Why Transform Correlations Before Pooling?
A correlation is constrained to the interval from \(-1\) to \(1\). Its sampling distribution can be skewed, especially when the true correlation is far from zero or the sample is small. Its standard error also depends on the correlation itself. These features make a correlation a poor candidate for applying ordinary pooled-estimate calculations directly on the raw scale.
Fisher’s \(z\) transformation maps any correlation strictly between \(-1\) and \(1\) to the full real line. On this transformed scale, the sampling distribution is often closer to normal and its variance has a simple approximation. The workflow is therefore: transform each imputation-specific correlation, pool the transformed estimates and their variances, then apply the inverse transformation to the pooled estimate and confidence limits.
Assumptions and Rubin Pooling on the Transformed Scale
Suppose there are \(m\) completed data sets, each containing the same \(n\) individuals for the pair of variables being correlated. Let \(r_j\) be the Pearson correlation in completed data set \(j\), and let \(q_j=\operatorname{atanh}(r_j)\). For an ordinary Pearson correlation from a sample with a reasonably well-behaved bivariate distribution, the usual complete-data variance approximation is \(u_j=1/(n-3)\). This approximation is most familiar under bivariate normal sampling; it is not an exact variance formula for every distribution or every weighted or complex-sample correlation.
The imputed data sets should be generated under a proper multiple-imputation procedure for the intended analysis. In particular, their between-imputation variation should represent uncertainty due to missing values, rather than just repeated calculations on one fixed completed data set. The correlation must be finite in every imputation: if any \(r_j=1\) or \(r_j=-1\), its Fisher transform is infinite and the routine calculation does not apply.
When the complete-data degrees of freedom can be treated as large, a common Rubin degrees-of-freedom approximation is $$ df=(m-1)\left(1+\frac{\bar U}{(1+1/m)B}\right)^2. $$ A two-sided confidence interval on the transformed scale is \(\bar q\pm t_{0.975,df}SE\) for a 95% interval. If a suitable finite complete-data degrees of freedom applies, use the relevant small-sample adjustment rather than assuming it is infinite. If \(B=0\), the displayed degrees-of-freedom expression is not used as written; the limiting inference is based on the within-imputation uncertainty and the applicable complete-data degrees of freedom.
Why Fisher’s Transformation Helps
The variance-stabilizing idea can be seen from a first-order variance argument. For a sample correlation \(r\) under standard large-sample conditions, its approximate variance is \((1-r^2)^2/(n-1)\). The derivative of \(z=\operatorname{atanh}(r)\) with respect to \(r\) is \(1/(1-r^2)\). Applying the first-order variance rule therefore gives an approximate transformed variance of $$ \operatorname{Var}(z)\ \approx\ \left(\frac{1}{1-r^2}\right)^2\frac{(1-r^2)^2}{n-1} =\frac{1}{n-1}. $$
This calculation shows why the transformation reduces the dependence of the variance on the correlation. The familiar refined approximation for the variance of Fisher’s \(z\) is \(1/(n-3)\), which is the value used in the examples below. The approximation is not a guarantee of normality in small samples or with extreme correlations; those are settings where diagnostics and possibly a different analysis are important.
Worked Example: Pooling Five Imputation-Specific Correlations
Suppose five hypothetical completed data sets each contain \(n=50\) individuals. Their correlations between two measurements are \(0.30,\ 0.40,\ 0.35,\ 0.50,\) and \(0.25\). Assume these are ordinary Pearson correlations, the transformed variance approximation is suitable, and the imputations are proper for the intended analysis.
Transform each correlation: the corresponding Fisher values are approximately \(0.3095,\ 0.4236,\ 0.3654,\ 0.5493,\) and \(0.2554\). Thus $$ \bar q=\frac{0.3095+0.4236+0.3654+0.5493+0.2554}{5}=0.3806. $$ The complete-data variance in each imputation is \(u_j=1/(50-3)=1/47=0.02128\), so \(\bar U=0.02128\). The sample variance of the five transformed estimates is \(B\approx0.01282\). Therefore $$ T=0.02128+\left(1+\frac{1}{5}\right)(0.01282) =0.02128+0.01538=0.03666, \qquad SE=\sqrt{0.03666}=0.1915. $$
Using the large-complete-data degrees-of-freedom approximation gives $$ df=4\left(1+\frac{0.02128}{1.2(0.01282)}\right)^2\approx22.72. $$ The 95% critical value is approximately \(t_{0.975,22.72}=2.071\). Hence the transformed interval is $$ 0.3806\pm2.071(0.1915)=[-0.0160,\ 0.7772]. $$ Back-transforming the pooled point estimate gives \(\tanh(0.3806)\approx0.3633\). Back-transforming the limits separately gives an approximate correlation-scale interval of \([-0.0160,\ 0.6511]\). The pooled correlation estimate is about \(0.363\); the interval is wide enough to include zero, so these calculations do not establish a nonzero correlation at the two-sided 5% level.
Back-Transforming and Interpreting the Interval
Because \(\tanh(q)\) is strictly increasing, back-transforming both ends of a transformed-scale interval preserves their order and coverage interpretation. Transform the pooled estimate and each confidence limit separately: $$ \hat r_{\mathrm{pooled}}=\tanh(\bar q),\qquad \left[\tanh\{\bar q-t_{0.975,df}SE\},\ \tanh\{\bar q+t_{0.975,df}SE\}\right]. $$
The correlation-scale interval is generally not symmetric around the back-transformed point estimate. That is expected: a symmetric interval on the \(z\) scale becomes asymmetric under the nonlinear inverse transformation. The limits remain within \((-1,1)\), as correlation limits must.
Worked Example: Large Between-Imputation Variation
Consider four hypothetical completed data sets, each with \(n=80\), giving correlations \(0.10,\ 0.25,\ 0.45,\) and \(0.60\). Assume the same Pearson-correlation and proper-imputation conditions used above. The Fisher values are approximately \(0.1003,\ 0.2554,\ 0.4847,\) and \(0.6931\), so \(\bar q\approx0.3834\) and \(B\approx0.06757\). Each within-imputation variance is \(1/(80-3)=1/77=0.01299\).
The total variance and standard error are $$ T=0.01299+\left(1+\frac{1}{4}\right)(0.06757) =0.09745,\qquad SE=\sqrt{0.09745}=0.3122. $$ The degrees of freedom are approximately $$ df=3\left(1+\frac{0.01299}{1.25(0.06757)}\right)^2=3.99. $$ Using \(t_{0.975,3.99}\approx2.776\), the transformed interval is \(0.3834\pm2.776(0.3122)\), or approximately \([-0.4833,\ 1.2501]\). The pooled correlation is \(\tanh(0.3834)\approx0.3657\), and the back-transformed interval is approximately \([-0.4489,\ 0.8484]\).
The sizeable \(B\) indicates substantial variation among imputations, and the small degrees of freedom reflect that uncertainty. Reporting only the pooled point estimate would conceal how imputation uncertainty affects precision. The interval is broad and includes zero; it is not evidence that the correlation is exactly zero.
Worked Example: Negative Correlations and Bounded Limits
Suppose four hypothetical completed data sets of \(n=30\) individuals give correlations \(-0.60,\ -0.30,\ 0.00,\) and \(0.20\). These are finite correlations, so the Fisher transformation is defined. Their transformed values are approximately \(-0.6931,\ -0.3095,\ 0.0000,\) and \(0.2027\), with \(\bar q\approx-0.2000\) and \(B\approx0.1525\). The within-imputation variance is \(1/27=0.03704\).
Thus \(T=0.03704+1.25(0.1525)\approx0.2277\), and \(SE\approx0.4771\). The Rubin degrees of freedom are approximately \(4.28\); using \(t_{0.975,4.28}\approx2.706\), the transformed interval is $$ -0.2000\pm2.706(0.4771)\approx[-1.491,\ 1.091]. $$ The pooled estimate is \(\tanh(-0.2000)\approx-0.1974\). The back-transformed limits are approximately \([-0.904,\ 0.797]\). A transformed limit may lie outside \([-1,1]\); this is not an error, because \(z\) is unbounded. Its back-transformation returns a valid correlation limit.
Diagnostics, Alternatives, and Limitations
Before interpreting a result, check the computations and the analysis conditions. In particular, confirm that every imputation uses the same intended pair of variables and sample definition, that each correlation is finite, and that \(n\) is the number of individuals contributing to that correlation in a completed data set. In ordinary MICE analyses, this is usually the same across completed data sets. If it is not, or if the correlation is weighted or otherwise nonstandard, do not mechanically use \(1/(n-3)\); obtain a defensible complete-data variance on the transformed scale.
- Inspect the transformed estimates. A very large \(B\), or one \(q_j\) far from the others, may signal sensitivity to imputed values, model instability, or a sparse or extreme data pattern. Check the imputation and analysis before treating the pooled result as reliable.
- Check boundary behavior. A correlation of exactly \(1\) or \(-1\) makes \(q_j\) infinite. Do not silently clip it to a convenient value. Investigate whether the variables are identical, whether there is a data or coding problem, or whether a method suited to the actual design is needed.
- Respect the approximation. The variance \(1/(n-3)\) is a conventional approximation for ordinary correlations, not a universal variance formula. Small samples, highly non-normal data, dependence among observations, or unusual sampling designs can undermine it.
Directly averaging the \(r_j\) values can be a descriptive summary, but it does not account for within-imputation and between-imputation uncertainty by Rubin’s rules. Pooling on the Fisher scale is the standard approach when the correlation-scale estimates can be transformed and the complete-data variance approximation is appropriate. If the scientific target is instead a model coefficient or another association measure, use an analysis and pooling scale appropriate to that estimand rather than treating a correlation as interchangeable with it.
| Approach | What it does | Main consideration |
|---|---|---|
| Fisher-\(z\) pooling | Transforms correlations, pools estimates and variances, then back-transforms. | Uses a variance approximation that may not suit every design or distribution. |
| Raw-scale averaging | Takes the arithmetic mean of the imputation-specific correlations. | Does not by itself provide Rubin-pooled uncertainty and can behave poorly near the bounds. |
| Model-based analysis of an association | Estimates a parameter defined by a specified statistical model. | May target a different quantity from a marginal correlation; the estimand must be clear. |
A common mistake is to pool the raw correlations and their standard errors as though the correlation were an unconstrained, constant-variance parameter. That reasoning ignores both the bounds and the changing sampling variance of \(r\). Transform first, pool \(q_j\) with its variance on the same scale, and only then back-transform the estimate and interval.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- What is the Fisher transform of a correlation \(r=0.50\), to four significant figures?
- For \(m=4\), \(n=43\), \(\bar q=0.20\), and \(B=0.0100\), use \(u_j=1/(n-3)\) to calculate \(T\) and \(SE\).
- A transformed 95% interval is \([-0.10,\ 0.50]\). What are the approximate correlation-scale limits?
- What should be done if one completed data set gives \(r=1\), and why is arbitrary clipping not an adequate fix?
Solutions. (1) \(q=\frac{1}{2}\log((1+0.50)/(1-0.50))=\frac{1}{2}\log(3)\approx0.5493\). (2) \(\bar U=1/(43-3)=0.02500\), so \(T=0.02500+(1+1/4)(0.0100)=0.03750\) and \(SE=\sqrt{0.03750}\approx0.1936\). (3) Apply \(\tanh\) to each limit: \(\tanh(-0.10)\approx-0.0997\) and \(\tanh(0.50)\approx0.4621\). The correlation-scale interval is approximately \([-0.0997,\ 0.4621]\). (4) The Fisher transform is infinite at \(r=1\). Investigate the data and the analysis to find why the perfect correlation occurred, then use a method appropriate to the situation; clipping changes the estimate without a principled variance or inference justification.