Bayesian factor analysis with uncertain functional constraints about factor loadings

Factors models are commonly used to analyze high-dimensional data in both single-study and multi-study settings. Bayesian inference for such models relies on Markov chain Monte Carlo (MCMC) methods, which scale poorly as the number of studies, observations, or measured variables increase. To address this issue, we propose new variational inference algorithms to approximate the posterior distribution of Bayesian latent factor models using the multiplicative gamma process shrinkage prior. The proposed algorithms provide fast approximate inference at a fraction of the time and memory of MCMC-based implementations while maintaining comparable accuracy in characterizing the data covariance matrix. We conduct extensive simulations to evaluate our proposed algorithms and show their utility in estimating the model for high-dimensional multi-study gene expression data in ovarian cancers. Overall, our proposed approaches enable more efficient and scalable inference for factor models, facilitating their use in high-dimensional settings. An R package VIMSFA implementing our methods is available on GitHub (github.com/blhansen/VI-MSFA). Supplementary materials for this article are available online.

The development of simulation-based methods, such as Markov chain Monte Carlo (MCMC), has contributed to an increased interest in the Bayesian framework as an alternative to deal with factor models. Many studies have used Bayesian factor analysis to explore gene expression data. We are particularly interested in the application of a sparse latent factor model (SLFM) based on sparsity priors (mixtures) to assess the significance of factors. The SLFM measures how strong the observed coherent expression pattern is in the data, which is an important source of information to evaluate gene activity. In the literature, this type of model has shown better results than other approaches intended for identification of patterns and metagene groups related to the underlying biology. However, a full Bayesian factor model relying on MCMC algorithms has an expensive computational cost, which makes it unattractive for general users. In this paper, we present the package slfm which uses C++ implementation via Rcpp to improve the computational performance of the SLFM within the widely used statistical tool R. We investigate real and simulated microarray data related to breast cancer.

Patient reported outcomes are gaining more attention in patient-centered health outcomes research and quality of life studies as important indicators of clinical outcomes, especially for patients with chronic diseases. Factor analysis is ideal for measuring patient reported outcomes. If there is heterogeneity in the patient population and when sample size is small, differential item functioning and convergence issues are challenges for applying factor models. Bayesian hierarchical factor analysis can assess health disparity by assessing for di˙erential item functioning, while avoiding convergence problems. We conducted a simulation study and used an empirical example with American Indian minorities to show thatffitting a Bayesian hierarchical factor model is an optimal solution regardless of heterogeneity of population and sample size.

SummaryAssessing the impact of an intervention by using time-series observational data on multiple units and outcomes is a frequent problem in many fields of scientific research. Here, we propose a novel Bayesian multivariate factor analysis model for estimating intervention effects in such settings and develop an efficient Markov chain Monte Carlo algorithm to sample from the high-dimensional and nontractable posterior of interest. The proposed method is one of the few that can simultaneously deal with outcomes of mixed type (continuous, binomial, count), increase efficiency in the estimates of the causal effects by jointly modeling multiple outcomes affected by the intervention, and easily provide uncertainty quantification for all causal estimands of interest. Using the proposed approach, we evaluate the impact that Local Tracing Partnerships had on the effectiveness of England’s Test and Trace programme for COVID-19.

The revised criteria for posttraumatic stress disorder (PTSD) in the fifth edition of the Diagnostic and Statistical Manual necessitated the development of new screening tools for youth, one of the most widely used of which is the UCLA Posttraumatic Stress Disorder Reaction Index for DSM-5 (RI-5). Thus far, the few studies that have investigated the RI-5's factor structure have supported a four-factor model. However, to date this research has been limited to youth with histories of exposure to single-event traumatic stressors, a significant limitation as evidence suggests many trauma-exposed youth report exposure to multiple types of traumatic stressors, or polyvictimization. It is imperative to determine the generalizability of previous factor models to specific populations which they are purported to represent. We investigated whether the RI-5's four-factor model replicated in a sample of 455 polyvictimized justice-involved adolescents. Initial confirmatory factor analysis demonstrated that the four-factor model did not converge. Therefore, we utilized Bayesian Structural Equations Modeling (BSEM) to determine why the previously proposed factor structure did not converge. The BSEM model suggested that the global factor structure was acceptable and did not require addition or subtraction of any factor or cross-factor loadings. However, small and moderate residual covariances resulted in model misspecification, suggesting there may be additional associations not captured by the current DSM-5 model for polyvictimized youth. Future work should continue examining the RI-5's factor structure in order to better understand whether the current results are unique and how measurements assessing DSM-5 PTSD symptom criteria perform in diverse trauma-exposed youth populations.

The factor analytic model, as here applied, conceives of a numerical quality judgment of a certain program played by a given reproducing device as a weighted sum of a measure of the quality of the different reproducing properties (e.g., purity of transients, full treble) possessed by the device in question. The weights constitute measures of the requirements made by the particular program on these properties. Factor analysis splits a raw-data matrix—consisting of, e.g., quality judgments for a number of combinations of program-loudspeaker systems—into two matrices: a factor loading matrix consisting of the weights and a factor-score matrix consisting of the quality of the reproducing properties. The rank of these matrices (number of factors) reflects the number of dimensions (properties) that implicitly enter into the listener's judgments. Four listeners judged, on a 7-point scale, the quality of 24 programs (music, speech, traffic noise, etc.) played on 10 sound reproducing systems of highly different general quality. The data were factor analyzed (component analysis of covariances), and factor loadings for the programs and factor scores for the loudspeaker systems were computed. Nine factors were extracted and rotated, and 7 of them tentatively interpreted (sound level, purity of transients, environmental information, bass boost, full-treble reproduction, high-treble relative midrange, disturbing directional effects). An attempt at validation showed good agreement between factor scores (reproducing properties) for the 4 listeners in spite of variation of preferences between listeners reflected in the factor loadings. Despite technical imperfections, it is concluded that factor analysis is a useful instrument for the assessment of acoustical properties.

To develop a short, flexible, and comprehensive tool to measure psychological distress in caregivers of malignant mesothelioma (MM) patients: the Mesothelioma Psychological Distress Tool-Caregivers version (MPDT-C). Based on a systematic review of the relevant literature, aspects associated with caregiver distress were derived. Expert researchers/clinicians developed a pool of items for each identified aspect. Content validity was assessed through a multiple mixed-methods approach. A multicenter study was conducted to explore the factorial structure of the 47-item MPDT-C through Bayesian factor analysis. The Bayesian exploratory factor analysis revealed an underlying three-factor structure. Factors were labeled Secondary Traumatic Stress, Engagement in Caring, and Meaningful Cognitive Restructuring. All scales showed sufficient reliability and corrected item-total correlations. Females scored higher than males for Engagement in Caring. Taking care of malignant mesothelioma patients is a stressful process that influences the caregiver's physical, emotional, and social wellbeing. Our study offers preliminary evidence in support of the adequate psychometric properties of the MPDT-C, and these should now be replicated. Results suggest that the MPDT-C is a reliable tool with which to detect the psychological distress of this traumatized population.

Sparse Bayesian factor analysis for structural damage detection under unknown environmental conditions

This study focuses on validating a Negative Interaction Scale using a Bayesian approach for confirmatory factor analysis. Considering the significant impact of negative interactions on couple relationships, both in terms of satisfaction and the mental and physical health of individuals, the aim was to provide a reliable and precise tool for measuring these behaviors in a specific cultural context, such as that of young and adult individuals in Peru. A non-probabilistic convenience sampling design was used, involving 313 participants, predominantly women, and a majority identifying as heterosexual. The average age was around 26 years, encompassing various types of couple relationships. The Negative Interaction Scale (NIS-5) and the Peruvian version of the Relationship Assessment Scale (RAS) were employed for data collection, using Bayesian statistical tools for analysis, particularly Bayesian confirmatory factor analysis (BCFA), offering advantages in flexibility, precision, and the evaluation of the proposed unidimensional structure. The NIS-5 demonstrated a unidimensional structure with excellent fit indices, moderate to strong factor loadings, and good internal consistency, supported by an omega reliability coefficient of 0.836. These results validate the unifactorial structure of the NIS-5, reflecting that negative interaction can be globally and uniquely interpreted within the study context. The study confirms the validity and reliability of the NIS-5 as an instrument for measuring negative interaction in couple relationships in Peru. It highlights the importance of considering negative interaction to improve relationship quality and suggests the implementation of programs or theoretical models to promote healthier relationships.

In this study, we have adopted an approach for objective optimization of the selection of band ratios in forming Advanced Space-borne Thermal Emission and Reflection Radiometer (ASTER) band ratio composites, based on factor analysis loadings and specific variances. The factor loadings and specific variances of all 72 possible spectral band ratios of the visible and near infrared (VNIR) and the short-wave infrared (SWIR) of ASTER data from Wadi Kid, southeastern Sinai Peninsula, Egypt, were utilized to construct two separate rankings of band ratios in an ascending manner in order to determine the best ASTER band ratio combinations for lithological mapping in the study area. Two ASTER band ratio composites ((band 6/band 3–band 1/band 3–band 9/band 5) and (8/6–8/7–4/7) in blue–green–red (BGR)) were built, based on the rankings of factor loadings and specific variances, respectively. These two composites were fused together based on a discrete wavelet transform (DWT) decomposition, which enables decomposition of an image into (high-pass) details and (low-pass) approximations at various levels of resolution, resulting in improved lithological discrimination of granitic and meta-sediments rock units and a reduction in the BGR imprints of topographic reliefs.

The genetic variance-covariance matrix (G) is a quantity of central importance in evolutionary biology due to its influence on the rate and direction of multivariate evolution. However, the predictive power of empirically estimated G-matrices is limited for two reasons. First, phenotypes are high-dimensional, whereas traditional statistical methods are tuned to estimate and analyse low-dimensional matrices. Second, the stability of G to environmental effects and over time remains poorly understood. Using Bayesian sparse factor analysis (BSFG) designed to estimate high-dimensional G-matrices, we analysed levels variation and covariation in 10,527 expressed genes in a large (n=563) half-sib breeding design of three-spined sticklebacks subject to two temperature treatments. We found significant differences in the structure of G between the treatments: heritabilities and evolvabilities were higher in the warm than in the low-temperature treatment, suggesting more and faster opportunity to evolve in warm (stressful) conditions. Furthermore, comparison of G and its phenotypic equivalent P revealed the latter is a poor substitute of the former. Most strikingly, the results suggest that the expected impact of G on evolvability-as well as the similarity among G-matrices-may depend strongly on the number of traits included into analyses. In our results, the inclusion of only few traits in the analyses leads to underestimation in the differences between the G-matrices and their predicted impacts on evolution. While the results highlight the challenges involved in estimating G, they also illustrate that by enabling the estimation of large G-matrices, the BSFG method can improve predicted evolutionary responses to selection.

Assessing model fit is a key component of structural equation modeling (SEM); however, measures of fit in Bayesian SEM remain limited. Recently, versions of frequentist fit indices have been adapted for use in Bayesian models, but the impact of prior information on these fit indices remains unknown. This simulation study investigates the performance of three fit indices (RMSEA, CFI, and TLI) in Bayesian confirmatory factor analysis (CFA) across a variety of prior specifications that included different degrees of informativeness and inaccuracy. Results show that Bayesian fit indices are impacted by prior choice, particularly when sample sizes are small. We discuss implications of assessing model fit with Bayesian fit indices and provide recommendations for applied researchers.

To elucidate the food intake structures of three meals, i.e., breakfast, lunch, and dinner, and also snacks between meals, we conducted nutrition surveys on 62 females living in Nakano-city (agricultural district) and 50 females living in Shirahama-town (fishery district). By means of descriptive statistical methods and factor analysis using 23 food group data for each meal, the main results obtained are as follows. 1. Nakano-city had higher mean values of rice, miso, and other vegetables, though these of confectioneries, fruits, seaweeds, and fishes were significantly higher in Shirahama-town. 2. Rice, bread, and eggs were mostly consumed at breakfast. By means of factor analysis, the first factor extracted is positively related to the intakes of rice, miso, pulses, and other vegetables, i.e., the "gtraditional food pattern", while the factor is negatively related to bread, oil and fats, fruits, milk and its products, i.e., the?"modern food pattern". 3. The first factor in lunch foods is almost the same as that in breakfast. The second factor is related to noodles and side dishes with positive loadings, and rice and miso with negative values. It is found that the difference of the second factor score is statistically signinif icant between the two districts. 4. At dinner, Rice, noodles, meats, and fishes are more comsumed but bread and eggs are less than those at breakfast and lunch. The first factor is related positively to rice and negatively to noodles, and the second factor is thought to be related to side dishes. The difference of means of the first factor score between the two districts is not less than but closed to five percent level, while the difference of the second factor score had highly statistical significance. 5. The first factor for sanck foods is related to (the leftovers of) side dishes of other meals. The second factor is related to bread and confectioneries with positive factor loadings and to noodles and seaweeds with negative values. Statistically significant dif f ereces are found in both of two factor scores between two districts. 6. For total daily food intakes, the first factor is related to rice and Miso with positive high factor loadings, and bread, noodles, confectioneries, and seaweeds with netagive values. The second factor is related to the side dishes such as vegetables, pulses, fruits, and fishes. statistically significant differences were found in both of two factor scores between the two districts. 7. To be more practical and effective for the further nutrition education at community level, the food intake patterns for each meal should be taken into consideration as well as regional features.

In this chapter, we study a partially linear model with autoregressive beta distributed errors [6] from the Bayesian point of view. Our proposal also provides a useful method to determine the optimal order of the autoregressive processes through an adaptive procedure using the conditional predictive ordinate (CPO) statistic [9].In this context, the linear predictor of the beta regression model \(g(\mu_{t})\) incorporates an unknown smooth function for the auxiliary time covariate t and a sequence of autoregressive errors ϵ t , i.e., $$\begin{aligned} g(\mu_t)=x_{t}^{\top}\beta+f(t)+ \varepsilon_t,\end{aligned}$$ for \(t=1,\ldots,T\), where x t is a \(k\times1\) vector of nonstochastic explanatory variable values and β is a \(k\times1\) fixed parameter vector. Furthermore, these models have a convenient hierarchical representation allowing to us an easily implementation of a Markov chain Monte Carlo (MCMC) scheme. We also propose to modify the traditional conditional predictive ordinate (CPO) to obtain what we call the autoregressive CPO, which is computed for each new observation using only the data from previous time periods.KeywordsBeta Regression ModelAutoregressive ErrorPartial Linear ModelConditional Predictive Ordinate (CPO)Unknown Smooth FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We focus on sparse modelling of high-dimensional covariance matrices using Bayesian latent factor models. We propose a multiplicative gamma process shrinkage prior on the factor loadings which allows introduction of infinitely many factors, with the loadings increasingly shrunk towards zero as the column index increases. We use our prior on a parameter-expanded loading matrix to avoid the order dependence typical in factor analysis models and develop an efficient Gibbs sampler that scales well as data dimensionality increases. The gain in efficiency is achieved by the joint conjugacy property of the proposed prior, which allows block updating of the loadings matrix. We propose an adaptive Gibbs sampler for automatically truncating the infinite loading matrix through selection of the number of important factors. Theoretical results are provided on the support of the prior and truncation approximation bounds. A fast algorithm is proposed to produce approximate Bayes estimates. Latent factor regression methods are developed for prediction and variable selection in applications with high-dimensional correlated predictors. Operating characteristics are assessed through simulation studies, and the approach is applied to predict survival times from gene expression data.