Example trial | Analysis with MCP-Mod | MCP step | Dose-response estimation | Power and sample size calculations at design stage | Calculating required sample size for a trial | References

Model definition | Linear model | Mathematical Details of Kenward-Roger method | Special Considerations for mmrm models | Derivative of the overall covariance matrix $\Sigma$ | Derivative of the $\Sigma^{-1}$ | Subjects with missed visits | Scenario under group specific covariance estimates | Scenario under weighted mmrm | Contracted linear covariance adjustment | Subject contributions with weights | Contraction of $Q$ | Contraction of $P$ | Inference | Degrees of freedom in contrast space | One-dimensional shortcut | Parameterization methods and Kenward-Roger | Implementation in mmrm | Spatial Exponential Derivatives | Spatial Gaussian Derivatives | References

Model definition | Linear model | Covariance matrix model | Unstructured covariance matrix | Grouped covariance matrix | Spatial covariance matrix | Maximum Likelihood Estimation | Weighted least squares estimator | Determinant and quadratic form | Restricted Maximum Likelihood Estimation | Completing the square | Objective function

Satterthwaite degrees of freedom for asymptotic covariance | One-dimensional contrast | Jacobian approach | Jacobian calculation | Connection to the scalar Kenward-Roger shortcut | Multi-dimensional contrast | Satterthwaite degrees of freedom for empirical covariance | Appendix: Trace identities | Cyclic invariance of trace | Trace and squared eigenvalues | References

Handling Exposure in Count Outcomes with Offsets | Example 1: Count outcome with offset | Customizing Prior Distributions | Model Preparation | Example 2: Using priors directly supported by brms | Example 3: Using the R2-D2 shrinkage prior | Example 4: Custom hierarchical prior (advanced) | Example 5: Coefficient-specific priors | Example 6: Hierarchical priors with shared variance (stanvars) | Stratification for nuisance parameters | Example 7: Fully stratified continuous one-way shrinkage model | Example 8: Stratified one-way survival model | Summary

Installation | Getting started | Data | Structure of the model class | Model setup | Logistic model with bivariate (log) normal prior | Advanced model specification | Obtaining the posterior | Plotting the model fit | Escalation Rules | Increments rules | Rules for next best dose recommendation | Cohort size rules | Stopping rules | Simulations | Examining single trial behavior | Simulating from a true scenario | Predicting the future course of the trial | Simulating 3+3 design outcomes | Dual-endpoint dose escalation designs | Dual-endpoint designs with a joint model | Dual-endpoint designs with separate models | References

Introduction | Model Specification and Fitting Step | Notation | Statistical model | Parameter grouping | Priors distributions for shrunken predictive terms | Normal priors for one-way shrinkage models | Regularized horseshoe priors for global shrinkage models | Model fitting | Standardization Step (G-Computation) | Comparison of One-way vs Global Shrinkage Models | References

Introduction | The Data | Choice of hyperprior parameters | One-way shrinkage models | One-way model for x1 | One-way model for x2 | One-way model for x3 | Forest plot of results from all one-way models | Global shrinkage model | Model fitting | Summary of subgroup effects | Global Model: Visualization | Comparing Multiple Models in One Plot | References

Introduction | Alpha Spending Functions | From Spending to Boundaries | Repeated and Sequential P-values | Case Study using repeated p-values and look_back = FALSE | Hypotheses and Graph | Observed P-values | Running the Procedure (look_back = FALSE) | Repeated p-values | Nominal boundaries | Comparison with Paper Values | Boundary table | Case Study using sequential p-values and look_back = TRUE | Trial Setup | Graph and Alpha Allocation | P-values and Information Fractions | Sequential P-values (look_back = TRUE) | Per-hypothesis look_back | When look_back makes a difference | Customizing Spending Functions: Spending Time | Monitoring: Adjusting for Changed Final Information | Additional Examples | Different Spending Functions per Hypothesis | Using Spending Functions from Other Packages | User-Defined Spending Functions | Summary | References

Introduction | Bonferroni-based procedures | Bonferroni test | Weighted Bonferroni test | Holm Procedure | Weighted Holm Procedure | Fixed sequence procedure | Fallback procedure | Serial gatekeeping procedure | Parallel gatekeeping procedure | Successive procedure | Hochberg-based procedures | Hochberg procedure | Simes-based procedures | Hommel procedure | Parametric procedures | Šidák test | Dunnett test | Weighted Dunnett test | Dunnett procedure | Reference

Introduction | Basic usage | Initial graph | Update graph | Perform graphical MCPs | Power simulations | References

Motivating example | Create the graph | Perform the graphical multiple comparison procedure | Adjusted p-values and rejections | Obtain final and intermediate graphs after rejections | Obtain possible orders of rejections | Obtain adjusted significance levels | Power simulation | Input: Marginal power for primary hypotheses | Input: Marginal power for secondary hypotheses | Input: Correlation structure to simulate test statistics | User-defined success criteria | Output: Simulate power | Reference

Introduction | 1. Alpha Spending Functions | Comparison with gsDesign | Comparison with rpact | Detailed Example | 2. Group Sequential Boundaries | 3. Correlation Matrix | 4. Repeated P-values | 5. Sequential P-values | Relationship Between Sequential and Repeated P-values | 6. Wang-Tsiatis Boundaries | Verification of boundary shape | Summary | References

Introduction | Example Dataset | Fit Initial Cox Model | Model-Free Tipping Point Analysis | Step-by-step walk through of the analyses using model-free imputation | Visualize Pooled Kaplan--Meier Curves | Visualize Tipping Point Graph | Summarize tipping point via Analysis Results Dataset (ARD) | Assess plausibility | Model-Based Tipping Point Analysis | Summary via Analysis Results Dataset (ARD) | Jump-to-Reference (J2R) as a Special Case of Model-based Imputation | Inspect the data | Fit the original Cox model | Run model-based imputation | J2R Results | References

Example | Using decider | Using crmPack | Comparison of parameter posteriors | Monte Carlo uncertainty of the differences | $$\operatorname{MCSE}(\hat\theta_{\text{decider}} -\hat\theta_{\text{crmPack}}) | Comparison of fit | Comparison of model code | decider | crmPack | Conclusion | References

Introduction | Single combination arm | Model specification | Model implementation | Design implementation | Other single agent models | Parallel monotherapy and combination arms | Customizations | Historical arms | Delayed arm opening | Disabling borrowing for arm decisions | Evaluating a fixed data scenario | References

Introduction | Framework | Algorithm | Examples | Standard components | No backfill cohorts | Simple backfill cohorts | More complex backfill cohorts | Simulations with backfill cohorts | Investigating single trial data | Investigating simulation results | Limitations | Backwards Compatibility | References

Introduction | How is this done? | Using knit_print in crmPack | Common customisations | Rendering complex classes | Design | Dose toxicity model | Stopping rule | Escalation rule | Use of placebo | Backfill cohorts | Dose recommendation | Cohort size | Observed data | Starting dose | Restoring console-like behaviour | Accessing the output of knit_print | Providing your own knit_print method | Class coverage

Abstract | Introduction | Framework | Using crmPack | Implementing a CRM trial | Dose escalation with safety and efficacy | Extending crmPack functionality | Summary | Acknowledgments | References

Introduction | Implementation | Ordinal data | The LogisticLogNormalOrdinal class | Model fitting | Rules classes for ordinal models | On the need for a diagonal covariance matrix | Some observations | Environment | References

Introduction | High level usage | Important information for usage | Information for debugging | Worked out example | Alternative: read user code from external file | Note | References

Example 1: Recommend a dose for the next cohort | Setting up the data | Structure of the model class | Obtain the posterior | Use ggmcmc to diagnose | Plot the model fit | prior mean curve | Escalation rules | Recommended dose for the next cohort | Example 2: Run a simulation to evaluate operating characteristics | Set up safety window and DADesign to be completed | Set up true curves | Perform the simulations | Interpret the simulation results

Simulation setting | Design definition | Simulation run

Introduction | The dose grid | The dose-toxicity model | The increment rule | The dose selection rule | The cohort size | The stopping rule | Trial definition | Analysing a trial | The single patient run-in | The first full cohort | The second full cohort | The third full cohort | The fourth full cohort | The fifth full cohort | Summarising the trial results | Note | References

Defining the design | The dose grid | The dose toxicity model | The escalation rules | The maximum increment | The NextBest rule for recommending the best dose for the next cohort | The cohort size | The stopping rules | The overall trial design | References

Introduction | Study definition | Incoherence and rigidity | Does the prior make sense? | Final observation | References

Example

Class and slot changes | Naming convention motivation | New classes | Renamed classes | Renamed slots | Moved dose and prob Functions from Slots to Methods | Generate data, define a model and get samples | Dose | Prob | New Random Number Generator settings for the MCMC | New no-argument constructors | Handling of NA or placebo returned as next dose | Evaluation of stopping rules at a specific dose | Further details in class and methods name changes | Classes | Methods | References

Introducing tidy methods to crmPack | Basic approach | Exceptions | Examples | Using tidy crmPack data | Cohort size | Environment

Introduction | Covariance and Correlation Matrices | Transformation to Variance Parameters | Unstructured (us) | Homogeneous (ad) and Heterogeneous Ante-dependence (adh) | Homogeneous (toep) and Heterogeneous Toeplitz (toeph) | Homogeneous (ar1) and Heterogeneous (ar1h) Autoregressive | Homogeneous (cs) and Heterogeneous (csh) Compound Symmetry | Spatial Covariance Structure | Spatial exponential (sp_exp) | Spatial Gaussian (sp_gau) | References

Background | G-Computation Estimator | Covariance Estimator of $\hat{\theta}_t$ | Implementation

Prediction of conditional mean | Mathematical Derivations | Implementation of predict | Parametric Sampling for Prediction Interval | Prediction of Conditional Mean for New Subjects | Simulate response | Conditional Simulation | Marginal Simulation | Implementation of simulate | Relationship Between predict and simulate Results | predict options | simulate options | Comparison with SAS

Overview | Relationship to reporter | Setup | Example 1: gtsummary summary input | TXT preview | HTML preview | Example 2: gtsummary cross-table input | Example 3: plain data frame input | Example 4: controlling column widths | Automatic widths | Narrow max_table_width | Manual column_widths | Example 5: manual pagination | Common layout controls | Output checklist

Data of the current study | Benchmark analysis with a diffuse prior | Constructing the robust MAP prior | Converting the prior for use in brms.mmrm | Fitting the Bayesian MMRM with the MAP prior | Multivariate mixture priors | Specifying a known prior | Constructing a prior from real data | References

Background | Candidate models | Multiple contrast test | Dose-response modelling | References

Preliminaries | For which types of study designs can I use MCP-Mod? | What is the difference between the original and generalized MCP-Mod, and what type of response can generalized MCP-Mod handle? | How many doses do we need to perform MCP-Mod? | How to determine the doses to be used for a trial using MCP-Mod? | How to set up the candidate set of models? | Can MCP-Mod be used in trials without placebo control? | Why are bounds used for the nonlinear parameters in the fitMod function? | Should model-selection or model-averaging be used for analysis? | Which model selection criterion should be used? | How to deal with intercurrent events and missing data? | Can MCP-Mod be used in trials with multiple treatment regimens? | What about dose-response estimates, when the MCP part was (or some of the model shapes were) not significant? | References

1 Retrieved dropout models in a nutshell | 2 Data simulation using function simulate_data() | 3 Estimators based on retrieved dropout models | 4 Implementation of the defined retrieved dropout models in rbmi | 4.1 Basic MAR model | 4.2 Retrieved dropout model 1 (RD1) | 4.3 Retrieved dropout model 2 (RD2) | 4.4 Brief summary of results | References

Introduction | Setup | Example Data | Define Variables | Validating Data | Catching Validation Errors | Summarising Missing Data | Missing by Visit | Subject Patterns | Summary by Treatment Group | Preparing ICE Data | Complete Workflow | Summary

Introduction | Prerequisites and Setup | Threshold-Based Responder (CHG > 3) | Analyse | Pool | Results | Clinical Cutoff Responder (CHG > 5) | Derive the New Endpoint | Pool and Display | Storing Results as ARD | Caveats

Introduction | Setup and Data | Data Preparation | Validation | Missingness Summary | rbmi Analysis Pipeline | Specify the Imputation Method | Fit the Imputation Model | Generate Imputed Datasets | Analyse Each Imputed Dataset | Pool Results | Tidying Results | Efficacy Table | Forest Plot | Treatment Difference Mode | LS Mean Display Mode | Binary/Responder Analysis

Introduction | Statistical Context | Step 1: Setup and Data Preparation | Step 2: Define Imputation Model | Step 3: Add Responder Variables | Step 4: Continuous Endpoint Analysis (CHG) | Step 5: Responder Endpoint Analysis (CRIT1FLN) | Define Analysis Function | Define Variables and Run Analysis | Final Notes | Efficient Storage | See Also

Introduction | The Storage Problem | Setup | Example with Package Data | Reducing Imputed Data | What's in the Reduced Data? | Expanding Back to Full Data | Verifying Data Integrity | Practical Workflow | Save Reduced Data | Load and Analyse | Storage Comparison | When to Use This Approach | Edge Cases | No Missing Data | Single Imputation | Summary

Introduction | Setup | Inspecting Draws with describe_draws() | Inspecting Imputations with describe_imputation() | MI Diagnostic Statistics in ARD | When Diagnostics Are Not Available | Learn More

Background and Data | Design stage | Analysis stage | Multiple comparisons | Dose-response estimation | How to adjust for covariates? | References

Background and data set | Candidate models | Analysis without covariates | Analysis with covariates | Avoiding problems with complete seperation and 0 responders | Considerations around optimal contrasts at design stage and analysis stage | Power and sample size considerations | References

Data Simulation Code | Calculate Mean Response from Emax Model | Generate Normal Mean Vector and Covariance Matrix | Simulate Longitudinal Outcomes | Cut Interim Data | Longitudinal Data Analysis | Completers Analysis | Repeated Measures Analysis | Generalized MCP-Mod | Final Analysis | Futility Interim Analysis | Simulation study | References

Perform multiple contrast test | Fit non-linear dose-response models here illustrated with Emax model | Calculate optimal designs, here illustrated for target dose (TD) estimation | References

Implementation | Last run | Convergence | Parameter coverage

Raw data | Models | Checking and troubleshooting | Summaries | Marginals | Predictions | Plots

Common elements | Data | Likelihood | Variance | Expected value of the control group | The decline models | The non-proportional decline model | The proportional decline model | The slowing models | The non-proportional slowing model | The proportional slowing model | References

Weighted Least Square (WLS) Empirical Covariance | Difference of Implementations | Proof of Identity | Proof for Covariance Estimator | Proof for Degrees of Freedom | Special Considerations in Implementations | Pseudo Inverse of a Matrix | Avoiding the Crossproduct of the G Matrix | References

Introduction | Package Structures | data | data-raw | design | SAS | TMB | inst | man | NAMESPACE | NEWS.md | R | README | simulations | src | chol_cache.h | covariance.h | derivatives.h | empirical.cpp | exports.cpp | jacobian.cpp | kr_comp.cpp | Makevars | mmrm.cpp | predict.cpp | test files | tmb.cpp and tmb_includes.h | utils.h | tests | vignettes | Other files | _pkgdown.yml | .gitignore | .lintr | .pre-commit-config.yaml | .Rbuildignore

Introduction to Type I/II/III Hypothesis Testing | Contained Effect | Type II Hypothesis Testing | Type III Hypothesis Testing | Hypothesis Testing in SAS | Special Considerations | Reference Levels | Example of Reference Levels | Why Model Covariate Order Changes My Testing in SAS? | Why mmrm Gives More Covariates Than SAS? | Excluding columns due to collinearity | Intercept-free models | Intercept-free Models with stats::model.matrix() | Intercept-free Models with PROC MIXED | Type-II Contrast Matrices in Intercept-free Models

Common Usage | Data Introduction | Obtain Treatment Effect for Continuous Outcomes Using the General Variance | Obtain Treatment Effect for Binary Outcomes | Obtain Treatment Effect for Counts | Using Different Covariate-Adaptive Randomization Schema | Obtain the Confidence Intervals for the Marginal Means and contrast

Power for multiple contrast test versus group sample size | Power versus treatment effect | Power under mis-specification | Sample size based on metrics other than power for the multiple contrast test | References

Introduction | Datasets | FEV Data | BCVA Data | Model Implementations | Ante-dependence (heterogeneous) | PROC GLIMMIX | mmrm | Ante-dependence (homogeneous) | Auto-regressive (heterogeneous) | gls | Auto-regressive (homogeneous) | glmmTMB | Compound symmetry (heterogeneous) | Compound symmetry (homogeneous) | Spatial exponential | Toeplitz (heterogeneous) | Toeplitz (homogeneous) | Unstructured | lmer | Benchmarking | Convergence Times | Marginal Treatment Effect Estimates Comparison | Impact of Missing Data on Convergence Rates | Session Information

Constructing an archetype | Informative priors | Modeling and analysis | All archetypes | Variations on archetypes

Introduction | Power simulations | Bonferroni tests | Hochberg tests | Simes tests | Parametric tests | Mixed tests of Bonferroni, Hochberg and Simes | Mixed tests of parametric and one of Bonferroni, Hochberg and Simes | Conclusions

Introduction | Asymptotic Covariance | Empirical Covariance | Jackknife Covariance | Bias-Reduced Covariance | Kenward-Roger Covariance

Introduction | General concepts | What is an average treatment effect? | How to estimate "unconditional / marginal" average treatment effect? | How to estimate the variance of the g-computation estimator of average treatment effect? | Analysis example | Comparing different implementations | Ge et al (2011) | Ye et al (2023) | References

Example data | Marginal means for clinical trials | Existing capabilities | How brms.mmrm estimates marginal means | How brm_marginal_draws() works | Subgroup analysis | References

Priors | Sampling | Imputation of missing outcomes | References

Raw data | Preprocessing | Formula | Priors | Model | Marginals | Visualization | Comparing models and data | Plotting draws | Comparing priors and posteriors | Appendix A: Contrasts | Appendix B: Imputation of missing outcomes | Imputation before model fitting | Imputation during model fitting | References

Simple | Change from baseline | Advanced | Prior | Posterior

About | Prerequisites | Data | Pre-processing | Descriptive statistics | Fitting MMRMs | Bayesian model | Frequentist model | Comparison | Extract estimates from Bayesian model | Extract estimates from frequentist model | Summary

About | Prerequisites | Data | Pre-processing | Descriptive statistics | Fitting MMRMs | Bayesian model | Frequentist model | Comparison | Extract estimates from Bayesian model | Extract estimates from frequentist model | Summary | Session info

Data | Formula | Models | Marginals | Model comparison | Visualization | References

Motivating example | Create a graph | Perform the graphical multiple comparison procedure based on the closure principle | Bonferroni tests | Obtain the closure | Obtain adjusted significance levels | Mixed procedures for graphical approaches | Parametric tests for primary hypotheses | Parametric tests for primary hypotheses and Simes tests for secondary hypotheses | Power calculation | Input: Marginal power for primary hypotheses | Input: Marginal power for secondary hypotheses | Input: Correlation structure to simulate test statistics | User-defined success criteria | Output: Calculate power | Reference

About | Conclusion | Setup | Subgroup scenario | Unstructured scenario | Autoregressive moving average scenario | Autoregressive scenario | Moving average scenario | Compound symmetry scenario | Diagonal scenario | References

Example using dummy data | References

1 Introduction | 2 Data and model specification | 3 Reference-based conditional mean imputation - frequentist inference | 3.1 Draws | 3.2 Impute | 3.3 Analyse | 3.4 Pool | 4 Reference-based conditional mean imputation - information-anchored inference | 4.1 Draws | 4.2 Imputation step including calculation of delta-adjustment | 4.3 Analyse | 4.4 Pool | References

Common usage | Common customizations | Extraction of model features | Lower level functions | Hypothesis testing | Tidymodels | Acknowledgments | References

General definition | MMRM special case | Example | Differences compared to SAS | References

Abstract | The basic linear mixed-effects model | Extending the basic linear mixed-effects model | The MMRM as a special case | Missing data | References

1 Introduction | 2 Data simulation using function simulate_data() | 3 Handling of observed post-ICE data in rbmi under reference-based imputation | 4 Efficiently changing reference-based imputation strategies | 5 Imputation under MAR with time-varying covariates | 6 Custom imputation strategies | 7 Custom analysis functions | 8 Sensitivity analyses: Delta adjustments and tipping point analyses | 8.1 Simple delta adjustments and tipping point analyses | 8.2 More flexible delta-adjustments using the dlag and delta arguments of delta_template() | References

1 Introduction | 2 The Data | 3 Draws | 4 Impute | 5 Analyse | 6 Pool | 7 Code

1 Scope of this document | 2 Introduction to estimands and estimation methods | 2.1 Estimands | 2.2 Alignment between the estimand and the estimation method | 3 Statistical methodology | 3.1 Overview of the imputation procedure | 3.2 Setting, notation, and missing data assumptions | 3.3 The base imputation model | 3.4 Imputation step | 3.5 \(\delta\)-adjustment | 3.6 Analysis step | 3.7 Pooling step for inference of (approximate) Bayesian MI and Rubin’s rules | 3.8 Bootstrap and jackknife inference for conditional mean imputation | 3.9 Pooling step for inference of the bootstrapped MI methods | 3.10 Comparison between the implemented approaches | 4 Mapping of statistical methods to rbmi functions | 5 Comparison to other software implementations | References