Factor Score Indeterminacy and Common Interpretive Errors

Why Factor Scores Need Careful Interpretation

A fitted factor model describes how a set of observed variables covaries through a smaller set of latent factors. That description does not, by itself, make each person’s factor values directly observable. For an individual with observed measurements, a scoring method produces an estimate of those unobserved values. Different methods can produce different estimates while using the same fitted model, because they use different criteria to turn the observed measurements into a point score.

This distinction matters in applications. A factor score may be used to rank individuals, compare groups, or enter a later analysis. Those uses can be misleading if the score is treated as an error-free measurement of a known latent value. We will keep three objects separate: the model’s latent random factor vector, an individual’s unobserved realization of that vector, and a score computed from that individual’s observed data.

Indeterminacy is not the same as rotational indeterminacy. Rotation concerns alternative factor coordinates and loading representations for the model. Even after a particular representation has been selected and held fixed, uncertainty about an individual’s factor values remains. Scoring methods then provide different ways to summarize that uncertainty or to approximate the latent values.

Conditional Factor Scores Under a Gaussian Model

A useful precise account is available when the model is Gaussian. Assume \(\eta\sim N(0,\Phi)\) and \(\epsilon\sim N(0,\Psi)\), independently, with \(\Phi\) and \(\Psi\) positive definite. The conditional distribution of \(\eta\) given \(z\) then has a mean and covariance. The mean is the regression score; the covariance describes uncertainty that remains after observing \(z\).

Proof. For fixed observed \(z\), the joint Gaussian density viewed as a function of \(\eta\) is proportional to $$ \exp\left\{-\frac{1}{2}\left[\eta'\Phi^{-1}\eta+ (z-\Lambda\eta)'\Psi^{-1}(z-\Lambda\eta)\right]\right\}. $$ Expanding the terms that involve \(\eta\) gives the quadratic form $$ \eta'A\eta-2\eta'\Lambda'\Psi^{-1}z, \qquad A=\Phi^{-1}+\Lambda'\Psi^{-1}\Lambda. $$ Since \(\Phi^{-1}\) is positive definite and \(\Lambda'\Psi^{-1}\Lambda\) is positive semidefinite, \(A\) is positive definite. Completing the square, with \(m=A^{-1}\Lambda'\Psi^{-1}z\), yields $$ \eta'A\eta-2\eta'\Lambda'\Psi^{-1}z =(\eta-m)'A(\eta-m)-m'Am. $$ The terms not involving \(\eta\) affect only the normalizing constant. The conditional density is therefore Gaussian with mean \(m\) and covariance \(A^{-1}\). This proves the first expression for the conditional mean and the conditional covariance.

To verify the other expression for the mean, note that \(A\Phi=I+\Lambda'\Psi^{-1}\Lambda\Phi\). Also, \(\Sigma\Psi^{-1}\Lambda=\Lambda\Phi\Lambda'\Psi^{-1}\Lambda+\Psi\Psi^{-1}\Lambda=\Lambda(\Phi\Lambda'\Psi^{-1}\Lambda+I)\). Consequently, \(\Sigma\Psi^{-1}\Lambda=\Lambda\Phi A\). Multiplying by \(\Sigma^{-1}\) on the left and by \(A^{-1}\) on the right gives \(\Psi^{-1}\Lambda A^{-1}=\Sigma^{-1}\Lambda\Phi\). Transposing this equality gives \(A^{-1}\Lambda'\Psi^{-1}=\Phi\Lambda'\Sigma^{-1}\), as required. Finally, since \(A\) is positive definite, its inverse is positive definite. The conditional covariance is therefore nonzero in every nonzero direction, so observing \(z\) does not determine an exact \(\eta\). \(\square\)

The theorem describes conditional uncertainty under a specified probability model; it does not say that all scoring methods have the same goal. The regression score is the conditional mean, which minimizes expected squared error under the stated Gaussian model. Bartlett scoring instead chooses factor values to fit the observed measurements by weighted least squares, without the same shrinkage toward the population mean.

Regression and Bartlett Scores Can Differ

For the common factor model with \(\Psi\) positive definite and \(\Lambda'\Psi^{-1}\Lambda\) invertible, the regression score and Bartlett score for an observed vector \(z\) are $$ \hat{\eta}_{R}=\Phi\Lambda'\Sigma^{-1}z, \qquad \hat{\eta}_{B}=(\Lambda'\Psi^{-1}\Lambda)^{-1}\Lambda'\Psi^{-1}z. $$ The regression score uses the factor distribution as well as the measurement model. Bartlett scoring minimizes the weighted squared residual \((z-\Lambda a)'\Psi^{-1}(z-\Lambda a)\) over candidate factor values \(a\). These objectives need not select the same value.

The difference is systematic in this example, not merely a rounding issue. The regression score is a conditional mean and is pulled toward the population mean of zero. Bartlett scoring imposes a different fitting criterion. Under this model, its weights satisfy \(W_B\Lambda=1\), where \(W_B=(\Lambda'\Psi^{-1}\Lambda)^{-1}\Lambda'\Psi^{-1}\). That property gives \(E[\hat{\eta}_B\mid\eta]=\eta\) when the model holds, but it does not make the realized score equal to the individual’s unobserved factor value.

What the Uncertainty Means for Individual Scores

In the numerical model above, the conditional variance can be computed directly: $$ \operatorname{Var}(\eta\mid z) =\left(1+\frac{337}{144}\right)^{-1} =\frac{144}{481}\approx0.299. $$ This variance is the same for every \(z\) in this Gaussian model. It shows that even after observing both measurements, uncertainty about the person’s latent factor remains. The regression score gives the center of the conditional distribution, not a claim that the true individual factor equals that center exactly.

Rotation Changes Coordinates, Not the Underlying Model

A second source of apparent disagreement is comparing scores expressed in different rotated factor coordinates. Suppose \(T\) is an invertible \(q\times q\) matrix and define \(\Lambda^*=\Lambda T\), \(\eta^*=T^{-1}\eta\), and \(\Phi^*=T^{-1}\Phi T^{-T}\). Then \(\Lambda^*\eta^*=\Lambda\eta\), and the common covariance representation is unchanged. A numerical score vector will generally have different entries in the new coordinates, even when it represents the same latent point.

Proof. The common covariance under the transformed representation is $$ \Lambda^*\Phi^*\Lambda^{*'} =\Lambda T(T^{-1}\Phi T^{-T})T'\Lambda' =\Lambda\Phi\Lambda'. $$ Since the uniqueness covariance is unchanged, the observed covariance \(\Sigma\) is unchanged. The transformed regression score is $$ \Phi^*\Lambda^{*'}\Sigma^{-1}z =(T^{-1}\Phi T^{-T})T'\Lambda'\Sigma^{-1}z =T^{-1}\Phi\Lambda'\Sigma^{-1}z =T^{-1}\hat{\eta}_R. $$ The conditional covariance transforms by applying the same linear transformation to the conditional factor vector: \(\operatorname{Var}(T^{-1}\eta\mid z)=T^{-1}\operatorname{Var}(\eta\mid z)T^{-T}\). Thus the entries change as coordinates should, while the represented point and model remain consistent. \(\square\)

Common Interpretive Errors

Factor scores are useful summaries, but their interpretation must match how they were computed. The following errors conflate the unobserved factor, its estimate, and the model used to obtain that estimate.

• Calling an estimated score the observed factor. The factor is latent; the score is calculated from observed measurements and a scoring rule. Under the Gaussian assumptions above, the conditional variance is positive, so the score is not an exact observation of the factor.
• Treating scores from different methods as interchangeable. Regression and Bartlett scores optimize or summarize different things. Their discrepancies do not, by themselves, indicate that one model has been fitted differently.
• Assuming regression-score variance equals factor variance. Regression scores are conditional means and are typically less variable than the latent factors. Their compressed range is a consequence of prediction under uncertainty, not proof that the factors themselves have a narrow range.
• Comparing rotated score entries without aligning coordinates. A score’s components depend on the factor coordinate system. Before comparing scores across rotations, transform them to a common orientation or compare quantities that are invariant to the coordinate change.
• Interpreting small score differences as certain rank differences. Two nearby point estimates do not establish that one person’s latent factor is truly higher. The estimates do not remove individual-level uncertainty.