Uncertainty In Ratios And Contrasts

Why Ratios and Contrasts Need Their Own Uncertainty Calculation

Population pharmacokinetic results are often interpreted through combinations of parameters rather than through a single estimate. A clearance ratio can express the relative clearance between two groups; a difference in typical clearance can express an absolute change in units such as L/h. Each quantity has its own sampling uncertainty. The standard errors of the component estimates alone are not enough: their covariance also matters.

The calculations below treat parameter estimates as a vector with an estimated covariance matrix. They rely on the estimates being approximately jointly normal in the region relevant to the calculation, and on the covariance matrix representing their joint sampling uncertainty. For a nonlinear quantity such as a ratio, a first-order approximation is also needed. These conditions should be considered before interpreting a resulting standard error or interval.

Contrasts: The Covariance Term in a Difference

Let \(\hat{\theta}_1\) and \(\hat{\theta}_2\) estimate two population parameters. Write their estimated variances as \(\widehat{\operatorname{Var}}(\hat{\theta}_1)\) and \(\widehat{\operatorname{Var}}(\hat{\theta}_2)\), and their estimated covariance as \(\widehat{\operatorname{Cov}}(\hat{\theta}_1,\hat{\theta}_2)\). The estimated contrast is \(\hat{\theta}_1-\hat{\theta}_2\). The variance rule for a difference gives

The covariance is subtracted because the second estimate has a negative coefficient in the contrast. Positive covariance means the estimates tend to move together, so their difference tends to vary less than it would if they varied independently. Negative covariance has the opposite effect. Omitting the covariance is justified only when the estimates are uncorrelated, or when an independence assumption is otherwise warranted.

Ratios on the Original Scale

For a ratio \(R=\theta_1/\theta_2\), the parameter estimates are transformed nonlinearly. The delta method approximates the variance of a smooth function of estimates by evaluating its gradient at the estimates and combining that gradient with the covariance matrix. For the ratio, the gradient with respect to \((\theta_1,\theta_2)\) is \((1/\theta_2,-\theta_1/\theta_2^2)\). Thus, for \(\hat{R}=\hat{\theta}_1/\hat{\theta}_2\),

The result is an approximation because the ratio is nonlinear. It is most dependable when the denominator is well away from zero relative to its uncertainty and the estimates are reasonably described by a joint normal approximation near their estimated values. If the component parameters have the same units, the ratio and its standard error are dimensionless. If their units differ, the ratio’s units are the numerator’s units divided by the denominator’s units.

Ratios on the Log Scale

For positive parameters, a ratio can instead be expressed as a difference of logarithms: \(\log(\theta_1/\theta_2)=\log(\theta_1)-\log(\theta_2)\). This is especially convenient when the model estimates log parameters or when a multiplicative comparison is the scientific target. If \(\hat{\eta}_1\) and \(\hat{\eta}_2\) estimate \(\log(\theta_1)\) and \(\log(\theta_2)\), respectively, then \(\hat{\eta}_1-\hat{\eta}_2\) estimates the log ratio. Its variance is the ordinary variance of a difference:

If the log-scale estimates are approximately jointly normal, an interval for the log ratio can be calculated on that scale and then exponentiated. Exponentiation turns the two endpoints into an interval for the ratio. This produces an interval that is positive and typically asymmetric around the ratio estimate. It does not remove uncertainty or make a weakly determined ratio precise.

Derivation and Assumptions

The formula for a ratio follows from the first-order Taylor approximation. Let \(g(\theta_1,\theta_2)=\theta_1/\theta_2\). Near the estimated parameter vector, a small estimation error changes \(g\) approximately by its gradient multiplied by the parameter error:

For any vector of estimates with covariance matrix \(\Sigma\), a linear combination with coefficient vector \(a\) has variance \(a^T\Sigma a\). Applying that rule to the gradient gives the first-order variance approximation \(\nabla g^T\Sigma\nabla g\). Since the gradient is evaluated using estimated parameters in practice, the calculated result is itself an estimate of the variance. This derivation explains both the covariance term and the approximation: higher-order changes in the nonlinear function are omitted.

Before using these formulas, check that the parameter estimates refer to the same model fit and that the variances and covariance come from the same estimated covariance matrix. A covariance matrix must be symmetric and positive semidefinite; in particular, for two estimates, the absolute covariance cannot exceed the product of their standard errors. The ratio approximation also requires a denominator sufficiently far from zero for a local linear approximation to be useful. For a log ratio, both underlying parameters must be positive, and the log-scale estimates must have a suitable approximate sampling distribution.

Diagnostics, Alternatives, and Limitations

A few checks catch many practical failures:

• Check scale and units. Variances have squared units, covariance has the product of the two units, and the final standard error has the units of the contrast or ratio.
• Check covariance signs and magnitude. For a difference or ratio, a positive covariance often reduces the variance because the second parameter enters with a negative derivative. The covariance must still be compatible with the two variances.
• Check whether a ratio is locally well behaved. If the denominator is near zero relative to its standard error, the ratio can be highly skewed or poorly approximated by a normal distribution. A symmetric interval from the delta method can then be misleading.
• Recompute the arithmetic. In ratio calculations, the powers of the denominator differ across terms. Write out each term separately before summing, and verify that the final variance is nonnegative.

The raw-scale delta method is direct and useful when the denominator is stable and the ratio estimate is reasonably close to normal. A log-scale interval is often attractive for positive parameters and multiplicative comparisons, particularly when the estimated log parameters are approximately normal. The two approaches need not yield identical intervals: they use different approximations and express uncertainty on different scales. When a denominator may approach zero, a Fieller-type interval can better reflect the resulting ratio uncertainty and may be unbounded. Resampling methods can also reflect nonlinear behavior, but depend on a suitable resampling scheme and successful model refitting.

Common mistake: “The covariance is positive, so I should add its contribution to the variance of a ratio.” This is wrong for the usual ratio \(\hat{\theta}_1/\hat{\theta}_2\): its derivative with respect to the denominator is negative, so the covariance contribution is \(-2\hat{\theta}_1\widehat{\operatorname{Cov}}(\hat{\theta}_1,\hat{\theta}_2)/\hat{\theta}_2^3\). With positive estimates and positive covariance, that term is negative. The sign follows from the function’s derivatives, not from a rule that every variance component must be added.