MCDA Value Function Types • valueJudgementCE

Introduction

Value functions are mathematical transformations that convert raw clinical outcome values into a standardized scale (typically 0-100) representing “value” or “desirability” from a decision-making perspective. The choice of value function reflects stakeholder preferences about how outcomes are valued.

Why Value Functions Matter

Consider an adverse event rate: - Linear: Going from 10% to 20% AE rate has the same negative value as 20% to 30% - Exponential: Going from 10% to 20% is concerning, but 20% to 30% is dramatically worse - Threshold: Below 15% is acceptable (value=100), above 15% is unacceptable (value=0)

The choice affects benefit-risk conclusions and should reflect clinical reality and stakeholder preferences.

Value Function Types

1. Linear Value Functions (Current Standard)

Description: Constant marginal value - each unit change has equal importance

Increasing direction (higher is better, e.g., efficacy): $v(x) = 100 \times \frac{x - \text{min}}{\text{max} - \text{min}}$

Decreasing direction (lower is better, e.g., adverse events): $v(x) = 100 \times \frac{\text{max} - x}{\text{max} - \text{min}}$

When to use: - Default choice (FDA/EMA recommendation) - No strong evidence of non-linear preferences - Transparency and simplicity are priorities - Most clinical outcomes

Advantages: - Simple and interpretable - Transparent calculations - Regulatory acceptance - Conservative (neutral) assumptions

Disadvantages: - May not reflect true stakeholder preferences - Assumes equal marginal value across range

2. Piecewise Linear Value Functions

Description: Different slopes in different regions, connected at breakpoints

When to use: - Clinical guidelines define severity categories (mild/moderate/severe) - Regulatory thresholds exist (acceptable/concerning/unacceptable) - Stakeholders value ranges differently - MCID (Minimally Clinically Important Difference) creates natural breakpoints

Example applications: - QoL scales with clinical interpretation bands - Lab values with normal/borderline/abnormal ranges - Dose levels with safety thresholds

Advantages: - Captures clinical interpretation thresholds - More flexible than linear - Still relatively transparent - Can match stakeholder preferences better

Disadvantages: - More parameters to justify - Arbitrary breakpoint placement needs rationale - More complex than linear

3. Exponential/Power Value Functions

Description: Curved relationship reflecting risk attitudes

Risk-averse (diminishing returns), b > 1: $v(x) = 100 \times \left(\frac{x - \text{min}}{\text{max} - \text{min}}\right)^b$

Risk-seeking (increasing returns), 0 < b < 1: $v(x) = 100 \times \left(\frac{x - \text{min}}{\text{max} - \text{min}}\right)^b$

When to use: - Stakeholder elicitation reveals non-linear preferences - Diminishing returns clinically meaningful (first 50% efficacy more valuable) - Severity increases exponentially (rare but catastrophic events) - Utility theory supports risk attitudes

Example applications: - Mortality (each life equally valuable - linear or power=1) - Quality of life (diminishing returns at high levels) - Rare serious AEs (exponential concern)

Advantages: - Captures risk attitudes - Single parameter (power) to adjust - Smooth and differentiable

Disadvantages: - Requires justification for non-linearity - Power parameter selection subjective - Less transparent than linear

4. Sigmoid (S-shaped) Value Functions

Description: Slow change at extremes, rapid change in middle

$v(x) = \frac{100}{1 + \exp(-k \times (x - \text{midpoint}))}$

When to use: - Threshold effects (outcome “tips” at clinical cutpoint) - Very low/high values matter less than mid-range - Psychological preferences show S-shaped patterns - Binary clinical decisions with gray zone

Example applications: - Biomarkers with clinical interpretation thresholds - PROs with “clinically meaningful change” bands - Dose-response with therapeutic window

Advantages: - Captures threshold effects - Smooth transition (no sharp jumps) - Biologically/clinically realistic for many outcomes

Disadvantages: - Two parameters (midpoint, steepness) to justify - Can be complex to explain - Requires clear rationale for threshold location

5. Step Functions (Discrete Categories)

Description: Discrete value levels for categorical outcomes

When to use: - Outcomes are inherently categorical (CTCAE grades) - Clinical practice uses discrete classifications - No meaningful distinction within categories - Regulatory guidance defines categories

Example applications: - CTCAE toxicity grades (1-5) - ECOG performance status (0-4) - Disease severity classifications

Advantages: - Matches clinical practice - Simple interpretation - Natural for categorical data

Disadvantages: - Ignores variation within categories - Sharp discontinuities may not reflect reality - Can be sensitive to threshold placement

Comparing Value Functions

Let’s visualize all function types for the same outcome:

Key observations: - Linear: Constant concern across all levels - Piecewise: Different concern in different ranges - Exponential: Increasing concern (risk-averse) - Sigmoid: Threshold effect at ~25% - Step: Categorical interpretation

Regulatory and Practical Considerations

FDA/EMA Guidance

Default recommendation: Linear value functions - Transparent and conservative - Neutral assumption (no risk attitude imposed) - Easiest to justify and explain

When non-linear may be acceptable: 1. Strong clinical/theoretical rationale 2. Validated through stakeholder elicitation 3. Pre-specified in analysis plan 4. Sensitivity analysis shows robustness 5. Documented justification

Selection Criteria

Clinical validity: - Does the function match clinical understanding? - Are there clinical thresholds or categories? - Do clinicians think linearly or non-linearly about this outcome?

Stakeholder preferences: - Elicit preferences using standard techniques - Choice experiments - Swing weighting - Direct function specification

Data requirements: - Linear: Only min/max thresholds - Piecewise: Breakpoints and slopes - Exponential: Power parameter - Sigmoid: Midpoint and steepness - Step: Category thresholds and values

Interpretability: - Can you explain it to regulators? - Can patients understand it? - Does it make intuitive sense?

Sensitivity Analysis

Always perform sensitivity analysis:

Recommendations

1. Start with Linear

Default choice unless clear rationale for non-linear
Easiest to justify and defend
Regulatory preference

2. Consider Non-linear When:

Clinical thresholds exist (→ piecewise or step)
Stakeholder preferences clearly non-linear (→ power or sigmoid)
Strong theoretical rationale (→ appropriate function type)
Validated through preference elicitation

3. Document Everything

Rationale for function choice
Parameter selection process
Stakeholder input
Sensitivity analyses
Clinical interpretation

4. Validate Choices

Face validity with clinicians
Stakeholder review
Sensitivity analysis
Comparison with alternative functions

5. Be Conservative

When in doubt, use linear
Simpler is better for communication
Consider audience (regulators, patients, clinicians)

References

Methodological

Thokala P, et al. (2016). Multiple criteria decision analysis for health care decision making. Value in Health, 19(1):1-13.
Keeney RL, Raiffa H. (1993). Decisions with Multiple Objectives: Preferences and Value Trade-Offs. Cambridge University Press.
Dyer JS, Sarin RK. (1979). Measurable multiattribute value functions. Operations Research, 27(4):810-822.

Regulatory

FDA. (2013). Structured Approach to Benefit-Risk Assessment in Drug Regulatory Decision-Making.
EMA. (2011). Benefit-Risk Methodology Project: Report of the BRMWP Task Force.
PROTECT. (2012). Work Package 5: Benefit-Risk Integration and Representation.

Applications

Mussen F, et al. (2007). Structured benefit-risk assessment for medicinal products. Pharmacoepidemiol Drug Saf, 16(S1):S2-15.
Mt-Isa S, et al. (2014). Balancing benefit and risk of medicines. Clinical Pharmacology & Therapeutics, 96(4):438-446.
Levitan B, et al. (2018). Application of the BRAT framework to case studies. Pharmacoepidemiol Drug Saf, 27(11):1223-1234.

Implementation Note

The current valueJudgementCE package implements linear value functions by default, which is consistent with FDA/EMA guidance and provides a transparent, conservative approach to benefit-risk assessment.

If you need to implement alternative value functions for specific applications, the examples in this vignette provide the mathematical framework. Always ensure proper documentation and justification for non-linear choices.

Package Functions for Value Function Visualization

The valueJudgementCE package now includes dedicated functions for visualizing linear value functions used in MCDA analyses:

Single Value Function Plot

Create a plot showing how raw values transform to normalized scores (0-100):

Compare Benefits vs Risks

Create side-by-side comparison showing the different normalization directions:

Visualize All MCDA Criteria

Create a multi-panel plot for all criteria in your clinical scales:

Compare Linear to Alternative Value Function Types

Create comparison plots showing how linear functions compare to other approaches (piecewise, exponential, sigmoid, step) for both benefits and risks:

This visualization helps stakeholders understand: - How the linear function (current standard, shown in gray) provides a neutral, transparent approach - How alternative functions would transform the same clinical data differently - Why regulatory agencies prefer linear functions (simplicity, transparency, no imposed risk attitudes) - The potential impact of choosing non-linear approaches on benefit-risk conclusions

Key observations: - Linear (Current Standard): Equal marginal value across all levels - regulatory preference - Piecewise Linear: Different slopes in different regions - useful when clinical thresholds exist - Exponential: Captures diminishing returns or increasing concern - requires stakeholder justification - Sigmoid: Threshold effects with smooth transitions - useful for binary clinical interpretations - Step: Discrete categories - matches categorical clinical practice (e.g., CTCAE grades)

These functions help communicate the MCDA normalization process to stakeholders and demonstrate the linear value function approach used throughout the package.