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A factor score is not the value of the factor — it is one of many possible estimates of it, and the three standard ways of computing that estimate (regression, Bartlett, and Anderson-Rubin) do not agree with each other, do not agree with the true factor, and trade off different properties against each other by design, not by accident. This is called factor score indeterminacy, and it is a property of the common factor model itself, not a symptom of a bad sample or a small dataset — it exists even when the model fits the population perfectly. This guide walks through how each estimation method is computed, what it optimizes and what it sacrifices, demonstrates the indeterminacy problem with a reproducible numeric example, and sets out the specific, common situations where an unweighted sum or mean score is the more defensible choice than any factor score at all.
What a Factor Score Actually Is
In exploratory or confirmatory factor analysis, the common factor (F) is a latent variable — it is never observed directly. What you observe are indicator variables (X) that F is assumed to influence, connected by factor loadings (λ) and item-specific unique variances (ψ). A factor score is a case-by-case numeric estimate of where each respondent or unit falls on F, built from a weighted combination of their observed indicator values.
The weights are not arbitrary. They come from solving an estimation problem, and which problem you solve determines which method you are using. The three methods below all use the same loadings and unique variances from the fitted model — they differ only in how they turn those parameters into a weight matrix.
The Three Standard Estimation Methods
Regression method (Thomson, 1951). Also called the Thomson method. Treats the factor score as the best linear predictor of F given X, in the ordinary least-squares sense — the weight matrix is R-1ΛΦ, where R is the indicator correlation matrix, Λ the loading matrix, and Φ the factor correlation matrix (identity if factors are orthogonal). It has the highest possible correlation with the true factor of any linear estimator — it is literally the maximum-validity choice. The cost: it is a regression prediction, so it is regressive — its variance is smaller than the true factor’s, extreme cases get pulled toward the mean, and in a multi-factor solution the estimate for one factor picks up extra correlation with the other true factors beyond what those factors actually correlate with each other. That last property is sometimes called a loss of univocality.
Bartlett method (Bartlett, 1937). A weighted (generalized) least-squares method: weight matrix Ψ-1Λ(Λ'Ψ-1Λ)-1. Its defining property is conditional unbiasedness — the expected value of the Bartlett estimate, given the true factor score, equals the true factor score. It also does not introduce spurious correlation with other true factors beyond what those factors actually share (see the worked comparison below). The cost is the reverse of the regression method’s: Bartlett scores are more variable than the true factor, not less, and factor-score estimates for correlated factors still correlate with each other more than could be attributed to sampling alone.
Anderson-Rubin method (Anderson & Rubin, 1956). A rescaled version of the Bartlett method — weight matrix Ψ-1Λ(Λ'Ψ-1RΨ-1Λ)-1/2 — built specifically to force the resulting factor-score estimates to be mutually uncorrelated (and unit-variance) with each other, regardless of how correlated the underlying factors actually are. This is useful when you need literally orthogonal scores to enter as regressors elsewhere. The cost is that this orthogonality is achieved by construction, not because it reflects reality: when the true factors are correlated, forcing the estimates apart lowers validity (correlation with the true factor) below both other methods, and the estimates end up carrying an uncontrolled, method-driven relationship to the other true factor instead.
Historical framing and the standard reference for comparing these three side by side: Jeff W. Grice, “Computing and Evaluating Factor Scores,” Psychological Methods, 6(4), 2001, 430–450.
Why Factor Scores Are Indeterminate — Even With a Perfectly Fitting Model
The properties above are about how the three methods differ from each other. A separate, more fundamental problem is that none of them recovers the one true factor score — there are infinitely many different sets of factor scores consistent with the same fitted model, and they can disagree with each other substantially. This is factor score indeterminacy, first formalized by Louis Guttman in 1955.
The key summary statistic is the squared multiple correlation of the true factor on the indicators, R² = Λ'R-1Λ (single-factor case). Its square root is the determinacy of the best possible (regression-method) estimate — the maximum correlation any linear estimator can have with the true factor. Guttman showed that the minimum possible correlation between two different factor-score estimates that both satisfy the model exactly is 2R² − 1. When R² is below 0.5, that bound is negative — meaning two analysts, both using a method that is fully consistent with the identical fitted model, could in principle end up with factor scores that are negatively correlated with each other.
Worked example (population-level, not sample data — the model parameters below are stated illustrative values, and every number that follows was computed directly from them by a small reproducible script implementing the standard matrix formulas, not estimated or eyeballed):
| Model | Loadings | R² | Determinacy (√R²) | Guttman minimum bound (2R²−1) |
|---|---|---|---|---|
| Weak measurement model (3 indicators) | 0.40, 0.45, 0.50 | 0.4375 | 0.661 | −0.125 |
| Strong measurement model (6 indicators) | 0.80, 0.75, 0.70, 0.75, 0.65, 0.75 | 0.8799 | 0.938 | 0.760 |
The weak model — three indicators with loadings in the 0.40–0.50 range, which is a genuinely common result in early-stage scale development, not a contrived worst case — produces a negative Guttman bound. That is the mathematical demonstration of the indeterminacy problem: it is not about sample size or measurement error in the ordinary sense, it is about how much information a given set of indicators carries about the factor in the first place. The strong model, with six well-loading indicators, pushes determinacy above 0.9 and the bound comfortably positive, which is the practical argument for why indeterminacy is a real but manageable problem in well-designed scales and a serious one in thin ones.
A Worked Comparison: Regression vs. Bartlett vs. Anderson-Rubin on the Same Model
To see the three methods actually diverge from each other requires a model with two or more correlated (oblique) factors — in a clean single-factor model, all three methods turn out to be proportional to each other and produce the same rank ordering of cases, differing only in scale. (This is algebraically necessary: with one factor, R-1Λ and Ψ-1Λ are scalar multiples of each other by the Sherman–Morrison identity, since R = ΛΛ’ + Ψ.) The choice of method only substantively matters once you have multiple, correlated factors.
The illustrative model below (disclosed population parameters, not real data): six indicators in simple structure, three loading only on Factor 1 (λ = 0.80, 0.70, 0.60) and three loading only on Factor 2 (λ = 0.75, 0.65, 0.55), with the two factors correlated at φ = 0.50. All figures below are exact output from the same reproducible script, computing each method’s weight matrix from the closed-form formulas above and then the implied covariance and correlation structure of the resulting scores.
| Method | Own-factor validity (F1 / F2) | Contamination with the other true factor (F1→trueF2 / F2→trueF1) | Correlation between the two estimates | Variance of F̂1 / F̂2 |
|---|---|---|---|---|
| Regression (Thomson) | 0.882 / 0.854 | 0.522 / 0.540 | 0.612 | 0.779 / 0.729 |
| Bartlett | 0.876 / 0.843 | 0.438 / 0.421 | 0.369 | 1.303 / 1.408 |
| Anderson-Rubin | 0.847 / 0.795 | 0.312 / 0.247 | 0.000 | 1.000 / 1.000 |
True factor correlation in this model: φ = 0.500. The pattern is exactly what the formulas predict, and it is worth reporting honestly rather than rounding into a tidier story: the regression method does have the highest own-factor validity of the three (0.882/0.854), as the theory guarantees, but its estimates are also the most correlated with each other (0.612, above the true 0.500) and the most contaminated by the other true factor. Anderson-Rubin forces the two estimates to exactly zero correlation by construction — useful if you need literally orthogonal scores for a downstream model — but pays for it with the lowest validity of the three and an uncontrolled, asymmetric relationship to the other true factor (0.312 one direction, 0.247 the other, rather than tracking the true 0.500 either way). Bartlett sits in between on validity and produces the lowest inter-estimate correlation of the two non-orthogonal methods (0.369), and its raw covariance with the other true factor comes out at exactly 0.500 in this model — matching φ precisely, which is the textbook “conditionally unbiased” property showing through — but because Bartlett’s own variance is inflated (1.303/1.408, not 1), that same relationship on the standardized correlation scale comes out lower, at 0.438/0.421, not 0.500. Unbiasedness is a covariance-scale property here, not a correlation-scale one; conflating the two is a common misreading of what “Bartlett scores are unbiased” actually guarantees.
When a Simple Sum or Mean Score Is the More Defensible Choice
None of this is an argument that factor scores are always the right tool. In several specific, common situations, an unweighted sum or mean of the item responses is the more defensible choice — not a shortcut taken because factor scores are hard to compute, but a considered decision with its own methodological justification.
- The indicators are close to tau-equivalent (roughly equal loadings). When item loadings on a factor are similar to each other, the optimal regression weights converge toward equal weights anyway — unit (differential) weighting loses little validity relative to true regression weighting, a robustness result that goes back to work on the general insensitivity of linear composites to weight perturbation (Wainer, 1976; Dawes, 1979). If your loadings already sit in a narrow band, a sum score approximates the “optimal” weighted score closely, at a fraction of the complexity and with none of the indeterminacy problem, because a sum score does not claim to estimate an unobserved variable — it is defined directly from the observed items.
- Determinacy is low. As the worked example above shows directly, a small item set with moderate loadings can produce a Guttman bound near zero or negative. In that regime, no factor-score method is trustworthy at the individual-case level, and a sum score at least has the honesty of not claiming to estimate something it demonstrably cannot pin down.
- The score needs to be portable across samples or studies. Factor score coefficients (the weight matrices above) are estimated from a specific sample’s covariance structure. Apply them to a new sample and, strictly, you are no longer computing the same score — the weights themselves would change if re-estimated on the new data. A sum or mean score is defined the same way in every sample by construction, which matters for scales meant to be scored consistently across sites, cohorts, or replications.
- You need a portable, human-interpretable metric. Clinical cut-scores, established norms, and practitioner-facing scoring rules are almost always built on sum or mean scores, not sample-specific factor-score weights, precisely because the scoring rule has to be reproducible by someone without access to the original covariance matrix.
- The measurement model isn’t the point of the analysis. If EFA or CFA was run to confirm a scale’s structure but the substantive research question only needs a summary score to correlate with other variables, introducing factor-score indeterminacy into that downstream analysis is an avoidable source of noise. A well-validated sum score, backed by an EFA/CFA run that confirms the structure is unidimensional and the loadings are reasonably even, is often the simpler and equally defensible choice.
Conversely, prefer an explicit factor-score method (or, better, a full structural equation model that avoids computing factor scores at all by relating indicators directly to outcomes) when loadings are markedly unequal, when the factor’s relationship to other variables is the actual research question, or when you specifically need one of the three methods’ distinguishing properties — maximum validity (regression), conditional unbiasedness (Bartlett), or literal orthogonality (Anderson-Rubin). Many methodologists go a step further and argue the indeterminacy problem is reason enough to avoid computing factor scores altogether in a latent-variable analysis, relating indicators to the outcome directly inside a structural model instead of extracting a score first and using it as a stand-in for the factor in a second step.
A Practical Decision Procedure
- Fit the measurement model (EFA to establish structure, CFA to confirm it) and check construct validity before scoring anything.
- Compute R² (the squared multiple correlation of the factor on its indicators) for each factor. Below roughly 0.7–0.8, treat any individual-case factor score as unreliable regardless of which method computed it — this is a determinacy problem, not something a different estimation method fixes.
- Check how uneven the loadings are. Close to tau-equivalent: a sum or mean score is a reasonable, low-complexity choice. Markedly uneven: a weighted approach (factor score or an explicit SEM) captures real information a sum score would discard.
- Decide what the score needs to do downstream. Entering it as a predictor that must be uncorrelated with a companion score: consider Anderson-Rubin. Needing an estimate that doesn’t spuriously inherit correlation with other factors: consider Bartlett. Wanting the single most accurate estimate of that respondent’s standing on the factor, and willing to accept regression to the mean: use the regression method. Needing a scoring rule that travels to a new sample or a clinical setting unchanged: use a sum or mean score, or avoid extracting a score at all and keep the factor inside a full SEM.
- Report the method used, explicitly. “Factor scores were computed” without naming the method is common in published research and is not reproducible — a reader (or replicator) cannot recompute the same numbers without knowing which of the three weight matrices was used.
Frequently Asked Questions
Are factor scores the same as component scores from principal component analysis?
No. Principal component scores are exact linear combinations of the observed variables with no indeterminacy, because a principal component is defined as a function of the observed data, not as an estimate of an unobserved latent variable. Factor scores are estimates of a latent common factor and inherit the indeterminacy problem described above precisely because the factor itself is never observed. Conflating the two is a common error when a paper describes running “PCA” but reports loadings and communalities that are actually from common factor analysis.
Which method do statistical packages use by default?
Defaults vary by software and version, and change between releases, so check your specific package’s current documentation rather than assuming — but the regression method is the most commonly offered default across mainstream statistical software, with Bartlett and Anderson-Rubin typically available as alternative options rather than defaults.
Does a higher Cronbach’s alpha mean lower factor score indeterminacy?
Cronbach’s alpha and factor determinacy are related but distinct: alpha is a reliability estimate for a sum score under a specific (tau-equivalent) measurement model, while determinacy (R² above) is specific to the factor-analytic model and to the particular set of loadings, not just their average size. High alpha with a few very unevenly loading items can still coexist with a comparatively low determinacy for the factor score itself; check both rather than assuming one implies the other.
Can two different factor-score methods disagree by a meaningful amount in practice?
Yes, and the worked comparison above shows it directly: on the same fitted model, the three methods produced own-factor validity ranging from 0.795 to 0.882 and inter-estimate correlations ranging from 0.000 to 0.612. Which method a paper used is not a footnote detail — it changes the numbers.
See also: Exploratory Factor Analysis, Confirmatory Factor Analysis (CFA), Construct Validity, Convergent and Discriminant Validity, Reliability in Research Measurement, Psychometrics, Item Response Theory, Differential Item Functioning, Research Methods.
