boinet (version 1.3.0)

fp.logit: Fractional Polynomial Logistic Regression for Dose-Efficacy Modeling

Description

Performs fractional polynomial (FP) logistic regression with two degrees of freedom to estimate the efficacy probabilities. This method provides a flexible alternative to standard polynomial regression for capturing non-linear dose-efficacy relationships. Fractional polynomials have steadily gained popularity as a tool for flexible parametric modeling of regression relationships and are particularly useful when the relationship between dose and response may not follow simple linear or quadratic patterns.

Usage

fp.logit(obs, n, dose)

Value

A numeric vector of estimated efficacy probabilities for each dose level, corresponding to the input dose vector. Values are bounded between 0 and 1. The estimates are derived from the best-fitting fractional polynomial model selected based on deviance criteria.

Arguments

obs

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

n

Numeric vector of the total number of patients treated at each dose level. Must be positive integers. Length must match obs and dose.

dose

Numeric vector of dose levels investigated. Should be positive values representing actual doses (not dose level indices). Length must match obs and n.

Details

Fractional polynomials extend conventional polynomials by allowing non-integer powers from a predefined set. All commonly used transformations such as the logarithmic, square, cubic, or reciprocal are embedded in the FP method. This approach is especially valuable in dose-finding studies where the true shape of the dose-response curve is unknown and may exhibit complex non-linear behavior.

Mathematical Framework: The fractional polynomial of degree m for a positive variable x takes the form: $$FP_m(x) = \beta_0 + \sum_{j=1}^{m} \beta_j H_j(x)$$

where \(H_j(x)\) represents transformed versions of x using powers from the set {-2, -1, -0.5, 0, 0.5, 1, 2, 3}, with 0 representing the natural logarithm.

Implementation Details: This function implements FP logistic regression with 2 degrees of freedom, using very liberal selection criteria (select=0.99999, alpha=0.99999) to avoid the closed testing procedure and focus on finding the best-fitting model.

References

• Royston, P., & Altman, D. G. (1994). Regression using fractional polynomials of continuous covariates: parsimonious parametric modelling. Journal of the Royal Statistical Society: Series C (Applied Statistics), 43(3), 429-467.

See Also

mfp for the underlying fractional polynomial fitting algorithm, glm for standard logistic regression, obd.select for optimal biological dose selection using the output from this function.

Examples

# Modeling efficacy probabilities in a dose-escalation study
dose_levels <- c(25, 50, 100, 200, 400)  # mg doses
efficacy_responses <- c(1, 3, 8, 12, 10)  # patients with efficacy
total_patients <- c(6, 6, 12, 15, 12)     # total patients per dose

# Fit fractional polynomial model
efficacy_probs <- fp.logit(obs = efficacy_responses,
                          n = total_patients,
                          dose = dose_levels)

# Display results
results <- data.frame(
  Dose = dose_levels,
  Observed_Rate = efficacy_responses / total_patients,
  FP_Predicted = round(efficacy_probs, 3)
)
print(results)