Calculating Prediction-Corrected Observations

From a Correction Rule to Calculated Values

In a prediction-corrected visual predictive check (pcVPC), each outcome is rescaled to the prediction scale chosen for its time bin. The calculation is simple, but it must be applied consistently: an observed value and its simulated counterparts must receive the same correction factor for the same record design. Otherwise, differences between observed and simulated summaries can reflect inconsistent scaling rather than model performance.

Let $Y_{ij}$ denote the observed outcome for subject $i$ at observation $j$, and let $P_{ij}$ be the positive model prediction associated with that record's design, such as its time, dose, and covariates. Let $b(j)$ identify the time bin containing observation $j$. Write $P^{ref}_{b(j)}$ for the positive reference prediction used for that bin. The calculation below concerns applying those predictions; it does not specify how a reference prediction should be selected.

Definition: The prediction-corrected observed outcome is the observed value multiplied by the ratio of its bin's reference prediction to its record-specific prediction: $$ Y^{pc}_{ij}=Y_{ij}\frac{P^{ref}_{b(j)}}{P_{ij}}. $$ For simulation replicate $m$, the corresponding corrected outcome is $$ Y^{pc,(m)}_{ij}=Y^{(m)}_{ij}\frac{P^{ref}_{b(j)}}{P_{ij}}. $$ Here $Y^{(m)}_{ij}$ is the simulated outcome at the matching record. The correction requires $P_{ij}\ne 0$; for the usual positive concentration scale, both $P_{ij}$ and $P^{ref}_{b(j)}$ must be positive.

The ratio $P^{ref}_{b(j)}/P_{ij}$ is a dimensionless scaling factor. A prediction above the bin reference gives a factor below one, so the outcome is scaled down; a prediction below the reference gives a factor above one, so the outcome is scaled up. The result has the same units as the original outcome. For example, if outcomes are measured in mg/L, corrected outcomes are also in mg/L.

Calculating Corrected Observations

Worked Example: Scaling Two Observations in One Bin

Suppose a time bin has reference prediction $P^{ref}=6.0$ mg/L. Two hypothetical records in that bin have predictions $P_1=10.0$ mg/L and $P_2=12.0$ mg/L, with observed outcomes $Y_1=8.0$ mg/L and $Y_2=15.0$ mg/L. All predictions are positive, as required for this concentration-scale correction.

For the first record, the scaling factor is $6.0/10.0=0.6000$. Therefore, $$ Y^{pc}_1=8.0\frac{6.0}{10.0}=8.0(0.6000)=4.8\text{ mg/L}. $$

For the second record, the factor is $6.0/12.0=0.5000$, so $$ Y^{pc}_2=15.0\frac{6.0}{12.0}=15.0(0.5000)=7.5\text{ mg/L}. $$

The uncorrected outcomes, 8.0 and 15.0 mg/L, have been expressed on the common 6.0 mg/L reference-prediction scale. This operation does not replace the observations with predictions: it rescales each observation in proportion to its own prediction.

The corresponding simulation calculation uses the same record-level scale factor. In a predictive check, simulated outcomes are generated for the observation design being checked. For a given record, use the prediction associated with that design and apply its correction factor to the observed value and to every matched simulated value. If the simulated records retain the same times, doses, and covariates as the observed records, the factor is identical across simulation replicates for that record.

Worked Example: Correcting Matched Simulated Outcomes

Consider a hypothetical record with $P=5.0$ mg/L and a bin reference prediction of $4.0$ mg/L. The observed outcome is $3.6$ mg/L. Three simulated outcomes at the same record design are $4.0$, $6.0$, and $2.0$ mg/L. The prediction ratio is $4.0/5.0=0.8000$.

The observed corrected outcome is $$ Y^{pc}=3.6(0.8000)=2.88\text{ mg/L}. $$

Apply that same ratio to each simulation: $$ Y^{pc,(1)}=4.0(0.8000)=3.20\text{ mg/L}, $$ $$ Y^{pc,(2)}=6.0(0.8000)=4.80\text{ mg/L}, $$ $$ Y^{pc,(3)}=2.0(0.8000)=1.60\text{ mg/L}. $$

The observed and simulated outcomes have all been transformed by 0.8000 because they correspond to the same design and prediction. Using a different factor for the simulations would make the comparison inconsistent.

What the Scaling Preserves

The correction can be understood by comparing an outcome with its prediction. If both $P_{ij}$ and $P^{ref}_{b(j)}$ are positive, then substitution gives

\frac{Y^{pc}_{ij}}{P^{ref}_{b(j)}}= \frac{Y_{ij}(P^{ref}_{b(j)}/P_{ij})}{P^{ref}_{b(j)}}= \frac{Y_{ij}}{P_{ij}}.

Thus the observed-to-prediction ratio is preserved while the prediction in the denominator is replaced by the common bin reference. Equivalently, the proportional deviation from prediction is preserved:

\frac{Y^{pc}_{ij}}{P^{ref}_{b(j)}}-1 = \frac{Y_{ij}}{P_{ij}}-1.

This is a direct algebraic result, not an assumption that correction removes all sources of variation. The transformation places values on a common prediction scale; residual variability and subject-to-subject variability remain in the corrected outcomes.

The reference prediction must be nonzero for the ratio identity to be defined, and positive for the usual interpretation as a common positive concentration scale. For instance, if $Y=1$, $P=2$, and $P^{ref}=0$, the correction formula gives $Y^{pc}=0$, but $Y^{pc}/P^{ref}$ is undefined. One must not claim the ratio identity in that case. Similarly, a zero or negative record-specific prediction is not a valid denominator for the standard multiplicative correction.

Worked Example: Checking the Ratio Property

Suppose a hypothetical record has $Y=9.0$ mg/L, $P=12.0$ mg/L, and $P^{ref}=6.0$ mg/L. Both predictions are positive. The correction factor is $6.0/12.0=0.5000$, so $$ Y^{pc}=9.0(0.5000)=4.5\text{ mg/L}. $$

On the original scale, the outcome-to-prediction ratio is $9.0/12.0=0.7500$. On the corrected scale it is $4.5/6.0=0.7500$. The proportional deviation from prediction is $-0.2500$, or $-25.00\%$, on either scale: $$ \frac{9.0}{12.0}-1=-0.2500 \qquad\text{and}\qquad \frac{4.5}{6.0}-1=-0.2500. $$

The equality holds here because both predictions are nonzero. The corrected value is not an estimate of the outcome that would necessarily have occurred under a different dose or regimen; it is a rescaled value for visual or numerical comparison.

Practical Calculation and Quality Checks

A reliable implementation separates record matching from arithmetic. First identify the bin and prediction for every observed record. Then calculate the correction factor and apply it to that observed value and its matched simulated values. Finally, check the resulting values and retain enough information to trace each correction back to its prediction and bin.

Match each record to its bin and prediction.
Confirm that the observed value, prediction, bin, and simulated outcomes refer to the same observation design. In particular, check time, dose, and relevant covariates.

Validate the predictions.
For the standard positive-scale correction, verify that the record prediction and bin reference prediction are both finite and positive. Investigate zeros, negative values, or missing predictions rather than silently dividing.

Calculate one factor and apply it consistently.
Compute $C_{ij}=P^{ref}_{b(j)}/P_{ij}$. Multiply the observed outcome and every matched simulated outcome by $C_{ij}$.

Check the output.
Confirm that corrected values have the outcome's units, and spot-check that $Y^{pc}_{ij}/P^{ref}_{b(j)}=Y_{ij}/P_{ij}$ when both predictions are positive. Check that simulation replicates were not assigned another record's factor.

Several patterns can signal a calculation problem. Unexpected jumps at bin boundaries may arise if records were assigned to the wrong bin or if different reference predictions were used inconsistently. Extremely large corrected values can result from a very small positive $P_{ij}$; they warrant checking the prediction and the applicability of multiplicative scaling. A corrected simulated distribution that differs from the observed distribution for no plausible model reason can reflect misaligned records or an incorrect factor, not necessarily a model defect.

The correction is a multiplicative rescaling. It is most interpretable when the prediction is meaningful and positive and when proportional deviations are the scale of interest. It does not repair a misspecified model, eliminate residual error, or guarantee that distributions from different designs become identical. It also does not justify replacing the model prediction with a rounded or unrelated value. Preserve precision during calculation and round only for reporting.

How Prediction Correction Differs from Raw Comparisons

A raw VPC compares observed and simulated outcomes on their original scale. When typical predictions differ substantially across records within a bin, the spread of raw outcomes can partly reflect those different prediction levels. A pcVPC instead rescales values to the bin reference prediction, making the comparison less dependent on unequal typical prediction levels. The calculation does not change which observations were collected or which simulations were generated.

This distinction matters when interpreting the result: correction changes the scale of the values, not the evidence in the underlying records. The raw and corrected views answer related but different descriptive questions. If predictions are already similar within bins, correction factors are near one and corrected values will be close to raw values. If predictions vary substantially, the transformation can visibly change the spread and locations of values. That change is expected from the formula and should not be mistaken for new information in the data.

A frequent mistake is to correct observed outcomes but leave simulated outcomes uncorrected, or to use a simulation-specific factor when the observed and simulated records share the same design. The resulting summaries no longer compare quantities on the same scale. The correction must be defined by the matched record design and applied to both sides of the predictive comparison.

Key takeaway: For each record, calculate $P^{ref}_{b(j)}/P_{ij}$, then multiply the observed outcome and every matched simulated outcome by that same factor. For the usual positive concentration scale, both predictions must be positive; the ratio-preservation identity also requires a nonzero reference prediction.

Check Your Understanding

Answer each question, then compare with the solutions below.

A record has $Y=7.2$ mg/L, $P=9.0$ mg/L, and $P^{ref}=6.0$ mg/L. Calculate its corrected outcome.
For the record in question 1, a matched simulation has $Y^{(m)}=10.5$ mg/L. What is its corrected value?
Why is the identity $Y^{pc}/P^{ref}=Y/P$ not defined when $P^{ref}=0$, even though the correction formula may return a numerical value?
What calculation error occurs if observed outcomes use one factor but matched simulated outcomes use another?

Solutions. (1) The factor is $6.0/9.0=0.6667$, so $Y^{pc}=7.2(0.6667)=4.8$ mg/L. (2) Apply the same matched-record factor: $10.5(0.6667)=7.0$ mg/L. (3) Division by zero is undefined, so the ratio identity cannot be evaluated; a zero reference is also not a positive common concentration scale. (4) The observed and simulated values are no longer expressed using the same transformation, so their comparison can reflect inconsistent scaling rather than predictive performance.