Worked Comparison of Dosing-History Encodings

Why Compare Dosing-History Encodings?

A repeated-dose PK/PD analysis can represent treatment history in several plausible ways. A current-dose predictor asks whether the regimen currently being taken is associated with response. A cumulative-dose predictor represents total administered amount. A recency-weighted predictor retains contributions from prior doses but discounts older ones. These are not interchangeable summaries: each encodes a different biological hypothesis and implies a different response trajectory after a dose change.

A useful comparison therefore does more than identify which model has the smallest residual error in the fitting data. It checks whether dose histories have been reconstructed consistently, clarifies what the parameters mean, evaluates predictions on records not used for fitting, and examines whether forecasts after regimen changes are credible. The calculations below use hypothetical data and simplified response models to make these contrasts explicit; a production analysis may require nonlinear PK/PD structure and subject-level random effects.

Definition: Let \(D_j\) be the amount administered during dosing interval \(j\), and let \(t_j\) be the time assigned to that dose history. At assessment time \(t\), a current-dose predictor \(Q(t)\) summarizes the regimen currently in effect; a cumulative-dose predictor is \(C(t)=\sum_{t_j\leq t}D_j\); and an exponentially recency-weighted predictor is \(W(t)=\sum_{t_j\leq t}D_j\exp[-\lambda(t-t_j)]\), where \(\lambda>0\) is a prespecified or estimated decay rate. The three quantities represent current regimen, total prior amount, and discounted prior amount, respectively.

Specify the Candidate Models Before Comparing Them

For a transparent worked comparison, let \(Y_i\) be a response measured in percentage points of reduction from baseline for record \(i\). Consider three linear mean models with a common residual-error structure:

\begin{aligned} \text{Current dose:}\quad &Y_i=\beta_{0,Q}+\beta_{1,Q}Q_i+\epsilon_i,\\ \text{Cumulative dose:}\quad &Y_i=\beta_{0,C}+\beta_{1,C}C_i+\epsilon_i,\\ \text{Recency weighted:}\quad &Y_i=\beta_{0,W}+\beta_{1,W}W_i+\epsilon_i. \end{aligned}

For this comparison, assume the records are independent across subjects, the dose history is correctly reconstructed, the conditional mean is adequately represented by the chosen linear function, and residuals have mean zero and a common variance. These assumptions are simplifications, not automatic properties of repeated-dose data. If several records come from the same subject, their dependence must be represented in the model or the validation split; treating all such records as independent can make uncertainty and apparent predictive performance misleading.

The slope units differ. If \(Y_i\) is in percentage points and \(Q_i\) is in mg/week, \(\beta_{1,Q}\) is percentage points per mg/week. The cumulative slope is percentage points per mg, and the weighted slope is also percentage points per mg of weighted history. A coefficient from one encoding cannot be compared numerically with a coefficient from another as if they represented the same exposure contrast.

Worked Example: Reconstructing a Recency-Weighted History

Suppose weekly dose amounts are \(D_1=350\) mg and \(D_2=700\) mg, administered at the starts of weeks 1 and 2. The response assessment is at the start of week 3. Let the history half-life be one week, so \(\lambda=\ln(2)/7\) per day. The dose from week 1 is 14 days old, and the dose from week 2 is 7 days old.

The cumulative amount by assessment is \(C=350+700=1{,}050\) mg. Each dose is discounted according to its age:

W=350\exp\left(-\frac{\ln(2)}{7}\times14\right) +700\exp\left(-\frac{\ln(2)}{7}\times7\right) =350(0.25)+700(0.50)=437.5\text{ mg}.

Thus the same record has a current regimen of 700 mg/week, cumulative history of 1,050 mg, and weighted history of 437.5 mg. The weighted value is lower than the cumulative amount because the older dose receives a smaller weight. It is not a measured concentration or an amount physically remaining in the body; its meaning depends on the specified weighting rule.

Fit and Evaluate on a Common Scale

Each candidate is fitted using the same response records and compatible error assumptions. To compare predictive performance, reserve subjects or later observations before fitting, then calculate predictions for that held-out set. The split must respect the data structure: when subjects contribute multiple records, hold out whole subjects if the goal is prediction for new subjects. If the goal is prediction of later outcomes for known subjects, a time-ordered split can instead test forecasting, provided information from the held-out future is not used to construct predictors or estimate parameters.

For held-out observations \(Y_i\) and predictions \(\hat{Y}_i\), the sum of squared prediction errors is \(\mathrm{SSE}_{\mathrm{test}}=\sum_i(Y_i-\hat{Y}_i)^2\). Mean squared prediction error is this sum divided by the number of held-out records. Both use the same response units after squaring, so lower values indicate closer predictions on that test set. A small test set gives an unstable comparison, and a single split may be sensitive to which subjects were reserved; repeated or grouped cross-validation can assess that instability.

Worked Example: Comparing Held-Out Predictions

Suppose hypothetical training data from six subjects produce these fitted models, with \(Q\) in mg/week and \(C,W\) in mg:

\hat{Y}_{Q}=8+0.020Q,\qquad \hat{Y}_{C}=8+0.000400C,\qquad \hat{Y}_{W}=8+0.000800W.

The coefficients are illustrative fitted estimates, not universal PK/PD constants. For two held-out subjects, suppose the observed responses and reconstructed histories are as follows.

Subject	Observed \(Y\) (percentage points)	\(Q\) (mg/week)	\(C\) (mg)	\(W\) (mg)
A	12.5	350	7,000	5,000
B	13.2	700	10,500	6,500

For subject A, the predictions are \(8+0.020(350)=15.0\), \(8+0.000400(7{,}000)=10.8\), and \(8+0.000800(5{,}000)=12.0\) percentage points. For subject B, they are \(22.0\), \(12.2\), and \(13.2\) percentage points, respectively. Squared prediction errors are then:

\begin{aligned} \mathrm{SSE}_{Q}&=(12.5-15.0)^2+(13.2-22.0)^2=6.25+77.44=83.69,\\ \mathrm{SSE}_{C}&=(12.5-10.8)^2+(13.2-12.2)^2=2.89+1.00=3.89,\\ \mathrm{SSE}_{W}&=(12.5-12.0)^2+(13.2-13.2)^2=0.25+0=0.25. \end{aligned}

On these two held-out records, the recency-weighted model has the lowest SSE and mean squared prediction error, \(0.25/2=0.125\) squared percentage points. This is evidence of better prediction for this particular test set, not proof that the weighted formulation is biologically correct or will be best in a new population. In particular, two test records provide little information about performance across a broad range of dose histories.

What a Dose-Change Forecast Reveals

Forecasts expose differences that can be obscured by similar fit statistics. Under the current-dose model, a change in \(Q\) changes the predicted mean immediately by \(\beta_{1,Q}\Delta Q\). Under the cumulative model, the predictor changes as additional dose is administered and never decreases, so the forecast depends on the entire prior total. Under exponential weighting, old contributions fade while new doses add weight. These are consequences of the predictor definitions; they do not establish the actual biological speed of response.

For regularly spaced intervals of length \(\Delta t\), the weighted history has a convenient update. If \(D_{\mathrm{new}}\) is the dose amount added in the new interval, then:

W_{\mathrm{new}}=W_{\mathrm{old}}\exp(-\lambda\Delta t)+D_{\mathrm{new}}.

This follows by separating the old doses from the new dose in the defining sum: each old term gains an additional time interval and is multiplied by \(\exp(-\lambda\Delta t)\), while the new dose has age zero and therefore weight \(\exp(0)=1\). The update is valid only when the chosen timing convention and dose amounts match the history definition.

Worked Example: Forecasting After a Weekly Dose Increase

Suppose a hypothetical patient has cumulative history \(C_{\mathrm{old}}=6{,}000\) mg and weighted history \(W_{\mathrm{old}}=6{,}000\) mg immediately before a weekly dose changes from 350 to 700 mg/week. Use the fitted models above, assume the weighted-history half-life is one week, and forecast one week later after 700 mg has been administered during that interval. The current-dose predictor is then \(Q=700\) mg/week.

The current-dose model predicts \(8+0.020(700)=22.0\) percentage points at the new regimen. The cumulative amount is \(C_{\mathrm{new}}=6{,}000+700=6{,}700\) mg, giving \(8+0.000400(6{,}700)=10.68\) percentage points. For the weighted history, the prior contribution is halved over one week:

W_{\mathrm{new}}=6{,}000(0.5)+700=3{,}700\text{ mg},\qquad \hat{Y}_{W}=8+0.000800(3{,}700)=10.96\text{ percentage points}.

The three predictions differ because each model encodes a different dose-response timing assumption. The current-dose model represents the full regimen change at once; the cumulative and weighted summaries preserve more of the earlier history. The forecasts are conditional model predictions, not causal effects of increasing dose and not guarantees about an individual response. A more realistic forecast may need an explicit PK/PD time course, uncertainty intervals, and the relevant subject-level covariates.

Diagnostics, Alternatives, and Limitations

Before interpreting a model comparison, verify that the candidate histories are aligned to the response assessment and are based on the intended dose record, such as administered rather than merely prescribed doses. Check the dose units, interval boundaries, missed doses, and decay-rate units. A factor-of-seven error in converting a daily decay rate to a weekly interval, for example, changes the weighted history substantially. Plot residuals against time, fitted values, and each candidate predictor; look for systematic trends, changing variance, and residual patterns associated with dose changes. Compare observed and predicted trajectories, not only a single aggregate score.

The encodings can be strongly correlated, especially when dose regimens change little. In that setting, fitted slopes may be unstable and their separate biological interpretations weak, even when predictions are similar. Check predictor ranges and parameter uncertainty; do not interpret small differences in test error as decisive when they are within the variation from resampling. An estimated decay parameter can also be weakly identified if the data contain few informative changes in dose or timing.

The main alternative is a mechanistic or semi-mechanistic PK/PD model that represents exposure and response dynamics more directly, rather than using a single history summary. Such models may be preferable when concentration data, known turnover processes, or delayed effects are central to the question, but they require suitable structure and data. A current-dose model is simpler and may be adequate when response tracks regimen rapidly. Cumulative dose is interpretable when total exposure is the substantive hypothesis, but it cannot represent fading effects without additional structure. Recency weighting offers a compact memory of past dosing, yet its kernel and decay rate are assumptions that must be justified or estimated. None of these summaries alone resolves confounding, adherence error, or extrapolation beyond observed dose histories.

Common pitfall: Selecting the model with the smallest training-set residual error and concluding that its history encoding is the true biological mechanism. More flexible histories can fit observed records well without predicting new subjects or dose changes reliably. Compare held-out predictions, inspect uncertainty and diagnostics, and describe the result as support for predictive adequacy under the tested conditions—not proof of mechanism or causation.

Check Your Understanding

Answer each question, then check the concise solutions below.

If the dose amount is 400 mg per interval and the weighted history has a one-interval half-life, what is the next weighted history after one interval if the previous value was 1,200 mg and a new 400 mg dose is added?
Why are slopes from a current-dose model and a cumulative-dose model not directly comparable numerically?
A candidate model has lower training SSE but higher held-out SSE than another candidate. Which has better performance on the held-out set, and what should not be concluded?
After a dose increase, which of the three simple predictors changes immediately to the new regimen value, and which retain information from earlier dosing?

Solutions. (1) The prior weighted value is halved, so \(W_{\mathrm{new}}=1{,}200(0.5)+400=1{,}000\) mg. (2) Their predictor units differ: the first slope is response per mg/week, while the second is response per mg of cumulative amount. (3) The candidate with lower held-out SSE performs better on that test set; this alone does not prove its biological mechanism or guarantee superior performance in other populations. (4) Current dose changes immediately to the new regimen; cumulative dose and recency-weighted history retain prior-dose contributions, although the weighted history discounts older contributions over time.