Rounding in Dose and Covariate Inputs

Why Input Rounding Deserves a Separate Check

A dose or covariate may look precise enough for a data table while still affecting a model calculation. The consequence depends on the model: a dose may enter a prediction proportionally, while a covariate may enter through a nonlinear function. A small input change can therefore alter a predicted concentration, an individual likelihood contribution, or—after refitting—the parameter estimates.

The key question is not whether a recorded value has been rounded. It is whether the rounding changes a quantity relevant to the analysis by enough to matter. Answering that requires comparing the values actually supplied to the model with defensible higher-precision values, while holding other choices fixed. If the input precision is genuinely unavailable, the comparison is a sensitivity analysis, not a recovery of the unknown exact value.

Definition: Input rounding is the replacement of an input value, such as dose or body weight, by a nearby value at a chosen precision. If a value is rounded to the nearest increment \(q\), the rounding error is at most \(q/2\), assuming ordinary nearest-value rounding. The effect of that error on model outputs depends on the model’s sensitivity to the input and on the other data and parameters.

From Input Changes to Prediction Changes

Let \(f(D,x,\theta)\) denote a model prediction for dose \(D\), covariate \(x\), and parameters \(\theta\). Write the rounded inputs as \(D+\Delta D\) and \(x+\Delta x\). If \(f\) is differentiable in the inputs near the values being considered, a first-order Taylor expansion gives a useful local approximation:

\Delta f \approx \frac{\partial f}{\partial D}\Delta D+ \frac{\partial f}{\partial x}\Delta x.

This approximation explains why input precision alone is not a measure of impact: the derivatives matter too. For a positive prediction, the corresponding relative change is approximately \(\Delta f/f\). When both dose and covariates change, their contributions can reinforce or offset one another. The approximation is local; a direct evaluation at both sets of inputs is preferable when the change is not small or the model is strongly nonlinear.

A related calculation applies to a likelihood contribution. If the individual log-likelihood is \(\ell(D,x,\theta)\), then, locally,

\Delta \ell \approx \frac{\partial \ell}{\partial D}\Delta D+ \frac{\partial \ell}{\partial x}\Delta x.

This is a diagnostic approximation, not a substitute for recomputing the likelihood with the rounded inputs. The likelihood can respond differently from the prediction: a small prediction change may matter more when the observation is precise or lies in a region where the likelihood changes steeply.

Worked Example: Rounding a Dose in a Proportional Prediction

Suppose a hypothetical one-compartment bolus model predicts concentration at time \(t\) using \(C(t)=D/V\,\exp(-kt)\). Assume complete bioavailability, fixed \(V=40\) L and \(k=0.12\) h\(^{-1}\), and evaluate the prediction at \(t=6\) h. Compare an input dose of 248 mg with the rounded dose 250 mg.

Assumptions. The model treats the dose as an instantaneous bolus, and the stated parameter values and time are held fixed. This example isolates the dose-input effect; it does not claim that this model describes a real study.

At 248 mg, the prediction is

C_{248}=\frac{248\ \mathrm{mg}}{40\ \mathrm{L}}\exp[-(0.12\ \mathrm{h}^{-1})(6\ \mathrm{h})] =6.2\ \mathrm{mg/L}\times\exp(-0.72) \approx 3.01786\ \mathrm{mg/L}.

At 250 mg, it is

C_{250}=\frac{250\ \mathrm{mg}}{40\ \mathrm{L}}\exp(-0.72) =6.25\ \mathrm{mg/L}\times\exp(-0.72) \approx 3.04220\ \mathrm{mg/L}.

The absolute change is about \(3.04220-3.01786=0.02434\) mg/L. Because this model is proportional to dose, the relative change is exactly the dose ratio:

\frac{C_{250}-C_{248}}{C_{248}}\times100 =\left(\frac{250}{248}-1\right)\times100 \approx 0.806\%.

Thus rounding from 248 mg to 250 mg increases this fixed-parameter prediction by about 0.806%. Whether that is material depends on the intended use, the observation scale, and the uncertainty in the model and data; the percentage alone does not settle that question.

Covariate Rounding Can Change a Likelihood Contribution

Covariates often enter through powers, exponentials, or other nonlinear relationships. Their rounding effect is therefore not generally proportional to the covariate change. For example, if a prediction is proportional to \(W^{-0.75}\), its local relative sensitivity to weight is \(-0.75\): a small percentage increase in weight produces approximately a 0.75-times-as-large percentage decrease in the prediction.

To connect a prediction change to a likelihood, consider an observation \(y\) with a normal error model: \(Y\sim N(\mu,\sigma^2)\), where \(\mu\) is the model prediction and \(\sigma^2\) is a fixed residual variance. The log-likelihood contribution is

\ell=-\frac{1}{2}\left[\log(2\pi\sigma^2)+\frac{(y-\mu)^2}{\sigma^2}\right].

The standardized squared residual \(Q=(y-\mu)^2/\sigma^2\) is the quadratic part of this contribution. With \(\sigma^2\) held fixed, a decrease in \(Q\) increases the log-likelihood by half the decrease in \(Q\). This comparison is meaningful here because both the error distribution and variance are explicitly held constant.

Worked Example: Rounding Weight in a Nonlinear Covariate Model

Consider a hypothetical prediction model \(\mu(W)=50/[4(W/70)^{0.75}]\) mg/L, with weight \(W\) in kg. The observed concentration is \(y=12.60\) mg/L, and the normal residual variance is fixed at \(\sigma^2=0.18\) \((\mathrm{mg/L})^2\). Compare using \(W=70.0\) kg with using a rounded covariate \(W=69.6\) kg.

Assumptions. The normal error model is used for this observation, the variance remains fixed, and only the weight input changes. The example treats both weights as plausible model inputs for comparison; it does not assert which one is the true unrounded weight.

At 70.0 kg, the prediction is \(50/4=12.5\) mg/L. Its quadratic contribution is

Q_{70}=\frac{(12.60-12.50)^2}{0.18} =\frac{0.0100}{0.18} \approx 0.05556.

At 69.6 kg, the prediction is \(50/[4(69.6/70)^{0.75}]\approx12.5538\) mg/L. Substitution using the unrounded model evaluation gives

Q_{69.6} =\frac{\left(12.60-\frac{50}{4(69.6/70)^{0.75}}\right)^2}{0.18} \approx 0.0118371 \approx 0.01184.

The quadratic term decreases by about \(0.05556-0.0118371=0.04372\). Since the variance is unchanged, the log-likelihood difference, rounded-input value minus the 70.0-kg value, is

\ell_{69.6}-\ell_{70} =-\frac{1}{2}(Q_{69.6}-Q_{70}) \approx \frac{0.04372}{2} =0.02186.

The rounded covariate gives a higher log-likelihood contribution for this observation under the stated model. That result does not show that the rounded weight is more accurate, nor does it establish that the model is better overall. It shows that even a modest covariate change can shift a likelihood contribution, and that the direction depends on the observation’s residual.

Refitting Is Necessary to Assess Parameter Effects

A change in a prediction at fixed parameters is not the same as a change in parameter estimates. To assess parameter effects, the model must be fitted again with the alternative inputs and the estimates compared. A simple Gaussian model makes the distinction explicit. For independent observations \(Y_i=\beta D_i+\varepsilon_i\), with \(\varepsilon_i\sim N(0,\sigma^2)\), the maximum-likelihood estimate of \(\beta\) is the least-squares slope through the origin:

\hat{\beta}=\frac{\sum_i D_i y_i}{\sum_i D_i^2}.

This expression follows by minimizing \(\sum_i(y_i-\beta D_i)^2\). Differentiating with respect to \(\beta\) gives \(-2\sum_iD_i(y_i-\beta D_i)=0\); rearranging yields the formula above, provided \(\sum_iD_i^2>0\). Thus changing the doses changes both the numerator and denominator of the estimate.

Worked Example: Dose Rounding Shifts a Fitted Slope

Suppose two hypothetical observations have doses \(D_1=100.4\) mg and \(D_2=199.6\) mg, with responses \(y_1=10.04\) and \(y_2=19.96\) in arbitrary response units. Fit the through-origin model \(y_i=\beta D_i+\varepsilon_i\) first using the stated doses, then using doses rounded to 100 and 200 mg.

Assumptions. The errors are independent normal with a common fixed variance, the model has no intercept, and the only change between fits is the dose input. Under these assumptions, maximizing the likelihood is equivalent to minimizing the residual sum of squares.

Using the stated doses,

\hat{\beta}_{\mathrm{stated}} =\frac{(100.4)(10.04)+(199.6)(19.96)}{100.4^2+199.6^2} =\frac{4992.032}{49920.32} =0.10000.

Using rounded doses,

\hat{\beta}_{\mathrm{rounded}} =\frac{(100)(10.04)+(200)(19.96)}{100^2+200^2} =\frac{4996}{50000} =0.09992.

The estimate decreases by \(0.00008\), or \(0.08\%\) relative to 0.10000. With the rounded-dose estimate, the fitted responses are \(100(0.09992)=9.992\) and \(200(0.09992)=19.984\). The residuals are 0.048 and \(-0.024\), so the residual sum of squares is \(0.048^2+(-0.024)^2=0.00288\). With the stated doses and \(\hat{\beta}=0.1\), both residuals are zero. If the common variance is fixed at \(0.01\), the rounded-input fit has a log-likelihood lower by \(0.00288/(2\times0.01)=0.144\) log units. This small example isolates the mechanism: changed inputs can shift both the fitted estimate and the maximized likelihood.

Diagnostics and Practical Choices

A useful assessment separates the input effect from unrelated changes in computation. Keep the model, observations, parameter values, and numerical settings fixed when first comparing predictions or individual contributions. Then refit from comparable starting values using each input version. Record both the fixed-parameter differences and the refitted differences; they answer different questions.

Identify the actual input precision.
Check the source records, units, transformations, and export formats. A value displayed with few digits may have been stored at higher precision, while a value with many digits may only be a derived representation.

Make a controlled comparison.
Evaluate the same prediction and likelihood contribution with the recorded value and a defensible alternative. Change one input at a time before testing combined changes.

Refit and compare relevant outputs.
Compare estimates, predictions, and likelihood values after refitting. Use the model’s units and the decision context to judge whether a difference matters, rather than relying on a single arbitrary threshold.

Check reproducibility.
Retain the input version and precision used, and verify that the comparison can be reproduced from the data and analysis code.

Warning signs include changes that are unexpectedly large relative to the input perturbation, discontinuous jumps when a covariate is varied smoothly, or a result that depends on hidden rounding during data import. A sharp response may be legitimate—for example, if a model contains a threshold—but it should be traced to the model structure rather than attributed automatically to optimizer failure. Conversely, nearly identical predictions do not guarantee identical estimates when the data weakly identify some parameters.

The main alternatives are to preserve the best available input precision, to perform a targeted sensitivity analysis over plausible input values, or to represent genuine uncertainty in an input explicitly if the model and analysis support doing so. These approaches are not interchangeable. Rounding all values to a convenient common precision can discard information; retaining extra digits cannot restore precision that was never measured. Sensitivity analysis reveals dependence on plausible alternatives but does not identify the unknown value.

Common Mistake and Limitations

A common mistake is to calculate that a rounded input changes a fixed-parameter prediction only slightly and then conclude that the fitted model is unaffected. That conclusion does not follow: refitting can redistribute changes across parameters, particularly when parameters are correlated or weakly identified. The correction is to distinguish the fixed-parameter comparison from the refitted comparison and to report both when parameter impact is the question.

Another mistake is to infer that the value producing the better likelihood is the more accurate input. The likelihood evaluates compatibility under the specified model and data; it does not validate the measurement record. Likewise, rounding uncertainty should not be treated as a new random error automatically. Whether a measurement error model is appropriate depends on how the input was generated and what information is available.

Key takeaway: Quantify rounding effects by comparing model outputs under clearly specified input values. For parameter impacts, refit the model; for likelihood impacts, state the observation model and what was held fixed. Judge materiality in context, and preserve the provenance and precision of the inputs.

Check Your Understanding

Try each question, then compare your reasoning with the solutions below.

In a model whose prediction is proportional to dose, what is the relative prediction change when dose increases from 398 mg to 400 mg?
A prediction is proportional to \(W^{-0.75}\). Near 70 kg, estimate the relative prediction change for a weight change of \(-0.5\) kg using a first-order approximation. State the direction.
For the normal-error example, why does the change in log-likelihood equal half the decrease in \(Q\) when the variance is fixed?
In the two-observation slope example, what fitted estimate results from using doses rounded to 100 and 200 mg, and why is a refit needed to answer the parameter-impact question?

Solutions. (1) Proportionality gives \((400/398-1)\times100=0.5025\%\), so the prediction increases by about 0.503%. (2) The first-order relative change is approximately \(-0.75(-0.5/70)=0.005357\), or a 0.536% increase in the prediction. This is a local approximation, not an exact bound. (3) The log-likelihood is \(-\tfrac12[\log(2\pi\sigma^2)+Q]\). With \(\sigma^2\) fixed, the log term is unchanged, so \(\Delta\ell=-\tfrac12\Delta Q\); a decrease in \(Q\) therefore raises \(\ell\) by half that decrease. (4) The rounded-dose estimate is \((100\times10.04+200\times19.96)/(100^2+200^2)=4996/50000=0.09992\). A fixed-parameter prediction comparison cannot show how the fitting criterion rebalances the estimate; the model must be refitted with the alternative inputs.