Dortmund

In this article the MCPMod package for the R programming environment will be introduced. It implements a recently developed methodology for the design and analysis of dose-response studies that combines aspects of multiple comparison procedures and modeling approaches (Bretz et al. 2005). The MCPMod package provides tools for the analysis of dose finding trials, as well as a variety of tools necessary to plan an experiment to be analyzed using the MCP-Mod methodology.


Introduction
In pharmaceutical drug development, dose-response studies typically have two main goals.The first goal is to establish that changes in dose lead to desirable changes in the (efficacy and/or safety) endpoint(s) of interest, the so-called proof-of-concept (PoC) step.Once such a dose-response signal has been shown, the second goal is then to select one or more "good" dose level(s) for the confirmatory Phase III studies, the so-called dose-finding step.
Traditionally these goals have been addressed either by using a multiple comparison procedure (MCP), or by using a modeling (Mod) approach.The MCP approach regards the dose as a qualitative factor and generally makes few, if any, assumptions about the underlying doseresponse relationship.However, inferences about the target dose are restricted to the discrete, possibly small, set of doses used in the trial.Within the modeling approach, a parametric (typically non-linear) functional relationship is assumed between dose and response.The dose is taken to be a quantitative factor, allowing greater flexibility for target dose estimation.The validity of the modeling approach, however, strongly depends on an appropriate dose-response model being pre-specified for the analysis.
In this paper we present the MCPMod package written in the R system for statistical computing (R Development Core Team 2009) and available from the Comprehensive R Archive Network at http://CRAN.R-project.org/package=MCPMod.The package implements a hybrid methodology, combining multiple comparison procedures with modeling techniques (called MCP-Mod procedure, (Bretz et al. 2005)).This approach provides the flexibility of modeling for dose estimation, while preserving the robustness to model misspecification associated with MCP. Figure 1 gives an overview of the MCP-Mod procedure.It starts by defining a set of candidate models M covering a suitable range of dose-response shapes.Each of the doseresponse shapes in the candidate set is tested using appropriate contrasts and employing MCP techniques that preserve the family-wise error rate (FWER).PoC is established when at least one of the model tests is significant.Once PoC is verified, either a "best" model or a weighted average of the set of significant models M * ⊆ M is used to estimate the dose-response profile and the target doses of interest.

Optimum contrast coefficients
Test for a significant dose-response signal Model selection or averaging Dose-response and target dose estimation As outlined above, the MCP-Mod procedure is performed in several steps: 1) calculation of contrast coefficients, representing the candidate model shapes, 2) conduct of a multiple contrast test, and, depending on the result, 3) a model selection step to fit (typically nonlinear) dose-response models and to estimate the target doses.Each individual step above can be implemented with the R statistical language, possibly using add-on packages available at the CRAN servers (http://CRAN.R-project.org/).However, it is desirable to have one package, which performs these steps automatically and also allows to design a trial for the MCP-Mod procedure.The MCPMod package provides these functionalities and the aim of this paper is to give a detailed description of the package.
For self containment of the paper we will first review the key features and statistical methods of the MCP-Mod procedure in Section 2, while the MCPMod package will be introduced and illustrated with examples in Section 3.
PoC and estimating target doses, we consider the one-way layout where ) denotes the mean response at dose d i for some dose-response model f (d, θ), n i denotes the number of patients allocated to dose d i , N = k i=1 n i is the total sample size, and ij denotes the error term for patient j within dose group i.Following Bretz et al. (2005), we note that most parametric dose-response models f (d, θ) used in practice can be written as where f 0 (d, θ * ) denotes the standardized model function, parameterized by the vector θ * .In this parameterization, θ 0 is a location and θ 1 a scale parameter such that only the parametervector θ * determines the shape of the model function.As seen later, it is sufficient to consider the standardized model f 0 instead of the full model f for the derivation of the optimal model contrasts.

MCP-Mod methodology
In this subsection we review the core elements of the MCP-Mod methodology.We start considering the basic MCP-Mod procedure for the analysis of a dose-response trial and then focus on design issues.For more information on the basic methodology see Bretz et al. (2005), for recommendations regarding the practical implementation and design aspects see Pinheiro, Bornkamp, and Bretz (2006a).

Analysis considerations
The motivation for MCP-Mod is based on the work by Tukey, Ciminera, and Heyse (1985), who recognized that the power of standard dose-response trend tests depends on the (unknown) dose-response relationship.They proposed to simultaneously use several trend tests and subsequently to adjust the resulting p−values for multiplicity.Bretz et al. (2005) formalized this approach and extended it in several ways.
Assume that a set M of M parameterized candidate models is given, with corresponding model functions f m (d, θ m ), m = 1, . . ., M, and parameters θ * m of the standardized models f 0 m (determining the model shapes).For each of the dose-response models in the candidate set we would like to test the hypothesis H m 0 : c m µ = 0, where c m = (c m1 , . . ., c mk ) is the optimal contrast vector representing model m, subject to k i=1 c mi = 0.Each of the dose-response models in the candidate set is hence tested using a single contrast test, where is the pooled variance estimate.Every single contrast test thus translates into a decision procedure to determine whether the given doseresponse shape is statistically significant, based on the observed data.
The contrast coefficients c m1 , . . ., c mk for the m-th model are chosen such that they maximize the power to detect the underlying model.It can be shown that these optimal contrast coefficients do not depend on the full parameter vector θ m of the model, but only the parameters in its standardized model function θ * m , which determine the model shape (see Bretz et al. (2005)) and the group sample sizes.Letting (µ 0 m1 , . . ., µ 0 ) , the i th entry of the optimal contrast c m for detecting the shape m is proportional to where μ = N −1 k i=1 µ 0 mi n i (Bornkamp (2006) p. 88, Casella and Berger (1990) p. 519).A unique representation of the optimal contrast can be obtained by imposing the regularity condition k i=1 c 2 mi = 1.The final detection of a significant dose-response signal (i.e., demonstrating PoC), is based on the maximum contrast test statistic Under the null hypothesis of no dose-response effect µ d 1 = ... = µ d k and under the distributional assumptions stated in equation (1), T 1 , . . ., T M jointly follow a central multivariate t distribution with N − k degrees of freedom and correlation matrix R = (ρ ij ), where Multiplicity adjusted critical values and p−values can be calculated using the identity of the sets [T max ≤ q] = [T 1 ≤ q, . . ., T M ≤ q], where q is a real number.As the joint distribution of (T 1 , . . ., T M ) is multivariate t, numerical integration routines for evaluation of multivariate t integrals, such as the randomized quasi-Monte Carlo methods of Genz and Bretz (2002) implemented in the R package mvtnorm (Genz, Bretz, Miwa, Mi, Leisch, Scheipl, and Hothorn 2009), can be used to compute the desired equicoordinate quantiles of the multivariate t distribution.PoC is hence established if T max ≥ q 1−α , where q 1−α is the multiplicity adjusted critical value at level 1 − α (i.e., the equicoordinate 1 − α quantile of the corresponding central multivariate t distribution).Furthermore, all dose-response shapes with contrast test statistics larger than q 1−α can be declared statistically significant at level 1 − α under strong control of the FWER.These models then form a reference set  2)), this can be exploited in the nls function by using the Golub-Pereyra algorithm (see Golub and Pereyra (2003) for a review of these methods).So we only need to derive starting values for the standardized model parameters θ * .Although we use automatic methods for finding good data-based starting values for θ * , convergence problems can occur, especially when the number of dose levels used in the trial is small compared to the parameters in the model function.In the case of non-convergence the 'best' of the remaining significant, converging models can be used for dose estimation, if any.An approach to partially overcome these convergence issues is to use box constraints on θ * .This will be implemented as an alternative in future versions of the package.
Once a dose-response model has been selected, one can proceed to estimate the target dose(s) of interest.One possible choice is the minimum effective dose (M ED), which is defined as the smallest dose ensuring a clinically relevant and statistically significant improvement over placebo (Ruberg 1995).Formally, where ∆ is the clinical relevance threshold.A common estimate for the M ED is where f (d) is the predicted mean response at dose d, and L(d) is the corresponding lower bound of the pointwise confidence intervals of level 1 − 2γ.Note that M ED corresponds to the M ED 2 estimator in Bretz et al. (2005), who found this estimator to be least biased compared to two other alternative estimates in a simulation study.A different target dose is the EDp which is defined as the smallest dose that gives a certain percentage p of the maximum effect δ max observed in (d 1 , d k ].Formally, where δ max = f max − f (d 1 ), and An estimate EDp is obtained by plugging the empirical estimates into the definition ( 5).
An alternative to selecting a single dose-response model is to apply model averaging techniques and produce weighted estimates across all models in M * for a given quantity ψ of interest.
In the context of dose-response analysis, the parameter ψ could for example be a target dose (M ED, EDp, . ..) or the mean responses at a specific dose Buckland, Burnham, and Augustin (1997) proposed to use the weighted estimate where ψ is the estimate of ψ under model for given weights w .The idea is thus to use estimates for the final data analysis which rely on the averaged estimates across all L models.Buckland et al. (1997) proposed the use of the weights which are defined in dependence of a common information criterion IC, such as AIC or BIC applied to each of the L models, and prior model weights p .If each model is given the same prior model weight, the p cancel out in equation ( 6).

Design considerations I: Power and sample size calculations
An important step at the planning phase of any clinical trial is to properly design the study in order to achieve the study objectives.Because dose finding studies have two major goals, PoC testing and dose estimation, different criteria can be used to design a study.Dette, Bretz, Pepelyshev, and Pinheiro (2008) derived optimal designs, which minimize the (asymptotic) variance of the M ED estimate.Using their approach, asymptotic confidence intervals for the M ED can be calculated, conditional on a selected model.At the planning stage one would then specify the maximum width of the confidence interval and calculate the sample size necessary to ensure a certain precision of the M ED estimate.
An alternative approach is to focus on calculating the sample size necessary to achieve a pre-specified power to detect PoC (Pinheiro et al. 2006a).We thus start by introducing the power calculation under a given specific model m from the candidate set M, generalize it afterwards to multiple models and finally focus on sample size calculation.
The power of the MCP procedure is determined by the distribution of T max under the alternative hypothesis that the m-th dose-response model is true.Under this assumption, the mean responses at the doses . The power to detect a dose-response signal (i.e., PoC) under model m for sample sizes n = (n 1 , . . ., n k ) is then P (max It follows from the properties of the multivariate t distribution and the assumptions in equation (1), that, under the m-th model, the contrast test statistics T 1 , . . ., T M are jointly distributed as non-central multivariate t with N − k degrees of freedom and correlation matrix R = (ρ ij ).The non-centrality parameter vector is δ m = (δ m1 , . . ., δ mM ) , where Again, the mvtnorm package can be used to calculate the necessary probabilities.
So far we have only considered the power calculation under a single model m.In practice we would rather account for the inherent model uncertainty.To this end, we would calculate the power for each of the M models from the candidate set M and aggregate the resulting values into a single combined measure of power, such as the (weighted) average, the minimum or a quantile.The sample size is then calculated as the smallest sample size ensuring a minimum combined power value, say π * , to detect PoC under the assumed set of dose-response mean vectors.We restrict ourselves to the case that either the allocation weights r i ≥ 0, subject to i r i = 1 or the allocation ratios ρ i relative to the dose group with the fewest patients, i.e. ρ i = r i / min(r i ) are prespecified.The group sample sizes n = (n 1 , . . ., n k ) can then be obtained from n i = N r i for allocation weights or from n i = ρ i n min , where n min is the smallest group sample size.Since the combined power is a monotone increasing function of N (or n min , if allocation ratios are specified) a unique smallest integer giving a power larger than π * exists (see also Pinheiro et al. (2006a)).The bisection search method can be used to obtain the sample size ensuring a pre-specified combined power π * .In practice, rounding techniques need to be applied to obtain integer sample sizes.

Design considerations II: Sensitivity analysis
In the derivations above we conditioned on the mean vectors µ m in equation ( 7) and hence on the parameters θ m = (θ m0 , θ m1 , θ * m ) .Since the sample size is calculated under this condition, it is critical that the model parameters are reliably determined.For the determination of location and scale parameters θ m0 and θ m1 , prior knowledge about the expected placebo response δ 0 and the maximum response δ max can be used at the design stage.It is typically straightforward to plug in these quantities into the model equations, assuming that θ * m is known and then solving for θ m0 and θ m1 , see Pinheiro et al. (2006a) for a more detailed description of this approach.
Based on prior knowledge about the shape of the model function, Pinheiro, Bretz, and Branson (2006b) discussed strategies to obtain guesstimates for the standardized model parameters θ * m .The elicitation of prior information for θ * m may impact both the design and the analysis of a dose finding study using the MCP-Mod methodology, as the guesstimates are used to obtain the optimal model contrasts at the MCP step, which in turn determine the effective power to detect PoC.Therefore, it is of importance to investigate the sensitivity of the procedure to misspecification of the parameters in the standardized models and, in particular, the impact it has on the effective power to detect PoC.Pinheiro et al. (2006a) considered different measures of loss in power associated with a misspecification of the standardized model parameters.One possibility, subsequently denoted as LP 1 , is to calculate the difference between the nominal power (the power obtained, when the guesstimate is correct) and the actual power (the power obtained, when the used guesstimate does not coincide with the true parameter), i.e.
LP 1 = nominal power − actual power. (8) Thus, LP 1 can be interpreted as the difference between the power that was intended for the study and the power one actually obtains.Alternatively, one could also calculate the difference between the power that could be achieved if the true parameter values were known at the design stage (potential power) and the actual power.This is denoted by LP 2 and hence LP 2 = potential power − actual power.
Graphical methods can be used to display the loss in power for a range of true standardized model parameters.From our experience the loss in power associated with misspecification of the parameters in the standardized model function is often negligible for reasonable candidate sets, because dose-response models with parameter vectors θ m deviating from the guesstimate θ * m are often detected from some other model in the candidate set.In cases where the loss in power is not acceptable, the inclusion of an additional model in the candidate set could be considered.

The MCPMod package
In this section we describe the R package MCPMod for implementing the MCP-Mod methodology.The package consists of two main parts (see also Figure 2).The first part contains several functions that are useful for planning a trial: calculation of the optimal contrasts and the critical value (planMM), the sample size (sampSize) or functions that support the selection of a 'good' candidate set and sensitivity analysis (guesst, plotModels, powerMM, LP).The second part consists of one main function named MCPMod that implements the full MCP-Mod approach for analysis of a given dose-response data set.

Preliminaries
Before illustrating the different functions in more detail we first describe how to specify the candidate set of models M for these functions.Table 1 gives an overview of the doseresponse models that are implemented (note that user-defined non-linear models can also be specified, see the package documentation for details).The candidate set of models needs to be specified as a list, where the list elements should be named according to the underlying dose-response model function (see Table 1: Dose-response models implemented in the MCPMod package.Column (*) lists for each model the parameters for which guesstimates are required and the order in which they need to be specified in the models list, while column ( ) lists the parameters, which fixed and not estimated.For the beta model Thus, if multiple model shapes from the same model class are to be used, the parameters are handed over as a matrix, for models having two parameters in the standardized model function, and as a vector for one-parameter standardized models.This general structure applies to all built-in models.Note that the linear-in-log and the beta models also contain a parameter (c and D, respectively, see Table 1) that is not estimated from the data but needs to be pre-specified.These parameters are not handed over via the candidate model list but via seperate arguments scal (corresponding to D) and off (corresponding to c) respectively to the top-level functions.

Planning Code
In this section we provide a brief overview of the functions guesst, plotModels, fullMod, planMM, powerMM, sampSize and LP.These functions are useful for designing a trial using MCP-Mod.For a detailed description of the arguments to the functions we refer to the documentation of the package.

Function guesst
The selection of suitable guesstimates and model shapes is a major aspect of the MCP-Mod methodology.Incorporating contrasts/models that are likely to be true (and excluding those that are very unlikely) can greatly improve the power of the methodology.The guesst function supports the translation of clinical knowledge available prior to the start of a study into the required guesstimates.The function calculates the guesstimates according to the percentage p * of the maximum effect that is achieved at a certain dose d * .Suppose, for example, we want to calculate a guesstimate for the ED 50 parameter from the E max model.If we expect a response of 90% at dose 0.2, the ED 50 guesstimate can be calculated by calling > guesst(d = 0.2, p = 0.9, model = "emax") ed50 0.02222222 With this guesstimate, the standardized model for the E max model is given by f 0 (d, ED 50 ) = d/(0.022222+ d), and the optimal contrast can be calculated from equation (3).For models with two standardized model parameter one (d * , p * ) pair is not sufficient to obtain guesstimates for the standardized model parameters.For example, for the logistic model we need to specify two pairs to obtain a guesstimate > guesst(d = c(0.05,0.2), p = c(0.2,0.9), model = "logistic") ed50 delta 0.1080279 0.0418583 In this example the standardized model function for the logistic model is given by f 0 (d, ED 50 , δ) = 1/{1 + exp[(0.1080279− d)/0.0418583]},from which the corresponding optimal contrast can be obtained.In a similar way one can obtain guesstimates with the guesst function for all built-in models.

Function plotModels
Before deciding for any particular candidate set of model shapes it is useful to display them graphically.This can be done with the plotModels function.Since the model shapes, specified in the models list, do not depend on the location (defined through the baseline effect) and scale (defined through the maximum effect) of the model, one additionally needs to specify those via the base and maxEff arguments.Using the candidate set mods2 defined above (and setting the scal parameter of the beta model equal to 1.2), a graphical representation can be obtained as follows (see Figure 3 for the output) > doses <-c(0, 0.05, 0.2, 0.6, 1) > plotModels(mods2, doses, base = 0, maxEff = 0.4, scal = 1.2)

Function fullMod
Similar to the plotModels function above, also other functions (powerMM, sampSize, LP) require information about the doses, the full model functions, i.e. the candidate model shapes, the baseline effect, the maximum effect and possible other additional parameters like off or scal.The fullMod function derives the full model functions (i.e. the location and scale parameters) for each model from the stated information (see Section 2.2) and packages this with the used dose levels into a fullMod object, which can then be used as an input parameter for the four above mentioned functions.When assuming the baseline effect 0 and the maximum effect 0.4 and using the candidate set mods2 (and setting the scal parameter of the beta model equal to 1.2) one can package this information via > doses <-c(0, 0.05, 0.2, 0.6, 1) > fmods2 <-fullMod(mods2, doses, base = 0, maxEff = 0.4, scal = 1.2) Dose Model means 0.0 0.1 0.2 0.3 0.4 q q q q q betaMod 0.0 0.2 0.4 0.6 0.8 1.0 q q q q q emax1 q q q q q emax2 0.0 0.2 0.4 0.6 0.8 1.0 q q q q q linear q q q q q logistic1 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 q q q q q logistic2 Figure 3: Model shapes for the selected candidate model set, produced with plotModels function.

Function planMM
The planMM function calculates the quantities necessary to conduct the multiple contrast test: The optimal model contrasts and their correlations (see equations ( 3) and ( 4)) and the critical value using the mvtnorm package.This information is returned in a planMM object.
The arguments alpha and twoSide determine the significance level and sidedness of the test.By default one-sided testing at level α = 0.025 is performed.The sample size allocations are handed over as a vector via the n argument (for balanced allocations a single number is sufficient).Assuming a balanced allocation of 20 patients per dose group, the candidate set mods2 and the doses from above, the planMM function can be called as follows > pM <-planMM(mods2, doses, n = 20, alpha = 0.05, twoSide = FALSE, scal = 1.2) > pM
The representation of the optimal contrast is unique as we imposed the condition of unit Euclidean length.In the output we then obtain the multiplicity adjusted critical value for the maximum contrast and finally the correlations of the contrasts.In this example some contrasts are quite highly correlated.For example, the correlation between emax2 and logistic1 is 0.956, indicating that both describe similar dose-response shapes, as can also be seen in Figure 3.The beta model contrast however, seems to be relatively different from the others.This is due to the fact that the beta model shape is, contrary to the other model shapes, not monotone.The contrasts can also be graphically displayed using the plot method for planMM objects (see Figure 4).

Dose
Contrast coefficients −0.5 0.0 0.5 0.0 0.2 0.4 0.6 0.8 1.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q linear emax1 emax2 betaMod logistic1 logistic2 Figure 4: Graphical display of optimal contrasts.

Function powerMM
The powerMM function is designed to calculate the power to detect the model shapes in the candidate set for different sample sizes.We need to hand over either an object of class fullMod or the doses, the baseline and the maximum effect via doses, base and maxEff and the standard deviation of the response via sigma.One can calculate the power for sample sizes ranging from lower to upper in stepsizes step.Summary functions can be used to combine the different power values for the different model shapes into one value, as described in Section 2.2.By default the minimum, the mean and the maximum power are calculated.
The resulting power values are returned as an object of class powerMM in a matrix.There exists also a plot method to display the results graphically.Using the information packaged in the fmods2 object from above one obtains the following result > pM <-powerMM(fmods2, sigma = 1, alpha = 0.05, lower = 10, upper = 110, + step = 10) > plot(pM, line.at= 0.9, models = "none")  In Figure 5 it can be seen that a mean power of 90 % is achieved with approximately 90 patients per dose.Note that the power can also be calculated for unbalanced but fixed allocations.The allocation ratios (or allocation weights, depending on the value of the typeN argument) then need to be supplied via the alRatio argument.In the plot above only the summary power values are displayed, although the plot method for powerMM allows the display of the power values for the individual candidate models as well.

Function LP
The LP function is designed to calculate the loss in power associated with misspecification of the guesstimates for one model in the candidate set.To illustrate the function we use a very simple candidate set consisting of only a linear and an E max shape and illustrate the calculation of LP 1 (see equation ( 8)).We select 0.15 as the guesstimate for the ED 50 parameter, and want to investigate the loss in power in the interval [0.03, 0.8] (specified via paramRange).Hence we calculate how much power we loose, if an alternative ED 50 value is true, but we selected 0.15 as our guesstimate.As before doses, base, maxEff (or an object of class fullMod) and sigma need to be specified together with the sample size.After calling the LP function we display the results using the associated plot method.The optional spldf argument determines the degrees of freedom for the spline that is used to smooth the power values in the plot.q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Figure 6: Difference of actual and nominal power for Emax model.
As seen from Figure 6 the loss in power is relatively large if a small ED 50 value is true.If the true ED 50 is equal to the guesstimate then the actual power and nominal power coincide.For ED 50 larger than the specified guesstimate we actually gain power, because the E max model becomes almost linear and is captured by the linear model included in the candidate set.

Analysis Code
The analysis functionalities are incorporated in one main function MCPMod, which implements the full MCP-Mod approach.According to the methodology described in Section 2, it consists of two main steps: (i) MCP-step (calculation of optimal contrasts, critical value, contrast test statistics and possibly p−values and selection of the set of significant models) and (ii) modeling step (model fitting, model selection/model averaging and dose estimation).
We now describe some of the more important arguments for the MCPMod function.For a complete description of the MCPMod function we refer to the online documentation.The doseresponse data set is handed over to the MCPMod function via the data argument.It should be handed over as a data frame containing two columns corresponding to the dose levels and the response values.The selModel argument determines how to select a dose estimation model out of the set of significant models (if there are any significant models).One can choose between the maximum contrast test statistic (the default option), the AIC, the BIC or model averaging based on either the AIC or the BIC (see Section 2.2).Another important argument is doseEst, which determines the dose estimator to be used.Three slightly different estimators for the M ED are currently implemented (see Bretz et al. (2005) for a detailed description of those three estimators, option "MED2" is the default value, corresponding to the estimator described in Section 2.2) as well as an estimator of the EDp.Additional parameters for the dose estimators (such as γ for M ED estimators (default: γ = 0.1) and p for the ED estimator (default: p = 0.5)) are handed over via the dePar argument.0.17 The summary output includes some information about important input parameters that were used when calling MCPMod.Then the output includes also the optimal contrasts and the contrast correlations together with the contrast test statistics, the multiplicity adjusted p−values and the critical value.Finally, information about the fitted dose-response model, its parameter estimates and the target dose estimate are displayed.
A graphical display of the dose-response model used for dose estimation can be obtained via the plot method for MCPMod objects.When complData = TRUE, the full dose-response data set is plotted instead of only the group means.The clinRel option determines whether the clinical relevance threshold should be displayed.q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q emax q Responses Model Predictions To illustrate the different options available for the MCPMod function we will now re-analyze the biom data set with different input parameters.Specifically, we will now apply model averaging techniques.The target dose is hence estimated as the weighted average of the dose estimates under the different significant models.The weights are determined via the AIC criterion (see equation ( 6)) with uniform prior weights (which is the default).The target dose we are now interested in is the ED95, which is the dose that achieves 95 % percent of the maximum effect.

MCPMod
PoC (alpha = 0.05, one-sided): yes Model with highest t-statistic: emax2 Models used for dose estimation: emax logistic linear betaMod Dose estimate: ED95% 0.669 The output of the print method now contains the four models selected for dose-response estimation as well as the model averaged ED95 estimate.We edited the output of the summary method here as there is some overlap with the previous call to the summary function.

Figure 5 :
Figure 5: Power to detect PoC under the assumed candidate set for different summary functions.

Figure 7 :
Figure 7: Fitted model with data set.
Standard information criteria like the AIC or BIC might also be used.The estimate of the model function is then obtained by calculating the least squares estimates for θ.For non-linear models iterative optimization techniques need to be used, such as those implemented in the nls function in R. As the non-linear models described here are partially linear (see equation ( Figure 2: Overview of main functions in the MCPMod package.

Table 1
) and the individual list entries should correspond to the required guesstimates or NULL if no guesstimates are needed.Suppose, for example, we want to include in our candidate set a linear model, an E max model and a logistic model.From the standardized model functions in Table1we see that we need to specify one guesstimate for the E max model (ED 50 parameter), two guesstimates for the logistic model (ED 50 and δ) and none for the linear model (since its standardized model function does not contain any unknown parameters).Suppose our guesstimate for the ED 50 parameter of the E In some cases one might want to include several model shapes per model class.For example, if the candidate model set includes two E max model shapes, two logistic model shapes, a beta model shape and a linear model shape the model list would look like Bretz et al. (2005)nce threshold ∆ is handed over via the clinRel argument.The pVal argument determines, whether multiplicity adjusted p−values for the multiple contrast test should be calculated or not (per default p-values are not calculated).To illustrate the MCPMod function we use the dose-response data set biom used byBretz et al. (2005)to illustrate the MCP-Mod methodology.The data result from a randomized doubleblind parallel group trial with a total of 100 patients being allocated to either placebo or one of four active doses coded as 0.05, 0.20, 0.60, and 1, with 20 patients per group.Here, we use the MED2 estimator with γ = 0.05 to estimate the M ED, the clinical threshold ∆ is set to 0.4 and the dose estimation model is selected according to the maximum contrast test statistic.Employing the candidate model set mods2 the results can be obtained by calling From the output we conclude that the maximum contrast is significant at one-sided level 0.05.Thus a significant dose-response relationship can be established, i.e., positive PoC.Furthermore we conclude that emax2 has the largest test statistic among all contrasts and consequently the E max model was used for the dose-estimation step.The M ED estimate is 0.17.The 90% in the M ED estimate refers to the confidence level of L(d) used in the dose estimator (see Section 2.2).A more detailed summary of the results is available via the summary method.