boinet (version 1.3.0)

multi.iso: Model Averaging of Multiple Unimodal Isotonic Regression

Description

Performs model averaging across multiple unimodal isotonic regression models to estimate efficacy probabilities. This approach addresses the uncertainty in the location of the optimal dose by considering all possible modes (peak response locations).

The method is particularly valuable when the dose-response relationship is expected to be unimodal (single peak) but the location of maximum efficacy is unknown. This commonly occurs with targeted therapies, immunotherapies, and biologics where efficacy may plateau or even decrease at very high doses due to off-target effects or immune suppression.

Usage

multi.iso(obs, n)

Value

The multi.iso returns a vector of estimated probabilities for each dose level.

Numeric vector of model-averaged estimated efficacy probabilities for each dose level. The length matches the input vectors, and values are bounded between 0 and 1. The estimates incorporate uncertainty about the mode location through AIC-weighted averaging across all possible unimodal models.

Arguments

obs

Numeric vector specifying the number of patients experiencing the event of interest at each dose level. Must be non-negative integers.

n

Numeric vector specifying the total number of patients treated at each dose level. Must be positive integers with the same length as obs.

Details

Unimodal Isotonic Regression: Unlike standard isotonic regression which assumes monotonic relationships, unimodal isotonic regression allows for:

• Increasing phase: Response increases up to a peak

• Peak response: Maximum efficacy at the mode

• Decreasing phase: Response decreases beyond the peak

• Constraint: Non-decreasing before mode, non-increasing after mode

Model Ensemble Approach: The function fits J separate unimodal models (where J = number of doses), each assuming the mode is at a different dose level:

• Model 1: Mode at dose 1 (monotonically decreasing)

• Model 2: Mode at dose 2 (increase then decrease)

• ...

• Model J: Mode at dose J (monotonically increasing)

Model Averaging: Given the location of the mode to be at dose level \(k\), a frequentist model averaging approach is used for estimating the efficacy probability at each dose level, where the weights are calculated based on a pseudo-likelihood on the unimodal isotonic regression.

References

• Turner, T. R., & Wollan, P. C. (1997). Locating a maximum using isotonic regression. Computational Statistics and Data Analysis, 25, 305-320.

• Tjort, N. L, & Claeskens, G. (2003). Frequentist model average estimators. Journal of the American Statistical Association, 98, 879-899.

See Also

ufit for underlying unimodal isotonic regression, pava for standard isotonic regression, fp.logit for alternative dose-response modeling approach, obd.select for dose selection using estimated probabilities.

Examples

# Basic unimodal model averaging
# Scenario: Targeted therapy with efficacy peak at intermediate dose

# Dose-response data showing unimodal pattern
observed_responses <- c(2, 6, 12, 8, 4)  # Peak at dose 3
total_patients <- c(8, 10, 15, 12, 9)

# Apply multi-unimodal isotonic regression
averaged_probs <- multi.iso(obs = observed_responses, n = total_patients)

# Compare with simple observed probabilities
simple_probs <- observed_responses / total_patients

# Display comparison
results <- data.frame(
  Dose = 1:5,
  Observed_Events = observed_responses,
  Total_Patients = total_patients,
  Simple_Probability = round(simple_probs, 3),
  MultiIso_Probability = round(averaged_probs, 3),
  Difference = round(averaged_probs - simple_probs, 3)
)

cat("Unimodal Model Averaging Results:\\n")
print(results)