Why Mean Imputation Changes Correlation
Correlation describes the direction and strength of a linear relationship relative to the variation in both variables. Mean-imputing missing values in one variable changes that variation: each imputed value is exactly at the mean, so it contributes no deviation from the mean. If the other variable is fully observed, the covariance also changes, but not enough to offset the reduction in the imputed variable’s standard deviation. Under a clear set of assumptions, the resulting correlation is therefore attenuated toward zero.
The result below is a population calculation, not an identity guaranteed for every realized sample. It isolates the effect of mean imputation when the missingness indicator is independent of both variables. In actual data, the mean used for imputation is usually estimated from observed values, and a particular missingness pattern may not reproduce the population relationship exactly.
Assume \(X\) and \(Y\) have finite, positive variances, \(0<q\leq 1\), and \(R\) is independent of \((X,Y)\). The imputation value in this population result is the true mean \(\mu_X\). These assumptions make the derivation precise; they are not claims that a real analysis knows the true mean or that its missingness mechanism is MCAR.
Why the Correlation Is Multiplied by the Square Root of the Observed Fraction
Write \(\rho=\operatorname{Corr}(X,Y)\). Because \(R\) is independent of \(X\), the imputed variable retains the original mean:
After centering at this mean, the deviation of an imputed value is zero. Algebraically, \(X^*-\mu_X=R(X-\mu_X)\), since \(R\) is either zero or one. Squaring this identity and using \(R^2=R\), then applying independence, yields:
For covariance with the fully observed \(Y\), centering both variables gives:
Correlation divides covariance by the product of standard deviations. The standard deviation of \(X^*\) is \(\sqrt{q}\) times that of \(X\), while \(Y\)'s standard deviation is unchanged. Thus, for \(q>0\),
This explains the attenuation: the covariance is multiplied by \(q\), but the denominator is multiplied by only \(\sqrt{q}\). Since \(0<q\leq1\), \(\sqrt{q}\leq1\). The sign of the correlation is unchanged, while its magnitude is reduced unless \(q=1\) or the original correlation is zero. The attenuation factor is the square root of the fraction observed, not the fraction missing.
Worked Example: Calculating Attenuation From Population Moments
Suppose a hypothetical population has \(\operatorname{Var}(X)=16\) points squared, \(\operatorname{Var}(Y)=9\) units squared, and \(\operatorname{Cov}(X,Y)=6\) point-units. Assume \(X\) is observed for \(q=0.75\) of individuals, \(Y\) is fully observed, and the observation indicator is independent of \((X,Y)\). Impute missing \(X\) values at \(\mu_X\).
First calculate the original correlation:
The imputed variance is \(q\operatorname{Var}(X)=0.75(16)=12\) points squared, and the imputed covariance is \(q\operatorname{Cov}(X,Y)=0.75(6)=4.5\) point-units. Therefore:
Equivalently, \(\sqrt{0.75}(0.5000)=0.4330\). The population correlation is positive both before and after imputation, but its magnitude falls from 0.5000 to 0.4330 under these assumptions.
Population Result Versus a Realized Sample
In a data set, analysts typically calculate the mean from observed \(X\) values and then calculate a sample correlation. That observed mean is not generally the population mean, and a realized pattern of missing rows need not have the balance implied by independence. Consequently, the theorem gives a population benchmark under MCAR, not an exact correction formula for every completed sample.
Worked Example: A Finite Sample Does Not Follow the Exact Factor
Consider this hypothetical complete data set. Suppose \(X\) is missing for rows 1 and 6, and the remaining \(X\) values are used to calculate its imputation mean. The values are illustrative, not from a study.
| Row | \(X\), before masking | \(Y\) | \(X\) after mean imputation |
|---|---|---|---|
| 1 | 1 | 2 | 3.5 |
| 2 | 2 | 1 | 2 |
| 3 | 3 | 4 | 3 |
| 4 | 4 | 3 | 4 |
| 5 | 5 | 6 | 5 |
| 6 | 6 | 5 | 3.5 |
The observed \(X\) mean is \((2+3+4+5)/4=3.5\). In the hypothetical complete data, both \(X\) and \(Y\) have mean \(3.5\). Their sums of squared deviations are each \(17.5\), and their sum of cross-products is \(14.5\). Using sample covariance and variances, each divided by \(n-1=5\), the original sample correlation is:
After imputation, the completed \(X\) values still have mean \(3.5\). Their sum of squared deviations is \(5\), and their sum of cross-products with the centered \(Y\) values is \(7\). The \(Y\) sum of squared deviations remains \(17.5\). Thus:
The observed fraction is \(q=4/6\), so multiplying the original sample correlation by \(\sqrt{q}\) would give \(\sqrt{4/6}(0.8286)=0.6765\), not \(0.7483\). This does not contradict the theorem: a finite realization is not the same as a population expectation, even when the observed mean happens to equal the complete-data mean. Here, mean imputation does reduce the sample correlation, but not by the theorem’s exact factor.
Assumptions, Diagnostics, and Alternatives
Independence of \(R\) and \((X,Y)\) is essential to the stated identities. If missingness depends on \(X\), on \(Y\), or on both, the observed rows may have different means or covariances from the full population. Then neither the variance and covariance multipliers nor the square-root correlation factor is guaranteed. A fully observed \(Y\) does not by itself make \(X\)'s missingness independent.
Useful checks include comparing the distributions of fully observed variables such as \(Y\) across rows with observed and missing \(X\), checking whether observed \(X\) values and relevant auxiliary variables suggest systematic differences, and comparing complete-case and imputed summaries. Such checks can reveal concerning patterns, but cannot establish MCAR from observed data alone. Also verify that \(q>0\) and that \(Y\) varies: if no \(X\) values are observed, \(X^*\) is constant; if \(Y\) is constant, correlation is undefined as well.
The main alternatives answer different questions and have different limitations. Complete-case analysis avoids assigning a single value to missing \(X\), but discards rows and can be biased or inefficient depending on the missingness process and analysis. Multiple imputation represents missing values with repeated plausible draws under a specified imputation model and carries imputation uncertainty into analysis; it requires suitable models and assumptions, and it is not automatically valid merely because it is multiple. Mean imputation is simple, but its artificial point mass and loss of spread make its completed-data correlation an unreliable stand-in for the underlying association.
Worked Applications and Boundary Cases
Worked Example: Attenuation of a Negative Correlation
Suppose a hypothetical population has \(\operatorname{Var}(X)=16\), \(\operatorname{Var}(Y)=25\), and \(\operatorname{Cov}(X,Y)=-8\), in the corresponding squared and product units. Let \(q=0.25\), and assume MCAR and population-mean imputation.
The original correlation is \(-8/\sqrt{16(25)}=-8/20=-0.4000\). Imputation gives \(\operatorname{Var}(X^*)=0.25(16)=4\) and \(\operatorname{Cov}(X^*,Y)=0.25(-8)=-2\). Hence:
The correlation remains negative, but its magnitude is halved. It would be incorrect to say that the association has reversed; only its magnitude has changed in this calculation.
Worked Example: The Zero-Observation Boundary
Let \(X\) and \(Y\) each have positive finite variance in the population, but suppose no \(X\) values are observed, so \(q=0\). Every completed \(X\) value is set to \(\mu_X\), meaning \(X^*=\mu_X\) for every individual.
Thus \(\operatorname{Var}(X^*)=0\), and since a constant has zero covariance with \(Y\), \(\operatorname{Cov}(X^*,Y)=0\). The correlation formula would require division by \(\sqrt{\operatorname{Var}(X^*)\operatorname{Var}(Y)}=0\), so \(\operatorname{Corr}(X^*,Y)\) is undefined. The expression \(\sqrt{q}\rho=0\) is not a defined correlation in this case; the correlation conclusion requires \(0<q\leq1\).
Check Your Understanding
Try each problem, then compare your reasoning with the solutions below.
- A population has \(\operatorname{Var}(X)=9\), \(\operatorname{Var}(Y)=4\), and \(\operatorname{Cov}(X,Y)=3\). If \(q=4/9\) and the assumptions of the theorem hold, calculate the original and imputed correlations.
- A population has \(\operatorname{Var}(X)=16\), \(\operatorname{Var}(Y)=25\), and \(\operatorname{Cov}(X,Y)=-8\). What is the imputed correlation when \(q=0.25\)?
- Why is correlation undefined when \(q=0\), even though the variance and covariance identities remain valid?
- Does a fully observed \(Y\) guarantee that mean imputation attenuates the population correlation by \(\sqrt{q}\)? State the required condition that may fail.
Solutions.
1. The original correlation is \(3/\sqrt{9(4)}=3/6=0.5000\). The attenuation factor is \(\sqrt{4/9}=2/3\), so the imputed correlation is \((2/3)(0.5000)=0.3333\). Equivalently, the imputed variance is \((4/9)(9)=4\), the imputed covariance is \((4/9)(3)=1.333\), and \(1.333/\sqrt{4(4)}=0.3333\).
2. The original correlation is \(-8/\sqrt{16(25)}=-0.4000\). Multiplying by \(\sqrt{0.25}=0.5\) gives an imputed correlation of \(-0.2000\).
3. When \(q=0\), every value of \(X^*\) equals \(\mu_X\), so \(\operatorname{Var}(X^*)=0\). Correlation divides covariance by the product of standard deviations, and that denominator is zero. The variance and covariance identities still describe zero variance and zero covariance, but they do not define a correlation.
4. No. The result requires the missingness indicator \(R\) to be independent of the pair \((X,Y)\), as well as positive finite variances and \(q>0\). If missingness depends on either variable, the covariance and variance identities need not hold.