Adjusted Pairwise Likelihood (APW)

• Adjusted Pairwise Likelihood (APW) is a pseudo-likelihood method that incorporates pairwise and second-order probability information with scalar or matrix adjustments to correct bias in complex sampling designs.
• It extends traditional composite likelihood by using weighting schemes, sandwich variance corrections, and moment-matching calibrations, ensuring computational feasibility and model-agnostic application.
• APW has proven effective in domains such as survey analysis, binary factor models, and phylogenetics, delivering nearly unbiased estimates, valid uncertainty quantification, and robust hypothesis testing.

Adjusted Pairwise Likelihood (APW) is a principled class of pseudo-likelihood methods that employ pairwise or second-order probability information and rigorous scalar, matrix, or normalization adjustments to restore correct frequentist properties in estimation and inference under complex dependence structures and informative or complex sampling designs. APW frameworks generalize ordinary pairwise composite likelihood via weighting schemes, sandwich (Godambe) variance corrections, and moment-matching calibrations, yielding computationally feasible, model-agnostic methods for consistent estimation, valid uncertainty quantification, and robust hypothesis testing in diverse domains ranging from survey analysis to phylogenetic dating.

1. Foundational Principles of Adjusted Pairwise Likelihood

APW arises from the recognized limitation of standard first-order, sampling-weighted pseudo-likelihood—commonly formulated by exponentiating likelihood contributions via marginal inclusion probabilities $w_i \propto 1/\pi_i$—which implicitly assumes attenuation of inclusion dependencies $\mathrm{Cov}(\delta_i,\delta_j) = O(1/N)$ as $N \to \infty$. In many practical settings (multi-stage clusters, household samples, interconnected molecular sequences), such attenuation fails and persistent dependencies bias inference for both frequentist and Bayesian target parameters. APW redresses this by integrating pairwise or second-order probabilities $\pi_{ij}$ and weights $w_{ij} = 1/\pi_{ij}$, yielding pseudo-likelihoods and posteriors whose theoretical properties extend to dependent, informative sampling designs and non-trivial dependence structures (Williams et al., 2017).

Formally, given inclusion indicators $\delta_i \in \{0,1\}$, parametric density $p(y_i|\theta)$, and observed sample $y_o = \{y_i:\delta_i=1\}$, the Adjusted Pairwise pseudo-likelihood is

$$L_2(\theta) = \prod_{i<j:\,\delta_i=\delta_j=1} [p(y_i|\theta)\,p(y_j|\theta)]^{w_{ij}}$$

which can be rearranged as

$$L_2(\theta) = \prod_{i:\,\delta_i=1} p(y_i|\theta)^{w_i^*}, \quad w_i^* \equiv \sum_{j\neq i:\delta_j=1} w_{ij}$$

with normalization $\sum_i w_i^* = n$ to control dispersion (Williams et al., 2017).

2. Construction and Variants of APW Across Domains

APW methodology extends beyond survey sampling to factor modeling for binary data and molecular phylogenetics:

• Survey Sampling and Binary Factor Models: For weighted binary responses $y_i^{(h)}$ and sample weights $w_h$, the APW log-likelihood leverages weighted empirical pairwise cell proportions $\hat p_{c_i c_j}^{(ij)}$ and model-implied probabilities $\pi_{c_i c_j}^{(ij)}(\theta)$:

$$\ell_{APW}(\theta) = \sum_{i<j}\sum_{c_i=0,1}\sum_{c_j=0,1} \hat p_{c_i c_j}^{(ij)}\,\log\,\pi_{c_i c_j}^{(ij)}(\theta)$$

with sampling weights entering exclusively in $\hat p_{c_i c_j}^{(ij)}$ (Jamil et al., 2023).

• Phylogenetic Inference: For DNA alignments $X$ across $M$ taxa, the APW framework uses pairwise composite likelihood

$$\ell_C(\theta) = \sum_{i<j}\log p_{ij}(D_{ij}|\theta)$$

and scalar magnitude adjustments $w_1, w_2$ derived from eigenvalues of the sensitivity and variability matrices, yielding

$$\ell_{APW_k}(\theta) = w_k\,\ell_C(\theta),\quad k=1,2$$

embedded within Bayesian MCMC for credible interval calibration (Ellison et al., 2 Dec 2025).

• General Composite-Likelihood Testing: For independent replicates $y_i$, the pairwise likelihood ratio statistic $W_{pw}$ is adjusted via first- and second-moment matching (Molenberghs & Verbeke, Satterthwaite type) or parameter-invariant rescaling (Pace–Salvan–Sartori), each requiring stable estimation of sensitivity $H$ and variability $J$ (Cattelan et al., 2014).

3. Asymptotic Theory and Consistency Properties

Posterior consistency for APW relies on higher-order sampling design restrictions:

• Nonzero Pairwise Probabilities: $\sup_\nu[1/\min_{i<j\in U}\pi_{ij}] < \infty$;
• Bounded 3rd-to-2nd Order Ratios: $$\sup_\nu\max_{i,k,\ell}|\,\pi_{ik\ell}/(\pi_{ik}\pi_{i\ell})-1\,| < \infty$$;
• Asymptotic Factorization of 4th Order: $$|\,\pi_{ikj\ell}/(\pi_{ik}\pi_{j\ell})-1\,| = O(1/N)$$;
• Constant Sampling Fraction: $n/N \to f \in (0,1)$;

These guarantee contraction of the APW pseudo-posterior on the population generating law $P_0$ at the usual rate $\xi_N \approx \log n/\sqrt{n}$ with respect to the sampling-weighted average Hellinger distance using $\pi_{ij}$ (Williams et al., 2017).

In composite likelihood approaches, the Godambe information $G(\theta) = H(\theta)\,J(\theta)^{-1}H(\theta)$ underpins asymptotic normality:

$$\sqrt{n}(\hat\theta_{APW}-\theta) \to_d N(0,G(\theta)^{-1})$$

where $H$ and $J$ subsume pairwise dependence and survey design (Jamil et al., 2023, Cattelan et al., 2014).

4. Practical Implementation and Computational Algorithms

Practical deployment of APW comprises several key steps:

Step	Action	Domain-specific remarks
Compute $\pi_{ij}$ or $\hat p_{c_i c_j}^{(ij)}$	Extract second-order probabilities or weighted cell proportions from sampling/design information	For cluster-sampling, restrict to within-cluster pairs (Williams et al., 2017, Jamil et al., 2023)
Form weights $w_{ij}=1/\pi_{ij}$ or apply moment-matching scalar $w_k$	Establish unnormalized or moment-matched magnitude adjustments	Eigenvalue-based $w_k$ for composite likelihood (Ellison et al., 2 Dec 2025, Cattelan et al., 2014)
Aggregate weights for each unit: $w_i^* = \sum_{j\neq i}w_{ij}$	Normalize for over/under-dispersion control, typically $\sum_i w_i^* = n$	Ensures pseudo-posterior proper scaling (Williams et al., 2017)
Substitute likelihood contributions in statistical code	Replace each log-likelihood term with weighted version, i.e., $w_i^*\log p(y_i|\theta)$ or scale composite likelihood by $w_k$	Directly compatible with MCMC engines (Stan, JAGS, NIMBLE) for Bayesian inference (Ellison et al., 2 Dec 2025)
Sandwich variance estimation	Estimate $H$ and $J$, preferably by simulation	Avoid plug-in methods unless $n$ is large; use simulation approach if model allows (Cattelan et al., 2014)
Estimating equations solution	Newton-Raphson, Fisher scoring, or BFGS algorithms; cost $O(p^2)$ per iteration	Precompute weighted cell proportions to maximize efficiency (Jamil et al., 2023)

Practical simulation evidence finds APW delivers unbiased point estimates, valid standard errors, and nominal coverage in survey, binary factor, and phylogenetic models—even in scenarios marked by dependence and informative design (Williams et al., 2017, Jamil et al., 2023, Ellison et al., 2 Dec 2025, Cattelan et al., 2014).

5. Variance Estimation, Test Statistics, and Goodness-of-Fit

Rigorous variance adjustment under APW is essential especially for clustered and complex sampling:

• Sandwich/Godambe Correction: Form $\widehat I$, $\widehat J_{\text{cluster}}$ at the cluster, stratum, or simulation level, yielding robust design-based $$\widehat G=\widehat I \widehat J_{\text{cluster}}^{-1} \widehat I$$ (Jamil et al., 2023, Cattelan et al., 2014).
• Goodness-of-Fit (GOF) Testing: Pearson-type moment-adjusted and Wald-type quadratic-form statistics reinterpret first and second-order margins and residuals under the APW. Notably, the Pearson statistic requires only diagonals and, paired with moment-based degrees-of-freedom correction, maintains correct type-I error for moderate $n$ and complex designs (Jamil et al., 2023).
• Adjusted CL Ratio Tests: Rescale $W_{pw}$ via moment-matching or parameter-invariant factors to restore a $\chi^2$ law:

$$W_{(1)} = W_{pw}/\kappa \sim \chi^2_p,\qquad W_{(2)} = W_{pw}/k \sim \chi^2_\nu$$

Satterthwaite-type and Pace–Salvan–Sartori adjustments are effective except for very small sample sizes with empirical estimation (Cattelan et al., 2014).

Simulation-based estimation of $H$ and $J$ yields reliable coverage and consistency for APW-enabled test statistics, whereas empirical ("plug-in") approaches require $n \gg 30$ for accuracy. Monte Carlo estimation is advised whenever simulation from the model is feasible (Cattelan et al., 2014).

6. Domain-Specific Illustration and Computational Impact

APW methods have demonstrated substantive efficacy and computational gains:

• Survey/Cluster Sampling: In household-based designs, APW eliminates residual bias from under-accounted within-cluster dependencies, outperforming equal or marginal weighting for estimating sub-population relationships (e.g., spouse substance use modeling), while preserving flexibility for fully Bayesian estimation (Williams et al., 2017).
• Binary Factor Analysis: For latent factor models under clustered, unequal-probability sampling, APW estimation greatly reduces bias and maintains valid GOF characteristics compared to unweighted approaches; computational cost scales as $O(p^2)$, supporting high-dimensional application (Jamil et al., 2023).
• Phylogenetics/Node-Age Estimation: APW1 and APW2, based on moment-matching to the true likelihood's test statistic, enable genome-scale Bayesian MCMC running up to 15× faster than the full likelihood with comparable coverage and robustness to fossil calibration uncertainty and prior misspecification (Ellison et al., 2 Dec 2025).

Empirical findings indicate APW recovers nearly unbiased estimators, valid confidence regions, and maintains credible interval coverage for moderate $n$ even under misspecified priors or misplacement of calibration points, thus providing calibration-robust, computationally tractable inference (Ellison et al., 2 Dec 2025).

7. Methodological Recommendations and Limitations

APW provides a "nearly automated estimation procedure applicable to any model specified by the data analyst," requiring only second-order probability or pairwise frequency calculations and standard numerical or MCMC routines (Williams et al., 2017). However, practitioners should heed:

• For composite likelihood test statistics, simulation-based estimation of sensitivity and variability matrices is preferred except in very large $n$ scenarios; empirical methods may underperform otherwise (Cattelan et al., 2014).
• In highly stratified or multistage samples, pairwise probabilities may only be computable for last-stage clusters; this approximation is sufficient for most applications (Williams et al., 2017).
• Moment-matching adjustments (APW1/APW2) correct only first (mean) and second (variance) moments; additional higher-moment excursions may not be fully captured, which is common in finite-sample composite likelihood settings (Cattelan et al., 2014, Ellison et al., 2 Dec 2025).

A plausible implication is that APW frameworks can serve as general-purpose estimation and testing engines whenever full likelihoods are computationally prohibitive and dependence or informative design effects are non-negligible. Their design-based flexibility and robust frequentist behavior make them particularly suitable for survey, psychometric, and phylogenetic applications involving high-dimensional or complex cluster structures (Williams et al., 2017, Jamil et al., 2023, Ellison et al., 2 Dec 2025, Cattelan et al., 2014).