Why MCP-Mod Uses Two Stages
A dose-ranging study can show that some doses differ from placebo without revealing which curve best describes the pattern. A shallow linear increase, a rapid early increase followed by a plateau, and a steep sigmoid may all be plausible before the data are observed. MCP-Mod addresses these two related but distinct tasks: it first tests for a dose-response signal across prespecified candidate shapes, then fits supported models to estimate dose-response features such as a target dose.
The first stage, MCP (multiple contrast procedure), protects the type I error rate when several candidate shapes are tested. The second stage, Mod (modeling), uses dose-response models to describe the data and support dose estimation. A significant contrast is evidence against the no-dose-response null for that contrast; it does not establish that its associated curve is the unique true model.
Candidate Shapes and Contrasts
Suppose a randomized study assigns participants to placebo and doses \(d_1,\ldots,d_K\), and the outcome is a continuous measure for which larger values indicate greater benefit. Let \(\bar{Y}_i\) denote the observed mean outcome at dose \(d_i\), with \(d_0=0\) for placebo. Before examining outcomes, investigators specify candidate mean-response shapes \(f_j(d)\). Examples include a linear trend, an Emax curve, and a sigmoid Emax curve:
Here \(ED_{50}\) is the dose at which the modeled dose-related component reaches half its maximum, and \(h\) controls the steepness of the sigmoid shape. Candidate parameter values such as \(ED_{50}\) and \(h\) must be chosen or prespecified in a defensible way; choosing them after inspecting the observed dose means can invalidate the intended multiplicity control unless that search is included in the testing procedure.
A candidate shape can be translated into a contrast. For a vector of shape values \(f_j(d_i)\), one convenient choice is to subtract its average across the \(K+1\) dose groups from each value. The resulting contrast coefficients sum to zero. Multiplying all coefficients by the same positive constant changes neither the standardized test statistic nor its p-value.
A positive \(C_j\) indicates that the observed means tend to align with an increasing version of the candidate shape. Under independent observations, equal group sizes \(n\), and a common within-group variance \(\sigma^2\), the contrast standard error is \(\sigma\sqrt{\sum_i c_{ij}^2/n}\). With unequal group sizes, the variance becomes \(\sigma^2\sum_i c_{ij}^2/n_i\).
Worked Example: Testing Three Candidate Shapes
Worked Example: Testing Three Candidate Shapes
Consider a hypothetical study with 40 participants in each of five groups: placebo and 1, 2, 4, and 8 mg. The outcome is a change score in arbitrary clinical units, with larger values favorable. Suppose the observed group means are 100, 106, 112, 114, and 115, respectively. Assume the common within-group standard deviation is 12 units, as specified for this teaching calculation.
| Dose (mg) | 0 | 1 | 2 | 4 | 8 |
|---|---|---|---|---|---|
| Observed mean | 100 | 106 | 112 | 114 | 115 |
For the linear candidate, centered and conveniently scaled coefficients are \((-3,-2,-1,1,5)\). They sum to zero, and the contrast estimate is:
Each group mean has variance \(12^2/40=3.6\) square units. Therefore:
For an Emax shape with \(ED_{50}=2\) mg, one possible scaled coefficient vector is \((-6.9,-1.9,0.6,3.1,5.1)\). It also sums to zero. Thus:
For a sigmoid Emax shape with \(ED_{50}=2\) mg and \(h=2\), use coefficients \((-41.5,-24.5,1,26.5,38.5)\). Then \(C_{\mathrm{sigmoid}}=813.5\) units, \(\sum_i c_i^2=4508\), and \(SE=\sqrt{3.6(4508)}=127.4\) units. The resulting statistic is \(Z=813.5/\sqrt{16228.8}=6.387\).
For a one-sided familywise significance level of 0.05, a simple Bonferroni procedure tests each contrast at \(0.05/3=0.01667\). The standard normal critical value is approximately 2.128. Each statistic exceeds that threshold, so all three candidate contrasts reject their individual null hypotheses under this multiplicity adjustment. This supports a dose-response signal across these candidate shapes; it does not show that all three curves are equally plausible or that any one is definitively correct.
Why the Multiplicity Adjustment Works
The contrast statistics are generally correlated because they use the same group means. A procedure that tested each contrast at 0.05 without adjustment would therefore not, in general, keep the probability of at least one false positive at 0.05. Bonferroni control is valid even when the tests are correlated: if there are \(M\) tests and each has type I error no greater than \(\alpha/M\), the probability of one or more false rejections is no greater than the sum of their individual error probabilities.
This bound follows from the union bound and does not require independence. It may be conservative, especially when candidate contrasts are highly correlated. A max-\(T\) procedure instead calibrates the distribution of the largest standardized contrast while accounting for their joint covariance; it can be more powerful when that joint distribution is estimated appropriately. The testing method, one- or two-sided direction, candidate shapes, and multiplicity plan should be specified before unblinded outcome comparisons.
The contrast construction has a simple justification. If the expected group means follow a candidate shape, their systematic differences across dose align with the centered shape values. The contrast weights sum to zero, so a common baseline level cancels. For independent group means, linearity of expectation and variance gives:
Under the no-dose-response null, all group means are equal, so \(E(C_j)=0\) because the coefficients sum to zero. Dividing the observed contrast by its standard error yields the standardized test statistic. This reasoning explains both the baseline cancellation and the standard error formula; it also shows why the independent-arm variance assumptions matter.
Model Fitting and Supported-Shape Selection
After MCP testing, the modeling stage fits the candidate dose-response models associated with the supported contrasts. A model adds a baseline and response parameters; for example, an Emax mean model is \(E(Y\mid d)=E_0+E_{\max}d/(ED_{50}+d)\). The model stage can compare fit and complexity using a criterion such as AIC, but the models should be checked for adequate fit rather than selected mechanically. AIC favors a balance between fit and the number of estimated parameters; it is not a test that the preferred model is true.
Worked Example: Comparing Supported Models
Suppose the candidate models from the preceding hypothetical study are fitted to its five group means. Treat the group-mean variance as known at 3.6 square units. Define \(Q\) as the weighted residual sum of squares, \(\sum_i(\bar{Y}_i-\widehat{\mu}_i)^2/3.6\). Suppose numerical fitting gives \(Q=14.88\) for the linear model, \(Q=5.14\) for Emax, and \(Q=4.00\) for sigmoid Emax. Assume the respective mean models estimate \(k=2\), \(k=3\), and \(k=4\) parameters. With known variance, the AIC values differ from \(Q+2k\) only by the same constant, which can be omitted for comparison.
Emax has the smallest AIC in this comparison, although its difference from sigmoid Emax is only 0.86. That small difference is not persuasive evidence that the Emax shape is uniquely correct. Residual plots, plausible parameter estimates, agreement with the observed dose pattern, and sensitivity to candidate specifications should also be considered. A fitted curve can be a useful working model even when the data do not sharply distinguish it from alternatives.
Estimating a Dose That Meets a Clinical Target
Once a model is judged suitable for the decision, a target dose can be calculated by solving the fitted response equation. The target must be clinically defined and expressed on the same response scale as the model. If a model is monotone increasing over the studied range, solving for the minimum dose that reaches the target is straightforward. If the target is outside the observed dose range, the calculation is extrapolation and should be treated as substantially less secure than interpolation.
Worked Example: Solving for a Target Dose
Suppose the selected Emax fit is \(E_0=99.8\) outcome units, \(E_{\max}=17\) units, and \(ED_{50}=1.8\) mg. A prespecified clinical target is a mean response of at least 112 units. The baseline model prediction is 99.8 units, so the model must add at least \(112-99.8=12.2\) units. Solve:
The plug-in estimate of the minimum target-attaining dose is therefore about 4.58 mg. It lies between the tested 4 and 8 mg doses, so it is an interpolation rather than an extrapolation. It is still a model-based estimate, not proof that every patient at that dose will reach the target, nor a complete dosing recommendation. Parameter uncertainty, model selection, safety constraints, and the planned clinical decision all matter.
Diagnostics, Alternatives, and Common Pitfalls
MCP-Mod is most useful when plausible dose-response shapes can be specified in advance and the study includes doses informative about their differences. Diagnostics should include the observed dose means with uncertainty intervals, fitted curves overlaid on those means, residual patterns, and checks for influential dose groups. Poor fit, implausible parameter estimates, a candidate family that misses the observed pattern, or a target dose driven by extrapolation are reasons to qualify or revisit the dose estimate. A plateau dose or minimum effective dose can be especially unstable when the studied doses do not bracket the relevant region.
The principal alternative is to fit one prespecified model directly, which can be simpler and more powerful if that model is correct but can miss a signal when it is wrong. Pairwise comparisons of each active dose with placebo answer dose-specific questions but do not use the dose ordering as efficiently as shape-based contrasts. A flexible model can reduce shape restrictions, but it can also use more parameters and require more data. MCP-Mod occupies a middle ground: it tests several planned shapes with multiplicity control, then uses model-based estimation. Its protections depend on prespecification and do not remove the need to assess model adequacy.
- Confusing signal detection with model identification. Rejecting several candidate contrast nulls supports a dose-response signal; it does not mean several distinct biological mechanisms have been established.
- Ignoring the candidate search. Adding shapes or tuning shape parameters after examining the outcomes can make nominal multiplicity control misleading unless the expanded search is accounted for.
- Reading a fitted target as exact. A target-dose calculation uses estimated parameters and a chosen model. Reporting only a point estimate conceals uncertainty and sensitivity to model choice.
- Overlooking the direction and endpoint. The one-sided increasing-trend test is appropriate only when larger values are beneficial and the increasing direction was justified. For an endpoint where lower is better, orient the contrasts accordingly.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- Why do contrast coefficients sum to zero, and what feature of the mean response does this remove?
- Three candidate contrasts are tested at familywise \(\alpha=0.05\) using Bonferroni control. What per-test significance level is used?
- In the worked model comparison, why does the sigmoid model not win simply because its \(Q\) is smallest?
- For \(E_0=100\), \(E_{\max}=20\), and \(ED_{50}=2\) mg, what dose gives a modeled response of 112 units?
- Name two reasons a target dose estimated by the modeling stage may not be reliable.
Solutions. (1) Zero-sum coefficients cancel a common baseline: if every dose mean is increased by the same amount, the contrast is unchanged. (2) \(0.05/3=0.01667\) per test. (3) The sigmoid model has an additional fitted parameter, and AIC penalizes complexity: its \(Q+2k=12.00\), compared with 11.14 for Emax. The small difference also does not establish that either model is true. (4) Solve \(100+20d/(2+d)=112\): \(20d=12(2+d)\), so \(8d=24\) and \(d=3\) mg. (5) Examples include a poor fit, an inadequate candidate family, sparse information near the target, extrapolation beyond tested doses, and uncertainty in model parameters or model choice.