Two-component Mixture Model Research Articles

Estimation of a Generalized Treatment Effect in a Control Group versus Treatment Group Design

A control group versus treatment group design is considered where the responses in the treatment group are modeled as a two-component mixture model that accounts for the possibility that only a fraction of the patients in the treated group will respond to the treatment. In this setting, the treatment effect is generalized to include both the fraction of treated patients that respond to the treatment and the magnitude of the response. Two alternative correlated and biased estimators are combined to yield an estimator that is preferable to either one of the estimators individually. The combined estimator is demonstrated on an illustrative blood pressure data set.

Cerebrospinal Fluid Homovanillic and 5-Hydroxyindoleacetic Acids in a Large Pediatric Population; Establishment of Reference Intervals and Impact of Disease and Medication.

Cerebrospinal fluid (CSF) homovanillic (HVA), and 5-hydroxyindoleacetic acids (5-HIAA) are biomarkers of neurological diseases affecting the dopaminergic and serotoninergic pathways. Establishing reference intervals for these metabolites faces the challenges of a lack of healthy controls and a negative correlation with age, making stratified intervals unrealistic. We propose a pipeline to determine continuous reference intervals for HVA and 5-HIAA using an indirect method. We also studied the confounding effects of different variables and explored the impact of antiepileptic and neuroleptic treatments on HVA and 5-HIAA values. The study used least squares regression to fit age-concentration curves from a cohort of pediatric patients (n = 1533), where the residuals represent metabolite values excluding age effect. Presuming that HVA and 5-HIAA primary deficiencies characterize a distinct subpopulation, we fitted a two-component finite mixture model in age-normalized data and set reference intervals at the central 95% of the nondeficient population. Patients with primary genetic deficiencies of HVA and/or 5-HIAA consistently fall outside the proposed continuous reference intervals. Using the new continuous reference intervals reduces the number of secondary deficiencies detected compared with using stratified values. No correlations were observed between CSF HVA and 5-HIAA values across the studied drug categories (antiseizure and neuroleptic medications). In addition, biopterin values positively influenced both metabolite concentrations. The proposed continuous reference intervals caused a substantial reduction in the number of secondary deficiencies detected, most of which demonstrated no conclusive correlations between the diseases and altered HVA and 5-HIAA values.

Statistical Framework for Modeling Asymmetrical Data with Dual Peaks

Objectives: To create a comprehensive framework that effectively identifies the most suitable model for asymmetrical data based on its unique characteristics. Methods: This study proposed a new model named Gompertz-Gumbel distribution (GGD) based on the results from the framework which utilizes various statistical tools, as well as information criteria. A dip test is used to check the modality of the data. To propose a new model, the finite mixture model concept was employed. The location, scale, shape, and weight parameters of the GGD were estimated using the maximum likelihood estimation method. Findings: The suggested framework exhibits superior performance in developing a suitable model for the asymmetrical data with dual peaks, resulting in the best fit for the data. To validate the effectiveness of the proposed model, it has been compared with various models like Gaussian models and two-component mixture models. The GGD's properties have also been determined. The various shapes of the GGD were also analyzed. Novelty: A novel framework is proposed to identify the appropriate model for the asymmetrical data with dual peaks that outperform the existing models. It shows the significance of the framework. Keywords: Lifetime distributions, Mixture models, Information Criteria, Goodness of fit, Asymmetrical data

ZIBGLMM: Zero-Inflated Bivariate Generalized Linear Mixed Model for Meta-Analysis with Double-Zero-Event Studies.

Double-zero-event studies (DZS) pose a challenge for accurately estimating the overall treatment effect in meta-analysis. Current approaches, such as continuity correction or omission of DZS, are commonly employed, yet these ad hoc methods can yield biased conclusions. Although the standard bivariate generalized linear mixed model can accommodate DZS, it fails to address the potential systemic differences between DZS and other studies. In this paper, we propose a zero-inflated bivariate generalized linear mixed model (ZIBGLMM) to tackle this issue. This two-component finite mixture model includes zero-inflation for a subpopulation with negligible or extremely low risk. We develop both frequentist and Bayesian versions of ZIBGLMM and examine its performance in estimating risk ratios (RRs) against the bivariate generalized linear mixed model and conventional two-stage meta-analysis that excludes DZS. Through extensive simulation studies and real-world meta-analysis case studies, we demonstrate that ZIBGLMM outperforms the bivariate generalized linear mixed model and conventional two-stage meta-analysis that excludes DZS in estimating the true effect size with substantially less bias and comparable coverage probability.

A novel skewed generalized normal distribution: properties, statistical inference, and its applications

In this paper, a novel skewed generalized normal distribution (NSGN), which encompasses several well-known distributions, is a two-component mixture model based on the generalized normal distribution. The NSGN distribution offers flexibility in modeling skewed data characterized by high kurtosis and strong skewness, making it highly applicable in finance, medicine, and engineering. The primary characteristics and properties of this new distribution are introduced, and the explicit expressions for the moments of order statistics are provided using recursive relations. The maximum likelihood estimation based on a profile likelihood approach, L-moments estimation, and a two-step estimation method combining Bayesian posterior maximum estimation with moment estimates are presented to estimate the four parameters of NSGN distribution. A simulation study compares the performance of these methods across different sample sizes and different parameters. The appropriateness of the proposed distribution has been tested by comparing it with several skewed distributions. Additionally, applications to real datasets showcase the beneficial properties of the NSGN for applied statistical research.

Posterior robustness with milder conditions: Contamination models revisited

Robust Bayesian linear regression is a classical but essential statistical tool. Although novel robustness properties of posterior distributions have been proved recently under a certain class of error distributions, their sufficient conditions are restrictive and exclude several important situations. In this work, we revisit a classical two-component mixture model for response variables, also known as contamination model, where one component is a light-tailed regression model and the other component is heavy-tailed. The latter component is independent of the regression parameters, which is crucial in proving the posterior robustness. We obtain new sufficient conditions for posterior (non-)robustness and reveal non-trivial robustness results by using those conditions. In particular, we find that even the Student-t error distribution can achieve the posterior robustness in our framework. A numerical study is performed to check the Kullback–Leibler divergence between the posterior distribution based on full data and that based on data obtained by removing outliers.

Deep Learning for Delineation of the Spinal Canal in Whole-Body Diffusion-Weighted Imaging: Normalising Inter- and Intra-Patient Intensity Signal in Multi-Centre Datasets.

Whole-Body Diffusion-Weighted Imaging (WBDWI) is an established technique for staging and evaluating treatment response in patients with multiple myeloma (MM) and advanced prostate cancer (APC). However, WBDWI scans show inter- and intra-patient intensity signal variability. This variability poses challenges in accurately quantifying bone disease, tracking changes over follow-up scans, and developing automated tools for bone lesion delineation. Here, we propose a novel automated pipeline for inter-station, inter-scan image signal standardisation on WBDWI that utilizes robust segmentation of the spinal canal through deep learning. We trained and validated a supervised 2D U-Net model to automatically delineate the spinal canal (both the spinal cord and surrounding cerebrospinal fluid, CSF) in an initial cohort of 40 patients who underwent WBDWI for treatment response evaluation (80 scans in total). Expert-validated contours were used as the target standard. The algorithm was further semi-quantitatively validated on four additional datasets (three internal, one external, 207 scans total) by comparing the distributions of average apparent diffusion coefficient (ADC) and volume of the spinal cord derived from a two-component Gaussian mixture model of segmented regions. Our pipeline subsequently standardises WBDWI signal intensity through two stages: (i) normalisation of signal between imaging stations within each patient through histogram equalisation of slices acquired on either side of the station gap, and (ii) inter-scan normalisation through histogram equalisation of the signal derived within segmented spinal canal regions. This approach was semi-quantitatively validated in all scans available to the study (N = 287). The test dice score, precision, and recall of the spinal canal segmentation model were all above 0.87 when compared to manual delineation. The average ADC for the spinal cord (1.7 × 10-3 mm2/s) showed no significant difference from the manual contours. Furthermore, no significant differences were found between the average ADC values of the spinal cord across the additional four datasets. The signal-normalised, high-b-value images were visualised using a fixed contrast window level and demonstrated qualitatively better signal homogeneity across scans than scans that were not signal-normalised. Our proposed intensity signal WBDWI normalisation pipeline successfully harmonises intensity values across multi-centre cohorts. The computational time required is less than 10 s, preserving contrast-to-noise and signal-to-noise ratios in axial diffusion-weighted images. Importantly, no changes to the clinical MRI protocol are expected, and there is no need for additional reference MRI data or follow-up scans.

Composite and Mixture Distributions for Heavy-Tailed Data—An Application to Insurance Claims

This research provides a comprehensive analysis of two-component non-Gaussian composite models and mixture models for insurance claims data. These models have gained attraction in actuarial literature because they provide flexible methods for curve-fitting. We consider 256 composite models and 256 mixture models derived from 16 popular parametric distributions. The composite models are developed by piecing together two distributions at a threshold value, while the mixture models are developed as convex combinations of two distributions on the same domain. Two real insurance datasets from different industries are considered. Model selection criteria and risk metrics of the top 20 models in each category (composite/mixture) are provided by using the ‘single-best model’ approach. Finally, for each of the datasets, composite models seem to provide better risk estimates.

Mixed-type defect pattern recognition in noisy labeled wafer bin maps

In semiconductor manufacturing, classification of defect patterns in wafer bin maps (WBMs) helps engineers detect process failures and identify their causes. In recent studies on WBMs, convolutional neural networks (CNNs) have demonstrated effective classification performance based on their high expressive power. However, previous studies have implicitly assumed that the labels of WBMs used for training CNNs are correct, even though labels are often incorrect. When trained on mislabeled data, CNNs with standard cross-entropy loss can easily overfit mislabeled samples, leading to poor generalization for testing data. To overcome this issue, we propose a novel training algorithm called sample bootstrapping. Sample bootstrapping identifies which samples have clean or noisy labels by using a two-component beta mixture model, and measures the uncertainty of each identified label. Then, samples with low uncertainty of their estimated labels are selected to build mini-batches via weighted random sampling. Finally, CNNs are trained on the selected mini-batches with dynamic bootstrapping loss. In this manner, we can correct only the samples that are highly likely to have noisy labels and prevent the risk of false correction of correctly labeled samples. Experiments on simulated and real datasets demonstrate the effectiveness of the proposed method.

Maximum likelihood estimation of the log-concave component in a semi-parametric mixture with a standard normal density

The two-component mixture model with known background density, unknown signal density, and unknown mixing proportion has been studied in many contexts. One such context is multiple testing, where the background and signal densities describe the distribution of the p-values under the null and alternative hypotheses, respectively. In this paper, we consider the log-concave MLE of the signal density using the estimator of Patra & Sen (2016) for the mixing probability. We show that it is consistent and converges at the global rate n−2/5. An EM-algorithm in combination with an active set algorithm implemented in the R-package logcondens was used to compute the log-concave MLE. When one is interested in estimation at a fixed point, a conjecture is made about the limit distribution of our estimator. The performance of our method is assessed through a simulation study.

A robust intelligent fault diagnosis method for rotating machinery under noisy labels

Despite achieving considerable success, the fault diagnosis methods will still be disturbed by noisy labels, this causes the model’s degradation and reduced diagnostic precision. Focused on solving the above issues, a robust intelligent fault diagnosis approach for rotating machinery under noisy labels is proposed. Firstly, we maintain two deep neural networks (DNNs) and alternatively execute parameters updating and models optimization by referring to the Co-teaching strategy, which can maximize filtering different error types and implement pre-training of DNNs. Secondly, adopting a two-component Gaussian mixture model (GMM) to fit training dataset’s cross-entropy (CE) loss and realize the clean and noisy labels division according to the threshold. Then, a data augmentation method called Mixup operation is employed in semi-supervised learning (SSL) to increase noise robustness and avoid error accumulation, subsequently, performing fine-tuning and correction for clean and noisy samples. Challenging experiments on a transmission gearbox dataset under different noisy labels levels show that the proposed method has robustness to noise and significantly surpasses other approaches, which provides an important reference value for accurate fault diagnosis for rotating equipment parts with noisy labels.

Statistical structural inference from edge weights using a mixture of gamma distributions

Abstract The inference of reliable and meaningful connectivity information from weights representing the affinity between nodes in a graph is an outstanding problem in network science. Usually, this is achieved by simply thresholding the edge weights to distinguish true links from false ones and to obtain a sparse set of connections. Tools developed in statistical mechanics have provided particularly effective ways to locate the optimal threshold so as to preserve the statistical properties of the network structure. Thermodynamic analogies together with statistical mechanical ensembles have been proven to be useful in analysing edge-weighted networks. To extend this work, in this article, we use a statistical mechanical model to describe the probability distribution for edge weights. This models the distribution of edge weights using a mixture of Gamma distributions. Using a two-component Gamma mixture model with components describing the edge and non-edge weight distributions, we use the Expectation–Maximization algorithm to estimate the corresponding Gamma distribution parameters and mixing proportions. This gives the optimal threshold to convert weighted networks to sets of binary-valued connections. Numerical analysis shows that it provides a new way to describe the edge weight probability. Furthermore, using a physical analogy in which the weights are the energies of molecules in a solid, the probability density function for nodes is identical to the degree distribution resulting from a uniform weight on edges. This provides an alternative way to study the degree distribution with the nodal probability function in unweighted networks. We observe a phase transition in the low-temperature region, corresponding to a structural transition caused by applying the threshold. Experimental results on real-world weighted and unweighted networks reveal an improved performance for inferring binary edge connections from edge weights.

Abstract 1489: Overexpression of ribosomal L22-like1 (RPL22L1) enhances cancer stem cell properties and promotes tumor progression in hepatocellular carcinoma (HCC)

Abstract The abnormality of ribosomal biogenesis is found to be involved in carcinogenesis and progression in various types of cancer. RPL22L1 gene is a highly homologous paralog of RPL22, both of which belong to the L22E family of ribosomal proteins and take part in encoding a cytoplasmic ribosomal protein that is a component of the 60S subunit of ribosomes. Our previous work has established a CRISPR/Cas9 genome-wide knockout screening profile and developed a two-component mixture model to identify cancer-specific essential genes in 436 cancer cell lines. Bioinformatics analysis reveals that RPL22L1 acts as a feature gene to predict cancer-specific essential genes in human cancer. Previous studies suggest that RPL22L1 is correlated with tumor progression and adverse prognosis in several cancer types, while its role in hepatocellular carcinoma (HCC) is unknown. In the present study, we found that RPL22L1 is highly expressed in embryonic stem cells (ES) and liver progenitor cells, which were acquired from a vitro hepatic differentiation model originating from ES cells. Hence, we hypothesized that RPL22L1 enhances cancer stem cell properties in HCC. Functional assays showed that RPL22L1 would promote sphere formation in vitro. qRT-PCR and Western Blotting results indicated that RPL22L1 could promote stemness-related gene expressions such as CD133, CD44, and SOX9 in HCC cell lines. Overexpression of RPL22L1 increased aldehyde dehydrogenase (ALDH) activity, which was decreased when RPL22L1 was silenced with short hairpin RNA (shRNA). Extreme limiting dilution assay showed that RPL22L1 overexpression stimulated self-renewal ability in immunodeficient mice. Furthermore, higher RPL22L1 expression can also promote cell proliferation and epithelial-mesenchymal-transition, as well as drug resistance to 5-Fluorouracil (5-FU) and cisplatin (CDDP). Silver nitrate staining demonstrated that RPL22L1 overexpression correlated with a larger size and number of nucleoli. The expression of RPL22L1 was also associated with adverse prognosis in HCC clinical samples. In conclusion, our study reveals aberrant ribosomal protein activity in HCC and the role of RPL22L1 in promoting HCC tumor progression. Citation Format: JinLin Huang, Zhenyang Guo, Jie Luo, Beilei Liu, Lanqi Gong, Xiaona Fang, Dora Lai-wan Kwong, Xinyuan Guan. Overexpression of ribosomal L22-like1 (RPL22L1) enhances cancer stem cell properties and promotes tumor progression in hepatocellular carcinoma (HCC) [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2023; Part 1 (Regular and Invited Abstracts); 2023 Apr 14-19; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2023;83(7_Suppl):Abstract nr 1489.

Rapid and precise tracking of water influx and efflux across cell membranes induced by a pulsed electric field.

Quantitative measurements of water content within a single cell are notoriously difficult. In this work, we introduce a single-shot optical method for tracking the intracellular water content, by mass and volume, of a single cell at video rate. We utilize quantitative phase imaging and a priori knowledge of a spherical cellular geometry, leveraging a two-component mixture model to compute the intracellular water content. We apply this technique to study CHO-K1 cells responding to a pulsed electric field, which induces membrane permeabilization and rapid water influx or efflux depending upon the osmotic environment. The effects of mercury and gadolinium on water uptake in Jurkat cells following electropermeabilization are also examined.

Population estimate and inter-island movement of sperm whales, Physeter macrocephalus, in the Eastern Caribbean Sea

When a population extends across international boundaries, management becomes more complex. This is especially true within a confined multinational area such as the Caribbean Sea. The population size of sperm whales in the Eastern Caribbean is estimated and the inter-island movements of individuals are quantified using a database of 1,394 photographic identifications taken between 1984 and 2006 by several research groups. A total of 194 individual sperm whales were identified off the leeward coasts of the islands of Dominica, Guadeloupe, Grenada, St. Lucia and Martinique. Population size was estimated using two-component finite mixture models. About 145 (95% CI=94- 219) sperm whales used Lesser Antillean waters in 1995 and this population appears to be growing slowly. There are differences amongst the individuals in their probability of identification. Of all individuals, 57 (29.4%) were identified during more than one year between 1995 and 2006. Long-term reidentification of associated females suggests that social units may be using the area for periods of at least 11 years. Twenty seven confirmed matches were made between islands, the majority (92.6%) of which were between Guadeloupe and Dominica, although there were two longer movements by single individuals between Dominica and the islands of St. Lucia and Grenada. High reidentification rates within the Lesser Antilles and no matches with identifications from nearby seas suggest the population in the Eastern Caribbean Sea is small and quite isolated. As such, we recommend that management actions be taken on a multi-island basis for the Eastern Caribbean, by encouraging the ratification of the SPAW protocols and that the current stock classification for the North Atlantic be reconsidered.

Sparse Estimation in Finite Mixture of Accelerated Failure Time and Mixture of Regression Models with R Package fmrs

Variable selection in large-dimensional data has been extensively studied in different settings over the past decades. In a recent article, Shokoohi et. al. [29, DOI:10.1214/18-AOAS1198] proposed a method for variable selection in finite mixture of accelerated failure time regression models for studies on time-to-event data to capture heterogeneity within the population and account for censoring. In this paper, we introduce the fmrs package, which implements the variable selection methodology for such models. Furthermore, as a byproduct, the fmrs package facilitates variable selection in finite mixture regression models. The package also incorporates a tuning parameter selection mechanism based on component-wise bic. Commonly used penalties, such as Least Absolute Shrinkage and Selection Operator, and Smoothly Clipped Absolute Deviation, are integrated into fmrs. Additionally, the package offers an option for non-mixture regression models. The C language is chosen to boost the optimization speed. We provide an overview of the fmrs principles and the strategies employed for optimization. Hands-on illustrations are presented to help users get acquainted with fmrs. Finally, we apply fmrs to a lung cancer dataset and observe that a two-component mixture model reveals a subgroup with a more aggressive form of the disease, displaying a lower survival time.

Semi-supervised learning from coarse histopathology labels

ABSTRACT Ultrasound imaging is commonly used to guide sampling the prostate tissue in transrectal biopsies, followed by detection of cancer through histopathological analysis and coarse labelling of sampled tissue. Ideally, the procedure should be improved by developing machine learning solutions that can identify the presence of cancer in ultrasound images to guide the biopsy procedure. Training a fully supervised learning model using coarse histopathology labels suffers from weakly annotated data which introduce label noise for each image pixel. To address this challenge, we propose a semi-supervised framework for learning with noisy labels. We leverage a two-component mixture model to cluster the training data into clean and noisy label samples based on their loss values. Then, during the semi-supervised training phase, we utilise the well-known MixMatch algorithm which incorporates consistency regularisation, entropy minimisation, and the Mixup regularisation as well as the cross-entropy loss function for noisy and clean sets, respectively. We evaluate the proposed framework with prostate ultrasound data obtained from 71 subjects, while sampling 264 biopsy cores. We achieve balanced accuracy, sensitivity, and specificity of 78.6%, 80.0%, and 77.1%, respectively. In a detailed comparison study, we demonstrate that our proposed framework outperforms the fully supervised method with state-of-the-art robust loss functions.

Spin it as you like: The (lack of a) measurement of the spin tilt distribution with LIGO-Virgo-KAGRA binary black holes

Context. The growing set of gravitational-wave sources is being used to measure the properties of the underlying astrophysical populations of compact objects, black holes, and neutron stars. Most of the detected systems are black hole binaries. While much has been learned about black holes by analyzing the latest LIGO-Virgo-KAGRA (LVK) catalog, GWTC-3, a measurement of the astrophysical distribution of the black hole spin orientations remains elusive. This is usually probed by measuring the cosine of the tilt angle (cosτ) between each black hole spin and the orbital angular momentum, with cosτ = +1 being perfect alignment. Aims. The LVK Collaboration has modeled the cosτ distribution as a mixture of an isotropic component and a Gaussian component with mean fixed at +1 and width measured from the data. We want to verify if the data require the existence of such a peak at cosτ = +1. Methods. We used various alternative models for the astrophysical tilt distribution and measured their parameters using the LVK GWTC-3 catalog. Results. We find that (a) augmenting the LVK model, such that the mean μ of the Gaussian is not fixed at +1, returns results that strongly depend on priors. If we allow μ > +1, then the resulting astrophysical cosτ distribution peaks at +1 and looks linear, rather than Gaussian. If we constrain −1 ≤ μ ≤ +1, the Gaussian component peaks at μ = 0.48−0.99+0.46 (median and 90% symmetric credible interval). Two other two-component mixture models yield cosτ distributions that either have a broad peak centered at 0.19−0.18+0.22 or a plateau that spans the range [ − 0.5, +1], without a clear peak at +1. (b) All of the models we considered agree as to there being no excess of black hole tilts at around −1. (c) While yielding quite different posteriors, the models considered in this work have Bayesian evidences that are the same within error bars. Conclusions. We conclude that the current dataset is not sufficiently informative to draw any model-independent conclusions on the astrophysical distribution of spin tilts, except that there is no excess of spins with negatively aligned tilts.

Nonparametric scanning tests of homogeneity for hierarchical models with continuous covariates.

In many applications of hierarchical models, there is often interest in evaluating the inherent heterogeneity in view of observed data. When the underlying hypothesis involves parameters resting on the boundary of their support space such as variances and mixture proportions, it is a usual practice to entertain testing procedures that rely on common heterogeneity assumptions. Such procedures, albeit omnibus for general alternatives, may entail a substantial loss of power for specific alternatives such as heterogeneity varying with covariates. We introduce a novel and flexible approach that uses covariate information to improve the power to detect heterogeneity, without imposing unnecessary restrictions. With continuous covariates, the approach does not impose a regression model relating heterogeneity parameters to covariates or rely on arbitrary discretizations. Instead, a scanning approach requiring continuous dichotomizations of the covariates is proposed. Empirical processes resulting from these dichotomizations are then used to construct the test statistics, with limiting null distributions shown to be functionals of tight random processes. We illustrate our proposals and results on a popular class of two-component mixture models, followed by simulation studies and applications to two real datasets in cancer and cariesresearch.

Sero-surveillance for IgG to SARS-CoV-2 at antenatal care clinics in three Kenyan referral hospitals: Repeated cross-sectional surveys 2020-21.

The high proportion of SARS-CoV-2 infections that have remained undetected presents a challenge to tracking the progress of the pandemic and estimating the extent of population immunity. We used residual blood samples from women attending antenatal care services at three hospitals in Kenya between August 2020 and October 2021and a validated IgG ELISA for SARS-Cov-2 spike protein and adjusted the results for assay sensitivity and specificity. We fitted a two-component mixture model as an alternative to the threshold analysis to estimate of the proportion of individuals with past SARS-CoV-2 infection. We estimated seroprevalence in 2,981 women; 706 in Nairobi, 567 in Busia and 1,708 in Kilifi. By October 2021, 13% of participants were vaccinated (at least one dose) in Nairobi, 2% in Busia. Adjusted seroprevalence rose in all sites; from 50% (95%CI 42-58) in August 2020, to 85% (95%CI 78-92) in October 2021 in Nairobi; from 31% (95%CI 25-37) in May 2021 to 71% (95%CI 64-77) in October 2021 in Busia; and from 1% (95% CI 0-3) in September 2020 to 63% (95% CI 56-69) in October 2021 in Kilifi. Mixture modelling, suggests adjusted cross-sectional prevalence estimates are underestimates; seroprevalence in October 2021 could be 74% in Busia and 72% in Kilifi. There has been substantial, unobserved transmission of SARS-CoV-2 in Nairobi, Busia and Kilifi Counties. Due to the length of time since the beginning of the pandemic, repeated cross-sectional surveys are now difficult to interpret without the use of models to account for antibody waning.