Why a Nonlinear Contrast Needs Its Own Uncertainty Calculation
A pooled coefficient vector can answer questions that are not themselves linear combinations of its entries. For example, a scientific contrast might be a ratio of two coefficients or a smooth function of several coefficients. The estimate is obtained by applying the function to the pooled vector, but its uncertainty depends on how uncertainty in each coefficient—and covariance between coefficients—moves the function. The delta method makes that propagation explicit.
This tutorial concerns inference for a smooth function of the pooled coefficient vector. It does not replace the pooling of the coefficients themselves. Rather, after pooling the vector and its covariance matrix, it uses a local linear approximation to quantify uncertainty in the nonlinear contrast.
The pooled covariance matrix \(\boldsymbol T\) includes both average sampling uncertainty within imputations and additional uncertainty associated with differences among the imputed-data estimates. Off-diagonal entries matter: coefficients can vary together, and a nonlinear function can be especially sensitive to that covariance.
Deriving the Delta-Method Variance
The approximation follows from a first-order Taylor expansion around the pooled estimate. Write \(\boldsymbol Q\) for the underlying coefficient vector and let \(\boldsymbol G=\nabla g(\bar{\boldsymbol Q})\). For values of \(\boldsymbol Q\) near \(\bar{\boldsymbol Q}\),
The first term is fixed at the estimate; the variable part of the approximation is the linear combination \(\boldsymbol G^\mathsf{T}(\boldsymbol Q-\bar{\boldsymbol Q})\). A linear combination with coefficient vector \(\boldsymbol G\) has variance \(\boldsymbol G^\mathsf{T}\operatorname{Var}(\boldsymbol Q)\boldsymbol G\). Replacing that covariance with the pooled estimate \(\boldsymbol T\) gives
This decomposition is useful for inference. The projected within-imputation variance is \(U_g=\boldsymbol G^\mathsf{T}\bar{\boldsymbol U}\boldsymbol G\); the projected between-imputation variance is \(B_g=\boldsymbol G^\mathsf{T}\boldsymbol B\boldsymbol G\). Applying the usual scalar Rubin degrees-of-freedom approximation gives
When a finite complete-data degrees-of-freedom adjustment is required by the analysis setting, an appropriate small-sample adjustment can be applied; the expression above is the basic large-complete-sample approximation. A confidence interval is formed as \(g(\bar{\boldsymbol Q})\pm t_{1-\alpha/2,\nu_g}\sqrt{\boldsymbol G^\mathsf{T}\boldsymbol T\boldsymbol G}\). Its coverage is approximate, relying both on the MI pooling approximation and on the local linearization.
Worked Calculations
Worked Example: A Ratio of Two Pooled Coefficients
Suppose five hypothetical imputations yield \(\bar{\boldsymbol Q}=(2,1)^\mathsf{T}\), with \(\bar{\boldsymbol U}=\begin{pmatrix}0.04&0.01\\0.01&0.09\end{pmatrix}\) and \(\boldsymbol B=\begin{pmatrix}0.01&0.004\\0.004&0.016\end{pmatrix}\). Consider \(g(Q_1,Q_2)=Q_1/Q_2\). Assume the coefficient vector is estimated on a scale where the second coefficient is meaningfully away from zero; here its pooled estimate is 1.
The contrast estimate and gradient are \(g(2,1)=2\) and \(\boldsymbol G=(1/Q_2,-Q_1/Q_2^2)^\mathsf{T}\), so \(\boldsymbol G=(1,-2)^\mathsf{T}\). Projecting the within-imputation covariance gives \(U_g=(1)(0.04)(1)+(-2)(0.09)(-2)+2(1)(0.01)(-2)=0.36\). Similarly, \(B_g=(1)(0.01)(1)+(-2)(0.016)(-2)+2(1)(0.004)(-2)=0.058\).
With \(m=5\), the total variance and standard error are \(\widehat{\operatorname{Var}}=0.36+1.2(0.058)=0.4296\) and \(SE=\sqrt{0.4296}=0.6554\). The relative increase is \(r_g=1.2(0.058)/0.36=0.1933\), giving \(\nu_g=4(1+1/0.1933)^2\approx152.4\). Using \(t_{0.975,152.4}\approx1.976\), the approximate 95% interval is \(2\pm1.976(0.6554)=(0.705,3.295)\). The ratio is dimensionless when the coefficients have the same units. This interval is an approximate range of values compatible with the data and model under the stated procedure, not a probability statement about the fixed contrast.
Worked Example: A Smooth Function of Two Coefficients
For a second hypothetical analysis, suppose \(\bar{\boldsymbol Q}=(1,2)^\mathsf{T}\), \(m=5\), and \(\bar{\boldsymbol U}=\begin{pmatrix}0.04&0.01\\0.01&0.09\end{pmatrix}\), \(\boldsymbol B=\begin{pmatrix}0.01&-0.002\\-0.002&0.016\end{pmatrix}\). Consider \(g(Q_1,Q_2)=\log(Q_1^2+Q_2^2)\), which is smooth wherever \(Q_1^2+Q_2^2>0\). The pooled vector is not at the excluded point.
The estimate is \(g(1,2)=\log(5)=1.609\). Its gradient is \(\boldsymbol G=\big(2Q_1/(Q_1^2+Q_2^2),2Q_2/(Q_1^2+Q_2^2)\big)^\mathsf{T}\); at \((1,2)\), this is \((0.4,0.8)^\mathsf{T}\).
The projected within variance is \(U_g=(0.4)^2(0.04)+(0.8)^2(0.09)+2(0.4)(0.8)(0.01)=0.0704\). The projected between variance is \(B_g=(0.4)^2(0.01)+(0.8)^2(0.016)+2(0.4)(0.8)(-0.002)=0.01056\). Therefore the total variance is \(0.0704+1.2(0.01056)=0.083072\), and the approximate standard error is \(\sqrt{0.083072}=0.2882\). The negative covariance reduces the projected between variance here; omitting covariance would produce a different result.
Worked Example: A Square-Root Contrast and Its Domain
Suppose four hypothetical imputations give a scalar pooled estimate \(\bar Q=0.36\), average within variance \(\bar U=0.0064\), and between variance \(B=0.0016\). The contrast is \(g(Q)=\sqrt Q\), and the relevant parameter is assumed positive. The pooled estimate is in the function’s smooth domain.
The estimate is \(\sqrt{0.36}=0.6\), and the gradient at the pooled estimate is \(g'(0.36)=1/(2\sqrt{0.36})=0.8333\). Thus \(U_g=(0.8333)^2(0.0064)=0.004444\), and \(B_g=(0.8333)^2(0.0016)=0.001111\). With \(m=4\), the total variance is \(0.004444+1.25(0.001111)=0.005833\), so \(SE=\sqrt{0.005833}=0.07638\).
Here \(r_g=1.25(0.001111)/0.004444=0.3125\) and \(\nu_g=3(1+1/0.3125)^2=52.92\). With \(t_{0.975,52.92}\approx2.006\), the approximate 95% interval is \(0.6\pm2.006(0.07638)=(0.447,0.753)\), in the square-root units of the original quantity. The calculation is appropriate only if positivity and local smoothness are credible; an estimate close to zero would warrant particular caution.
Assumptions, Checks, and Alternatives
The calculation assumes that the same target coefficient vector and estimand are used in every imputation, that each \(\boldsymbol U_j\) estimates the within-imputation covariance appropriately, and that the imputations support valid Rubin pooling for that vector. It also assumes \(g\) is differentiable in a neighborhood of the pooled estimate and that the first-order approximation is adequate over the uncertainty range. The gradient must be evaluated at \(\bar{\boldsymbol Q}\), and the full pooled covariance matrix—not just its diagonal—must be available.
Confirm that the pooled estimate lies in a region where the contrast is defined and differentiable. Ratios require a nonzero denominator; logarithms require a positive argument.
Verify that coefficient order is consistent in \(\bar{\boldsymbol Q}\), every \(\boldsymbol U_j\), and \(\boldsymbol B\). Confirm that covariance terms have not been dropped.
Inspect whether plausible variation in the coefficients would move the contrast through regions of sharply changing slope, a boundary, or a singularity. Large curvature relative to the uncertainty suggests the first-order result may be unreliable.
Give the contrast estimate, delta-method standard error, degrees of freedom or interval method, and the scale and units of the contrast. Avoid presenting a symmetric interval as if it had exact coverage for a highly asymmetric nonlinear quantity.
A common error is to calculate the contrast gradient but use only the diagonal elements of \(\boldsymbol T\), reasoning that each coefficient has its own standard error. That is incorrect whenever the gradient has multiple nonzero entries and the coefficients are correlated: the variance contains the cross-product terms. Another error is to transform the endpoints of a coefficient interval and call the result a delta-method interval. The delta method instead propagates covariance through the gradient; endpoint transformation is a different procedure and does not generally give the same interval.
There is also an estimand distinction. This calculation estimates \(g(\bar{\boldsymbol Q})\) and approximates its variance using the gradient at \(\bar{\boldsymbol Q}\). Pooling the transformed imputation-specific quantities \(g(\hat{\boldsymbol Q}_j)\) instead targets a pooled transformed estimate, which can differ because nonlinear functions do not generally commute with averaging. Curvature can make that difference material. The delta method is attractive because it uses the pooled vector and its covariance directly, but it is not a universal substitute for a method designed for a different transformed estimand or a strongly nonlinear sampling distribution.
| Approach | What it estimates or uses | Main caution |
|---|---|---|
| Delta method from the pooled vector | \(g(\bar{\boldsymbol Q})\), with variance approximated by \(\boldsymbol G^\mathsf{T}\boldsymbol T\boldsymbol G\) | Can be inaccurate under strong curvature or near a boundary |
| Pool transformed estimates | Imputation-specific values \(g(\hat{\boldsymbol Q}_j)\) and their associated uncertainty | Generally targets a different pooled quantity; each transformed estimate and variance must be valid |
| Direct nonlinear model or resampling-based inference | A method tailored to the nonlinear target or its sampling distribution | Requires suitable modeling or resampling assumptions and can involve more computation |
The delta method is most useful when the contrast is smooth over the region supported by the data, the gradient is stable, and a first-order covariance-based summary is appropriate. It may be unreliable when the denominator of a ratio is near zero, the gradient is very large, the estimated contrast lies near a boundary, or the distribution of the contrast is visibly asymmetric. In these situations, report the limitation and consider an approach that respects the contrast’s domain and sampling behavior.
Check Your Understanding
Answer each question, then compare your reasoning with the concise solutions below.
- For a smooth scalar contrast \(g(\boldsymbol Q)\), which vector is used to project the pooled covariance matrix, and where is it evaluated?
- Why can omitting the off-diagonal covariance terms change the delta-method variance?
- For \(g(Q)=\log Q\), what domain condition is required at the pooled estimate, and what is the derivative?
- Does pooling \(g(\hat{\boldsymbol Q}_j)\) generally produce the same estimate as calculating \(g(\bar{\boldsymbol Q})\)? Why or why not?
- Name one warning sign that a first-order delta-method interval may be unreliable.
Solutions. (1) Use \(\boldsymbol G=\nabla g(\bar{\boldsymbol Q})\), the gradient evaluated at the pooled vector. (2) Cross-products of gradient entries and coefficient covariances contribute to \(\boldsymbol G^\mathsf{T}\boldsymbol T\boldsymbol G\). (3) The pooled estimate must be positive, and \(d(\log Q)/dQ=1/Q\). (4) No; generally \(g\) does not commute with averaging, so curvature can make the two estimates differ. (5) Examples include a denominator near zero, a pooled estimate near a boundary, or a rapidly changing gradient over the uncertainty range.