Quartimax, Equamax, and Other Orthogonal Criteria

Why Orthogonal Criteria Differ

An orthogonal rotation changes how a fitted loading matrix allocates loadings across factors while preserving the model's common covariance representation. The rotation does not change a variable's communality, but it can make the loading pattern easier—or harder—to describe. Quartimax, varimax, and equamax are different rules for preferring some rotated patterns over others.

All three belong to the orthomax family. Their differences are easiest to understand by separating two questions: does a criterion reward a variable for loading strongly on just one factor, or does it reward each factor for having a few strong loadings and many weak ones? Quartimax emphasizes the first question, varimax the second, and equamax combines the two weightings.

This states the convention explicitly because criterion formulas can differ in their normalizing constants or in whether an algorithm minimizes the negative of the displayed objective. Here, a rotation with a larger value of the stated objective is preferred by that criterion. The loading matrix is not row-normalized before calculating these formulas.

What the Three Criteria Reward

For variable \(i\), its communality is \(h_i^2=\sum_j\lambda_{ij}^2\). Under an orthogonal rotation, its row of loadings is multiplied by an orthogonal matrix, so its squared length is unchanged:

Thus, a criterion cannot increase a variable's total communality by rotating; it can only change how that fixed amount is distributed across its factor loadings. Quartimax directly rewards concentration within each row through fourth powers. In contrast, varimax evaluates squared loadings column by column, rewarding a factor with a mixture of relatively large and small loadings across variables.

Proof. Put \(x_j=\lambda_{ij}^2\), so \(x_j\geq 0\) and \(\sum_jx_j=h_i^2\). The quartimax contribution from row \(i\) is \(\sum_jx_j^2\). Since \[ \left(\sum_jx_j\right)^2=\sum_jx_j^2+2\sum_{j<k}x_jx_k\geq \sum_jx_j^2, \] it is at most \(h_i^4\). Equality holds when every product \(x_jx_k\) for \(j\ne k\) is zero—that is, when at most one loading in the row is nonzero.

For factor \(j\), let \(x_i=\lambda_{ij}^2\) and \(c_j=\sum_i x_i\), which is fixed for this comparison. Its varimax contribution is \(\sum_i x_i^2-c_j^2/p\). The subtracted term is constant when \(c_j\) is fixed, so maximizing the contribution is equivalent to maximizing \(\sum_i x_i^2\). For nonnegative \(x_i\) with fixed sum, concentrating the sum increases this quantity: transferring an amount \(\delta\) from \(x_b\) to \(x_a\), where \(x_a\geq x_b\), changes the sum of squares by \[ (x_a+\delta)^2+(x_b-\delta)^2-x_a^2-x_b^2 =2\delta(x_a-x_b)+2\delta^2\geq 0 \] whenever \(0\leq\delta\leq x_b\). This proves the stated preference. \(\square\)

The proof describes the criteria's tendencies, not a promise that an entire matrix can attain all of these separate maxima at once. A rotation couples the rows and columns: the same orthogonal transformation must act on every variable. Varimax's columnwise reward commonly produces factors with a few strong loadings and many weak ones. Quartimax can instead favor assigning each variable mainly to one factor, but this can yield a broad general factor on which many variables load. Equamax uses an intermediate orthomax weighting; it does not guarantee a perfect balance or a unique, universally best pattern.

How Equamax Fits the Orthomax Family

The parameter \(\gamma\) controls the penalty for a factor column with a large sum of squared loadings. When \(\gamma=0\), there is no such penalty, giving quartimax. When \(\gamma=1\), the penalty centers the squared loadings within each column, giving varimax. Equamax sets \(\gamma=p/(2m)\), so its weighting depends on the number of variables relative to the number of factors.

To derive the displayed form, substitute \(\gamma=p/(2m)\) into the orthomax definition. Its column penalty coefficient becomes \(\gamma/p=(p/(2m))/p=1/(2m)\). The first term sums the fourth powers over all entries, so the expression follows directly. In particular, equamax does not always define a numerically distinct criterion: when \(p=2m\), its parameter equals the varimax parameter.

Worked Example: Calculating All Three Criteria

Worked Example: Quartimax and a Variable's Loading Pattern

Worked Example: Varimax and Factor Differentiation

Interpreting and Comparing the Criteria

All three objectives are unchanged by permuting factor columns or changing the sign of any factor: fourth powers and column sums of squares are unaffected. General orthogonal rotations, however, can change the objectives. Although the row communalities are preserved, the entrywise fourth-power sum and the distribution of squared loadings within columns need not be.

The practical choice is about the pattern of simplicity desired, not about which criterion has the largest numerical value. Quartimax puts direct emphasis on simple rows—variables with relatively few substantial loadings—and can encourage a general factor. Varimax focuses on simple columns—factors with relatively few substantial loadings. Equamax retains the quartimax fourth-power term while applying a column penalty whose strength depends on \(p\) and \(m\).

A common mistake is to say that quartimax, varimax, and equamax all simply “maximize sparsity,” then treat their objective values as interchangeable. That skips the essential distinction: quartimax has no column penalty, while varimax and equamax do. The correction is to calculate the specified objective, check its parameter and normalization, and describe the pattern it rewards. A larger quartimax value is not evidence that a solution is better by the varimax criterion.