Many correlated variables reduced to a few underlying components. What PCA actually does, why the default rule for choosing how many components to keep is the wrong one, and how to report it in APA 7.
Principal components analysis takes a set of correlated variables and rewrites them as a smaller set of uncorrelated components, each a weighted combination of the originals. The first component is the direction along which the data varies most, the second is the largest remaining direction perpendicular to it, and so on.
Each component carries an eigenvalue, which is the amount of variance it accounts for expressed in units of single variables. An eigenvalue of 9.49 across 18 variables means that component alone carries as much variance as roughly nine and a half of the original variables, or 52.7% of the total.
PCA is not factor analysis, though the two are routinely confused. PCA is a mathematical repackaging of observed variance with no model behind it: components are outputs, built from the items. Factor analysis posits latent variables that cause the item responses and separates shared variance from item-specific error. If your question is "what underlying construct produces these answers," you want exploratory factor analysis. If it is "can I compress these columns," PCA is the right tool.
The run opens with four figures: components above eigenvalue 1 by the Kaiser criterion, the number of variables in the analysis, the cases used with listwise deletion named, and the variance carried by the first component.
The Components card then lists every component with its eigenvalue, percent of variance, and cumulative percent.
Read the whole column, not the headline tile. That is where the real decision is.
The Kaiser criterion keeps every component with an eigenvalue above 1, on the logic that such a component explains more than a single variable would. It is the default in most software and it is known to over-extract.
A worked example from a real 18-variable run makes the point better than the argument does:
PC1 = 9.488 (52.7%) · PC2 = 1.351 (7.5%) · PC3 = 1.298 (7.2%) · PC4 = 0.601 (3.3%)
Kaiser says three components. But the first carries 52.7% on its own, the second and third barely clear the threshold at 1.35 and 1.30, and then the sequence falls off a cliff to 0.601. There is no meaningful gap between PC3 and PC4 in substance, only in relation to an arbitrary line at 1.0.
A scree reading, looking for the elbow where eigenvalues level off, says one dominant component. That is a materially different conclusion from the one the default rule produces, and it is the more defensible reading of these numbers.
Use cumulative variance as a third check, and above all use theory. Components you cannot interpret are not components you should retain.
A caution specific to scale data. If your items include reverse-worded questions that were never rescored, they will group together and can appear as a genuine additional component. Check the scoring before interpreting structure. See reverse-coded items.
Lay the data out one row per case, one column per variable, all numeric.
Open Data Reduction and choose Principal Components, then select your variables in the Setup panel on the right.
Watch the case count. Listwise deletion drops any case missing a single variable, so a wide analysis can lose a substantial share of the sample.
Use the action bar to add the result to a report, copy the table, or copy the APA text.
Report the number of variables and cases, the extraction decision and the rule behind it, the eigenvalues, and the cumulative variance.
A principal components analysis of 18 items (N = 250) produced a dominant first component (eigenvalue = 9.49) accounting for 52.7% of the variance. Although three components exceeded the Kaiser criterion, the scree plot and the sharp drop after the third eigenvalue supported a one-component solution.
State the retention rule you used and why. Reporting "three components were extracted" without naming the rule leaves a reader unable to judge the decision.
Where Kaiser and the scree disagree, say so and justify your choice. That disagreement is common and disclosing it is more credible than reporting whichever answer suited the argument.
Do not describe PCA output as factors, and do not describe components as latent variables. They are weighted composites of the items you supplied.
Every result carries a ReliCheck Intelligence card with one action: explain this result in plain language. It runs entirely on your Mac with Apple Intelligence. No upload, no API key, nothing to configure. The model only explains the result Quanta has already computed and validated. It never computes a statistic, and it never applies a decision on your behalf.
From the action bar you can add the result to a report, copy the table, or copy the APA text. Exported reports carry the APA table, its note, and a caution specific to that analysis beneath it.
Quanta's factor analysis and PCA are validated against an independent Python implementation and against closed-form values, recovering known simple structure. The engine is pure Swift with no R or Python at runtime. Full record on the validation page.
Data Reduction is part of every Quanta subscription, $9.99 a month or $79.99 a year after the 30-day trial. One price for everyone. See pricing.
The analysis, the report builder, and the on-device explanations all work with the network off. Your dataset is never uploaded.
McDonald's omega reliability from a single-factor model · Cronbach's alpha which cannot detect dimensionality · Correlation matrix what PCA is computed from · the complete list on the analyses page.