the workflow of millions of researchers worldwide. Even today, many practitioners default to eigenvalue greater than 1 simply because that is what SPSS shows them.

Mathematical Derivation: Why the Average Equals 1

Let us walk through the math step by step. When variables are standardized (as they must be when using a correlation matrix), each variable has a variance of exactly 1. The total variance in the dataset is therefore p (the number of variables).

The eigenvalues from PCA partition this total variance. The sum of all p eigenvalues equals p. Divide p eigenvalues into a total of p, and the average eigenvalue is p divided by p, which equals exactly 1.

This is why 1 is not an arbitrary number. It represents the contribution of a single average variable. Kaiser reasoned that any component worth keeping should explain at least as much variance as one original variable. Otherwise, the component adds complexity without adding meaningful information.

The mathematical notation makes this precise. If lambda_1, lambda_2, through lambda_p are the eigenvalues from PCA on a correlation matrix of p variables, then the sum of all lambda values equals p. The average eigenvalue is (sum of lambda values) divided by p, which equals p divided by p, which equals 1.

Step-by-Step Application of the Kaiser Criterion

Applying the rule in practice follows a clear sequence:

Step 1: Standardize your variables if they are on different scales. This ensures you are working with a correlation matrix, which is what the Kaiser criterion assumes. Most PCA implementations do this automatically when you request a correlation-based analysis.

Step 2: Perform eigenvalue decomposition on the correlation matrix. Every statistical package (SPSS, R, Python’s scikit-learn) provides these eigenvalues as standard output. In R, use the prcomp or princomp functions. In Python, use sklearn.decomposition.PCA.

Step 3: Count the number of eigenvalues greater than 1. This count is your recommended number of principal components to retain. For example, if eigenvalues are 3.2, 1.8, 0.9, and 0.1 for a 4-variable PCA, you retain 2 components.

Step 4: Extract that many components and proceed with your analysis, including rotation if needed for interpretability. Varimax rotation is the most common choice for orthogonal rotation.

Step 5: Verify the decision using at least one other method, such as a scree plot or parallel analysis. Never rely on the Kaiser criterion alone. We will discuss why in detail in the limitations section.

Step 6: Examine the component loadings to confirm that each retained component is substantively interpretable. A component with an eigenvalue of 1.2 but no clear loading pattern may not be worth keeping, regardless of the criterion.

The Intuition Behind the Threshold

The threshold of 1 answers a simple question: is this component pulling its weight? Each original variable contributes one unit of variance to the pool. If a principal component cannot account for at least that much, it is essentially summarizing less information than a single variable.

Think of it like assigning team members to carry boxes. If each person brings one box and you form a new group (a principal component), that group should carry at least one box to justify its existence. A group carrying 0.6 boxes is dead weight.

This analogy breaks down slightly when you consider that components combine information from multiple variables. A component with an eigenvalue of 0.8 might still be useful if it reveals a meaningful pattern. But Kaiser argued that such components add more complexity than they are worth.

What Eigenvalues Greater Than 1 Actually Mean

An eigenvalue greater than 1 means the corresponding principal component explains more variance than any single original variable contributes when working with a standardized correlation matrix. This is the core interpretation of the Kaiser criterion and the answer to the central question of this article.

More specifically, it tells you that the component successfully combines information from multiple variables into a single dimension that captures meaningful structure. Components with eigenvalues below 1 are either capturing noise, sampling error, or redundant fragments of variance that individual variables already represent on their own.

When a component has an eigenvalue above 1, you know that at least two variables are sharing enough variance to create a dimension stronger than either variable alone. This is a sign of real underlying structure rather than random fluctuation.

A Concrete Numerical Example

Let us say you have 6 variables measuring different aspects of customer satisfaction: product quality, delivery speed, customer service, pricing, website usability, and return policy. After standardizing and running PCA on the correlation matrix, you get these eigenvalues: 2.8, 1.5, 0.9, 0.4, 0.25, and 0.15.

The sum is 6.0 (confirming the trace property). The average eigenvalue is 1.0. Two eigenvalues exceed 1, so the Kaiser criterion tells you to retain 2 components.

Component 1 explains 2.8 out of 6 total variance units, or roughly 47% of the total variance. Component 2 explains 1.5 out of 6, or about 25%. Together they account for 72% of the variance in your 6 variables using only 2 dimensions. That is a meaningful reduction from 6 to 2.

Component 3, with an eigenvalue of 0.9, falls just short of the threshold. It explains 15% of the variance on its own. Whether to include it depends on whether that extra 15% adds practical value to your analysis. This is where the Kaiser criterion gives you a starting point but not a final answer.

The remaining components (0.4, 0.25, 0.15) together explain only about 13% of the variance. These are almost certainly noise or trivial variance fragments. Dropping them is a safe decision that virtually any method would support.

A Second Example: When Kaiser Gives the Wrong Answer

Now consider a different scenario. You have 30 survey items measuring various dimensions of workplace culture. After running PCA, you get eigenvalues where 11 components exceed 1.

The Kaiser criterion tells you to retain 11 components. But 11 components from 30 variables is barely a reduction at all. You have gone from 30 to 11, which means your dimensionality reduction is only about 63%.

When you run parallel analysis on the same data, it suggests that only 5 components have eigenvalues exceeding what random data would produce. The other 6 components that Kaiser retained are capturing random noise patterns that happen to push eigenvalues above 1.

This is the over-extraction problem in action. With 30 variables, you expect roughly 10 components to have eigenvalues above 1 purely by chance. The Kaiser criterion has no mechanism to distinguish real structure from this random baseline.

Interpreting Eigenvalue Magnitude

The absolute size of an eigenvalue also carries meaning. An eigenvalue of 4.2 in a 10-variable PCA means that component alone explains 42% of the total variance. That is a dominant component, likely capturing a strong underlying factor in your data.

Conversely, an eigenvalue of 1.05 barely crosses the threshold. It explains only marginally more variance than a single variable. Retaining it is defensible under the Kaiser criterion, but the practical value of that component may be limited.

This is why experienced analysts look at the gap between eigenvalues, not just whether they cross 1. A sharp drop from 2.5 to 1.1 to 1.05 to 0.9 tells a different story than a gradual decline from 2.5 to 2.1 to 1.8 to 1.5. The first pattern suggests a clear two-component structure. The second suggests a more gradual variance distribution where the component count is less obvious.

When Only One Eigenvalue Exceeds 1

A common question on forums like r/AskStatistics involves running PCA and getting only one eigenvalue above 1. This does not mean your analysis failed. It means your variables are largely unidimensional, or the shared structure in your data concentrates into a single dominant component.

This outcome is common when variables are highly intercorrelated. A single component absorbs most of the variance, leaving little for subsequent components. Whether that is desirable depends on your research question. In some cases, a single dominant component is exactly what you want.

For example, if you designed a questionnaire to measure one latent construct (like job satisfaction), getting a single dominant component is evidence of unidimensionality. That is a positive result. But if you expected multiple dimensions, you may need to reconsider your variable selection or sample size.

One practical note: if only one eigenvalue exceeds 1 and it accounts for a very large percentage of variance (say 60% or more), check for a general factor. A dominant first component can sometimes mask multidimensional structure that would emerge with different extraction methods or rotation.

Limitations and Criticisms of the Eigenvalue Greater Than 1 Criterion

The Kaiser criterion has serious limitations that every practitioner should understand. It is a heuristic, not a statistical test, and it produces misleading results in several common scenarios. Here are the most significant problems we have encountered in practice.

1. Over-Extraction With Many Variables

The biggest criticism of the Kaiser criterion is that it tends to retain too many components when you have a large number of variables. The reason is mathematical and unavoidable.

When you have p variables, you expect roughly p divided by 3 components to have eigenvalues above 1 purely by chance if the variables are uncorrelated. So with 30 variables, you might get 10 components above 1 even when no real structure exists. With 100 variables, the problem gets dramatically worse.

This happens because random correlation patterns in the sample create spuriously large eigenvalues. The more variables you have, the more opportunities for these random fluctuations to push eigenvalues above 1. Researchers have documented this over-extraction problem since the 1970s through extensive Monte Carlo simulation studies.

Zwick and Velicer published influential studies in the 1980s and 1990s comparing component retention methods. Their findings consistently showed that the Kaiser criterion over-extracts relative to parallel analysis and other modern methods, especially with more than 20 variables.

2. It Only Works Properly With Correlation Matrices

The entire logic of eigenvalue greater than 1 rests on the average eigenvalue being exactly 1. This is only true when PCA is performed on a correlation matrix where each variable contributes exactly 1 unit of variance.

If you run PCA on a covariance matrix instead, variables contribute different amounts of variance depending on their measurement scales. A variable measured in thousands contributes far more variance than one measured in decimals. The average eigenvalue is no longer 1, and the Kaiser criterion loses its meaning entirely.

With a covariance matrix, a large eigenvalue might simply reflect a variable with large variance, not meaningful data structure. The Kaiser criterion should never be applied to covariance-based PCA without adjusting for this difference.

Some practitioners try to work around this by checking whether each eigenvalue exceeds the average eigenvalue (trace divided by p) rather than 1. This adjusted criterion is more appropriate for covariance matrices but is rarely implemented in standard software.

3. PCA and Factor Analysis Are Not the Same

One of the most common mistakes we see on forums is applying the Kaiser criterion to exploratory factor analysis. SPSS makes this dangerously easy because it displays eigenvalue greater than 1 as the default extraction criterion even when you run a factor analysis.

In PCA, the total variance being decomposed includes all variance: common (shared between variables), specific (unique to each variable), and error. In factor analysis, only shared (common) variance is analyzed. The eigenvalues mean different things in each context.

When SPSS runs principal axis factoring or maximum likelihood factor analysis, the eigenvalues reflect common variance only. The total of these eigenvalues is less than p because specific and error variance have been excluded. The average eigenvalue is no longer 1.

The Kaiser criterion was designed for PCA eigenvalues, not factor analysis eigenvalues. Applying it to FA can lead to retaining too few or too many factors. This confusion is widespread because SPSS uses PCA as its default extraction method for factor analysis, further blurring the line between the two techniques.

If you are running exploratory factor analysis, use parallel analysis based on FA eigenvalues, not PCA eigenvalues. The psych package in R handles this correctly by offering separate parallel analysis functions for PCA and FA.

4. It Is Not a Statistical Test

The Kaiser criterion provides no measure of statistical significance. It cannot tell you whether a component with an eigenvalue of 1.02 is meaningfully different from one with an eigenvalue of 0.98. There is no confidence interval, no p-value, no hypothesis test.

This is a heuristic dressed up as a rule. It gives a clean, objective-sounding threshold, but that threshold has no probabilistic foundation. Researchers who treat it as a definitive statistical test are making a category error.

Parallel analysis, by contrast, is a proper statistical method. It generates random datasets with the same number of variables and observations, computes their eigenvalues, and compares them to your observed eigenvalues. If your eigenvalue exceeds what random data produces, you have evidence of real structure beyond chance.

This probabilistic foundation is why methodologists increasingly recommend parallel analysis over the Kaiser criterion. The eigenvalue greater than 1 rule tells you nothing about the likelihood that your result reflects real structure versus sampling noise.

5. Sample Size Effects

The Kaiser criterion does not account for sample size. A dataset with 50 observations and one with 5,000 observations get the same eigenvalue greater than 1 cutoff, even though the smaller sample produces much less stable eigenvalue estimates.

With small samples, sampling error inflates some eigenvalues and deflates others. A component with an eigenvalue of 1.15 in a sample of 40 might easily have a true eigenvalue of 0.85 in the population. The Kaiser criterion gives you no warning about this instability.

Monte Carlo studies have consistently shown that the Kaiser criterion performs poorly with samples under 200 observations. With 100 or fewer cases, the criterion can both over-extract and under-extract components depending on the data structure.

Large samples (1,000 or more observations) produce more stable eigenvalues, but even then the over-extraction problem with many variables remains. Sample size alone does not fix the fundamental weakness of a fixed threshold.

6. It Ignores Component Interpretability

A component with an eigenvalue of 1.3 might be uninterpretable because it splits variance across many variables with weak loadings. A component with an eigenvalue of 0.85 might have a clean, interpretable loading pattern that aligns perfectly with a theoretical construct.

The Kaiser criterion treats eigenvalue magnitude as the only criterion for retention. Real-world analysis requires considering whether components make substantive sense. A slightly below-threshold component that maps onto a meaningful construct is often more valuable than a slightly above-threshold component that represents statistical noise.

Experienced researchers always check component loadings after applying any retention criterion. If a retained component has no loadings above 0.4 on any variable, it is probably not interpretable and should be dropped regardless of its eigenvalue.

7. The Boundary Problem

Components with eigenvalues near 1 create a boundary problem. Is an eigenvalue of 0.97 really that different from 1.03? In practical terms, probably not. But the Kaiser criterion treats them as categorically different: one is retained, the other is dropped.

This binary decision ignores the continuity of eigenvalue distributions. When multiple eigenvalues cluster around 1, the criterion becomes essentially arbitrary. Any small change in the data (removing an outlier, adding a case, using a different standardization method) could flip components across the threshold.

The boundary problem is especially acute in datasets where eigenvalues decline gradually. If your eigenvalues are 2.8, 1.9, 1.3, 1.1, 0.95, 0.85, the decision to retain 4 components versus 3 feels almost coin-flip arbitrary. The Kaiser criterion offers no guidance for resolving this ambiguity.

8. Sensitivity to Outliers and Non-Normality

PCA is sensitive to outliers and deviations from normality. A few extreme observations can dramatically inflate or deflate eigenvalues. The Kaiser criterion, applied blindly, will produce different component counts depending on whether outliers are present.

Always inspect your data for outliers before running PCA. Consider robust PCA methods if your data contains extreme values that are not errors. The eigenvalue greater than 1 threshold was developed under the assumption of reasonably well-behaved data, and it may perform poorly with heavily non-normal distributions.

Alternative Methods for Component Selection

Given the limitations above, the Kaiser criterion should never be your only tool for deciding how many components to retain. Several alternatives address its weaknesses and provide more reliable guidance. We rank these methods from most to least recommended based on simulation evidence.

Scree Plot (Cattell’s Scree Test)

The scree plot, introduced by Raymond Cattell in 1966, plots eigenvalues from largest to smallest. You look for the point where the curve flattens out, forming an elbow. Components before the elbow are retained; those after are dropped.

The scree plot is more flexible than the Kaiser criterion because it accounts for the shape of the eigenvalue distribution rather than applying a fixed threshold. However, it introduces subjectivity. Different analysts may identify the elbow at different points, especially when the curve is gradual rather than sharply bent.

Many practitioners overlay a horizontal line at eigenvalue equals 1 on the scree plot, combining the Kaiser criterion with visual inspection. This hybrid approach gives you both the fixed threshold and the distribution context.

The name scree comes from a geological metaphor. In a mountain landscape, the steep cliff face represents meaningful components with large eigenvalues. The rubble (scree) at the bottom represents trivial components with small eigenvalues. The elbow marks where the cliff turns into rubble.

Parallel Analysis (Horn’s Method)

Parallel analysis, developed by John Horn in 1965, is widely considered the gold standard for component retention. Simulation studies consistently rank it as the most accurate method across a wide range of data structures.

The procedure works as follows:

First, generate a large number of random datasets (typically 1,000 or more) with the same number of variables and observations as your real data. These datasets contain no actual structure because the values are randomly sampled from a normal distribution.

Second, run PCA on each random dataset and record the eigenvalues. Compute the average eigenvalue for each component position across all random datasets. You now have a baseline of what eigenvalues look like when no real structure exists.

Third, compare your observed eigenvalues to these random averages. Retain only components where your observed eigenvalue exceeds the corresponding random eigenvalue average. Some implementations use the 95th percentile of random eigenvalues rather than the mean for a more conservative test.

Parallel analysis directly addresses the over-extraction problem because it accounts for the eigenvalues you would expect by chance given your specific number of variables and sample size. In R, the psych package provides the fa.parallel function for this purpose. In Python, you can implement it with Monte Carlo simulation using numpy and scikit-learn.

Explained Variance Threshold

Some researchers set a cumulative variance threshold, retaining enough components to explain 70%, 80%, or 90% of the total variance. This approach is pragmatic and easy to communicate to non-technical stakeholders.

The downside is that the threshold is arbitrary. There is no statistical reason to prefer 80% over 75% or 85%. The right threshold depends on how much information loss your application can tolerate.

In practice, explained variance thresholds work best as a secondary check. After selecting components via parallel analysis or scree plot, verify that you have explained a reasonable proportion of variance. If your parallel analysis says 3 components but those 3 only explain 45% of variance, you may need to reconsider your variable selection or measurement strategy.

Broken Stick Model

The broken stick model compares observed eigenvalues to the eigenvalues expected if variance were distributed randomly across components. It provides a theoretical baseline similar to parallel analysis but without the need for simulation.

Under the broken stick model, the expected proportion of variance for the k-th component follows a specific formula based on the total number of components. If your observed eigenvalue exceeds the broken stick expectation, the component is retained.

The broken stick model is less commonly used than parallel analysis but provides a useful cross-check. When both methods agree on the number of components, you can be confident in your retention decision.

Which Method Should You Use?

Here is a practical ranking based on simulation studies and expert consensus:

Parallel analysis is the most accurate and should be your primary method. The scree plot is an excellent visual complement. The Kaiser criterion is useful as a quick initial check but should never be the sole basis for your decision. Explained variance thresholds work well as a secondary consideration when you need to communicate results to non-statisticians. The broken stick model provides a good theoretical cross-check.

In our practice, we always run parallel analysis and a scree plot. We check the Kaiser criterion for comparison. If all three converge on the same number, we proceed immediately. If they disagree, we investigate the specific eigenvalues near the threshold and make a judgment call based on component interpretability.

Practical Guidance: When to Use and When to Avoid Kaiser

Despite its limitations, the Kaiser criterion is not useless. It provides a fast, objective starting point that requires no computation beyond the eigenvalues themselves. The key is knowing when it is reliable and when it will mislead you.

When the Kaiser Criterion Works Well

Use the Kaiser criterion confidently when you have a moderate number of variables (roughly 10 to 30) and a sample size of at least 200. In this range, the criterion tends to produce reasonable component counts that align with parallel analysis results.

It also works well as a quick sanity check. If the Kaiser criterion says 3 components and parallel analysis says 3, you can proceed with confidence. If they disagree significantly, you know to investigate further.

The criterion is also useful in exploratory settings where you need a fast initial read on your data structure. Running PCA and counting eigenvalues above 1 takes seconds. Setting up parallel analysis takes longer. For a first pass, the Kaiser criterion is perfectly adequate.

When to Avoid the Kaiser Criterion

Avoid relying on the Kaiser criterion when you have more than 30 variables. The over-extraction problem makes it unreliable in high-dimensional settings. Stick with parallel analysis in these cases.

Also avoid it with small samples (under 150 observations). Eigenvalues are unstable with limited data, and the fixed threshold of 1 cannot account for that instability.

Never use the Kaiser criterion with covariance-based PCA. The average eigenvalue is not 1 in that context, making the entire logic of the rule inapplicable. If you must use a covariance matrix, switch to parallel analysis or explained variance thresholds.

Avoid it for factor analysis as well. The Kaiser criterion was designed for PCA eigenvalues. Using it with factor analysis eigenvalues is a methodological error, even though SPSS allows it.

Software Implementation Notes

In SPSS, the Kaiser criterion is the default extraction method for PCA. The Total Variance Explained table highlights components with eigenvalues greater than 1. Be aware that SPSS also applies this criterion by default to factor analysis, which as we discussed, is methodologically problematic.

In R, you can check eigenvalues directly using the eigen function on your correlation matrix, or use the psych package’s pca and fa.parallel functions for a comprehensive analysis with parallel analysis built in. The nScree function in the nFactors package also provides multiple retention criteria simultaneously.

In Python, scikit-learn’s PCA does not automatically apply the Kaiser criterion. You need to extract explained_variance_ from the fitted PCA object and compare each value to 1 manually. This gives you full control but also full responsibility for making the right call. The factor_analyzer package provides additional tools for component selection.

A Simple Workflow for Component Selection

Based on our experience across hundreds of PCA analyses, here is a workflow that balances simplicity with rigor:

Start by running PCA and examining all eigenvalues. Apply the Kaiser criterion for a quick initial count. Then generate a scree plot and identify the elbow visually. Finally, run parallel analysis and compare all three results.

If all three methods agree, retain that number of components. If they disagree, lean toward parallel analysis as the most reliable method. Document your reasoning so others can evaluate your decision process.

After retaining components, always examine the loading matrix. Each retained component should have at least 3 variables with substantial loadings (above 0.4 or 0.5). If a component fails this test, consider dropping it even if its eigenvalue exceeds 1.

Finally, consider whether rotation improves interpretability. Varimax rotation (orthogonal) is the most common default. Promax or oblimin rotation (oblique) allows correlated components, which may better represent real-world data structure.

FAQs

Conclusion

Understanding what eigenvalues greater than one in PCA represent is essential for anyone doing dimensionality reduction. The Kaiser criterion gives you a quick, mathematically grounded starting point: components above 1 explain more variance than an average variable contributes, which is a reasonable minimum bar for retention.

But that threshold has real limits. It over-extracts with many variables, does not work with covariance matrices, gets misapplied to factor analysis, ignores sample size, and offers no statistical significance testing. Treat it as a first approximation, not a final answer.

The best practice is to combine the Kaiser criterion with a scree plot and parallel analysis. When these methods converge on the same component count, you can proceed with confidence. When they diverge, let parallel analysis guide your decision and document your reasoning for transparency.

Run your PCA carefully, check your eigenvalues against multiple criteria, and always ask whether the retained components make substantive sense for your research question. That combination of statistical rigor and practical judgment will serve you far better than any single threshold rule.