1. Assessing MCP-Mod relative to pairwise comparisons and trend tests in dose-ranging design and analysis
Fang Liu, Anran Wang, Meihua Wang, Man Jin, Akshita Chawla, Pranab Kumar Mitra, Sammy Yuan, Robin Mogg
Aug 1, 2018

2. Background
Importance of adequately understanding the dose-response relationship is well-recognized in drug development
Inadequate design and analysis of dose finding studies continues to plague many development programs
Consequences: limited understanding of dose-response relationship, failed Ph 3 trials, and post-marketing dose adjustments
Analysis of dose-response studies typically have two primary strategies, each with shortcomings when applied separately
Multiple comparisons procedures (MCP)
Model-based approaches (MOD)
MCP-Mod combines principles of each approach under a single approach

3. Motivations for MCP-MOD
Traditional methods
Multiple comparison procedures:
E.g.: pairwise comparison, trend test, or multiple comparison procedure (MCP) (Bretz et al 2010)
Qualitative dose level
Pros: few assumptions about the dose-response relationship
Cons: inferences are restricted to the dose level tested
Model-based approaches:
E.g.: Emax, Sigmoid Emax Models, or Pinheiro et al, 2006a
Quantitative dose level
Parametric functional relationship between dose and response: full profile over all possible dose levels
Pros: More flexibility for target dose estimation
Cons: Pre-specification of a single dose response model at design stage is difficult and risky (model uncertainty). No rigid pre-specification of how models are selected (potentially overfitting data)

4. Motivations for MCP-MOD
Hybrid method: Combining MCP and MOD into one procedure
Keep flexibility of modeling dose-response relationship (by Mod) while preserve FWER (family-wise error rate) (by MCP)
The European Medicines Agency published a Qualification Opinion for the MCPMod approach in 2014, in which it wrote, “The MCP-Mod method is efficient in the sense that it uses the available data better than the commonly applied pairwise comparison.”
FDA Division of Pharmacometrics issued a Determination letter supporting use of MCP-Mod.
The European Medicines Agency published a Qualification Opinion for the MCPMod approach in 2013, in which it wrote, “The MCP-Mod method is efficient in the sense that it uses the available data better than the commonly applied pairwise comparison.”

5. MCP-MOD D.1 D.2 D.3 D.4 A.1 Pre-specified from historical data
Calculating optimal contrast coefficients
D.3
D.4
A.1

6. MCP-MOD: Design Stage (I)
Response Y
Normal response: continuous
Generalized cases: binary, count, time to event
Dose: K dose level (quantitative)
placebo= 𝑑 1 , active dose= 𝑑 2 , 𝑑 3 ,…, 𝑑 𝑘
Identify a set of 𝑀 parameterized candidate models
𝑓 𝑚 𝑑 𝑖 , 𝛉 𝑚 = θ 0 + θ 1 𝑓 𝑚 0 ( 𝑑 𝑖 , 𝛉 𝑚 0 )
where 𝑖=1,…,𝑘; 𝑚=1,…,𝑀; 𝜃 0 is location parameter, 𝜃 1 is a scale parameter; 𝑓 𝑚 0 ( 𝑑 𝑖 , 𝛉 𝑚 0 ) denotes the standardized model function; 𝜃 0 , 𝜃 1 and 𝛉 𝑚 0 should be prespecified.

7. MCP-MOD: Design Stage (II)
For each model f m d, θ m , construct a contrast test to detect a positive dose effect using optimal coefficients, i.e., test the null hypothesis of no dose-response:
H 0 m : C m T u=0, m=1, 2,…M
where u=(u d 1 …, u d k ) is the mean response vector at each dose level.
c m = c m1 , …, c mk , The optimal contrast vector satisfies
i=1 k c mi =0 ,
Maximize the power to test the contrast hypothesis
Test statistics for model m: T m = i=1 k c mi Y i s i=1 k c mi 2 / n i ,
s 2 is the pooled variance estimate of σ 2
To control the FWER of testing M models simultaneously, use the maximum contrast test statistics: T max = max T 1 ,…, T M , Where (T 1 , …, T m ) ~ central multivariate t distribtuion when all M null hypotheses are assumed to be true.
Let q 1−α denote the multiplicity adjusted critical value at level 1−α derived from the multivariate t distribution (Tukey, et al , 1985)
Pr T max > q 1−α =1−Pr T 1 < q 1−α , …, T M < q 1−α =α
Every single contrast test translates into a decision procedure to determine whether the given dose response shape is statistically significant, based on the observed data.

8. MCP-Mod: Analysis Stage
Test dose-response PoC under model uncertainty: multiple contrast test
If the observed T max > q 1−α , then we can reject H 0 and establish PoC;
Models with T i > q 1−α are declared significant and kept for model selection, denoted by M ∗ ={ M 1 , .., M L };
If no model is significant, then no PoC
Select the best model:
Either select a single model from the significant models in M ∗ based on AIC, BIC or maximum contrast test statistics.
Or apply model averaging techniques Weighted estimates across all the significant models are produced for the quantities of interest (Buckland et al., 1997)
Based on dose response modelling approaches, the selected model is used to fit the observed data and estimate model parameter θ and target dose (such as the MED):
MED = min d∈( d 1 , d k ] p d > p d 1 +∆
where Δ is the clinical relevance threshold, p d =f(d, θ) denote predicted response at dose d

9. Trend Test (I)
To detect a linear trend in the response corresponding to three dose scales. Suppose d1,…, dn are the doses to be examined with d1=0 being control.
Arithmetic scale: Xi=di, for i=1,.., k
Ordinal scale: Xi=i
Logarithmic scale: Xi=log(di), for i>1 and
𝑋1=𝑙𝑜𝑔𝑑2− 𝑑2−𝑑1 𝑑3−𝑑2 (𝑙𝑜𝑔𝑑3−𝑙𝑜𝑔𝑑2)
At each scale Xi, testing the slope of a regression line is equivalent to testing the contrast among the treatment means , where yi is the response at each dose level; mici are the contrast coefficient corresponding to Xi; ; mi is the sample size at di.
Multiplicity adjustment of the 3 test statistics for the three dose scalings using trivariate t- distribution to control FWER.

10. Simulation Objectives
Performance of these 3 methods were evaluated through a simulation study
Pairwise comparisons
Trend test: select model based on AIC, Tmax or using model average technique.
MCP-Mod: select model based on AIC, Tmax or using model average technique.
Evaluation Criteria:
Power to detect POC and type-I error
Bias and MSE of the estimated MED
We first review our previous simulations study. We would like to compare the performance of these three methods as we mentioned, pairwise comparisons, trend test and MCPMod under different scenarios. We want to investigate the differences between these methods in power to detect dose-response, type I error, bias and mean squared error of the MED estimates.
Thanks Fang. So after a very exciting introduction of the background story and methods, we will then go to some interesting numerical experiments. We would like to compare the performance of these three methods as we mentioned, pairwise comparisons, trend test and MCPMod under different scenarios. We want to investigate the differences between these methods in power to detect dose-response, type I error, bias and mean squared error of the MED estimates.

11. Simulation Setup 4-arm, 5-arm and 6-arm parallel design
Placebo response rate: 0.35; Maximum response rate: 1.0;
clinically relevant effect vs. placebo: Δ = (0.2, 0.4, 0.55)
5,000 simulations each
8 true dose response profiles including monotonic and non-monotonic curves to evaluate power: Emax, exponential, linear, linear log, quadratic, beta model, logistic model, Step-2 model. Use the first 4 models or first 6 models as model candidates for MCPMod
Using flat true model (no dose-response relationship) to evaluate type-I error.
We considered three designs with different numbers of dose levels. Here we have 4-arm, 5-arm and 6-arm designs. For the 4 arm design, we have 3 doses plus placebo, and for 5 arm, 4 doses plus placebo, and so on. The dose range is from 0 to 1.
And 6 true models were considered in this simulation study, they included 4 monotonic curves and 2 non-monotonic curves. They are emax model, exponential model, linear model, linear model with log transformed dose, quadratic model and beta model. And For MCPMod, these six models are the candidate models as well.
We ran 5000 simulations for each combination of design, true model and delta .

12. Dose Response Relationship
Six-arm design
In addition to the previous six true models, two models, logistic model and Step-2 model as shown here were added to the true model settings in the simulation.
We take the six-arm design for example here, these plots show the mean functions for the. The y-axis is the response, x-axis is the dose range, and the points here are the doses for the design. The true MEDs are also marked here. The red reference lines show the MEDs for delta 0.4 and the green lines show the MEDs for delta 0.2 for different models.

13. Power to Detect Dose-Response Signal and Type I Error
We still used the previous 6 models as the candidate models for MCPMod. OK. These are the simulations results of power and type I error. We first look at the power to detect the dose-response. Here the rows are different true models, and the columns are different methods under different designs of dose arms. The best performance was highlighted here with red.
So MCPMod has higher power to detect the dose-response than pairwise comparisons or the Dunnett’s test for all the designs with different numbers of dose arms.
And MCPMod has the best power for quadratic model and beta model, emax model in general, and has the best power for linear-log models in the 5-arm design and 4-arm design. (In the six-arm design, actually the differences between the trend test and the MCPMod and for emax and linear log model are very small, compared to the MC standard errors of these values that are around to In summary, the power tends to be improved with more curved mean functions for MCPMod)
And for trend test, due to non-monotonic shapes, it tends to have lower power for quadratic model and beta model, even worse than the pairwise comparison in the 4 dose arm design. But it has the best power for linear model and exponential model. It tends to have better power when response are monotonic.
The relatively lower power of MCPMod compared to the trend test can be due to the number of candidate models in the MCP test, for the trend test, the multiplicity adjustment only includes three models while the MCPMod adjusts for the six models in this simulation. We also tried to only include 4 candidate models for MCPMod, and the power of MCPMod get improved.in general.
For MCPMod and trend test, adding more dose arms increases their power in general. And the power of pairwise comparisons is in general lower with more active arms due to the multiplicity adjustments by Dunnett’s test.
Then we look at the results for Type I error. These three methods control multiplicity well. They all have Type I error around 0.05, the values slightly above 0.05 can be due to the simulation error. The MC standard error here is around

14. MED Estimation (Δ =0.2)
These tables show the results of MSE and bias of MED estimates when delta equals to 0.2. MCPMod has best MSE and bias among all the methods for all the scenarios.

15. MED Estimation (Δ =0.55)
These tables show the results of MSE and bias of MED estimates when delta equals to 0.2. MCPMod has best MSE and bias among all the methods for all the scenarios.

16. Conclusions
For MCP-Mod and trend test, power to detect the dose-response relationship increases when more doses are included in the trial.
Trend test is comparable to MCP-Mod, in terms of power to detect the dose-response relationship. However, trend test has larger bias and MSE when estimating MED.
For MCP-Mod, MED estimate using model average technique has smaller MSE, while comparable bias comparing with Tmax and AIC model selection methods.
MCP-Mod with 4 candidate models is slightly more powerful than MCP-Mod with 6 candidate models when the true model is one of the four candidate models. However, power is much less when the true model is not part of the 4 candidate models, but part of the 6 candidate models.
So our findings are that for MCPMod and trend test, when we have more dose arms included in the trial, the power to detect dose-response get improved, whereas for the pairwise comparison, the power would go down with more doses.
Power of dose selection depends on the dose-response shape. When the true model is exponential or linear, the trend test has the best power among these three methods, but performance is relatively worse in non-monotonic models.
In terms of precision of the MED estimation, it seems that the Model based approach is better than non-modeling approach in general. MCPMod offers greater flexibility in identifying MED, and has the best MSE and bias of the MED estimates in general while it is able to have reasonable power as well.
Great. This is our last slide. Any questions there?
(Dose selection is a more difficult problem than PoC detecting, usually we get sample sizes that have sufficient power for PoC, but we do not give enough precision for estimation of MED.)
Thank you so much everyone! Thanks so much Fang Robin Frank Sammy Wen for your guidance and suggestions on this project.

17. Appendix
CHMP (2014) Qualification Opinion of MCP-Mod as an efficient statistical methodology for model-based design and analysis of Phase II dose finding studies under model uncertainty
ICH-E4 Harmonized Tripartite Guideline (1994), “Dose-response Information to support drug registration”
Tukey, J.W., Ciminera, J.L., and Heyse, J. F. (1985), Testing the statistical certainty of a response to increasing doses of a drug. Biometrics 41:295–301
Zhang, Donghui (2001), Testing the Trend of a Response Curve to an Increasing Sequence of Doses:A SAS Macro to Automate the Analysis. SAS Proceedings from PharmaSUG.