Pairwise Imputation Patterns and Inconsistent Correlations

Why Pairwise Correlations Can Conflict

A correlation matrix summarizes relationships among several variables at once. Its entries are not independent calculations: they must be capable of describing one joint distribution. Missing data can break that connection when each correlation is calculated from a different set of rows. Each pairwise calculation may look reasonable on its own, yet the collection of results may be impossible as a single correlation matrix.

This issue is easy to overlook because software often reports pairwise correlations without warning that the correlations use different subsets. The resulting numbers can be useful summaries of the available pairs, but they must not automatically be treated as a coherent description of all variables together.

Pairwise deletion means calculating each pairwise statistic from its pairwise-complete subset; it is not an imputation method. Pair-specific imputation is different: it fills missing values separately for each pairwise calculation. In either case, when the calculations use different subsets or different pair-specific fills, the results need not come from one common completed data set.

The key mathematical constraint is positive semidefiniteness. A valid correlation matrix \(R\) has diagonal entries 1 and satisfies \(a^\mathsf{T}Ra \geq 0\) for every vector \(a\). For variables observed together on a common set of rows, this follows because \(a^\mathsf{T}Ra\) is the sample variance of a linear combination of standardized variables, up to a positive scaling factor. A matrix assembled from pairwise calculations on different subsets has no such common-data argument guaranteeing this property.

A Three-Variable Consistency Check

For three variables, write the correlation matrix as

A necessary condition for this matrix to be positive semidefinite is that its determinant be nonnegative. Expanding the determinant gives

This condition has a useful interpretation. If both \(X\) and \(Z\) correlate strongly and positively with \(Y\), then \(X\) and \(Z\) cannot simultaneously have a very strong negative correlation in one coherent joint distribution. A negative determinant signals that the three pairwise values cannot all be correlations from the same common data structure.

Pair-Specific Means Are Not One Joint Imputation

The same example also shows why imputation performed separately for each pair can be ambiguous. For the \(X,Y\) calculation, the observed \(X\) values are 0 and 2, with mean \((0+2)/2=1\). For the \(X,Z\) calculation, the observed \(X\) values are 10 and 12, with mean \((10+12)/2=11\). Thus the pair-specific mean for \(X\) is 1 in one calculation and 11 in another.

Those means are not contradictory as summaries of different subsets. But if an analyst treats them as though they were one imputed value for each missing \(X\), the procedure has no single completed data set behind it. Each pair may be analyzed using a different set of rows and a different fill rule. Consequently, the resulting pairwise correlations need not satisfy the mathematical constraints that apply to correlations calculated together from one data set.

What a Common Set of Rows Guarantees

When all correlations are calculated from the same complete rows, they come from the same columns of observed values. Standardizing each column and forming the sample correlation matrix then gives a Gram matrix: each entry is an inner product between two standardized columns. For any vector \(a\), the quadratic form is the squared length of the corresponding linear combination of those columns, scaled by a nonnegative constant. It therefore cannot be negative. This is the reason a common-data correlation matrix is positive semidefinite.

Listwise deletion uses only rows complete on every variable in the analysis. It can therefore yield a coherent matrix when there are enough complete rows and each variable has nonzero variation. But it may discard many observations, and the resulting complete-case analysis can be biased if the inclusion of a row in the complete subset is related to the data in a way relevant to the analysis. The assumptions governing missingness matter; matrix coherence alone does not establish that the estimates represent the desired population.

A single jointly completed data set also gives a coherent sample correlation matrix, provided its variables have nonzero variation. That algebraic coherence does not guarantee that the imputed values are plausible or that the correlations are unbiased. In particular, a coherent matrix can still reflect distortion introduced by the imputation procedure. Multiple imputation addresses missing-data uncertainty differently, but it does not make pairwise incompatibilities irrelevant: summaries and analyses must still be defined and combined appropriately.

Diagnostics, Alternatives, and Limitations

Before interpreting pairwise correlations in an analysis with missing data, check how each entry was computed. In particular, determine the pair-specific sample size, whether the same rows were used across pairs, and whether any imputation was performed separately for each pair. For three variables, calculate the determinant as a simple consistency check. For larger matrices, check positive semidefiniteness using the matrix eigenvalues or an equivalent numerical procedure. Small negative eigenvalues can sometimes result from rounding; substantial negative values indicate incompatibility that needs investigation.

A coherent matrix is a mathematical requirement for a joint correlation summary, not a test of whether missingness is harmless. Pairwise deletion can be biased under some missingness patterns, and even estimates that pass a matrix check may be imprecise or biased. The earlier material on missingness mechanisms provides the framework for considering whether the missingness process threatens the target of inference.

The main alternatives answer different needs:

• Pairwise deletion retains available rows for each pair, but can produce incomparable sample sizes and an incoherent matrix.
• Listwise deletion uses a common complete-case sample and yields a coherent matrix, but may lose substantial information and can be biased under relevant missingness mechanisms.
• One joint imputation creates a coherent completed data set, but coherence alone does not account for uncertainty about missing values or guarantee accurate relationships.

No option is justified solely because it produces a valid-looking matrix. The analysis should make clear which observations contribute to each estimate, what missingness assumptions are being made, and what population or association the results are intended to describe.

Common Mistakes

• Checking each correlation only against the range \([-1,1]\). Every entry in an impossible matrix can still lie within this range. Check joint matrix constraints, such as the determinant condition for three variables.
• Treating pairwise deletion as imputation. Pairwise deletion omits rows separately for each pair; it does not fill missing cells. Pair-specific imputation is a distinct procedure and may also fail to define one joint data set.
• Assuming a valid matrix proves valid inference. Positive semidefiniteness establishes mathematical coherence, not lack of bias, adequate precision, or a correct missingness model.
• Calling unequal pairwise sample sizes proof of inconsistency. Unequal counts indicate different amounts of available information, but the matrix can still be positive semidefinite. Check the matrix itself.