Probabilistic Factorial Experimental Design

• Probabilistic factorial experimental design is a method that uses probability-based criteria to systematically plan, allocate, and analyze multifactor experiments.
• It incorporates models such as generalized linear models and criteria like D-optimality and EW D-optimality to efficiently estimate treatment effects and minimize variance.
• The approach leverages robust numerical algorithms, including lift-one and exchange methods, to optimize design allocations in complex, high-dimensional settings.

Probabilistic factorial experimental design refers to the systematic use of probability-based principles and constructions in the planning, allocation, and analysis of factorial experiments. Unlike classical deterministic designs, probabilistic approaches explicitly incorporate uncertainty, model structure, or randomization—either in the generation of experimental runs, the allocation of experimental resources, or in the inferential analyses of results. These principles enable efficient estimation of treatment effects, robust inference, and greater flexibility in complex or high-dimensional experimental settings.

1. Probabilistic Structure in Factorial Experimental Design

Probabilistic factorial experimental design encompasses the randomization of experimental assignments, the use of probabilistic criteria for design optimality, and inference frameworks that directly leverage the structure of probability models. In $2^k$ factorial designs with binary responses, the response variable is modeled via a generalized linear model (GLM), where

$\eta_i = \mathbf{x}_i' \boldsymbol{\beta}$

and the mean response is linked to the linear predictor through a function $g$: $\pi_i = E(Y|\mathbf{x}_i) = g^{-1}(\eta_i)$. The information contributed by each design point depends not only on the allocation $p_i$ but also on $w_i$, related to the Fisher information, and itself a probabilistic function of model parameters and responses: $$w_i = \left( \frac{d\pi_i}{d\eta_i} \right)^2 / [\pi_i(1 - \pi_i)]$$ The allocation proportions $\mathbf{p}$ thus determine, in conjunction with the probabilistic model, the asymptotic information matrix $M(\mathbf{p}) = X' W X$.

The optimal design is inherently probabilistic, as $w_i$ encapsulates distributional properties of the response and the dependence on parameter values or their distribution—linking the notion of "optimal allocation" to the underlying probability model of data generation.

2. Criteria for Design Optimality and Their Probabilistic Foundations

The most widely used probabilistic design criterion in the context of factorial designs is D-optimality. This principle seeks a design $\mathbf{p}$ that maximizes the determinant of the information matrix: $\max_{\mathbf{p}} |X' W X|$ Given that $W$ encodes the Fisher information contributed by each experimental condition—derived from the underlying probability model—D-optimality reflects the goal of minimizing the generalized variance of the parameter estimates. In models with binary responses, the determinant $|X' W X|$ is a homogeneous polynomial in the $p_i w_i$, summing over all $(d+1)$-point support sets of the design.

Further, the general equivalence theorem is adapted, leading to precise necessary and sufficient conditions for local D-optimality in the $2^k$ binary-response setting, but formulated in terms of determinants and avoiding matrix inversion.

Bayesian and expectation-weighted (EW) versions of D-optimality extend this to settings where model parameters themselves are random, with known or assumed prior distributions: $$\mathbf{p}_{\text{Bayes}} = \arg\max_{\mathbf{p}} E_{\boldsymbol\beta} [ \log | X' W(\boldsymbol\beta, \mathbf{p}) X | ]$$ The EW D-optimality criterion instead replaces the random $W$ by its expected value under the prior: $$\mathbf{p}_{EW} = \arg\max_{\mathbf{p}} | X' E(W) X |$$ Since $\log |X' W X|$ is concave in $W$, Jensen's inequality ensures the EW criterion upper bounds the true Bayesian criterion and often offers a computationally efficient approximation.

3. Algorithmic Construction and Numerical Search Methods

Implementation of probabilistic factorial experimental designs for binary responses involves efficient numerical algorithms, owing to the complexity of the determinant criterion. Two primary algorithms are employed:

(a) Lift-one Algorithm: Iteratively, for each design point $i$, the algorithm proposes changes to $p_i$, maximizes the objective in a one-dimensional slice, and updates the allocation vector. This method exploits the structure of the determinant as a polynomial in the $p_i$ to allow for analytically tractable maximization in each iteration.

(b) Exchange Algorithm: This algorithm redistributes allocation mass between pairs $(i,j)$, maximizing the objective with respect to $(p_i, p_j)$ (and possibly their integer-valued counterparts). Both approaches are proven to converge under reasonable regularity conditions and offer dramatically improved computational performance compared to generic non-linear optimization.

The practical upshot is these algorithms enable the construction of optimal or nearly-optimal designs even for high-dimensional ($k$ large) factorial spaces, where enumeration is infeasible.

4. Role of Priors and Robustness: EW D-Optimality and Uniform Designs

Uncertainty in model parameters ($\boldsymbol\beta$) can be addressed via prior distributions. Under symmetric priors (e.g., $\beta_i$ uniform on $[-a,a]$), the expected contributions $E(w_i)$ become equal across design points, rendering the uniform design EW D-optimal. If prior information indicates effect directionality (asymmetry), EW D-optimality adapts the design allocations accordingly, improving efficiency without major computation over the locally optimal (fixed-parameter) solution.

Robustness is formally quantified using the efficiency loss measure: $$R(\mathbf{p}, \mathbf{w}) = 1 - \left( \frac{|X' W X|_{\mathbf{p}}}{|X' W X|_{\mathbf{p}_w}} \right)^{1/(d+1)}$$ Uniform designs are maximally robust with respect to unknown effect directions; EW D-optimal designs maintain robustness if prior knowledge indicates probable effect orientation.

When fractionality is imposed (due to prohibitive run size), the optimal fractional design may diverge from classical regular fractions: D-optimality is achieved only for equal $w_i$, and with variance across $w_i$, support points of the optimal fraction are selected according to their expected contribution to Fisher information, not regular fraction patterns.

5. Applicability, Recommendations, and Implications

Probabilistic factorial design principles and their algorithmic implementation are directly applicable to factorial experiments with binary outcomes in medicine, engineering, and the social sciences. The formalism efficiently accommodates uncertainty in parameter values, prior information, and practical run constraints. For practitioners:

• EW D-optimal designs are recommended with prior information on effect sizes or directions.
• Uniform designs are advised for maximum robustness when prior knowledge is absent.
• When using fractional designs, select points by maximizing the expected information criterion, not solely by regular aliasing rules.

The theoretical and algorithmic advances enable more reliable, efficient experimental designs with binary responses, offering robust inference against parameter misspecification and maximizing statistical power under finite resource constraints.

6. Summary Table: Core Quantities and Formulas

Quantity	Formula
Information Matrix	$M(\mathbf{p}) = X' W X$
$w_i$ for GLM	$w_i = (d\pi_i/d\eta_i)^2 / [\pi_i(1 - \pi_i)]$
D-optimality criterion	$\max_{\mathbf{p}} | X' W X |$
Determinant as polynomial	$$|X' W X| = \sum_{I} |X[I]|^2 \prod_{i \in I} p_i w_i$$, $|I| = d+1$
EW D-optimal criterion	$\max_{\mathbf{p}} | X' E(W) X |$
Loss of efficiency	$$R(\mathbf{p}, \mathbf{w}) = 1 - \left( |X'WX|_{\mathbf{p}} / |X'WX|_{\mathbf{p}_w} \right)^{1/(d+1)}$$

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Probabilistic factorial experimental designs for binary responses harmonize classical optimal design theory, Bayesian/statistical robustness approaches, and scalable algorithmic implementation, supporting the efficient, theoretically grounded planning of complex experiments under model-based uncertainty and practical limitations.